Christer Carlsson and Robert Fullér Possibility for Decision
Studies in Fuzziness and Soft Computing, Volume 270 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail:
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Christer Carlsson and Robert Fullér
Possibility for Decision A Possibilistic Approach to Real Life Decisions
ABC
Authors Christer Carlsson IAMSR Abo Akademi University, Joukahainengatan 3-5A, Abo, FI-20520, Finland Email: christer.carlsson@abo.fi
ISBN 978-3-642-22641-0
Robert Fullér IAMSR Abo Akademi University, Joukahainengatan 3-5A, Abo, FI-20520, Finland Email: rfuller@abo.fi
e-ISBN 978-3-642-22642-7
DOI 10.1007/978-3-642-22642-7 Studies in Fuzziness and Soft Computing
ISSN 1434-9922
Library of Congress Control Number: 2011933573 c 2011 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com
Preface
This monograph summarizes the authors’ works in the the first decade of the 21st century on possibility distributions and decisions. The book is organized as follows. It begins, in Chapter ’Concepts and Issues’, with a short historical survey of development of fuzzy thinking and progresses through an analysis of the extension principle and averaging operators. In probability theory the expected value of functions of random variables plays a fundamental role in defining the basic characteristic measures of probability distributions. For example, the variance, covariance and correlation of random variables can be computed as the expected value of their appropriately chosen real-valued functions. In possibility theory we can use the principle of expected value of functions on fuzzy sets to define variance, covariance and correlation of possibility distributions. Marginal probability distributions are determined from the joint one by the principle of ’falling integrals’ and marginal possibility distributions are determined from the joint possibility distribution by the principle of ’falling shadows’. Probability distributions can be interpreted as carriers of incomplete information [203], and possibility distributions can be interpreted as carriers of imprecise information. In 1987 Dubois and Prade [126] defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. They also showed that this expectation remains additive in the sense of addition of fuzzy numbers. In Chapter ’A Normative View on Possibility Distributions’ we will discuss the weighted lower possibilistic and upper possibilistic mean values, crisp possibilistic mean value and variance of fuzzy numbers, which are consistent with the extension principle and with the well-known defintions of expectation and variance in probability theory. We will use the principle of average value of appropriately chosen real-valued functions to define possibilistic mean value, variance, covariance and correlation of possibility distributions. A function f : [0, 1] → R is said to be a weighting function if f is non-negative, monotone increasing and satisfies the following normalization condition 01 f (γ )d γ = 1. Different weighting functions can give different (case-dependent) importances to level-sets of possibility distributions. We can define the mean value (variance) of a possibility distribution as the f -weighted average of the probabilistic mean values (variances) of the respective
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uniform distributions defined on the γ -level sets of that possibility distribution. A measure of possibilistic covariance (correlation) between marginal possibility distributions of a joint possibility distribution can be defined as the f -weighted average of probabilistic covariances (correlations) between marginal probability distributions whose joint probability distribution is defined to be uniform on the γ -level sets of their joint possibility distribution [157]. The process of information aggregation appears in many applications related to the development of intelligent systems. In 1988 Yager introduced a new aggregation technique based on the ordered weighted averaging operators (OWA) [338]. The determination of ordered weighted averaging (OWA) operator weights is a very important issue of applying the OWA operator for decision making. One of the first approaches, suggested by O’Hagan, determines a special class of OWA operators having maximal entropy of the OWA weights for a given level of orness; algorithmically it is based on the solution of a constrained optimization problem. In 2001, using the method of Lagrange multipliers, Full´er and Majlender solved this constrained optimization problem analytically and determined the optimal weighting vector [153], and in 2003 they computed the exact minimal variability weighting vector for any level of orness [155]. In Chapter ’OWA Operators in Multiple Criteria Decisions’ we give a short survey of some later works that extend and develop these models. In traditional investment planning investment decisions are usually taken to be now-or-never, which the firm can either enter into right now or abandon forever. The decision on to close/not close a production plant has been understood to be a similar now-or-never decision for two reasons: (i) to close a plant is a hard decision and senior management can make it only when the facts are irrefutable; (ii) there is no future evaluation of what-if scenarios after the plant is closed. However, it is often possible to postpone,modify or split up a complex decision in strategic components, which can generate important learning effects and therefore essentially reduce uncertainty. If we close a plant we lose all alternative development paths which could be possible under changing conditions; on the other hand, senior management may have a difficult time with shareholders if they continue operating a production plant in conditions which cut into its profitability as their actions are evaluated and judged every quarter. In these cases we can utilize the idea of real options. The new rule, derived from option pricing theory, is that we should only close the plant now if the net present value of this action is high enough to compensate for giving up the value of the option to wait. Because the value of the option to wait vanishes right after we irreversibly decide to close the plant, this loss in value is actually the opportunity cost of our decision. In Chapter ’Fuzzy Real Options for Strategic Decisions’ we will use fuzzy real option models for the problem of closing/not closing a production plant in the forest products industry sector. In standard portfolio models uncertainty is equated with randomness, which actually combines both objectively observable and testable random events with subjective judgments of the decision maker into probability assessments. A purist on theory would accept the use of probability theory to deal with observable random events, but would frown upon the transformation of subjective judgments to
Preface
VII
probabilities. The use of probabilities has another major drawback: the probabilities give an image of precision which is unmerited - we have found cases where the assignment of probabilities is based on very rough, subjective estimates and then the subsequent calculations are carried out with a precision of two decimal points. This shows that the routine use of probabilities is not a good choice. The actual meaning of the results of an analysis may be totally unclear - or results with serious errors may be accepted at face value. In Chapter ’Portfolio Selection with Imprecise Future Data’ we have explored the use of possibility theory as a substituting conceptual framework. A significant amount of the literature on grid computing addresses the problem of resource allocation on the grid [37, 113, 116, 227, 241]. The presence of disparate resources that are required to work in concert within a grid computing framework increases the complexity of the resource allocation problem. Jobs are assigned either through scavenging, where idle machines are identified and put to work, or through reservation in which jobs are matched and pre-assigned with the most appropriate systems in the grid for efficient workflow. In grid computing a resource provider [RP] offers resources and services to other Grid users based on agreed service level agreement [SLA]. During the execution of a computing task the accomplishment of the SLA has the highest priority, which is why an RP often is using resource allocation models to safeguard against expected node failures. When spare resources at the RP’s own site are not available outsourcing will be an adequate solution for avoiding SLA violations. In Chapter ’Risk Assessment in Grid Computing’ using predictive probabilities and possibilities for modeling node failures we show a risk assessment model for SLA violations in the presence of spare resources. Fuzzy ontologies have been proposed as a solution for addressing semantic meaning in an uncertain and inconsistent world. As with fuzzy logic, reasoning is approximate rather than precise. The aim is to avoid the theoretic pitfalls of monolithic ontologies, facilitate interoperability between different and independent ontologies [112], and provide flexible information retrieval capabilities. The Knowledge Mobilization project (KNOWMOBILE) has been a joint effort by Institute for Advanced ˚ Akademi University and VTT Technical ReManagement Systems Research, Abo search Centre of Finland. Its goal was to better ”mobilize” knowledge stored in heterogeneous databases for users with various backgrounds, geographical locations and situations. The working hypothesis of the project was that fuzzy mathematics combined with domain-specific data models, in other words, fuzzy ontologies, would help manage the uncertainty in finding information that matches the user’s needs. In this way, the Knowledge Mobilization project places itself in the domain of knowledge management. In Chapter ’Knowledge Mobilization’, drawing heavily on Tommila, Hirvonen and Pakonen [313]; Hirvonen et al. [185]; and Carlsson, Full´er and Fedrizzi [95] we will describe an industrial demonstration of fuzzy ontologies in information retrieval in the paper industry where problem solving reports are annotated with keywords and then stored in a database for later use. It is widely assumed that the future of mobile telephony will rely on mobile services. In Chapter ’Mobile Value Services’ we will position our research in design science and new service design approaches and will discuss relevant theories with
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regard to barriers and benefits, adoption, acceptance and empirical research in order to point out the first differences between mobile services and mobile value services. As an example of the design and implementation of mobile value services we will work through the Bomarsund Mobile Guide. Furthermore, we will discuss two prototypes, MobiFish and Travel MoCo that were developed and implemented. Finally, we will present some evidence of the development of mobile TV in Finland and some other countries. ˚ Abo, Finland April 2011
Christer Carlsson Robert Full´er
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
Concepts and Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Extension Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Averaging Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Possibility Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 7 12 16 21 24
3
A Normative View on Possibility Distributions . . . . . . . . . . . . . . . . . . . . 3.1 Possibilistic Mean Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Possibilistic Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Possibilistic Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Possibilistic Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Non-interactive Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Perfect Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Joint Distribution: (1 − x − y) . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Joint Distribution: (y-x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Ball-Shaped Joint Possibility Distribution . . . . . . . . . . . . . . . 3.5.6 Mere Shadows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Zero Correlation and Non-interactivity . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Possibilistic Correlation Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 A Linear Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 A Nonlinear Relationship √ ............................ 3.8.3 Joint Distribution: (1 − x − y) . . . . . . . . . . . . . . . . . . . . . . .
27 28 32 35 36 37 38 42 44 44 45 47 49 50 51 53 55 56 56 58 60
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3.8.4 Ball-Shaped Joint Distribution √ √. . . . . . . . . . . . . . . . . . . . . . . . 3.8.5 Joint Distribution: (1 − x − y) . . . . . . . . . . . . . . . . . . . . . 3.9 A Normative View on Quasi Fuzzy Numbers . . . . . . . . . . . . . . . . . . 3.9.1 Exponential Membership Function . . . . . . . . . . . . . . . . . . . . . 3.10 Addition and Subtraction of Interactive Fuzzy Numbers . . . . . . . . .
61 63 64 70 73
4
OWA Operators in Multiple Criteria Decisions . . . . . . . . . . . . . . . . . . . 4.1 Obtaining OWA Operator Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Constrained OWA Aggregations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Recent Advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 OCA Functions for Group Settlement Modeling . . . . . . . . . . . . . . . . 4.4.1 Optimal Unconditional Settlement . . . . . . . . . . . . . . . . . . . . . 4.4.2 Optimal Settlement under Budget Constraints . . . . . . . . . . . 4.4.3 Optimal Settlement under Flexible Budget Constraints . . . . 4.5 Olympic OWA Operators for Modeling Group Decisions . . . . . . . . 4.5.1 Group Decisions under Fuzzy Budget Constraints . . . . . . . .
77 77 82 84 88 90 93 94 96 99
5
Fuzzy Real Options for Strategic Decisions . . . . . . . . . . . . . . . . . . . . . . . 5.1 A Probabilistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Fuzzy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 A Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Closing Production Plants - When and Where . . . . . . . . . . . . . . . . . . 5.4.1 Production, Exports, Jobs and Investments . . . . . . . . . . . . . . 5.4.2 The Production Plant and Future Scenarios . . . . . . . . . . . . . . 5.4.3 Closing/Not Closing a Plant: A Case Study . . . . . . . . . . . . . 5.4.4 A Binomial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 A Fuzzy Interval Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 A Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.7 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 A Fuzzy Pay-Off Method for Real Option Valuation . . . . . . . . . . . . 5.6 Optimal R&D Project Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101 102 104 105 106 108 111 114 115 116 119 121 124 128
6
Portfolio Selection with Imprecise Future Data . . . . . . . . . . . . . . . . . . . 133 6.1 Possibilistic Choice of Portfolios with Highest Utility Score . . . . . . 133 6.2 Recent Advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7
Risk Assessment in Grid Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Risk Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Predictive Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Predictive Nature of Bayesian Inference . . . . . . . . . . . . . . . . . . . 7.4 Predictive Probabilities in Grid Computing Management . . . . . . . . . 7.5 A Hybrid Approach to Computing a Measure of Success . . . . . . . . . 7.6 Predictive Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Predictive Possibilities in Grid Computing Management . . . . . . . . .
145 147 149 149 152 156 160 162
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Knowledge Mobilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Ontologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Description Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Knowledge Mobilization Project . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Fuzzy Ontology for Process Industry . . . . . . . . . . . . . . . . . . . 8.3.2 The Keyword Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 A Minimax Approach to Assess Keyword Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Demo Architecture and Implementation . . . . . . . . . . . . . . . . 8.4 An Approximate Reasoning Approach to Rank the Results of Fuzzy Queries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
167 168 169 171 173 175
Mobile Value Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Design Science and Mobile Service Design . . . . . . . . . . . . . . . . . . . . 9.2 Mobile Services and Mobile Value Services . . . . . . . . . . . . . . . . . . . 9.3 Design and Implementation of Mobile Value Services . . . . . . . . . . . 9.4 Designing Mobile Value Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 A Business Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˚ 9.6 The New Interactive Media (NIM) ALand Project . . . . . . . . . . . . . . 9.6.1 Stakeholder and User Involvement . . . . . . . . . . . . . . . . . . . . . 9.6.2 User Acceptance Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.5 Discussion, Limitations and Conclusions . . . . . . . . . . . . . . . 9.7 Will Mobile TV Be a Value Service for Consumers? . . . . . . . . . . . . 9.7.1 The Braudel Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 A Brief Review: The Mobile TV Development and User Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.3 Early Experience from Finland . . . . . . . . . . . . . . . . . . . . . . . . 9.7.4 The Use of Mobile TV in Finland - The Consumer Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.5 Data Analysis and the Results . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Mobile Value Services: Conclusions and the Next Steps . . . . . . . . .
185 187 189 193 196 200 201 205 205 208 209 210 213 214
178 179 180
215 218 218 219 221 223
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Chapter 1
Introduction
Hard decisions for a management team are those decisions which will have significant economic, financial, political and/or emotional consequences for the team and the company they serve. Hard decisions are normally difficult to make and this is made even harder if the decision situation is complex (i.e. there are many interdependent elements), the information about the decision alternatives and their consequences is imprecise and/or uncertain and the environment (or the context) unstable, dynamic and not well known. If a team or a group should make the decisions the group members may have different opinions about the alternatives and the risks or outcomes of the consequences. In the modern business world, which is dominated by real-time information readily available in abundance through the World Wide Web and by the notion that decisions need to be made quickly as otherwise the competition (or opposition, or whatever antagonistic force) will prevail, there is a growing tendency to make fast and bad decisions. Following Carlsson [92] in this Chapter we will take another route - we will try to show that groups can make fast and good decisions with the help of some recent and fairly exciting analytical tools that are imbedded in good and easy to use software [299]. We will support our argument with data and experiences from a real world case - the hard decision on the closing/not closing of a paper plant in the UK where there are several opposing and competing views: the responsibility to the shareholders is a good argument for closing the plant, the responsibility to the employees and the community where the plant has been operating for nearly a century is a good argument for not closing the plant. Then we have the overall market situation and the profitability development for the European forest industry, the differences in management styles in Finland and the UK, the different results skillful people get with different analytical tools and the different market trends people believe in (with or without the use of foresight methods). Still the management team needs to find a good (or preferably the best) decision to recommend to the board of directors - a good decision can be explained in logical and analytical terms with a good support of facts and can be explained with rational arguments; the best decision is simply dominating any other alternative that can be discussed or tested. The management team needs a bit more than that - they need to be able to understand all the C. Carlsson and R. Full´er: Possibility for Decision, STUDFUZZ 270, pp. 1–6. springerlink.com © Springer-Verlag Berlin Heidelberg 2011
2
1 Introduction
alternatives and their consequences, they need to be able to analyze and understand the alternatives with all the data that is available, they need to have a reasonable foresight into the coming markets, they need to be able to discuss the issues and the alternatives in terms they can understand jointly and they need to come to a consensus on what they should be doing. The situation is close to the situation worked on by Ackermann and Eden [3] where they develop ways for assisting managers who have to negotiate the resolution of messy, complex and/or strategic problems. We worked with the management team during an 18 month period and both followed the processes they went through and tried to support them with good analytical tools as best we could. We gained a fairly good understanding of how management works with hard decisions and how they formed consensus as a group - this is the story we will be telling in this chapter. Academic outsiders need a conceptual framework and a basis for forming an understanding of the processes they are going to work with. This was our starting point. The early support for hard decisions was developed with the theory and methods of OR [Operations (or Operational) Research]. This was a major movement for rational decision making in the 1960’s through 1980’s but its origins go back to the late 1930’s. OR is striving for rational decision-making - it is searching for and (if possible) using the best alternative, i.e. the one maximizing/minimizing an objective function. It differed from classical economic theory by assuming that full information is not available - thus there is certainty, risk or uncertainty on available alternatives and the outcomes of selecting among the alternatives. Operational research works with the assumption that a context could change in a systematic or random manner, and that the changes in most cases will impact the set of alternatives. The context may in some cases change as a function of the decision-making process itself, i.e. the decision makers will influence the context by starting a decision process. The first target of OR was to find good methods for solving operational and tactical problems but the scope was inevitably broadened to include also strategic problems as the methods gained acceptance among senior management. The development of Operations Research was supported by a developing theory as sets of problems were recognized and classified as generic: resource allocation, assignment, transportation, networking, inventory, queuing, scheduling, etc. Then, in the next phase, generic problems became the basis for modeling, problem-solving and decision-making theories: guidance for better, more effective actions in a complex environment. Then, finally, as computing power was developed the OR methods became increasingly more popular as non-professionals could use the methods for handling large, complex and difficult problems. Russell Ackoff in 1978 [2] was the first to warn against putting too much faith in the OR. He introduced a classification in (i) well-structured problems that can be dealt with using OR modeling theory and (ii) ill-structured problems- the rest, i.e. all the problems in real life decisionmaking. Then he concluded that there are no problems, only abstract constructs to bring OR modeling theory into play; his conclusion was that problem-solving theory is not useful for any practical purposes if it is building on OR. In 1970 Bellman and Zadeh actually showed similar results [22]. They assume that all the elements which define a decision context are not strictly given and may
1 Introduction
3
evolve during the decision process, which gives a more flexible approach than the one used in OR. Then they developed a variation of the traditional optimization models with the proposal that there need not be any strict differences between constraints and objective functions. Their conclusion is that if we want to support an evolving decision process we need new and other tools than OR - but we should keep the focus and the power of a theory which have been tested and proved many times over the years. Some years later Zadeh [352, 353] introduced soft decision-making: at some point there will be a trade-off between precision and relevance: if we increase the precision of our methods and models we will reach a point where the results we get will be irrelevant as guidance for practical decision making - on the other hand, if we need to get relevant guidance for decision making we will also reach some point where we will have to give up on precision. There are several reasons for this conclusion which may appear paradoxical for users of classical OR theory: (i) the facts about the problem and its context are normally not completely known; (ii) the data is imprecise, incomplete and/or frequently changing; (iii) the core of the problem is too complex to be adequately understood with OR theory; (iv) the dynamics of the problem context requires a problem solving process in real time (or almost real time); and, (v) knowledge and experience (own or developed by others) are necessary for building a theory to deal with ill-structured problems. Mathematical models are also used as part of the negotiation support systems Kersten and Lai [213] work on. The precision/relevance trade-off started the development of soft computing which is where we now work on building new and better theory to cope with hard problems with smart computing methods and intelligent computing technology. As we now have introduced soft computing we will next describe a context - the forest industry. The forest industry, and especially the paper making companies, has experienced a radical change of market since the change of the millennium. Especially in Europe the stagnating growth in paper sales and the resulting overcapacity have led to decreasing paper prices, which have been hard to raise even to compensate for increasing costs. Other drivers to contribute to the misery of European paper producers have been steadily growing energy costs, growing costs of raw material and the EUR/USD exchange rate which is unfavorable for an industry which still has to invoice a large part of its customers in USD and to pay its costs in Euros. The result has been a number of restructuring measures, such as closedowns of individual paper machines and production units. Additionally, a number of macroeconomic and other trends have changed the competitive and productive environment of paper making. The current industrial logic of reacting to the cyclical demand and price dynamics with operational flexibility is losing edge because of shrinking profit margins. Simultaneously, new growth potential is found in the emerging markets of Asia, especially in China, which more and more attracts the capital invested in paper production. This imbalance between the current production capacity in Europe and the better expected return on capital invested in the emerging markets represents new challenges and uncertainties for the paper producers that are different from traditional management paradigms in the forest products industry.
4
1 Introduction
The Finnish forest industry has earlier enjoyed a productivity lead over its competitors. The lead is primarily based on a high rate of investment and the application of the most advanced technologies. Investments and growth are now curtailed by the long distance separating Finland from the large, growing markets as well as the availability and price of raw materials. Additionally, the competitiveness of Finnish companies has suffered because costs here have risen at a faster rate than in competing countries. The paper plant in UK - which is owned by a Finnish forest industry multinational and is the context for the main part of our discussion - has a somewhat different situation: advanced technology was brought in a number of years ago which improved the cost structure and the plant is in the middle of its domestic market with export a very small part of the revenue but the plant has not been profitable for a number of years. Finnish energy policy has a major impact on the competitiveness of the forest industry. The availability and price of energy, emissions trading and whether wood raw material is produced for manufacturing or energy use will affect the future success of the forest industry. If sufficient energy is available, basic industry can invest in Finland. The UK does not differ significantly from Finland in terms of the investment climate for the basic industry. We have now outlined the context; let us turn to the decision problems we will have to tackle. In decisions on how to use existing resources the challenges of changing markets become a reality when senior management has to decide how to allocate capital to production, logistics and marketing networks, and has to worry about the return on capital employed.The networks are interdependent as the demand for and the prices of fine paper products are defined by the efficiency of the customer production processes and how well suited they are to market demand; the production should be cost effective and adaptive to cyclic (and sometimes random) changes in market demand; the logistics and marketing networks should be able to react in a timely fashion to market fluctuations and to offer some buffers for the production processes. Closing or not closing a production plant is often regarded as an isolated decision, without working out the possibilities and requirements of the interdependent networks, which in many cases turn out to be a mistake. Profitability analysis has usually had an important role as the threshold phase and the key process when a decision should be made on closing or not closing a production plant. Economic feasibility is a key factor but more issues are at stake. There is also the question of what kind of profitability analysis should be used and what results we can get by using different methods. Senior management worriesand should worry - about making the best possible decisions on the close/not close situations as their decisions will be scrutinized and questioned regardless of what that decision is going to be. The shareholders will react negatively if they find out that share value will decrease (closing a profitable plant, closing a plant which may turn profitable, or not closing a plant which is not profitable, or which may turn unprofitable) and the trade unions, local and regional politicians, the press etc. will always react negatively to a decision to close a plant almost regardless of the reasons. The idea of optimality of decisions comes from normative decision theory [59]. The decisions made at various levels of uncertainty can be modeled so that the
1 Introduction
5
ranking of various alternatives can be readily achieved, either with certainty or with well-understood and non-conflicting measures of uncertainty. However, the real life complexity, both in a static and dynamic sense, makes the optimal decisions hard to find many times. What is often helpful is to relax the decision model from the optimality criteria and to use sufficiency criteria instead. Modern profitability plans are usually built with methods that originate in neoclassical finance theory. These models are by nature normative and may support decisions that in the long run may be proved to be optimal but may not be too helpful for real life decisions in a real industry setting as conditions tend to be not so well structured as shown in theory and - above all - they are not repetitive (a production plant is closed and this cannot be repeated under new conditions to get experimental data). In practice and in general terms, for profitability planning a good enough solution is many times both efficient, in the sense of smooth management processes, and effective, in the sense of finding the best way to act, as compared to theoretically optimal outcomes. Moreover, the availability of precise data for a theoretically adequate profitability analysis is often limited and subject to individual preferences and expert opinions. Especially, when cash flow estimates are worked out with one number and a risk- adjusted discount factor, various uncertain and dynamic features may be lost. The case for good enough solutions is made in fuzzy set theory [59] : at some point there will be a trade-off between precision and relevance, in the sense that increased precision can be gained only through loss of relevance and increased relevance only through the loss of precision. In a practical sense, many theoretically optimal profitability models are restricted to a set of assumptions that hinder their practical application in many real world situations. Let us consider the traditional Net Present Value (NPV) model - the assumption is that both the microeconomic productivity measures (cash flows) and the macroeconomic financial factors (discount factors) can be readily estimated several years ahead, and that the outcome of the project is tradable in the market of production assets without friction. In other words, the model has features that are unrealistic in a real world situation. Having now set the scene, the problem we will address is the decision to close - or not to close - a UK production plant in the forest products industry sector. The plant we will use as an example is producing fine paper products, it is rather aged, the paper machines were built a while ago, the raw material is not available close by, energy costs are reasonable but are increasing in the near future, key domestic markets are close by and other (export) markets (with better sales prices) will require improvements in the logistics network. This is how the decision problem was described to us - the management team did not use precise figures and did not have them readily available, which made us believe that the joint understanding was formed in these imprecise terms. The intuitive conclusion is, of course in the same imprecise terms, that we have a sunset case and senior management should make a simple, macho decision and close the plant. On the other hand we have the UK trade unions, which are strong, and we have pension funds commitments until 2013 which are very strict, and we have long-term energy contracts which are expensive to get out of. Finally, by closing the
6
1 Introduction
plant we will invite competitors to fight us in the UK markets we have served for more than 50 years and which we cannot serve from other plants at any reasonable cost. We learned that intuitive decision making gives inferior results to systematic analytical decision processes - we found out that the possibilities formed with analytical models simply were not known before and that they represented solutions with surprising and positive consequences. Following our findings we will show that these decision processes will not be possible without effective information systems support. We have used the case of the aging fine paper mill as a context and background for the work we have carried out in chapters 2-4 to build the instruments that are needed to come to terms with the problems we have to face, understand and solve. The case itself is worked out in detail in chapter 5 where we also work out the details of the fuzzy real options methods we have developed and tested with a number of industrial applications. In chapter 6 the tools are further explored for portfolio selection problems. The theory and the instruments we developed in chapters 2-4 are then applied to risk assessment in grid computing (in chapter 7) where we show that the classical Bayes models can be extended with predictive possibilities; then we move on to use the results for fuzzy ontology (chapter 8) and the finally as a basis for mobile value services (chapter 9).
Chapter 2
Concepts and Issues
2.1
Fuzzy Sets
Fuzzy sets were introduced by Zadeh [351] in 1965 to represent/manipulate data and information possessing nonstatistical uncertainties. It was specifically designed to mathematically represent uncertainty and vagueness and to provide formalized tools for dealing with the imprecision intrinsic to many problems. Fuzzy sets serve as a means of representing and manipulating data that was not precise, but rather fuzzy. Some of the essential characteristics of fuzzy logic relate to the following [356]: (i) In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning; (ii) In fuzzy logic, everything is a matter of degree; (iii) In fuzzy logic, knowledge is interpreted a collection of elastic or, equivalently, fuzzy constraint on a collection of variables; (iv) Inference is viewed as a process of propagation of elastic constraints; and (v) Any logical system can be fuzzified. There are two main characteristics of fuzzy systems that give them better performance for specific applications: (i) Fuzzy systems are suitable for uncertain or approximate reasoning, especially for systems with mathematical models that are difficult to derive; and (ii) Fuzzy logic allows decision making with estimated values under incomplete or uncertain information. Definition 2.1.1 [351] Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function μA : X → [0, 1], and μA (x) is interpreted as the degree of membership of element x in fuzzy set A for each x ∈ X. It should be noted that the terms membership function and fuzzy subset get used interchangeably and frequently we will write simply A(x) instead of μA (x). The family of all fuzzy (sub)sets in X is denoted by F (X). Fuzzy subsets of the real line are called fuzzy quantities. Let A be a fuzzy subset of X; the support of A, denoted supp(A), is the crisp subset of X whose elements all have nonzero membership grades in A. A fuzzy subset A of a classical set X is called normal if there exists an C. Carlsson and R. Full´er: Possibility for Decision, STUDFUZZ 270, pp. 7–25. springerlink.com © Springer-Verlag Berlin Heidelberg 2011
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2 Concepts and Issues
x ∈ X such that A(x) = 1. Otherwise A is subnormal. An α -level set (or α -cut) of a fuzzy set A of X is a non-fuzzy set denoted by [A]α and defined by [A]α = {t ∈ X |A(t) ≥ α }, if α > 0 and cl(suppA) if α = 0, where cl(suppA) denotes the closure of the support of A. A fuzzy set A of X is called convex if [A]α is a convex subset of X for all α ∈ [0, 1]. Definition 2.1.2 A fuzzy number A is a fuzzy set of the real line with a normal, (fuzzy) convex and upper semi-continuous membership function of bounded support. The family of fuzzy numbers will be denoted by F . Let A be a fuzzy number. Then [A]γ is a closed convex (compact) subset of R for all γ ∈ [0, 1]. Let us introduce the notations a1 (γ ) = min[A]γ and a2 (γ ) = max[A]γ . In other words, a1 (γ ) denotes the left-hand side and a2 (γ ) denotes the right-hand side of the γ -cut. It is easy to see that if α ≤ β then [A]α ⊃ [A]β . Furthermore, the left-hand side function a1 : [0, 1] → R is monotone increasing and lower semicontinuous, and the right-hand side function a2 : [0, 1] → R is monoton decreasing and upper semi-continuous. we will use the notation [A]γ = [a1 (γ ), a2 (γ )]. The support of A is the open interval (a1 (0), a2 (0)). If A is not a fuzzy number then there exists an γ ∈ [0, 1] such that [A]γ is not a convex subset of R. Fig. 2.1 Triangular fuzzy number
1
a-
a
a+
Definition 2.1.3 A fuzzy set A is called triangular fuzzy number with peak (or center) a, left width α > 0 and right width β > 0 if its membership function has the following form ⎧ a−t ⎪ ⎪ 1− if a − α ≤ t ≤ a ⎪ ⎪ ⎪ α ⎨ t −a A(t) = 1− if a ≤ t ≤ a + β ⎪ ⎪ ⎪ β ⎪ ⎪ ⎩ 0 otherwise and we use the notation A = (a, α , β ). It can easily be verified that [A]γ = [a − (1 − γ )α , a + (1 − γ )β ], ∀γ ∈ [0, 1].
2.1 Fuzzy Sets
9
The support of A is (a − α , b + β ). A triangular fuzzy number with center a may be seen as a fuzzy quantity ”x is approximately equal to a”. Definition 2.1.4 A fuzzy set A is called trapezoidal fuzzy number with tolerance interval [a, b], left width α and right width β if its membership function has the following form ⎧ ⎪ ⎪ 1 − a − t if a − α ≤ t ≤ a ⎪ ⎪ ⎪ α ⎪ ⎪ ⎪ ⎨1 if a ≤ t ≤ b A(t) = ⎪ t −b ⎪ ⎪ 1− if a ≤ t ≤ b + β ⎪ ⎪ β ⎪ ⎪ ⎪ ⎩ 0 otherwise and we use the notation
A = (a, b, α , β ).
(2.1)
It can easily be shown that [A]γ = [a − (1 − γ )α , b + (1 − γ )β ] for all γ ∈ [0, 1]. The support of A is (a − α , b + β ).
Fig. 2.2 Trapezoidal fuzzy number 1
a-
a
b
b+
A trapezoidal fuzzy number may be seen as a fuzzy quantity ”x is approximately in the interval [a, b]”. Definition 2.1.5 Any fuzzy number A ∈ F can be described as ⎧ ⎪ a − t ⎪ ⎪ L if t ∈ [a − α , a] ⎪ ⎪ ⎪ α ⎪ ⎪ ⎪ ⎪ ⎨1 if t ∈ [a, b] A(t) = ⎪ ⎪ t − b) ⎪ ⎪ R if t ∈ [b, b + β ] ⎪ ⎪ β ⎪ ⎪ ⎪ ⎪ ⎩ 0 otherwise where [a, b] is the peak or core of A, L : [0, 1] → [0, 1] and R : [0, 1] → [0, 1] are continuous and non-increasing shape functions with L(0) = R(0) = 1 and R(1) = L(1) = 0. We call this fuzzy interval of LR-type and refer to it by A = (a, b, α , β )LR
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2 Concepts and Issues
The support of A is (a − α , b + β ). Definition 2.1.6 Let A = (a, b, α , β )LR be a fuzzy number of type LR. If a = b then we use the notation (2.2) A = (a, α , β )LR and say that A is a quasi-triangular fuzzy number. Furthermore if L(x) = R(x) = 1 − x, then instead of A = (a, b, α , β )LR we write A = (a, b, α , β ). Let A and B are fuzzy subsets of a classical set X = 0. / We say that A is a subset of B if A(t) ≤ B(t) for all t ∈ X. Furthermore, A and B are said to be equal, denoted A = B, if A ⊂ B and B ⊂ A. We note that A = B if and only if A(x) = B(x) for all x ∈ X. The intersection of A and B is defined as (A ∩ B)(t) = min{A(t), B(t)} = A(t) ∧ B(t), ∀t ∈ X. The union of A and B is defined as (A ∪ B)(t) = max{A(t), B(t)} = A(t) ∨ B(t), ∀t ∈ X. The complement of a fuzzy set A is defined as (¬A)(t) = 1 − A(t), ∀t ∈ X. A fuzzy set r¯ in the real line is said to be a fuzzy point, if its membership function is defined by 1 if z = r, r¯(z) = 0 if z = r. That is, r¯ is nothing else but the characteristic function of the singleton {r}. Triangular norms were introduced by Schweizer and Sklar [292] to model distances in probabilistic metric spaces. In fuzzy sets theory triangular norms are extensively used to model logical connective and. Definition 2.1.7 A mapping T : [0, 1]× [0, 1] → [0, 1] is said to be a triangular norm (t-norm for short) iff it is symmetric, associative, non-decreasing in each argument and T (a, 1) = a, for all a ∈ [0, 1]. In other words, any t-norm T satisfies the properties: T (x, y) = T (y, x), ∀x, y ∈ [0, 1] (symmetricity) T (x, T (y, z)) = T (T (x, y), z), ∀x, y, z ∈ [0, 1] (associativity) T (x, y) ≤ T (x , y ) if x ≤ x and y ≤ y (monotonicity) T (x, 1) = x, ∀x ∈ [0, 1] (one identy) These axioms attempt to capture the basic properties of set intersection. The basic t-norms are: • minimum (or Mamdani [246]): TM (a, b) = min{a, b}, • Łukasiewicz: TL (a, b) = max{a + b − 1, 0} • product (or Larsen [222]): TP (a, b) = ab
2.1 Fuzzy Sets
11
• weak: TW (a, b) =
min{a, b} if max{a, b} = 1 0
otherwise
• Hamacher [176]: Hγ (a, b) =
ab , γ ≥0 γ + (1 − γ )(a + b − ab)
(2.3)
All t-norms may be extended, through associativity, to n > 2 arguments. A t-norm T is called strict if T is strictly increasing in each argument. A t-norm T is said to be Archimedean iff T is continuous and T (x, x) < x for all x ∈ (0, 1). Every Archimedean t-norm T is representable by a continuous and decreasing function f : [0, 1] → [0, ∞] with f (1) = 0 and T (x, y) = f −1 ( min{ f (x) + f (y), f (0)} ). The function f is the additive generator of T . A t-norm T is said to be nilpotent if T (x, y) = 0 holds for some x, y ∈ (0, 1). The operation intersection can be defined by the help of triangular norms. Definition 2.1.8 Let T be a t-conorm. The T -intersection of A and B is defined as (A ∩ B)(t) = T (A(t), B(t)), ∀t ∈ X . Triangular conorms are extensively used to model logical connective or. Definition 2.1.9 A mapping S : [0, 1] × [0, 1] → [0, 1] is said to be a triangular conorm (t-conorm) if it is symmetric, associative, non-decreasing in each argument and S(a, 0) = a, for all a ∈ [0, 1]. In other words, any t-conorm S satisfies the properties: S(x, y) = S(y, x) (symmetricity) S(x, S(y, z)) = S(S(x, y), z) (associativity) S(x, y) ≤ S(x , y ) if x ≤ x and y ≤ y (monotonicity) S(x, 0) = x, ∀x ∈ [0, 1] (zero identy) If T is a t-norm then the equality S(a, b) := 1 − T (1 − a, 1 −b), defines a t-conorm and we say that S is derived from T . The basic t-conorms are: • • • •
maximum: SM (a, b) = max{a, b} Łukasiewicz: SL (a, b) = min{a + b, 1} probabilistic: SP (a, b) = a + b − ab strong:
max{a, b} if min{a, b} = 0 ST RONG(a, b) = 1 otherwise
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2 Concepts and Issues
• Hamacher: HORγ (a, b) =
a + b − (2 − γ )ab , γ ≥0 1 − (1 − γ )ab
The operation union can be defined by the help of triangular conorms. Definition 2.1.10 Let S be a t-conorm. The S-union of A and B is defined as (A ∪ B)(t) = S(A(t), B(t)), ∀t ∈ X .
2.2
The Extension Principle
In order to use fuzzy numbers and relations in any intelligent system we must be able to perform arithmetic operations with these fuzzy quantities. In particular, we must be able to to add, subtract, multiply and divide with fuzzy quantities. The process of doing these operations is called fuzzy arithmetic. we will first introduce an important concept from fuzzy set theory called the extension principle. We then use it to provide for these arithmetic operations on fuzzy numbers. In general the extension principle pays a fundamental role in enabling us to extend any point operations to operations involving fuzzy sets. In the following we define this principle. Definition 2.2.1 (Zadeh’s extension principle) Assume X and Y are crisp sets and let f be a mapping from X to Y , f : X → Y , such that for each x ∈ X , f (x) = y ∈ Y . Assume A is a fuzzy subset of X, using the extension principle, we can define f (A) as a fuzzy subset of Y such that supx∈ f −1 (y) A(x) if f −1 (y) = 0/ f (A)(y) = (2.4) 0 otherwise where f −1 (y) = {x ∈ X | f (x) = y}. If f (x) = λ x and A ∈ F then we will write f (A) = λ A. Especially, if λ = −1 then we have (−1A)(x) = (−A)(x) = A(−x), x ∈ R. It should be noted that Zadeh’s extension principle is nothing else but a straightforward generalization of set-valued functions (see [235] for details). The extension principle can be generalized to n-place functions using the sup-min operator. Definition 2.2.2 Let X1 , X2 , . . . , Xn and Y be a family of sets. Assume f is a mapping f : X1 × X2 × · · · × Xn → Y, that is, for each n-tuple (x1 , . . . , xn ) such that xi ∈ Xi , we have f (x1 , x2 , . . . , xn ) = y ∈ Y.
2.2 The Extension Principle
13
Let A1 , . . . , An be fuzzy subsets of X1 , . . . , Xn , respectively; then the (sup-min) extension principle allows for the evaluation of f (A1 , . . . , An ). In particular, f (A1 , . . . , An ) = B, where B is a fuzzy subset of Y such that f (A1 , . . . , An )(y) = sup{min{A1 (x1 ), . . . , An (xn )} | x ∈ f −1 (y)} if f −1 (y) = 0/ 0 otherwise.
(2.5)
For n = 2 then the sup-min extension principle reads f (A1 , A2 )(y) =
sup
{A1 (x1 ), A2 (x2 )}.
f (x1 ,x2 )=y
Example 2.2.1 Let f : X × X → X be defined as f (x1 , x2 ) = x1 + x2 , i.e. f is the addition operator. Suppose A1 and A2 are fuzzy subsets of X. Then using the supmin extension principle we get f (A1 , A2 )(y) = sup min{A1 (x1 ), A2 (x2 )} x1 +x2 =y
(2.6)
and we use the notation f (A1 , A2 ) = A1 + A2 . Example 2.2.2 Let f : X × X → X be defined as f (x1 , x2 ) = x1 − x2 , i.e. f is the subtraction operator. Suppose A1 and A2 are fuzzy subsets of X. Then using the sup-min extension principle we get f (A1 , A2 )(y) = sup min{A1 (x1 ), A2 (x2 )}, x1 −x2 =y
and we use the notation f (A1 , A2 ) = A1 − A2 . The sup-min extension principle for n-place functions is also a straightforward generalization of set-valued functions. Namely, let f : X1 × X2 → Y be a function. Then the image of a (crisp) subset (A1 , A2 ) ⊂ X1 × X2 by f is defined by f (A1 , A2 ) = { f (x1 , x2 ) | x1 ∈ A and x2 ∈ A2 } and the characteristic function of f (A1 , A2 ) is
χ f (A1 ,A2 ) (y) = sup{min{ χA1 (x), χA2 (x)} | x ∈ f −1 (y)}. Then replacing the characteristic functions by fuzzy sets we get Zadeh’s sup-min extension principle for n-place functions (2.5). Let A = (a1 , a2 , α1 , α2 )LR , and B = (b1 , b2 , β1 , β2 )LR , be fuzzy numbers of LRtype. Using the sup-min extension principle we can verify the following rules for addition and subtraction of fuzzy numbers of LR-type. A + B = (a1 + b1 , a2 + b2, α1 + β1 , α2 + β2)LR
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2 Concepts and Issues
A − B = (a1 − b2 , a2 − b1 , α1 + β2 , α2 + β1 )LR . In particular if A = (a1 , a2 , α1 , α2 ) and B = (b1 , b2 , β1 , β2 ) are fuzzy numbers of trapezoidal form then A + B = (a1 + b1, a2 + b2 , α1 + β1 , α2 + β2 )
(2.7)
A − B = (a1 − b2, a2 − b1 , α1 + β2, α2 + β1 ).
(2.8)
If A = (a, α1 , α2 ) and B = (b, β1 , β2 ) are fuzzy numbers of triangular form then A + B = (a + b, α1 + β1 , α2 + β2),
A − B = (a − b, α1 + β2 , α2 + β1 )
and if A = (a, α ) and B = (b, β ) are fuzzy numbers of symmetric triangular form then A + B = (a + b, α + β ),
A − B = (a − b, α + β ),
λ A = (λ a, |λ |α ).
Let A and B be fuzzy numbers with [A]α = [a1 (α ), a2 (α )] and [B]α = [b1 (α ), b2 (α )]. Then it can easily be shown that [A + B]α = [a1 (α ) + b1 (α ), a2 (α ) + b2 (α )], [A − B]α = [a1 (α ) − b2 (α ), a2 (α ) − b1 (α )], [λ A]α = λ [A]α ,
where [λ A]α = [λ a1 (α ), λ a2 (α )] if λ ≥ 0 and [λ A]α = [λ a2 (α ), λ a1 (α )] if λ < 0 for all α ∈ [0, 1], i.e. any α -level set of the extended sum of two fuzzy numbers is equal to the sum of their α -level sets. We note here that from sup min{A1 (x1 ), A2 (x2 )} = sup min{A1 (x1 ), A2 (−x2 )},
x1 −x2 =y
x1 +x2 =y
it follows that the equality A1 − A2 = A1 + (−A2 ) holds. However A − A is defined by the sup-min extension principle as (A − A)(y) = sup min{A(x1 ), A(x2 )}, y ∈ R x1 −x2 =y
which turns into [A − A]α = [a1 (α ) − a2 (α ), a2 (α ) − a1 (α )], which is generally not a fuzzy point. Theorem 2.2.1 (Nguyen, [266]) Let f : X → X be a continuous function and let A be fuzzy numbers. Then [ f (A)]α = f ([A]α ) where f (A) is defined by the extension principle (2.4) and f ([A]α ) = { f (x) | x ∈ [A]α }.
2.2 The Extension Principle
15
If [A]α = [a1 (α ), a2 (α )] and f is monoton increasing then from the above theorem we get [ f (A)]α = f ([A]α ) = f ([a1 (α ), a2 (α )]) = [ f (a1 (α )), f (a2 (α ))]. Theorem 2.2.2 (Nguyen, [266]) Let f : X × X → X be a continuous function and let A and B be fuzzy numbers. Then [ f (A, B)]α = f ([A]α , [B]α ), where f ([A]α , [B]α ) = { f (x1 , x2 ) | x1 ∈ [A]α , x2 ∈ [B]α }. Let f (x, y) = xy and let [A]α = [a1 (α ), a2 (α )] and [B]α = [b1 (α ), b2 (α )] be two fuzzy numbers. Applying Theorem 2.2.2 we get [ f (A, B)]α = f ([A]α , [B]α ) = [A]α [B]α . However the equation [AB]α = [A]α [B]α = [a1 (α )b1 (α ), a2 (α )b2 (α )] holds if and only if A and B are both nonnegative, i.e. A(x) = B(x) = 0 for x ≤ 0. In the definition of the extension principle one can use any t-norm for modeling the conjunction operator. Definition 2.2.3 Let T be a t-norm and let f be a mapping from X1 × X2 × · · · × Xn to Y , Assume (A1 , . . . , An ) is a fuzzy subset of X1 × X2 × · · · × Xn , using the extension principle, we can define f (A1 , A2 , . . . , An ) as a fuzzy subset of Y such that f (A1 , A2 , . . . , An )(y) =
sup{T (A1 (x), . . . , An (x)) | x ∈ f −1 (y)} if f −1 (y) = 0/ 0
otherwise
(2.9)
This is called the sup-T (or generally sup-t-norm) extension principle. Specially, if T is a t-norm and ∗ is a binary operation on R then ∗ can be extended to fuzzy quantities in the sense of the sup-T extension principle as A1 ∗ A2 (z) = sup T A1 (x1 ), A2 (x2 ) , z ∈ R. x1 ∗x2 =z
For example, if A and B are fuzzy numbers, TP (u, v) = uv is the product t-norm and f (x1 , x2 ) = x1 + x2 is the addition operation on the real line then the sup-product extended sum of A and B, called product-sum and denoted by A + B, is defined by
16
2 Concepts and Issues
f (A, B)(y) = (A + B)(y) = sup T (A1 (x1 ), A2 (x2 )) x1 +x2 =y
= sup A1 (x1 )A2 (x2 ). x1 +x2 =y
Let symbol F (X ) denote the family of all fuzzy subsets of a set X . When X is a topological space, we denote by F (X , K ) the set of all fuzzy subsets of X having upper semicontinuous, compactly-supported membership function. The following theorem illustrates that, if we use an arbitrary t-norm T in Zadeh’s extension principle, then we obtain results similar to those of Nguyen. Theorem 2.2.3 (Full´er and Keresztfalvi, [149]) Let X, Y , Z be locally compact topological spaces. If f : X × Y → Z is continuous and the t-norm T is upper semicontinuous, then the following equality holds for each A ∈ F (X , K ) and B ∈ F (Y, K )
[ f (A, B)]α =
f ([A]ξ , [B]η ), α ∈ (0, 1],
T (ξ ,η )≥α
where f (A, B) is defined by the sup-T extension principle (2.9).
2.3
Averaging Operators
In a decision process the idea of trade-offs corresponds to viewing the global evaluation of an action as lying between the worst and the best local ratings. This occurs in the presence of conflicting goals, when a compensation between the corresponding compatibilities is allowed. Averaging operators realize trade-offs between objectives, by allowing a positive compensation between ratings. An averaging (or mean) operator M is a function M : [0, 1]× [0, 1] → [0, 1] satisfying the following properties • • • • •
M(x, x) = x, ∀x ∈ [0, 1], (idempotency) M(x, y) = M(y, x), ∀x, y ∈ [0, 1], (commutativity) M(0, 0) = 0, M(1, 1) = 1, (extremal conditions) M(x, y) ≤ M(x , y ) if x ≤ x and y ≤ y (monotonicity) M is continuous
It is easy to see that if M is an averaging operator then min{x, y} ≤ M(x, y) ≤ max{x, y}, ∀x, y ∈ [0, 1] An important family of averaging operators is formed by quasi-arithmetic means M(a1 , . . . , an ) = f −1
1 n f (a ) ∑ i n i=1
2.3 Averaging Operators
17
This family has been characterized by Kolmogorov as being the class of all decomposable continuous averaging operators. For example, the quasi-arithmetic mean of a1 and a2 is defined by f (a1 ) + f (a2 ) M(a1 , a2 ) = f −1 . 2 The most often used mean operators are √ • geometric mean: xy, • median: ⎧ ⎪ ⎨ y if x ≤ y ≤ α , med(x, y, α ) = α if x ≤ α ≤ y, ⎪ ⎩ x if α ≤ x ≤ y, where α ∈ (0, 1). • harmonic mean:
2xy , x+y
• dual of harmonic mean:
x + y − 2xy , 2−x−y
x+y , 2 • dual of geometric mean: 1 − (1 − x)(1 − y), p x + y p 1/p • generalized p-mean: for p ≥ 1. 2 • arithmetic mean:
The concept of ordered weighted averaging (OWA) operators was introduced by Yager in 1988 [338] as a way for providing aggregations which lie between the maximum and minimums operators. The structure of this operator involves a nonlinearity in the form of an ordering operation on the elements to be aggregated. The OWA operator provides a new information aggregation technique and has already aroused considerable research interest [49, 344]. Definition 2.3.1 ([338]) An OWA operator of dimension n is a mapping F : Rn → R, that has an associated weighting vector W = (w1 , w2 , . . . , wn )T such as wi ∈ [0, 1], 1 ≤ i ≤ n, and w1 + · · · + wn = 1. Furthermore F(a1 , . . . , an ) = w1 b1 + · · · + wn bn =
n
∑ w jb j,
j=1
where b j is the j-th largest element of the bag a1 , . . . , an . A fundamental aspect of this operator is the re-ordering step, in particular an aggregate ai is not associated with a particular weight wi but rather a weight is associated with a particular ordered position of aggregate. When we view the OWA weights as a column vector we will find it convenient to refer to the weights with the low indices as weights at the top and those with the higher indices with weights at the
18
2 Concepts and Issues
bottom. It is noted that different OWA operators are distinguished by their weighting function. In [338] Yager pointed out three important special cases of OWA aggregations: • F ∗ : In this case W = W ∗ = (1, 0 . . . , 0)T and F ∗ (a1 , . . . , an ) = max{a1 , . . . , an }, • F∗ : In this case W = W∗ = (0, 0 . . . , 1)T and F∗ (a1 , . . . , an ) = min{a1 , . . . , an }, a1 + · · · + an • FA : In this case W = WA = (1/n, . . . , 1/n)T and FA (a1 , . . . , an ) = . n A number of important properties can be associated with the OWA operators. we will now discuss some of these. For any OWA operator F holds F∗ (a1 , . . . , an ) ≤ F(a1 , . . . , an ) ≤ F ∗ (a1 , . . . , an ). Thus the upper an lower star OWA operator are its boundaries. From the above it becomes clear that for any F min{a1 , . . . , an } ≤ F(a1 , . . . , an ) ≤ max{a1 , . . . , an }. The OWA operator can be seen to be commutative. Let a1 , . . . , an be a bag of aggregates and let {d1 , . . . , dn } be any permutation of the ai . Then for any OWA operator F(a1 , . . . , an ) = F(d1 , . . . , dn ). A third characteristic associated with these operators is monotonicity. Assume ai and ci are a collection of aggregates, i = 1, . . . , n such that for each i, ai ≥ ci . Then F(a1 , . . . , an ) ≥ F(c1 , c2 , . . . , cn ), where F is some fixed weight OWA operator. Another characteristic associated with these operators is idempotency. If ai = a for all i then for any OWA operator F(a1 , . . . , an ) = a. From the above we can see the OWA operators have the basic properties associated with an averaging operator. Example 2.3.1 A window type OWA operator takes the average of the m arguments around the center. For this class of operators we have ⎧ 0 if i < k ⎪ ⎪ ⎪ ⎨ 1 wi = (2.10) if k ≤ i < k + m ⎪ m ⎪ ⎪ ⎩ 0 if i ≥ k + m In order to classify OWA operators in regard to their location between and and or, a measure of orness, associated with any vector W is introduced by Yager [338] as follows 1 n orness(W ) = ∑ (n − i)wi. n − 1 i=1 It is easy to see that for any W the orness(W ) is always in the unit interval. Furthermore, note that the nearer W is to an or, the closer its measure is to one; while the nearer it is to an and, the closer is to zero. It can easily be shown that orness(W ∗ ) = 1, orness(W∗ ) = 0 and orness(WA ) = 0.5. A measure of andness is
2.3 Averaging Operators
19
defined as andness(W ) = 1 − orness(W ). Generally, an OWA operator with much of nonzero weights near the top will be an orlike operator, that is, orness(W ) ≥ 0.5, and when much of the weights are nonzero near the bottom, the OWA operator will be andlike, that is, andness(W ) ≥ 0.5. Example 2.3.2 Let W = (0.8, 0.2, 0.0)T . Then orness(W ) = 1/3(2 × 0.8 + 0.2) = 0.6, and andness(W ) = 1 − orness(W ) = 1 − 0.6 = 0.4. This means that the OWA operator, defined by F(a1 , a2 , a3 ) = 0.8b1 + 0.2b2 + 0.0b3 = 0.8b1 + 0.2b2, where b j is the j-th largest element of the bag a1 , a2 , a3 , is an orlike aggregation. In [338] Yager defined the measure of dispersion (or entropy) of an OWA vector by n
disp(W ) = − ∑ wi ln wi . i=1
We can see when using the OWA operator as an averaging operator disp(W ) measures the degree to which we use all the aggregates equally. If F is an OWA aggregation with weights wi the dual of F denoted F R , is an OWA aggregation of the same dimention where with weights wRi , wRi = wn−i+1 . We can easily see that if F and F R are duals then disp(F R ) = disp(F), orness(F R ) = 1 − orness(F) = andness(F). Thus is F is orlike its dual is andlike. Example 2.3.3 Let W = (0.3, 0.2, 0.1, 0.4)T and W R = (0.4, 0.1, 0.2, 0.3)T . Then orness(F) = 1/3(3 × 0.3 + 2 × 0.2 + 0.1) ≈ 0.466, orness(F R ) = 1/3(3 × 0.4 + 2 × 0.1 + 0.2) ≈ 0.533. The actual type of aggregation performed by an OWA operator depends upon the form of the weighting vector. A number of approaches have been suggested for obtaining the associated weights, i.e., quantifier guided aggregation [338, 339], exponential smoothing [39, 144] and learning [346]. An important application of the OWA operators is in the area of quantifier guided aggregations [338]. Assume {A1 , . . . , An }, is a collection of criteria. Let x be an object such that for any criterion Ai , Ai (x) ∈ [0, 1] indicates the degree to which this criterion is satisfied by x. If we want to find out the degree to which x satisfies ”all the criteria” denoting this by D(x), we get following Bellman and Zadeh [22]: D(x) = min{A1 (x), . . . , An (x)}.
(2.11)
20
2 Concepts and Issues
In this case we are essentially requiring x to satisfy ”A1 and A2 and · · · and An ”. If we desire to find out the degree to which x satisfies ”at least one of the criteria”, denoting this E(x), we get E(x) = max{A1 (x), . . . , An (x)}. In this case we are requiring x to satisfy ”A1 or A2 or · · · or An ”. In many applications rather than desiring that a solution satisfies one of these extreme situations, ”all” or ”at least one”, we may require that x satisfies most or at least half of the criteria. Drawing upon Zadeh’s concept [355] of linguistic quantifiers we can accomplish these kinds of quantifier guided aggregations. Definition 2.3.2 A quantifier Q is called • regular monotonically non-decreasing if Q(0) = 0,
Q(1) = 1,
if r1 > r2 then Q(r1 ) ≥ Q(r2 ).
• regular monotonically non-increasing if Q(0) = 1,
Q(1) = 0,
if r1 < r2 then Q(r1 ) ≥ Q(r2 ).
• regular unimodal if ⎧ 0 if r = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ monotone increasing if 0 ≤ r ≤ a ⎨ if a ≤ r ≤ b, 0 < a < b < 1 Q(r) = 1 ⎪ ⎪ ⎪ ⎪ monotone decreasing if b ≤ r ≤ 1 ⎪ ⎪ ⎩ 0 if r = 1 With ai = Ai (x) the overall valuation of x is FQ (a1 , . . . , an ) where FQ is an OWA operator. The weights associated with this quantified guided aggregation are obtained as follows i i−1 wi = Q −Q , i = 1, . . . , n. (2.12) n n Theorem 2.3.1 ([339]) If we construct wi via the method (2.12) we always get w1 + · · · + wn = 1 and wi ≥ 0 for any function Q : [0, 1] → [0, 1], satisfying the conditions of a regular nondecreasing quantifier. We call any function satisfying the conditions of a regular non-decreasing quantifier an acceptable OWA weight generating function. Let us look at the weights generated from some basic types of quantifiers. The quantifier, for all Q∗ , is defined such that 0 for r < 1, Q∗ (r) = 1 for r = 1.
2.4 Possibility Distributions
21
Using our method for generating weights wi = Q∗ (i/n) − Q∗ ((i − 1)/n) we get 0 for i < n, wi = 1 for i = n. This is exactly what we previously denoted as W∗ . For the quantifier there exists we have 0 for r = 0, ∗ Q (r) = 1 for r > 0. In this case we get w1 = 1,
wi = 0, for i = 1.
This is exactly what we denoted as W ∗ . Consider next the quantifier defined by Q(r) = r. This is an identity or linear type quantifier. In this case we get wi = Q
i i−1 i i−1 1 −Q = − = . n n n n n
This gives us the pure averaging OWA aggregation operator. Recapitulating using the approach suggested by Yager if we desire to calculate FQ (a1 , . . . , an ) for Q being a regular non-decreasing quantifier we proceed as follows: • Calculate
i i−1 wi = Q −Q . n n
• Calculate FQ (a1 , . . . , an ) = w1 b1 + · · · + wn bn . where bi is the ith largest of the a j . For example, the weights of the window-type OWA operator given by equation (2.10) can be derived from the quantifier ⎧ 0 if r ≤ (k − 1)/n ⎪ ⎪ ⎪ ⎨ (k − 1 + m) − nr Q(r) = 1 − if (k − 1)/n ≤ r ≤ (k − 1 + m)/n ⎪ m ⎪ ⎪ ⎩ 1 if (k − 1 + m)/n ≤ r ≤ 1
2.4
Possibility Distributions
Fuzzy numbers can be considered as possibility distributions [351, 354]. If A ∈ F is a fuzzy number and x ∈ R a real number then A(x) can be interpreted as the degree of possiblity of the statement ”x is A”. Let a, b ∈ R ∪ {−∞, ∞} with a ≤ b, then the degree of possibility that A ∈ F takes its value from [a, b] is defined by [354]
22
2 Concepts and Issues
Pos(A ∈ [a, b]) = max A(x). x∈[a,b]
The degree of necessity that A ∈ F takes its value from [a, b] is defined by Nec(A ∈ [a, b]) = 1 − Pos(A ∈ / [a, b]). A fuzzy set C in Rn is said to be a joint possibility distribution of fuzzy numbers Ai ∈ F , i = 1, . . . , n, if it satisfies the relationship max C(x1 , . . . , xn ) = Ai (xi )
x j ∈R, j =i
(2.13)
for all xi ∈ R, i = 1, . . . , n. Furthermore, Ai is called the ith marginal possibility distribution of C, and the projection of C on the ith axis is Ai for i = 1, . . . , n. Let C denote a joint possibility distribution of A, B ∈ F . Then C should satisfy the relationships max C(x1 , y) = A(x1 ), y∈R
and, max C(y, x2 ) = B(x2 ) y∈R
for all x1 , x2 ∈ R. If Ai ∈ F , i = 1, . . . , n and C is their joint possibility distribution then the relationships C(x1 , . . . , xn ) ≤ min{A1 (x1 ), . . . , An (xn )}, or, equivalently,
[C]γ ⊆ [A1 ]γ × · · · × [An ]γ
hold for all x1 , . . . , xn ∈ R and γ ∈ [0, 1]. Definition 2.4.1 Fuzzy numbers Ai ∈ F , i = 1, . . . , n are said to be non-interactive if their joint possibility distribution C satisfies the relationship C(x1 , . . . , xn ) = min{A1 (x1 ), . . . , An (xn )}, or, equivalently,
[C]γ = [A1 ]γ × · · · × [An ]γ
hold for all x1 , . . . , xn ∈ R and γ ∈ [0, 1]. If A, B ∈ F are non-interactive then their joint membership function is defined by A × B, where (A × B)(x, y) = min{A(x), B(y)} for any x, y ∈ R. It is clear that in this case any change in the membership function of A does not effect the second marginal possibility distribution and vice versa. On the other hand, A and B are said to be interactive if they can not take their values independently of each other [127].
2.4 Possibility Distributions
23
Definition 2.4.2 [67] Let C be the joint possibility distribution of (marginal possibility distributions) A1 , . . . , An ∈ F , and let f : Rn → R be a continuous function. Then fC (A1 , . . . , An ) ∈ F , will be defined by fC (A1 , . . . , An )(y) =
sup y= f (x1 ,...,xn )
C(x1 , . . . , xn ).
(2.14)
Note 2.1. If A1 , . . . , An are non-interactive, that is, their joint possibility distribution is defined by C(x1 , . . . , xn ) = min{A1 (x1 ), . . . , An (xn )}, then (2.14) turns into the extension principle (2.4) introduced by Zadeh in 1965 [351], f (A1 , . . . , An )(y) = sup min{A1 (x1 ), . . . , An (xn )}. y= f (x1 ,...,xn )
Furthermore, if C(x1 , . . . , xn ) = T (A1 (x1 ), . . . , An (xn )), where T is a t-norm then we get the t-norm-based extension principle, fC (A1 , . . . , An )(y) =
sup y= f (x1 ,...,xn )
T (A1 (x1 ), . . . , An (xn )).
We have the following lemma, which can be interpreted as a generalization of Nguyen’s theorem [266]. Theorem 2.4.1 [67] Let A1 , . . . , An ∈ F be fuzzy numbers, let C be their joint possibility distribution, and let f : Rn → R be a continuous function. Then, [ fC (A1 , . . . , An )]γ = f ([C]γ ), for all γ ∈ [0, 1]. Although possibility measure has been widely used, it has no self-duality property. This was the main motivation behind the concept of credibility measure which was first defined in [225], where the authors used this subclass of fuzzy measures to define the expected value of a fuzzy random variable ξ . Credibility theory was founded by Liu [228], and later, Li and Liu [224] gave the following four axioms as a sufficient and necessary condition for a credibility measure (Θ is a nonempty set and P(Θ ) is the power set of Θ ): 1. 2. 3. 4.
Cr{Θ } = 1; Cr is is a Choquet capacity: Cr{C} ≤ Cr{D}, if C ⊂ D; Cr isself-dual: Cr{C} + Cr{Θ \ C} = 1 for any C ∈ P(Θ ); Cr{ i Ci } ∧ 0.5 = supi Cr{Ci } for any {Ci } with Cr{Ci } ≤ 0.5.
It is easy to see that Cr{0} / = 0, and 0 ≤ Cr{C} ≤ 1 for any C ∈ P(Θ ). The credibility measure is subadditive (see [228]),
24
2 Concepts and Issues
Cr{C ∪ D} ≤ Cr{C} + Cr{D} for any C, D ∈ P(Θ ). To establish the connection between a fuzzy variable and a credibility measure, both defined on the credibility space (Θ , P(Θ ), Cr), we can see a fuzzy variable, A, as a function from this space to the set of real numbers, and its membership function can be derived from the credibility measure by μ (x) = min{2Cr{A = x}, 1}, for any x ∈ R. We call {A ∈ B} a fuzzy event, where B is a set of real numbers. However, in practice a fuzzy variable is specified by its membership function. In this case we can calculate the credibility of fuzzy events by the credibility inversion theorem [228]: Let A be a fuzzy variable with membership function μ . Then for any set B of real numbers, we have 1 Cr{A ∈ B} = sup μ (x) + 1 − sup μ (x) . 2 x∈B x/ ∈B With this formula it is possible to interpret the credibility in terms of the possibility and necessity measure, since Pos(B) = sup μ (x) and Nec(B) = 1 − sup μ (x). x/ ∈B
x∈B
Using this two measures, the theorem can be formulated as Cr{B} =
Pos(B) + Nec(B) . 2
(2.15)
We should note here that if one defines the credibility measure using the equation (2.15), then Li and Liu proved that this is equivalent to the definition in terms of the four axioms given above [224].
2.5
Probability Distributions
In probability theory, the dependency between two random variables can be characterized through their joint probability density function. Namely, if X and Y are two random variables with probability density functions fX (x) and fY (y), respectively, then the density function, fX,Y (x, y), of their joint random variable (X ,Y ), should satisfy the following properties R
fX,Y (x,t)dt = fX (x),
R
fX,Y (t, y)dt = fY (y)
(2.16)
for all x, y ∈ R. Furthermore, fX (x) and fY (y) are called the the marginal probability density functions of random variable (X ,Y ). X and Y are said to be independent if fX,Y (x, y) = fX (x) fY (y) holds for all x, y. The expected (or mean) value of random variable X is defined as M(X ) = x fX (x)dx, R
2.5 Probability Distributions
25
and if g is a function of X then the expected value of g(X ) can be computed as M(g(X )) =
R
g(x) fX (x)dx.
Furthermore, if h is a function of X and Y then the expected value of h(X ,Y ) can be computed as M(h(X ,Y )) =
R2
h(x, y) fX,Y (x, y)dxdy.
Especially, M(X +Y ) = M(X ) + M(Y ),
(2.17)
that is, the expected value of X and Y can be determined according to their individual density functions (that are the marginal probability functions of random variable (X ,Y )). The key issue here is that the joint probability distribution vanishes from the right-hand side of (2.17) - even if X and Y are not independent - because of the principle of ’falling integrals’ (2.16). Let a, b ∈ R ∪ {−∞, ∞} with a ≤ b, then the probability that X takes its value from [a, b] is computed by b
P(X ∈ [a, b]) =
a
fX (x)dx.
The covariance between two random variables X and Y is defined as cov(X ,Y ) = M (X − M(X ))(Y − M(Y )) = M(XY ) − M(X )M(Y ), and if X and Y are independent then cov(X ,Y ) = 0, since M(XY ) = M(X )M(Y ). The variance of random variable X is defined by cov(X , X ), that is 2
var(X ) = M(X ) − (M(X )) = 2
x fX (x)dx −
2
R
R
2 x fX (x)dx
.
It should be noted that var(X ) is nothing else but the expected value of (X − M(X ))2 . For any random variables X ,Y and real numbers λ , μ ∈ R the following relationship holds var(λ X + μ Y ) = λ 2 var(X ) + μ 2 var(Y ) + 2λ μ cov(X ,Y ). The correlation coefficient between X and Y is defined by cor(X ,Y ) = and it is clear that −1 ≤ cor(X ,Y ) ≤ 1.
cov(X ,Y )
, var(X )var(Y )
Chapter 3
A Normative View on Possibility Distributions
In probability theory the expected value of functions of random variables plays a fundamental role in defining the basic characteristic measures of probability distributions. For example, the variance, covariance and correlation of random variables can be computed as the expected value of their appropriately chosen real-valued functions. In possibility theory we can use the principle of expected value of functions on fuzzy sets to define variance, covariance and correlation of possibility distributions. Marginal probability distributions are determined from the joint one by the principle of ’falling integrals’ and marginal possibility distributions are determined from the joint possibility distribution by the principle of ’falling shadows’. Probability distributions can be interpreted as carriers of incomplete information [203], and possibility distributions can be interpreted as carriers of imprecise information. In 1987 Dubois and Prade [126] defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. They also showed that this expectation remains additive in the sense of addition of fuzzy numbers. In possibility theory we can use the principle of average value of appropriately chosen real-valued functions to define mean value, variance, covariance and correlation of possibility distributions. A function f : [0, 1] → R is said to be a weighting function if f is non-negative, monotone increasing and satisfies the following normalization condition 01 f (γ )d γ = 1. Different weighting functions can give different (case-dependent) importances to level-sets of possibility distributions. We can define the mean value (variance) of a possibility distribution as the f -weighted average of the probabilistic mean values (variances) of the respective uniform distributions defined on the γ -level sets of that possibility distribution. A measure of possibilistic covariance (correlation) between marginal possibility distributions of a joint possibility distribution can be defined as the f -weighted average of probabilistic covariances (correlations) between marginal probability distributions whose joint probability distribution is defined to be uniform on the γ -level sets of their joint possibility distribution [157]. We should note here that the choice of uniform probability distribution on the level sets of possibility distributions is not without reason. We suppose that each point of a given level set is equally possible and then we apply Laplace’s principle of Insufficient Reason: if elementary events are equally possible, C. Carlsson and R. Full´er: Possibility for Decision, STUDFUZZ 270, pp. 27–76. springerlink.com © Springer-Verlag Berlin Heidelberg 2011
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3 A Normative View on Possibility Distributions
they should be equally probable (for more details and generalization of principle of Insufficient Reason see [129], page 59). The idea of equipping the alpha-cuts with a uniform probability is not new and refers to early ideas of simulation of fuzzy sets by Yager [337], and possibility/probability transforms by Dubois et al [128] as well as the pignistic transform of Smets [302]. In this Chapter, following [54, 70, 157, 165]), we will introduce the concepts of possibilistic mean value, variance, covariance and correlation.
3.1
Possibilistic Mean Value
The possibilistic mean (or expected value), variance, covariance and correlation were originally defined from the measure of possibilistic interactivity (as shown in [70, 157]) but for simplicity, we will present the concept of possibilistic mean value, variance, covariance and possibilistic correlation in a probabilistic setting and point out the fundamental difference between the standard probabilistic approach and the possibilistic one. Let A ∈ F be fuzzy number with [A]γ = [a1 (γ ), a2 (γ )] and let Uγ denote a uniform probability distribution on [A]γ , γ ∈ [0, 1]. Recall that the probabilistic mean value of Uγ is equal to M(Uγ ) =
a1 (γ ) + a2 (γ ) , 2
and its probabilistic variance is computed by var(Uγ ) =
(a2 (γ ) − a1(γ ))2 . 12
In 2001 Carlsson and Full´er [54] defined the possibilistic mean (or expected) value of fuzzy number A as E(A) =
1 0
M(Uγ )2γ dγ =
1 a1 (γ ) + a2 (γ ) 0
2
2γ d γ =
1 0
(a1 (γ ) + a2(γ ))γ dγ ,
where Uγ is a uniform probability distribution on [A]γ for all γ ∈ [0, 1]. In 1986 Goetschel and Voxman introduced a method for ranking fuzzy numbers as [169] A ≤ B ⇐⇒
1 0
γ (a1 (γ ) + a2 (γ )) d γ ≤
1 0
γ (b1 (γ ) + b2 (γ )) d γ
As was pointed out by Goetschel and Voxman this definition of ordering was motivated in part by the desire to give less importance to the lower levels of fuzzy numbers. In the terminology introduced by Carlsson and Full´er, the ordering by Goetschel and Voxman can be written as A ≤ B ⇐⇒ E(A) ≤ E(B).
3.1 Possibilistic Mean Value
29
We note further that from the equality
E(A) =
1 0
1
γ (a1 (γ ) + a2 (γ ))d γ =
0
γ·
a1 (γ ) + a2(γ ) dγ 2 1 0
γ dγ
,
it follows that E(A) is nothing else but the level-weighted average of the arithmetic means of all γ -level sets, that is, the weight of the arithmetic mean of a1 (γ ) and a2 (γ ) is just γ . Example 3.1.1 If A = (a, α , β ) is a triangular fuzzy number with center a, leftwidth α > 0 and right-width β > 0 then a γ -level of A is computed by [A]γ = [a − (1 − γ )α , a + (1 − γ )β ], ∀γ ∈ [0, 1], Then, E(A) =
1 0
γ [a − (1 − γ )α + a + (1 − γ )β ]d γ = a +
β −α . 6
When A = (a, α ) is a symmetric triangular fuzzy number we get E(A) = a. Let A ∈ F be fuzzy number with [A]γ = [a1 (γ ), a2 (γ )], γ ∈ [0, 1]. A function f : [0, 1] → R is said to be a weighting function if f is non-negative, monoton increasing and satisfies the following normalization condition 1 0
f (γ )d γ = 1.
(3.1)
Definition 3.1.1 ([154]) We define the f -weighted possibilistic mean (or expected) value of fuzzy number A as E f (A) =
1 a 1 (γ ) + a 2 (γ ) 0
2
f (γ )d γ .
(3.2)
It should be noted that if f (γ ) = 2γ , γ ∈ [0, 1] then E f (A) =
1 a1 (γ ) + a2 (γ ) 0
2
2γ d γ =
1 0
[a1 (γ ) + a2 (γ )] γ d γ = E(A).
That is the f -weighted possibilistic mean value defined by (3.2) can be considered as a generalization of possibilistic mean value introduced earlier by Carlsson and Full´er [54]. From the definition of a weighting function it can be seen that f (γ ) might be zero for certain (unimportant) γ -level sets of A. So by introducing different weighting functions we can give different (case-dependent) importances to γ -levels sets of fuzzy numbers.
30
3 A Normative View on Possibility Distributions
Note 3.1. The f -weighted possibilistic mean value of fuzzy number A coincides with the value of A (with respect to reducing function f ) introduced by Delgado, Vila and Woxman in ([122], page 127). In fact, their paper has inspired us to introduce the notation of f -weighted possibilistic mean. Let us introduce a family of weighting function defined by 1 if γ ∈ (0, 1] one(γ ) = a if γ = 0 where a ∈ [0, 1] is an arbitrary real number. Then, Eone (A) =
1 a1 ( γ ) + a2 ( γ ) 0
2
× one(γ )d γ =
1 a 1 (γ ) + a 2 (γ )
2
0
dγ .
(3.3)
Note 3.2. We note here that the one-weighted possibilistic mean value defined by (3.3) coincides with the generative expectation of fuzzy numbers introduced by Chanas and M. Nowakowski in ([99], page 47). Definition 3.1.2 ([154]) Let f be a weighting function and let A ∈ F be fuzzy number with [A]γ = [a1 (γ ), a2 (γ )], γ ∈ [0, 1]. Then we define the f -weighted intervalvalued possibilistic mean of A as + M f (A) = [M − f (A), M f (A)],
where M− f (A) = and M+ f (A) =
1 0
1 0
a1 (γ ) f (γ )d γ , a2 (γ ) f (γ )d γ .
The following two theorems can directly be proved using the definition of f weighted interval-valued possibilistic mean. Theorem 3.1.1 Let A, B ∈ F two non-interactive fuzzy numbers and let f be a weighting function, and let λ be a real number. Then M f (A + B) = M f (A) + M f (B), M f (λ A) = λ M f (A), where the non-interactive sum of fuzzy numbers A and B is defined by the sup-min extension principle 2.6. Note 3.3. The f -weighted possibilistic mean of A, defined by (3.2), is the arithmetic mean of its f -weighted lower and upper possibilistic mean values, i.e. E f (A) =
M f − (A) + M + f (A) 2
.
(3.4)
3.1 Possibilistic Mean Value
31
Theorem 3.1.2 Let A and B be two non-interactive fuzzy numbers, and let λ ∈ R. Then we have E f (A + B) = E f (A) + E f (B),
E f (λ A) = λ E f (A),
where the non-interactive sum of fuzzy numbers A and B is defined by the sup-min extension principle 2.6 We will show an important relationship between the interval-valued probabilistic mean D(A) = [E∗ (A), E ∗ (A)] introduced by Dubois and Prade in [126] and the f + weighted interval-valued possibilistic mean M f (A) = [M − f (A), M f (A)] for any fuzzy number with strictly decreasing shape functions. An LR-type fuzzy number A can be described with the following membership function: ⎧ q −u ⎪ ⎪L − if q− − α ≤ u ≤ q− ⎪ ⎪ α ⎪ ⎪ ⎪ ⎨1 if u ∈ [q− , q+ ] A(u) = u − q+ ⎪ ⎪ ⎪ R if q+ ≤ u ≤ q+ + β ⎪ ⎪ β ⎪ ⎪ ⎩ 0 otherwise where [q− , q+ ] is the peak of fuzzy number A; q− and q+ are the lower and upper modal values; L, R : [0, 1] → [0, 1] with L(0) = R(0) = 1 and L(1) = R(1) = 0 are non-increasing, continuous functions. We will use the notation A = (q− , q+ , α , β )LR . Hence, the closure of the support of A is exactly [q− − α , q+ + β ]. If L and R are strictly decreasing functions then the γ -level sets of A can easily be computed as [A]γ = [q− − α L−1 (γ ), q+ + β R−1 (γ )], γ ∈ [0, 1]. The lower and upper probability mean values of the fuzzy number A are computed by [126] E∗ (A) = q− − α
1
L(u)du, 0
E ∗ (A) = q+ + β
1
R(u)du. 0
The f -weighted lower and upper possibilistic mean values are computed by
(3.5)
32
3 A Normative View on Possibility Distributions
M− f (A) = =
1
q− − α L−1 (γ ) f (γ )d γ
0
1 0
q− f (γ )d γ −
= q− − α M+ f (A) = =
1 0
1
0
0
α L−1 (γ ) f (γ )d γ
L−1 (γ ) f (γ )d γ , (3.6)
q+ + β R−1(γ ) f (γ )d γ
0
1
1
q+ f (γ )d γ +
= q+ + β
1 0
1 0
β R−1 (γ ) f (γ )d γ
R−1 (γ ) f (γ )d γ .
We can state the following theorem. Theorem 3.1.3 ([154]) Let f be a weighting function and let A be a fuzzy number of type LR with strictly decreasing and continuous shape functions. Then, the f weighted interval-valued possibilistic mean value of A is a subset of the intervalvalued probabilistic mean value, i.e. M f (A) ⊆ D(A).
3.1.1
Illustrations
Example 3.1.2 Let f (γ ) = (n + 1)γ n and let A = (a, α , β ) be a triangular fuzzy number with center a, left-width α > 0 and right-width β > 0 then a γ -level of A is computed by [A]γ = [a − (1 − γ )α , a + (1 − γ )β ], ∀γ ∈ [0, 1]. Then the power-weighted lower and upper possibilistic mean values of A are computed by M− f (A) =
1 0
[a − (1 − γ )α ](n + 1)γ nd γ
= a(n + 1) = a−
1 0
γ n d γ − α (n + 1)
1 0
(1 − γ )γ nd γ
α , n+2
and, M+ f (A) =
1 0
[a + (1 − γ )β ](n + 1)γ nd γ
= a(n + 1) = a+
1 0
β , n+2
γ d γ + β (n + 1) n
1 0
(1 − γ )γ n d γ
3.1 Possibilistic Mean Value
and therefore,
That is,
33
α β M f (A) = a − ,a + . n+2 n+2 1 α β β −α E f (A) = a− +a+ = a+ . 2 n+2 n+2 2(n + 2)
So,
β −α lim E f (A) = lim a + = a. n→∞ n→∞ 2(n + 2)
Example 3.1.3 Let A = (a, b, α , β ) be a fuzzy number of trapezoidal form with peak [a, b], left-width α > 0 and right-width β > 0, and let f (γ ) = (n + 1)γ n, n ≥ 0. A γ -level of A is computed by [A]γ = [a − (1 − γ )α , b + (1 − γ )β ], ∀γ ∈ [0, 1], then the power-weighted lower and upper possibilistic mean values of A are computed by M− f (A) =
1 0
[a − (1 − γ )α ](n + 1)γ nd γ
= a(n + 1) = a−
1 0
γ d γ − α (n + 1) n
1 0
(1 − γ )γ nd γ
α , n+2
and, M+ f (A)
=
1 0
[b + (1 − γ )β ](n + 1)γ nd γ
= b(n + 1) = b+ and therefore,
That is,
1 0
γ n d γ + β (n + 1)
1 0
(1 − γ )γ n d γ
β , n+2
α β M f (A) = a − ,b+ n+2 n+2
1 α β a+b β −α E f (A) = a− +b+ = + . 2 n+2 n+2 2 2(n + 2)
34
3 A Normative View on Possibility Distributions
So,
lim E f (A) = lim
n→∞
n→∞
a+b β −α a+b + = . 2 2(n + 2) 2
Example 3.1.4 Let f (γ ) = (n + 1)γ n , n ≥ 0 and let A = (a, α , β ) be a triangular fuzzy number with center a, left-width α > 0 and right-width β > 0 then α β α β M f (A) = a − ,a+ ⊂ D(A) = a − , a + n+2 n+2 2 2 and for n > 0 we have E f (A) = a +
β −α β −α ¯ = D(A) = a+ . 2(n + 2) 4
Example 3.1.5 Let A = (a, b, α , β ) be a fuzzy number of trapezoidal form and let 1 f (γ ) = (n − 1) √ −1 , n 1−γ where n ≥ 2. It is clear that f is a weighting function with f (0) = 0 and lim f (γ ) = ∞.
γ →1−0
Then the f -weighted lower and upper possibilistic mean values of A are computed by 1 1 − M f (A) = [a − (1 − γ )α ](n − 1) √ − 1 dγ n 1−γ 0 1 = a − α (n − 1) (1 − γ )1−1/n − (1 − γ ) d γ 0 1 1 = a − α (n − 1) − 2 − 1/n 2 = a−
(n − 1)α , 2(2n − 1)
3.2 Possibilistic Variance
35
and M+ f (A)
1 = [b + (1 − γ )β ](n − 1) √ − 1 dγ n 1−γ 0 1 = b + β (n − 1) (1 − γ )1−1/n − (1 − γ ) d γ 0 1 1 = b + β (n − 1) − 2 − 1/n 2 = b+
and therefore
1
(n − 1)β , 2(2n − 1)
(n − 1)α (n − 1)β M f (A) = a − ,b + . 2(2n − 1) 2(2n − 1)
That is, 1 (n − 1)α (n − 1)β E f (A) = a− +b+ 2 2(2n − 1) 2(2n − 1) =
a + b (n − 1)(β − α ) + . 2 4(2n − 1)
So, lim E f (A) = lim
n→∞
n→∞
=
a + b (n − 1)(β − α ) + 2 4(2n − 1)
a+b β −α + . 2 8
Note 3.4. When A is a symmetric fuzzy number then the equation E f (A) = E(A) holds for any weighting function f . In the limit case, when A = (a, b, 0, 0) is the characteristic function of interval [a, b], the f -weighted possibilistic and probabilistic interval-valued means are equal, D(A) = M f (A) = [a, b].
3.2
Possibilistic Variance
Definition 3.2.1 ([157]) The f -weighted possibilistic variance of A ∈ F can be written as Var f (A) =
1 0
var(Uγ ) f (γ )d γ =
1 (a2 (γ ) − a1 (γ ))2 0
12
f (γ )d γ .
where Uγ is a uniform probability distribution on [A]γ and var(Uγ ) denotes the variance of Uγ .
36
3 A Normative View on Possibility Distributions
If f (γ ) = 2γ then the f -weighted possibilistic variance is said to be a possibilistic variance of A, denoted by Var(A), and is defined by Var(A) =
1 0
var(Uγ )2γ dγ =
1 6
1 0
(a2 (γ ) − a1(γ ))2 γ dγ ,
where Uγ is a uniform probability distribution on [A]γ and var(Uγ ) denotes the variance of Uγ . For variance and covariance of fuzzy random variables the reader can consult, e.g. Puri and Ralescu [281], and Feng, Hu and Shu [142].
3.2.1
Illustrations
Example 3.2.1 If A = (a, α , β ) is a triangular fuzzy number then 2 1 1 γ a + β (1 − γ ) − (a − α (1 − γ )) d γ 6 0 (α + β )2 = . 72
Var(A) =
Example 3.2.2 Let A = (a, b, α , β ) be a trapezoidal fuzzy number and let f (γ ) = (n + 1)γ n be a weighting function. Then, Var f (A) = (n + 1)
2 1 a2 (γ ) − a1 (γ ) 0
2
γ ndγ
n+1 1 [(b − a) + (α + β )(1 − γ )]2 γ n d γ 4 0 1 1 n+1 = (b − a)2 γ n d γ + 2(b − a)(α + β ) (1 − γ )γ nd γ 4 0 0 1 + (α + β )2 (1 − γ )2 γ n d γ 0 2 n + 1 (b − a) 2(b − a)(α + β ) 2(α + β )2 = + + 4 n+1 (n + 1)(n + 2) (n + 1)(n + 2)(n + 3) =
(b − a)2 (b − a)(α + β ) (α + β )2 + + 4 2(n + 2) 2(n + 2)(n + 3) 2 b−a α +β (n + 1)(α + β )2 = + + . 2 2(n + 2) 4(n + 2)2(n + 3) =
3.3 Possibilistic Covariance
37
So, lim Var f (A) = lim
n→∞
n→∞
=
b−a α + β 2 (n + 1)(α + β )2 + + 2 2(n + 2) 4(n + 2)2(n + 3)
b−a . 2
Note 3.5. In 2001 Carlsson and Full´er [54] originally introduced the possibilistic variance of fuzzy numbers as Var(A) =
1 2
1 0
(a2 (γ ) − a1 (γ ))2 γ dγ ,
and in 2003 Full´er and Majlender [154] introduced the f -weighted possibilistic variance of A by Var f (A) =
3.3
1 4
1 0
(a2 (γ ) − a1 (γ ))2 f (γ )d γ .
Possibilistic Covariance
In 2004 Full´er and Majlender [157] introduced a measure of possibilistic covariance between marginal distributions of a joint possibility distribution C as the expected value of the interactivity relation between the γ -level sets of its marginal distributions. In 2005 Carlsson, Full´er and Majlender [70] showed that the possibilistic covariance between fuzzy numbers A and B can be written as the weighted average of the probabilistic covariances between random variables with uniform joint distribution on the level sets of their joint possibility distribution C. Definition 3.3.1 ([157]) The f -weighted measure of possibilistic covariance between A, B ∈ F , (with respect to their joint distribution C), can be written as Cov f (A, B) =
1 0
cov(Xγ ,Yγ ) f (γ )d γ ,
where Xγ and Yγ are random variables whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1]. Now we show how the possibilistic variance can be derived from possibilistic covariance. Let A ∈ F be fuzzy number with [A]γ = [a1 (γ ), a2 (γ )] and let Uγ denote a uniform probability distribution on [A]γ , γ ∈ [0, 1]. First we compute the level-wise covariances by
38
3 A Normative View on Possibility Distributions
cov(Uγ ,Uγ ) = M(Uγ2 ) − (M(Uγ ))2
a2 ( γ )
a2 ( γ ) 1 xdx a2 (γ ) − a1(γ ) a1 (γ ) a1 (γ ) a21 (γ ) + a1(γ )a2 (γ ) + a22 (γ ) a1 (γ ) + a2 (γ ) 2 = − 3 2
=
1 a2 (γ ) − a1(γ )
x2 dx −
2
a21 (γ ) − 2a1 (γ )a2 (γ ) + a22 (γ ) 12 (a2 (γ ) − a1 (γ ))2 = , 12 =
and we get Var f (A) = Cov f (A, A) = =
3.3.1
1 0
cov(Uγ ,Uγ ) f (γ )d γ
1 (a2 (γ ) − a1(γ ))2 0
12
f (γ )d γ .
Illustrations
If A and B are non-interactive, i.e. C = A × B. Then [C]γ = [A]γ × [B]γ , that is, [C]γ is rectangular subset of R2 for any γ ∈ [0, 1]. Then Xγ , the first marginal probability distribution of a uniform distribution on [C]γ = [A]γ × [B]γ , is a uniform probability distribution on [A]γ (denoted by Uγ ) and Yγ , the second marginal probability distribution of a uniform distribution on [C]γ = [A]γ × [B]γ , is a uniform probability distribution on [B]γ (denoted by Vγ ) that is Xγ and Yγ are independent. So, cov(Xγ ,Yγ ) = cov(Uγ ,Vγ ) = 0, for all γ ∈ [0, 1], and, therefore, we have Cov f (A, B) =
1 0
cov(Xγ ,Yγ ) f (γ )dγ =
1 0
cov(Uγ ,Vγ ) f (γ )dγ = 0.
If A and B are non-interactive then Cov f (A, B) = 0 for any weighting function f . Note 3.6. We should emphasize here that the inclusion of the weighting function f does not play any crucial role in our theory, since by setting f (γ ) = 1 for all γ ∈ [0, 1], f could be eliminated from the definition. Example 3.3.1 Now consider the case when A(x) = B(x) = (1 − x) · χ[0,1](x)
3.3 Possibilistic Covariance
39
for x ∈ R, that is, [A]γ = [B]γ = [0, 1 − γ ] for γ ∈ [0, 1]. Suppose that their joint possibility distribution is given by F(x, y) = (1 − x − y) · χT (x, y), where T = {(x, y) ∈ R2 |x ≥ 0, y ≥ 0, x + y ≤ 1}. This situation is depicted on Fig. 3.1, where we have shifted the fuzzy sets to get a better view of the situation. Fig. 3.1 Illustration of joint possibility distribution F
Fig. 3.2 Partition of [F]γ
It is easy to check that A and B are really the marginal distributions of F. A γ -level set of F is computed by [F]γ = {(x, y) ∈ R2 |x ≥ 0, y ≥ 0, x + y ≤ 1 − γ }.
40
3 A Normative View on Possibility Distributions
The density function of a uniform distribution on [F]γ can be written as ⎧ 1 ⎪ ⎪ if (x, y) ∈ [F]γ ⎨ dxdy f (x, y) = ⎪ [F]γ ⎪ ⎩ 0 otherwise in details, f (x, y) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
1
if (x, y) ∈ [F]γ
1−γ 1−γ −x
dxdy 0
0
0
that is, f (x, y) =
otherwise ⎧ ⎨
2 if (x, y) ∈ [F]γ (1 − γ )2 ⎩ 0 otherwise
The marginal functions are obtained as f1 (x) = [F]γ
1 dxdy
1−γ −x 0
and f2 (y) = [F]γ
1 dxdy
1−γ −y 0
⎧ ⎨ 2(1 − γ − x) if 0 ≤ x ≤ 1 − γ dy = (1 − γ )2 ⎩ 0 otherwise ⎧ ⎨ 2(1 − γ − y) if 0 ≤ y ≤ 1 − γ dx = (1 − γ )2 ⎩ 0 otherwise
Furthermore, the probabilistic expected values of marginal distributions of Xγ and Yγ are equal to (1 − γ )/3 see (Fig. 3.2). Really, 2 M(Xγ ) = (1 − γ )2 2 M(Yγ ) = (1 − γ )2
1−γ 0
1−γ 0
x(1 − γ − x)dx = (1 − γ )/3. y(1 − γ − y)dy = (1 − γ )/3.
And the covariance between Xγ and Yγ is positive on H1 and H4 and negative on H2 and H3 . In this case we get (see Fig. 3.2 for a geometrical interpretation), cov(Xγ ,Yγ ) = M(Xγ Yγ ) − M(Xγ )M(Yγ ) = [F]γ
1 dxdy
[F]γ
xydxdy
3.3 Possibilistic Covariance
41
1
− [F]γ
dxdy
[F]γ
1
xdxdy × [F]γ
dxdy
[F]γ
ydxdy.
That is, cov(Xγ ,Yγ ) =
1−γ 1−γ −x 2 (1 − γ )2 × xydxdy − . (1 − γ )2 9 0 0
cov(Xγ ,Yγ ) =
2 × (1 − γ )2
cov(Xγ ,Yγ ) =
1−γ 0
x(1 − γ − x)dx −
(1 − γ )2 . 9
(1 − γ )2 (1 − γ )2 (1 − γ )2 − =− . 12 9 36
Therefore we get Cov f (A, B) = −
1 1 (1 − γ )2 f (γ )d γ , 36 0
and Var f (A) = Var f (B) =
1 12
1 0
(1 − γ )2 f (γ )d γ .
Example 3.3.2 Let C be a joint possibility distribution (see Fig 3.3) defined from non-symmetrical marginal distributions of quasi-triangular form A and B by C(x, y) = TW (A(x), B(y)), where TW denotes the weak t-norm, that is, min{x, y} if max{x, y} = 1 TW (x, y) = 0 otherwise
Fig. 3.3 A non-symmetrical γ -level set of C (a, b)
Then with the notations a1 = a1 (γ ), a2 = a2 (γ ), b1 = b1 (γ ), b2 = b2 (γ ) we get cov(Xγ ,Yγ ) =
(b2 − b1 )(a2 − a1 )(a2 + a1 − 2a)(2b − b1 − b2 ) . 2(b2 − b1 + a2 − a1 )
42
3 A Normative View on Possibility Distributions
and their covariance is computed by Cov(A, B) =
1 (b2 − b1)(a2 − a1)(a2 + a1 − 2a)(2b − b1 − b2 )
2(b2 − b1 + a2 − a1 )
0
2γ d γ
that is, Cov(A, B) = −
1 (b2 − b1 )(a2 − a1 )(a2 + a1 − 2a)(b2 + b1 − 2b)
(b2 − b1 + a2 − a1 )
0
γ dγ
If the γ -level set of B is symmetrical, i.e., 2b − b1 − b2 = 0, or the γ -level set of A is symmetrical, i.e., a2 + a1 − 2a = 0 then cov(Xγ ,Yγ ) = 0. where a is the maximizing point of A and b is the maximizing point of B.
3.4
Possibilistic Correlation Coefficient
The f -weighted possibilistic correlation coefficient of A, B ∈ F (with respect to their joint distribution C) is defined by Full´er, Mezei and V´arlaki [161]
ρ f (A, B) =
1 0
where
ρ (Xγ ,Yγ ) =
ρ (Xγ ,Yγ ) f (γ )dγ
(3.7)
cov(Xγ ,Yγ ) var(Xγ ) var(Yγ )
and, where Xγ and Yγ are random variables whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1]. For any joint distribution C and for any f we have, −1 ≤ ρ f (A, B) ≤ 1. In other words, the f -weighted is nothing else, but the f -weighted average of the probabilistic correlation coefficients ρ (Xγ ,Yγ ) for all γ ∈ [0, 1]. Since ρ f (A, B) measures an average index of interactivity between the level sets of A and B, we sometimes call this measure as the index of interactivity between A and B. Note 3.7. There exist several other ways to define correlation coefficient for fuzzy numbers, e.g. Liu and Kao [226] used fuzzy measures to define a fuzzy correlation coefficient of fuzzy numbers and they formulated a pair of nonlinear programs to find the α -cut of this fuzzy correlation coefficient, then, in a special case, Hong [191] showed an exact calculation formula for this fuzzy correlation coefficient.
3.4 Possibilistic Correlation Coefficient
43
Vaidyanathan [319] introduced a new measure for the correlation coefficient between triangular fuzzy variables called credibilistic correlation coefficient. In 2005 Carlsson, Full´er and Majlender [70] defined the f -weighted possibilistic correlation of A, B ∈ F , (with respect to their joint distribution C) as Cov f (A, B) ρ old . f (A, B) = Var f (A) Var f (B)
(3.8)
where Uγ is a uniform probability distribution on [A]γ and Vγ is a uniform probability distribution on [B]γ , and Xγ and Yγ are random variables whose joint probability distribution is uniform on [C]γ for all γ ∈ [0, 1]. If [C]γ is convex for all γ ∈ [0, 1] then −1 ≤ ρ old f (A, B) ≤ 1 for any f . The main drawback of the definition of the former index of interactivity (3.8) is that it does not necessarily take its values from [−1, 1] if some level-sets of the joint possibility distribution are not convex. For example, consider a joint possibility distribution defined by C(x, y) = 4x · χT (x, y) + 4/3(1 − x) · χS(x, y), where,
T = (x, y) ∈ R2 | 0 ≤ x ≤ 1/4, 0 ≤ y ≤ 1/4, x ≤ y ,
and,
S = (x, y) ∈ R2 | 1/4 ≤ x ≤ 1, 1/4 ≤ y ≤ 1, y ≤ x .
Furthermore, we have, [C]γ = (x, y) ∈ R2 | γ /4 ≤ x ≤ 1/4, x ≤ y ≤ 1/4 (x, y) ∈ R2 | 1/4 ≤ x ≤ 1 − 3/4γ , 1/4 ≤ y ≤ x . We can see that [C]γ is not a convex set for any γ ∈ [0, 1) (see Fig. 3.4). Fig. 3.4 Not convex γ -level set
(3.9)
44
3 A Normative View on Possibility Distributions
Then the marginal possibility distributions of (3.9) are computed by (see Fig. 3.5), ⎧ if 0 ≤ x ≤ 1/4 ⎪ ⎨ 4x, 4 A(x) = B(x) = (1 − x), if 1/4 ≤ x ≤ 1 ⎪ ⎩3 0, otherwise After some computations we get ρ old f (A, B) ≈ 1.562 for the weighting function f (γ ) = 2γ . We get here a value bigger than one since the variance of the first marginal distributions, Xγ , exceeds the variance of the uniform distribution on the same support.
Fig. 3.5 Marginal distribution A
3.5
Illustrations
In this Section we will show five important examples for the possibilistic correlation coefficient.
3.5.1
Non-interactive Fuzzy Numbers
If A and B are non-interactive then their joint possibility distribution is defined by C = A × B. Since all [C]γ are rectangular and the probability distribution on [C]γ is defined to be uniform we get cov(Xγ ,Yγ ) = 0, for all γ ∈ [0, 1]. So Cov f (A, B) = 0 and ρ f (A, B) = 0 for any weighting function f .
3.5 Illustrations
45
Fig. 3.6 The case of non-interactive marginal distributions
3.5.2
Perfect Correlation
Fuzzy numbers A and B are said to be in perfect correlation, if there exist q, r ∈ R, q = 0 such that their joint possibility distribution is defined by [70] C(x1 , x2 ) = A(x1 ) · χ{qx1 +r=x2 } (x1 , x2 ) = B(x2 ) · χ{qx1+r=x2 } (x1 , x2 ),
(3.10)
where χ{qx1 +r=x2 } , stands for the characteristic function of the line {(x1 , x2 ) ∈ R2 |qx1 + r = x2 }. In this case we have [C]γ = (x, qx + r) ∈ R2 x = (1 − t)a1(γ ) + ta2 (γ ),t ∈ [0, 1] where [A]γ = [a1 (γ ), a2 (γ )]; and [B]γ = q[A]γ + r, for any γ ∈ [0, 1], and, finally, B(x) = A
x−r , q
for all x ∈ R. Furthermore, A and B are in a perfect positive [see Fig. 3.7] (negative [see Fig. 3.8]) correlation if q is positive (negative) in (3.10). If A and B have a perfect positive (negative) correlation then from ρ (Xγ ,Yγ ) = 1 (ρ (Xγ ,Yγ ) = −1) [see [70] for details], for all γ ∈ [0, 1], we get ρ f (A, B) = 1 (ρ f (A, B) = −1) for any weighting function f .
46
Fig. 3.7 Perfect positive correlation
Fig. 3.8 Perfect negative correlation
3 A Normative View on Possibility Distributions
3.5 Illustrations
3.5.3
47
Joint Distribution: (1 − x − y)
Consider the case, when A(x) = B(x) = (1 − x) · χ[0,1] (x), for x ∈ R, that is [A]γ = [B]γ = [0, 1 − γ ], for γ ∈ [0, 1]. Suppose that their joint possibility distribution is given by F(x, y) = (1 − x − y) · χT (x, y), where T = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 . A γ -level set of F is computed by [F]γ = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 − γ . This situation is depicted on Fig. 3.9, where we have shifted the fuzzy sets to get a better view of the situation.
Fig. 3.9 Illustration of joint possibility distribution F
The density function of a uniform distribution on [F]γ can be written as ⎧ ⎧ 2 ⎨ 1 ⎨ γ , if (x, y) ∈ [F] , if (x, y) ∈ [F]γ dxdy f (x, y) = = (1 − γ )2 [F]γ ⎩ ⎩ 0 otherwise 0 otherwise
48
3 A Normative View on Possibility Distributions
The marginal functions are obtained as ⎧ ⎨ 2(1 − γ − x) , if 0 ≤ x ≤ 1 − γ f1 (x) = (1 − γ )2 ⎩ 0 otherwise ⎧ ⎨ 2(1 − γ − y) , if 0 ≤ y ≤ 1 − γ f2 (y) = (1 − γ )2 ⎩ 0 otherwise We can calculate the probabilistic expected values of the random variables Xγ and Yγ , whose joint distribution is uniform on [F]γ for all γ ∈ [0, 1]: 2 M(Xγ ) = (1 − γ )2
1−γ 0
x(1 − γ − x)dx =
1−γ 3
y(1 − γ − y)dy =
1−γ . 3
and, M(Yγ ) =
2 (1 − γ )2
1−γ 0
We calculate the variations of Xγ and Yγ with the formula var(X ) = M(X 2 )−M(X)2 : M(Xγ2 )
2 = (1 − γ )2
1−γ 0
x2 (1 − γ − x)dx =
(1 − γ )2 6
and, var(Xγ ) = M(Xγ2 ) − M(Xγ )2 =
(1 − γ )2 (1 − γ )2 (1 − γ )2 − = . 6 9 18
And similarly we obtain var(Yγ ) =
(1 − γ )2 . 18
Using that M(Xγ Yγ ) =
2 (1 − γ )2
1−γ 1−γ −x 0
0
xydydx =
(1 − γ )2 , 12
(1 − γ )2 , 36 we can calculate the probabilistic correlation of the random variables: cov(Xγ ,Yγ ) = M(Xγ Yγ ) − M(Xγ )M(Yγ ) = −
ρ (Xγ ,Yγ ) =
cov(Xγ ,Yγ ) 1 =− . 2 var(Xγ ) var(Yγ )
3.5 Illustrations
49
And finally the f -weighted possibilistic correlation of A and B:
ρ f (A, B) =
1 0
1 1 − f (γ )d γ = − . 2 2
We note here that using the former definition (3.8) we would obtain ρ old f (A, B) = −1/3 for the correlation coefficient (see [70] for details).
3.5.4
Joint Distribution: (y-x)
Now consider the case when A(1 − x) = B(x) = x · χ[0,1] (x) for x ∈ R, that is, [A]γ = [0, 1 − γ ] and [B]γ = [γ , 1], for γ ∈ [0, 1]. Let E(x, y) = (y − x) · χS (x, y), where S = {(x, y) ∈ R2 |x ≥ 0, y ≤ 1, y − x ≥ 0}. This situation is depicted on Fig. 3.10, where we have shifted the fuzzy sets to get a better view of the situation. A γ -level set of E is computed by [E]γ = {(x, y) ∈ R2 |x ≥ 0, y ≤ 1, y − x ≥ γ }.
Fig. 3.10 ρ f (A, B) = 1/2
In this case, the probabilistic expected value of marginal distribution Xγ is equal to (1 − γ )/3 and the probabilistic expected value of marginal distribution of Yγ is equal to 2(1 − γ )/3 see (Fig. 3.11). And the covariance between Xγ and Yγ is positive on H1 and H4 and negative on H2 and H3 . After some calculations (see Fig. 3.11) we get ρ f (A, B) = 1/2, for any weighting function f .
50
3 A Normative View on Possibility Distributions
Fig. 3.11 Partition of [E]γ
3.5.5
Ball-Shaped Joint Possibility Distribution
Consider the case, when A(x) = B(x) = (1 − x2 ) · χ[0,1](x), √ for x ∈ R, that is [A]γ = [B]γ = [0, 1 − γ ], for γ ∈ [0, 1]. Suppose that their joint possibility distribution is given by: C(x, y) = (1 − x2 − y2 ) · χT (x, y), where
T = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x2 + y2 ≤ 1 .
A γ -level set of C is computed by [C]γ = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x2 + y2 ≤ 1 − γ . The density function of a uniform distribution on [F]γ can be written as ⎧ ⎧ 4 ⎨ 1 ⎨ , if (x, y) ∈ [C]γ , if (x, y) ∈ [C]γ dxdy f (x, y) = γ = (1 − γ )π [C] ⎩ ⎩ 0 otherwise 0 otherwise The marginal functions are obtained as ⎧ ⎨ 4 1 − γ − x2 , if 0 ≤ x ≤ 1 − γ f1 (x) = ⎩ (1 − γ )π 0 otherwise
3.5 Illustrations
51
⎧ ⎨ 4 1 − γ − y2 , if 0 ≤ y ≤ 1 − γ f2 (y) = ⎩ (1 − γ )π 0 otherwise We can calculate the probabilistic expected values of the random variables Xγ and Yγ , whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1] : 4 M(Xγ ) = (1 − γ )π 4 M(Yγ ) = (1 − γ )π
1 − γ − x2 dx =
√ 4 1−γ 3π
1 − γ − y2 dx =
√ 4 1−γ . 3π
√1−γ
x
0
√1−γ
y
0
We calculate the variations of Xγ and Yγ with the formula var(X ) = M(X 2 ) − M(X)2 : √1−γ 4 1−γ 2 M(Xγ ) = x2 1 − γ − x2 dx = (1 − γ )π 0 4 var(Xγ ) = M(Xγ2 ) − M(Xγ )2 =
1 − γ 16(1 − γ ) (1 − γ )(9π 2 − 64) − = . 4 9π 2 36π 2
And similarly we obtain var(Yγ ) =
(1 − γ )(9π 2 − 64) . 36π 2
Using that cov(Xγ ,Yγ ) = M(Xγ Yγ ) − M(Xγ )M(Yγ ) =
(1 − γ )(9π − 32) , 18π 2
we can calculate the probabilisctic correlation of the reandom variables:
ρ (Xγ ,Yγ ) =
cov(Xγ ,Yγ ) 2(9π − 32) = ≈ −0.302. (9π 2 − 64) var(Xγ ) var(Yγ )
And finally the f -weighted possibilistic correlation of A and B:
ρ f (A, B) =
3.5.6
1 2(9π − 32) 0
(9π 2 − 64)
f (γ )d γ =
2(9π − 32) . (9π 2 − 64)
Mere Shadows
Suppose that the joint possibility distribution of A and B is defined by, ⎧ ⎪ ⎨ A(x) if y = 0 C(x, y) = B(y) if x = 0 ⎪ ⎩0 otherwise
52
3 A Normative View on Possibility Distributions
where,
A(x) = B(x) = (1 − x) · χ[0,1](x),
for x ∈ R. Then a γ -level set of C is computed by [C]γ = (x, 0) ∈ R2 | 0 ≤ x ≤ 1 − γ (0, y) ∈ R2 | 0 ≤ y ≤ 1 − γ . Since all γ -level sets of C are degenerated, i.e. their integrals vanish, we calculate everything as a limit of integrals. We calculate all the quantities with the γ -level sets: γ [C]δ = (x, y) ∈ R2 | 0 ≤ x ≤ 1 − γ , 0 ≤ y ≤ δ
(x, y) ∈ R2 | 0 ≤ y ≤ 1 − γ , 0 ≤ x ≤ δ . First we calculate the expected value and variance of Xγ and Yγ :
1
M(Xγ ) = lim
γ [C]δ
δ →0 [C]γ dxdy δ
1
M(Xγ2 ) = lim
δ →0 [C]γ dxdy δ
var(Xγ ) = M(Xγ2 ) − M(Xγ )2 =
γ [C]δ
xdx =
x2 dx =
1−γ , 4 (1 − γ )2 , 6
(1 − γ )2 (1 − γ )2 5(1 − γ )2 − = . 6 16 48
Because of the symmetry, the results are the same for Yγ . We need to calculate their covariance, 1 M(Xγ Yγ ) = lim xydydx = 0, γ δ →0 [C]γ dxdy δ
Using this we obtain, cov(Xγ ,Yγ ) = −
[C]δ
(1 − γ )2 , 16
and for the correlation,
ρ (Xγ ,Yγ ) =
cov(Xγ ,Yγ ) 3 =− . 5 var(Xγ ) var(Yγ )
Finally we obtain the f -weighted possibilistic correlation:
ρ f (A, B) =
1 0
3 3 − f (γ )d γ = − . 5 5
In this extremal case, the joint distribution is unequivocally constructed from the knowledge that C(x, y) = 0 for positive x, y.
3.6 Zero Correlation and Non-interactivity
53
Fig. 3.12 Illustration of [C]0.4
3.6
Zero Correlation and Non-interactivity
We emphasize here that zero correlation does not always imply non-interactivity. Let A, B ∈ F be fuzzy numbers, let C be their joint possibility distribution, and let γ ∈ [0, 1]. Suppose that [C]γ is symmetrical, i.e. there exists a ∈ R such that C(x, y) = C(2a − x, y), for all x, y ∈ [C]γ (the line defined by {(a,t)|t ∈ R} is the axis of symmetry of [C]γ ). In this case cov(Xγ ,Yγ ) = 0. Indeed, let H = {(x, y) ∈ [C]γ |x ≤ a}, then
[C]γ
xydxdy =
H
xdxdy =
[C]γ
H
x + (2a − x) dxdy = 2a dxdy,
H
ydxdy = 2
[C]γ
xy + (2a − x)y dxdy = 2a ydxdy,
H
ydxdy,
[C]γ
H
dxdy = 2
dxdy, H
therefore, we obtain
[C]γ
− [C]γ
1 dxdy
[C]γ
1
cov(Xγ ,Yγ ) =
dxdy
xdxdy [C]γ
[C]γ
1 dxdy
xydxdy
[C]γ
ydxdy = 0.
54
3 A Normative View on Possibility Distributions
Fig. 3.13 A case of ρ f (A, B) = 0 for interactive fuzzy numbers
For example, let G be a joint possibility distribution with a symmetrical γ -level set, i.e., there exist a, b ∈ R such that G(x, y) = G(2a − x, y) = G(x, 2b − y) = G(2a − x, 2b − y), for all x, y ∈ [G]γ , where (a, b) is the center of the set [G]γ , In Fig. 3.13, the joint possibility distribution is defined from symmetrical marginal distributions as G(x, y) = TW (A(x), B(y)), where TW denotes the weak t-norm. Consider now joint possibility distributions that are derived from given marginal distributions by aggregating their membership values. Namely, let A, B ∈ F . We will say that their joint possibility distribution C is directly defined from its marginal distributions if C(x, y) = T (A(x), B(y)), x, y ∈ R, where T : [0, 1] × [0, 1] → [0, 1] is a function satisfying the properties max T (A(x), B(y)) = A(x), ∀x ∈ R,
(3.11)
max T (A(x), B(y)) = B(y), ∀y ∈ R,
(3.12)
y
and x
3.7 Possibilistic Correlation Ratio
55
for example a triangular norm. In this case the joint distribution depends barely on the membership values of its marginal distributions, and the covariance (and, consequently, the correlation) between its marginal distributions will be zero whenever at least one of its marginal distributions is symmetrical. Theorem 3.6.1 ([68]) Let A, B ∈ F and let their joint possibility distribution C be defined by C(x, y) = T (A(x), B(y)), for x, y ∈ R, where T is a function satisfying conditions (3.11) and (3.12). If A is a symmetrical fuzzy number then Cov f (A, B) = 0, for any fuzzy number B, aggregator T , and weighting function f . Really, if A is a symmetrical fuzzy number with center a such that A(x) = A(2a − x) for all x ∈ R then, C(x, y) = T (A(x), B(y)) = T (A(2a − x), B(y)) = C(2a − x, y), that is, C is symmetrical. Hence, considering the results obtained above we have cov(Xγ ,Yγ ) = 0, and, therefore, Cov f (A, B) = 0, for any weighting function f .
3.7
Possibilistic Correlation Ratio
In this Section generalizing the probabilistic correlation ratio we will introduce a correlation ratio for marginal possibility distributions of joint possibility distributions. In statistics, the correlation ratio is a measure of the relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample. The correlation ratio was originally introduced by Karl Pearson [273] as part of analysis of variance and it was extended to random variables by Andrei Nikolaevich Kolmogorov [219] as,
η 2 (X |Y ) =
D2 [E(X |Y )] , D2 (X)
where X and Y are random variables. If X and Y have a joint probability density function, denoted by f (x, y), then we can compute η 2 (X |Y ) using the following formulas E(X |Y = y) =
∞
−∞
x f (x|y)dx
and D2 [E(X |Y )] = E(E(X |y) − E(X ))2 ,
56
3 A Normative View on Possibility Distributions
and where, f (x|y) =
f (x, y) . f (y)
The correlation ratio measures the functional dependence between X and Y . It takes on values between 0 (no functional dependence) and 1 (purely deterministic dependence). It is worth noting that if E(X |Y = y) is linear function of y (i.e. there is a linear relationship between random variables E(X |Y ) and Y ) this will give the same result as the square of the correlation coefficient, otherwise the correlation ratio will be larger in magnitude. It can therefore be used for judging non-linear relationships. Also note that the correlation ratio is asymmetrical by nature since the two random variables fundamentally do not play the same role in the functional relationship; in general, η 2 (X |Y ) = η 2 (Y |X ). Following Full´er, Mezei and V´arlaki [162] we will introduce a correlation ratio for marginal possibility distributions of joint possibility distributions. Definition 3.7.1 [162] Let us denote A and B the marginal possibility distributions of a given joint possibility distribution C. Then the f -weighted possibilistic correlation ratio of marginal possibility distribution A with respect to marginal possibility distribution B is defined by
η 2f (A|B) =
1 0
η 2 (Xγ |Yγ )g(γ )dγ
(3.13)
where Xγ and Yγ are random variables whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1], and η 2 (Xγ |Yγ ) denotes their probabilistic correlation ratio. So the f -weighted possibilistic correlation ratio of the fuzzy number A on B is nothing else, but the f -weighted average of the probabilistic correlation ratios η 2 (Xγ |Yγ ) for all γ ∈ [0, 1].
3.8
Illustrations
In this section we will compute the possibilistic correlation ratio for joint some possibility distributions defined on the unit square.
3.8.1
A Linear Relationship
Consider the case, when A(x) = B(x) = (1 − x) · χ[0,1](x), for x ∈ R, that is [A]γ = [B]γ = [0, 1 − γ ], for γ ∈ [0, 1]. Suppose that their joint possibility distribution is given by C(x, y) = (1 − x − y) · χT (x, y), where T = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 .
3.8 Illustrations
57
Then we have [C]γ = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 − γ . The density function of a uniform distribution on [C]γ is ⎧ 2 ⎨ if (x, y) ∈ [C]γ f (x, y) = (1 − γ )2 ⎩0 otherwise The marginal functions are obtained as ⎧ ⎨ 2(1 − γ − x) f1 (x) = (1 − γ )2 ⎩ 0 ⎧ ⎨ 2(1 − γ − y) f2 (y) = (1 − γ )2 ⎩ 0
if 0 ≤ x ≤ 1 − γ otherwise if 0 ≤ y ≤ 1 − γ otherwise
For the correlation ration we need to calculate the conditional probability, distribution: E(X |Y = y) =
1−γ −y 0
x f (x|y)dx =
1−γ −y f (x, y)
x
0
f2 (y)
dx =
1−γ −y , 2
where 0 ≤ x ≤ 1 − γ . The next step is to calculate the variation of this distribution: D2 [E(X |Y )] = E(E(X|y) − E(X ))2 = =
1−γ 1−γ −y 0
(
2
−
1 − γ 2 2(1 − γ − y) ) 3 (1 − γ )2
(1 − γ )2 . 72
Using the relationship
(1 − γ )2 , 18 we obtain that the probabilistic correlation of Xγ on Yγ is D2 (Xγ ) =
1 η 2 (Xγ |Yγ ) = . 4 From this the f -weighted possibilistic correlation ratio of A with respect to B is,
η 2f (A|B) =
1 1 0
4
1 f (γ )dγ = . 4
We recall that the f -weighted normalized measure of interactivity between A ∈ F and B ∈ F (with respect to their joint distribution C) is defined by [161]
58
3 A Normative View on Possibility Distributions
ρ f (A, B) = where
1 0
ρ (Xγ ,Yγ ) f (γ )dγ
cov(Xγ ,Yγ ) ρ (Xγ ,Yγ ) = . var(Xγ ) var(Yγ )
and where Xγ and Yγ are random variables whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1], and ρ (Xγ ,Yγ ) denotes their probabilistic correlation coefficient. In this simple case
η 2f (A|B) = η 2f (B|A) = [ρ f (A, B)]2 , since E(Xγ |Yγ = y) is a linear function of y. Really, in this case we have, 1−γ −y 1−γ y 1−γ = − + 2 6 3 2 1−γ 1 1 1−γ = − y− − × 3 2 2 3 1−γ 1 1−γ = − y− = E(Xγ ) − ρ (Xγ ,Yγ )(y − E(Yγ )). 3 2 3
E(Xγ |Yγ = y) =
3.8.2
A Nonlinear Relationship
Consider the case, when A(x) = (1 − x2) · χ[0,1](x), for x ∈ R, that is [A]γ
√
B(x) = (1 − y) · χ[0,1](y),
= [0, 1 − γ ], [B]γ = [0, 1 − γ ], for γ ∈ [0, 1]. Suppose that their joint possibility distribution is given by: C(x, y) = (1 − x2 − y) · χT (x, y), where
T = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x2 + y ≤ 1 .
A γ -level set of C is computed by [C]γ = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x2 + y ≤ 1 − γ . The density function of a uniform distribution on [C]γ can be written as ⎧ ⎧ 3 ⎨ 1 ⎨ γ γ if (x, y) ∈ [C] 3 if (x, y) ∈ [C] dxdy f (x, y) = γ = 2 2(1 − γ ) [C] ⎩ ⎩ 0 otherwise 0 otherwise
3.8 Illustrations
59
The marginal functions are obtained as ⎧ 2 ⎨ 3(1 − γ − x ) if 0 ≤ x ≤ √1 − γ 3 f1 (x) = ⎩ 2(1 − γ ) 2 0 otherwise ⎧ √ ⎨ 3 1 − γ − y if 0 ≤ y ≤ 1 − γ 3 f2 (y) = 2 ⎩ 2(1 − γ ) 0 otherwise For the correlation ration we need to calculate the conditional probability distribution, E(Y |X = x) =
1−γ −x2 0
y f (y|x)dy =
1−γ −x2
y 0
f (x, y) 1 − γ − x2 dy = , f1 (x) 2
where 0 ≤ y ≤ 1 − γ . The next step is to calculate the variation of this distribution: D2 [E(Y |X )] = E(E(Y |x) − E(Y ))2 = =
√1−γ 1 − γ − x2 0
(
2
−
2(1 − γ ) 2 3(1 − γ − x2 ) ) dx 3 5 2(1 − γ ) 2
2(1 − γ ) . 175 2
Using the relationship
12(1 − γ )2 , 175 we obtain that the probabilistic correlation ratio of Yγ with respect to Xγ is D2 (Yγ ) =
1 η 2 (Yγ |Xγ ) = . 6 From this the f -weighted possibilistic correlation ratio of B with respect to A is,
η 2f (B|A) = Similarly, from D2 [E(X|Y )] =
1 1 0
1 f (γ )dγ = . 6 6
3(1 − γ ) , and from 320 D2 (Xγ ) =
we obtain,
η 2f (A|B)
=
19(1 − γ ) , 320
1 3 0
19
f (γ )dγ =
3 . 19
60
3 A Normative View on Possibility Distributions
That is η 2f (B|A) = η 2f (A|B).
3.8.3
√ Joint Distribution: (1 − x − y)
Consider the case, when √ A(x) = (1 − x) · χ[0,1](x), B(x) = (1 − y) · χ[0,1](y), for x ∈ R, that is [A]γ = [0, (1 − γ )2 ], [B]γ = [0, 1 − γ ], for γ ∈ [0, 1]. Suppose that their joint possibility distribution is given by: √ C(x, y) = (1 − x − y) · χT (x, y), where
√ T = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 .
A γ -level set of C is computed by √ [C]γ = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 − γ . The density function of a uniform distribution on [C]γ can be written as ⎧ ⎧ 3 ⎨ 1 ⎨ if (x, y) ∈ [C]γ if (x, y) ∈ [C]γ dxdy f (x, y) = γ = (1 − γ )3 [C] ⎩ ⎩0 otherwise 0 otherwise The marginal functions are obtained as ⎧ √ ⎨ 3(1 − γ − x) if 0 ≤ x ≤ (1 − γ )2 f1 (x) = (1 − γ )3 ⎩ 0 otherwise ⎧ ⎨ 3(1 − γ − y)2 if 0 ≤ y ≤ 1 − γ f2 (y) = (1 − γ )3 ⎩ 0 otherwise For the correlation ration we need to calculate the conditional probability distribution, √ 1−γ −√x 1−γ −√x f (x, y) 1−γ − x E(Y |X = x) = y f (y|x)dy = y dy = , f1 (x) 2 0 0 where 0 ≤ y ≤ 1 − γ . The next step is to calculate the variation of this distribution:
3.8 Illustrations
61
D2 [E(Y |X)] = E(E(Y |x) − E(Y ))2 √ √ (1−γ )2 1 − γ − x 1 − γ 2 3(1 − γ − x) = ( − ) dx 2 4 (1 − γ )3 0 =
(1 − γ )2 . 80
Using the relationship
3(1 − γ )2 , 80 we obtain that the probabilistic correlation ratio of Yγ with respect to Xγ is D2 (Yγ ) =
1 η 2 (Yγ |Xγ ) = . 3 From this the f -weighted possibilistic correlation ratio of B with respect to A is,
η 2f (B|A) = Similarly, from D2 [E(X|Y )] =
1 1 0
3(1 − γ )4 , and from 175 D2 (Xγ ) =
we obtain:
η 2f (A|B) =
3.8.4
1 f (γ )dγ = . 3 3
37(1 − γ )4 , 700
1 12 0
37
f (γ )dγ =
12 . 37
Ball-Shaped Joint Distribution
Consider the case, when A(x) = B(x) = (1 − x2 ) · χ[0,1](x), √ for x ∈ R, that is [A]γ = [B]γ = [0, 1 − γ ], for γ ∈ [0, 1]. Suppose that their joint possibility distribution is ball-shaped, that is, C(x, y) = (1 − x2 − y2 ) · χT (x, y), where
T = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x2 + y2 ≤ 1 .
A γ -level set of C is computed by [C]γ = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x2 + y2 ≤ 1 − γ .
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3 A Normative View on Possibility Distributions
The density function of a uniform distribution on [C]γ can be written as ⎧ ⎧ 4 ⎨ 1 ⎨ γ if (x, y) ∈ [C] if (x, y) ∈ [C]γ dxdy f (x, y) = = (1 − γ )π [C]γ ⎩ ⎩0 otherwise 0 otherwise The marginal functions are obtained as ⎧ ⎨ 4 1 − γ − x2 if 0 ≤ x ≤ 1 − γ f1 (x) = ⎩ (1 − γ )π 0 otherwise ⎧ ⎨ 4 1 − γ − y2 if 0 ≤ y ≤ 1 − γ f2 (y) = ⎩ (1 − γ )π 0 otherwise For the correlation ration we need to calculate the conditional probability distribution: √1−γ −x2 √1−γ −x2 f (x, y) 1 − γ − x2 E(Y |X = x) = y f (y|x)dy = y dy = , f1 (x) 2 0 0 √ where 0 ≤ y ≤ 1 − γ . The next step is to calculate the variation of this distribution: D2 [E(Y |X )] = E(E(Y |x) − E(Y ))2 √ √1−γ 1 − γ − x2 4 1 − γ 2 4 1 − γ − x2 = ( − ) dx 2 3π π (1 − γ ) 0 =
(1 − γ )(27π 2 − 256) . 144π 2
Using the relationship D2 (Yγ ) =
(1 − γ )(9π 2 − 64) , 36π 2
we obtain that the probabilistic correlation ratio of Yγ with respect to Xγ is
η 2 (Yγ |Xγ ) =
27π 2 − 256 . 36π 2 − 256
Finally, we get that the f -weighted possibilistic correlation ratio of B with respect A is,
η 2f (B|A) =
1 27π 2 − 256 0
36π 2 − 256
g(γ )dγ =
27π 2 − 256 . 36π 2 − 256
3.8 Illustrations
3.8.5
63
√ √ Joint Distribution: (1 − x − y)
√ Consider the case, when A(x) = B(x) = (1 − x) · χ[0,1] (x), for x ∈ R, that is [A]γ = [B]γ = [0, (1 − γ )2 ], for γ ∈ [0, 1]. Suppose that their joint possibility distribution is given by: √ √ C(x, y) = (1 − x − y) · χT (x, y), where
√ √ T = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 .
A γ -level set of C is computed by √ √ [C]γ = (x, y) ∈ R2 | x ≥ 0, y ≥ 0, x + y ≤ 1 − γ . The density function of a uniform distribution on [C]γ can be written as ⎧ ⎧ 6 ⎨ 1 ⎨ if (x, y) ∈ [C]γ if (x, y) ∈ [C]γ f (x, y) = = (1 − γ )4 [C]γ dxdy ⎩ ⎩ 0 otherwise 0 otherwise The marginal functions are obtained as ⎧ √ ⎨ 6(1 − γ − x)2 if 0 ≤ x ≤ (1 − γ )2 f1 (x) = (1 − γ )4 ⎩ 0 otherwise ⎧ √ ⎨ 6(1 − γ − y)2 if 0 ≤ y ≤ (1 − γ )2 4 f2 (y) = (1 − γ ) ⎩ 0 otherwise For the correlation ration we need to calculate the conditional probability distribution, √ (1−γ −√x)2 (1−γ −√x)2 f (x, y) (1 − γ − x)2 E(Y |X = x) = y f (y|x)dy = y dy = , f1 (x) 2 0 0 where 0 ≤ y ≤ (1 − γ )2 . The next step is to calculate the variation of this distribution: D2 [E(Y |X )] = E(E(Y |x) − E(Y ))2 √ √ (1−γ )2 (1 − γ − x)2 (1 − γ )2 2 6(1 − γ − x)2 = ( − ) dx 2 5 (1 − γ )4 0 =
19(1 − γ )4 . 1400
Using the relationship D2 (Yγ ) =
9(1 − γ )4 , 350
64
3 A Normative View on Possibility Distributions
we obtain that the probabilistic correlation of Yγ with respect to Xγ is,
η 2 (Yγ |Xγ ) =
19 . 36
That is, the f -weighted possibilistic correlation ratio of B with respect to A is,
η 2f (B|A) =
3.9
1 19
36
0
f (γ )dγ =
19 . 36
A Normative View on Quasi Fuzzy Numbers
A quasi fuzzy number A is a fuzzy set of the real line with a normal, fuzzy convex and continuous membership function satisfying the limit conditions [57] lim μA (t) = 0,
t→∞
lim μA (t) = 0.
t→−∞
A quasi triangular fuzzy number is a quasi fuzzy number with a unique maximizing point. Furthermore, we call Q the family of all quasi fuzzy numbers. Quasi fuzzy numbers can also be considered as possibility distributions [127].
Fig. 3.14 A quasi triangular fuzzy number with membership function e−|x|
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-1
0
1
2
3
4
5
t
The possibilistic mean (or expected value), variance and covariance can be defined from the measure of possibilistic interactivity (as shown in [70, 157, 161]) but for simplicity, we will present the concept of possibilistic mean value, variance, covariance in a pure probabilistic setting. Let A ∈ F be fuzzy number with [A]γ = [a1 (γ ), a2 (γ )] and let Uγ denote a uniform probability distribution on [A]γ , γ ∈ [0, 1]. Recall that the probabilistic mean value of Uγ is equal to M(Uγ ) =
a1 (γ ) + a2 (γ ) , 2
3.9 A Normative View on Quasi Fuzzy Numbers
65
and its probabilistic variance is computed by var(Uγ ) =
(a2 (γ ) − a1(γ ))2 . 12
The f -weighted possibilistic mean value (or expected value) of A ∈ F is defined as [154] 1 1 a 1 (γ ) + a 2 (γ ) E f (A) = E(Uγ ) f (γ )d γ = f (γ )d γ , 2 0 0 where Uγ is a uniform probability distribution on [A]γ for all γ ∈ [0, 1]. If f (γ ) = 1 for all γ ∈ [0, 1] then we get E f (A) =
1 0
E(Uγ ) f (γ )d γ =
1 a1 (γ ) + a2 (γ )
2
0
dγ .
Now we will extend the concept of possibilistic mean value to the family of quasi fuzzy numbers. Definition 3.9.1 The f -weighted possibilistic mean value of A ∈ Q is defined as E f (A) =
1 0
E(Uγ ) f (γ )d γ =
1 a 1 (γ ) + a 2 (γ ) 0
2
f (γ )d γ ,
where Uγ is a uniform probability distribution on [A]γ for all γ > 0. The value of E f (A) does not depend on the boundedness of the support of A. The possibilistic mean value is originally defined for fuzzy numbers (i.e. quasi fuzzy numbers with bounded support). If the support of a quasi fuzzy number A is unbounded then its possibilistic mean value might even not exist. However, for a symmetric quasi fuzzy number A we get E f (A) = a, where a is the center of symmetry, for any weighting function f . Now we will characterize the family of quasi fuzzy numbers for which it is possible to calculate the possibilistic mean value. First we show an example for a quasi triangular fuzzy number that does not have a mean value. Example 3.9.1 Consider the following quasi triangular fuzzy number ⎧ if x ≤ 0 ⎨ 0 1 μA (x) = ⎩√ if 0 ≤ x x+1 In this case a1 (γ ) = 0,
a2 ( γ ) =
1 − 1, γ2
and its possibilistic mean value can not be computed, since the following integral does not exist (not finite),
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3 A Normative View on Possibility Distributions
E(A) = =
1 a1 (γ ) + a2 (γ ) 0
1 1 0
2
2γ dγ
1 1 − 1 γ d γ = − γ dγ . γ2 γ 0
Note 3.8. This example is very important: if the membership √ function of the quasi fuzzy number tends to zero slower than the function 1/ x then it is not possible to calculate the possibilistic mean value, (clearly, the value of the integral will be infinitive), otherwise the possibilistic mean value does exist. To show this, suppose that there exists ε > 0, such that the membership function of quasi fuzzy number A satisfies the property,
μA (x) = O(x− 2 −ε ) 1
if x → +∞. This means that there exists and x0 ∈ R such that,
μA (x) ≤ Mx− 2 −ε , 1
if x > x0 and where M is a positive real number. So the possibilistic mean value of A is bonded from above by 1 1
M − 2 −ε multiplied by the possibilistic mean value of a quasi fuzzy number with membership 1 function x− 2 −ε plus an additional constant (because of the properties of a quasi fuzzy number we know that the interval [0, x0 ] accounts for a finite value in the integral). Suppose that,
⎧ 0 if x < 0 ⎪ ⎨ if 0 ≤ x ≤ 1 μA (x) = 1 ⎪ ⎩ − 1 −ε x 2 if x ≥ 1
A similar reasoning holds for negative fuzzy numbers with membership function 1 (−x)− 2 −ε . Then we get, a1 (γ ) = 0, and since
1 1
a 2 (γ ) = γ − 2 − ε ,
ε − 12 = 1, ε + 12
3.9 A Normative View on Quasi Fuzzy Numbers
67
we can calculate the possibilistic mean value of A as, E(A) = =
1 a1 (γ ) + a2 (γ )
2
0
1 − ε − 12 1 0
γ
ε+ 2
2γ dγ =
1 − 0
γ
1 ε + 12
γ dγ
1 1 1 1 dγ = (ε + ) γ ε + 2 = ε + 1/2 2 0
Theorem 3.9.1 If A is a non-symmetric quasi fuzzy number then E f (A) exists if and only if there exist real numbers ε , δ > 0 , such that, 1 μA (x) = O x− 2 −ε , if x → +∞ and
1 μA (x) = O (−x)− 2 −δ ,
if x → −∞. Note 3.9. If we consider other weighting functions, we need to require that μA (x) = 1 O(x−1−ε ), when x → +∞ (in the worst case, when f (γ ) = 1, is the critical growth γ rate.) Example 3.9.2 Consider the following quasi triangular fuzzy number, ⎧ ⎨ 0 if x ≤ 0 μA (x) = 1 ⎩ if 1 ≤ x x+1 In this case we have, a1 (γ ) = 0,
a2 (γ ) =
1 − 1, γ
and its possibilistic mean value is, E(A) = =
1 a1 ( γ ) + a 2 ( γ ) 0
1 0
2
2 γ dγ =
1 1 0
γ
− 1 γ dγ
(1 − γ )dγ = 1/2.
This example is very important since the volume of A can not be normalized since ∞ 0 μA (x)dx does not exist. In other words, μA can not be considered as a density function of any random variable. Now we will extend the concept of possibilistic variance to the family of quasi fuzzy numbers. Definition 3.9.2 The measure of f -weighted possibilistic variance of a quasi fuzzy number A is the f -weighted average of the probabilistic variances of the respective
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3 A Normative View on Possibility Distributions
Fig. 3.15 Quasi triangular fuzzy number 1/(x + 1), x≥0
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0.5
0.25
0
2.5
5
7.5
10
12.5
15
17.5
2
uniform distributions on the level sets of A. That is, the f -weighted possibilistic variance of A is defined by Var f (A) =
1 0
var(Uγ ) f (γ )dγ =
1 (a2 (γ ) − a1 (γ ))2 0
12
f (γ )dγ .
where Uγ is a uniform probability distribution on [A]γ for all γ > 0. The value of Var f (A) does not depend on the boundedness of the support of A. If f (γ ) = 2γ then we simple write Var(A). From the definition it follows that in this case we can not make any distinction between the symmetric and non-symmetric case. And it is also obvious, since in the definition we have the square of the a1 (γ ) and a2 (γ ) functions, that the decreasing rate of the membership function has to be the square of the mean value case. We can conclude: Theorem 3.9.2 If A is a quasi fuzzy number then Var(A) exists if and only if there exist real numbers ε , δ > 0, such
μA (x) = O(x−1−ε ) if x → +∞ and
μA (x) = O((−x)−1−δ ),
if x → −∞. Note 3.10. If we consider other weighting functions, we need to require that
μA (x) = O(x−2−ε ), 1 when x → +∞ (in the worst case, when f (γ ) = 1, √ is the critical growth rate.) γ Example 3.9.3 Consider again the quasi triangular fuzzy number, ⎧ ⎨ 0 if x ≤ 0 μA (x) = 1 ⎩ if 1 ≤ x x+1
3.9 A Normative View on Quasi Fuzzy Numbers
69
In this case we have, a1 (γ ) = 0,
a2 (γ ) =
1 − 1, γ
and its possibilistic variance does not exist since 1 (a2 (γ ) − a1 (γ ))2 0
12
2 γ dγ =
1 (1/γ − 1)2 0
12
2γ dγ = ∞.
Now we will extend the concept of possibilistic covariance to the family of quasi fuzzy numbers. Definition 3.9.3 The f -weighted measure of possibilistic covariance between A, B ∈ Q, (with respect to their joint distribution C), is defined by, Cov f (A, B) =
1 0
cov(Xγ ,Yγ ) f (γ )d γ ,
where Xγ and Yγ are random variables whose joint distribution is uniform on [C]γ for any γ > 0. It is easy to see that the possibilistic covariance is an absolute measure in the sense that it can take any value from the real line. To have a relative measure of interactivity between marginal distributions Full´er, Mezei and V´arlaki introduced the normalized covariance in 2010 (see [161]). A normalized f -weighted index of interactivity of A, B ∈ F (with respect to their joint distribution C) is defined by
ρ f (A, B) =
1 0
where
ρ (Xγ ,Yγ ) =
ρ (Xγ ,Yγ ) f (γ )dγ
cov(Xγ ,Yγ ) var(Xγ ) var(Yγ )
and, where Xγ and Yγ are random variables whose joint distribution is uniform on [C]γ for all γ ∈ [0, 1]. In other words, the ( f -weighted) index of interactivity is nothing else, but the f -weighted average of the probabilistic correlation coefficients ρ (Xγ ,Yγ ) for all γ ∈ [0, 1]. It is clear that for any joint possibility distribution this correlation coefficient always takes its value from interval [−1, 1], since ρ (Xγ ,Yγ ) ∈ [−1, 1] for any γ ∈ [0, 1] and 01 f (γ )d γ = 1. Since ρ f (A, B) measures an average index of interactivity between the level sets of A and B, we may call this measure as the f -weighted possibilistic correlation coefficient. Now we will extend the concept of possibilistic correlation to the family of quasi fuzzy numbers. Definition 3.9.4 The f -weighted possibilistic correlation coefficient of A, B ∈ Q (with respect to their joint distribution C) is defined by
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3 A Normative View on Possibility Distributions
ρ f (A, B) =
1 0
where
ρ (Xγ ,Yγ ) =
ρ (Xγ ,Yγ ) f (γ )dγ
cov(Xγ ,Yγ ) var(Xγ ) var(Yγ )
and, where Xγ and Yγ are random variables whose joint distribution is uniform on [C]γ for any γ > 0.
3.9.1
Exponential Membership Function
Now we will calculate the possibilistic mean value and variance of a quasi triangular fuzzy number defined by the membership function e−x , x ≥ 0, which can also be seen as a density function of a standard exponential random variable. In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate. Fig. 3.16 Quasi triangular fuzzy number and density function of an exponential random variable with parameter one: e−x , x ≥ 0.
1
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0.25
-0.4
0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
Consider the following quasi triangular fuzzy number
0 if x < 0 μA (x) = e−x if x ≥ 0
From 0∞ μA (x)dx = 1 it follows that μA can also be considered as the density function of a standard exponential random variable (with parameter one). It is wellknown that the mean value and the variance of this probability distribution is equal to one. In the fuzzy case we have, a1 (γ ) = 0,
a2 (γ ) = − ln γ ,
and its possibilistic mean value is E(A) =
1 a1 (γ ) + a2 (γ ) 0
2
2γ d γ =
1 0
1 −(ln γ )γ dγ = , 4
3.9 A Normative View on Quasi Fuzzy Numbers
71
and its possibilistic variance is, 1 (a2 (γ ) − a1(γ ))2
2γ d γ 12 1 (− ln γ )2 1 = γ dγ = . 6 24 0
Var(A) =
0
Let C be the joint possibility distribution, defined by the membership function,
μC (x, y) = e−(x+y) , x ≥ 0, y ≥ 0, of quasi fuzzy numbers A and B with membership functions
μA (x) = e−x , x ≥ 0,
and
μB (y) = e−y , y ≥ 0.
In other words, the membership function of C is defined by a simple multiplication (by Larsen t-norm [222]) of the membership values of μA (x)and μB (y), that is, μC (x, y) = μA (x) × μB (y). The γ -cut of C can be computed by [C]γ = {(x, y) | x + y ≤ − ln γ ; x, y ≥ 0}. Then M(Xγ ) = M(Yγ ) = − M(Xγ2 ) = M(Yγ2 ) =
ln γ , 3
(ln γ )2 , 6
and, var(Xγ ) = M(Xγ2 ) − M(Xγ )2 (ln γ )2 (ln γ )2 − 6 9 2 (ln γ ) = . 18 =
Similarly we obtain, var(Yγ ) =
(ln γ )2 . 18
Furthermore, M(Xγ Yγ ) =
(ln γ )2 , 12
cov(Xγ ,Yγ ) = M(Xγ Yγ ) − M(Xγ )M(Yγ ) = −
(ln γ )2 , 36
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3 A Normative View on Possibility Distributions
we can calculate the probabilistic correlation by
ρ (Xγ ,Yγ ) =
cov(Xγ ,Yγ ) 1 =− . 2 var(Xγ ) var(Yγ )
That is, ρ (Xγ ,Yγ ) = −1/2 for any γ > 0. Consequently, their possibilistic correlation coefficient is, ρ f (A, B) = −1/2 for any weighting function f . On the other hand, in a probabilistic context, μC (x, y) = μA (x) × μB (y) = e−(x+y) can be also considered as the joint density function of independent exponential marginal probability distributions with parameter one. That is, in a probabilistic context, their (probabilistic) correlation coefficient is equal to zero. Note 3.11. The probabilistic correlation coefficient between two standard exponential marginal probability distributions can not go below (1 − π 2/6). Really, the lower limit, denoted by τ , can be computed from,
τ=
∞ ∞ 0
=−
1 − e−x − e−y )+ − (1 − e−x)(1 − e−y ) dxdy
0 ∞ ∞ 0
0
e−x e−y dxdy +
(1 − e−x − e−y )+ dxdy
0<x, 0
= −1 +
(2e−x − 1)dxdy
0<x, 0
using the substitutions u = e−x , v = e−y we get, 2 1 τ = −1 + − dudv u uv u<1, v<1, u+v>1
= −1 + = −1 +
1 1 1 0
u
1 1 2u + log(1 − u) 2 − dvdu = 1 + du v u 0
0
1 log(1 − u) 0
u
du =
1 ∞ k−1 u
∑
0 k=1
k
du
∞
1 π2 = 1 − . 2 6 k=1 k
= 1− ∑
In the case of possibility distributions there is no known lower limit [177]. If the joint possibility distribution C is given by the minimum operator (Mamdani t-norm [246]), μC (x, y) = min{ μA (x), μB (y)} = min{e−x , e−y }, x ≥ 0, y ≥ 0, then A and B are non-interactive marginal possibility distributions and, therefore, their possibilistic correlation coefficient equal to zero.
3.10 Addition and Subtraction of Interactive Fuzzy Numbers
3.10
73
Addition and Subtraction of Interactive Fuzzy Numbers
In this Section we will consider additions of interactive fuzzy numbers. The interactivity relation between fuzzy numbers will be defined by their joint possibility distribution. Following Carlsson, Full´er and Majlender [67] we will show that Nguyen’s theorem remains valid in this environment. We will give explicit formulas for the γ -level sets of the extended sum of two interactive fuzzy numbers in perfect correlation. We will show that (i) the interactive sum A + B of two fuzzy numbers A and B having a correlation coefficient minus one, where B(x) = (−A)(x) = A(−x) for all x ∈ R, is equal to fuzzy zero; (ii) the interactive difference A − B, of two fuzzy numbers A and B having a correlation coefficient one and having identical membership functions, is equal to fuzzy zero. Properties of additions of interactive fuzzy numbers, when their joint possibility distribution is defined by a t-norm have been extensively studied in the literature [14, 125, 150, 190, 211]. It is clear that if C(x1 , . . . , xn ) = T (A1 (x1 ), . . . , An (xn )) then the joint possibility distribution is defined directly and pointwise from the membership values of its marginal possibility distributions by an aggregation operator. However, the interactivity relation between fuzzy numbers may be given by a more general joint possibility distribution, which can not be directly defined from the membership values of its marginal possibility distributions by any aggregation operator. In this Section, following [67], we will consider some properties of the addition operator on perfectly correlated fuzzy numbers, where the interactivity relation is given by their joint possibility distribution. We recall that numbers A and B are said to be in perfect correlation, if there exist q, r ∈ R, q = 0 such that their joint possibility distribution is defined by C(x1 , x2 ) = A(x1 ) · χ{qx1 +r=x2 } (x1 , x2 ) = B(x2 ) · χ{qx1+r=x2 } (x1 , x2 ),
(3.14)
where χ{qx1 +r=x2 } , stands for the characteristic function of the line {(x1 , x2 ) ∈ R2 |qx1 + r = x2 }. In this case we have, [C]γ = (x, qx + r) ∈ R2 x = (1 − t)a1(γ ) + ta2 (γ ),t ∈ [0, 1] where [A]γ = [a1 (γ ), a2 (γ )], and [B]γ = q[A]γ + r, for any γ ∈ [0, 1], and, finally, their membership functions satisfy the following property, x−r B(x) = A , q for all x ∈ R. Now let us consider the extended addition of interactive fuzzy numbers A and B that are in perfect correlation,
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3 A Normative View on Possibility Distributions
(A + B)(y) = sup C(x1 , x2 ). y=x1 +x2
That is,
(A + B)(y) = sup A(x1 ) · χ{qx1 +r=x2 } (x1 , x2 ). y=x1 +x2
Fig. 3.17 The correlation coefficient between A and B is -1
Then from (2.14) and (3.14) we find, [A + B]γ = cl{x1 + x2 ∈ R|A(x1 ) > γ , qx1 + r = x2 } = cl{(q + 1)x1 + r ∈ R|A(x1 ) > γ } = (q + 1)cl{x1 ∈ R|A(x1 ) > γ } + r = (q + 1)[A]γ + r,
that is,
[A + B]γ = (q + 1)[A]γ + r,
(3.15)
for all γ ∈ [0, 1]. If q = −1 then (see Fig. 3.17) [B]γ = −[A]γ + r, for all γ ∈ [0, 1], then A + B will be a crisp number. Really, from (3.15) we get [A + B]γ = 0 × [A]γ + r = [r, r]γ = {r}, for all γ ∈ [0, 1].
3.10 Addition and Subtraction of Interactive Fuzzy Numbers
75
If q = −1 and r = 0, i.e. A(x) = (−B)(x) = B(−x), ∀x ∈ R, then from (3.15) we get (A + B)(z) =
0 if z = 0 1 if z = 0
that is, A + B = 0¯ where 0¯ denotes a fuzzy point with support {0}. Note 3.12. The interactive sum A + B, of two fuzzy numbers A and B having a cor¯ relation coefficient -1 and with A(x) = (−B)(x) = B(−x), ∀x ∈ R, is equal to 0. Let us consider now the subtraction operator for interactive fuzzy numbers A and B, where their joint possibility distribution is defined by (3.14). (A − B)(y) = sup C(x1 , x2 ). y=x1 −x2
That is,
(A − B)(y) = sup A(x1 ) · χ{qx1 +r=x2 } (x1 , x2 ). y=x1 −x2
Then for a γ -level set of A − B we get, [A − B]γ = cl{x1 − x2 ∈ R|A(x1 ) > γ , qx1 + r = x2 } = (1 − q)[A]γ − r for all γ ∈ [0, 1]. In particular if q = 1, i.e. [B]γ = [A]γ + r, ∀γ ∈ [0, 1] then
[A − B]γ = −[r, r]γ = −{r},
that is, the fuzziness of A − B vanishes. Note 3.13. If q = 1 and r = 0 we have A(x) = B(x), for x ∈ R and
C(x, y) = A(x)χ{x=y} (x, y) = B(y)χ{x=y} (x, y)
for all x, y ∈ R and from (3.15) we get ¯ A − B = 0,
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3 A Normative View on Possibility Distributions
for all z ∈ R. The interactive difference A − B, of two fuzzy numbers A and B having ¯ a correlation coefficient 1 and having identical membership functions, is equal to 0. Question 1 Let C be a joint possibility distribution with marginal possibility distributions A and B. On what conditions will the equality A +C B = A + B, hold, where the interactive sum of A and B is defined by, (A +C B)(y) = sup C(x1 , x2 ), y=x1 +x2
and the non-interactive sum of A and B is defined by, (A + B)(y) = sup min{A(x1 ), B(x2 )}? y=x1 +x2
Note 3.14. Let A and B be fuzzy numbers, where the membership function of B is defined by x−r B(x) = A , q for any x ∈ R, then for any q > 0 we find [A + B]γ = [A]γ + [B]γ
= [A]γ + q[A]γ + r = (q + 1)[A]γ + r = [A +C B]γ .
for all γ ∈ [0, 1]. So,
A +C B = A + B.
that is, the membership function of the interactive sum of fuzzy numbers with correlation coefficient one (defined by (2.14) and (3.14)) is equal to the membership function of their non-interactive sum (defined by their sup-min convolution).
Chapter 4
OWA Operators in Multiple Criteria Decisions
The process of information aggregation appears in many applications related to the development of intelligent systems. In 1988 Yager introduced a new aggregation technique based on the ordered weighted averaging operators (OWA) [338]. The determination of ordered weighted averaging (OWA) operator weights is a very important issue of applying the OWA operator for decision making. One of the first approaches, suggested by O’Hagan, determines a special class of OWA operators having maximal entropy of the OWA weights for a given level of orness; algorithmically it is based on the solution of a constrained optimization problem. In 2001, using the method of Lagrange multipliers, Full´er and Majlender solved this constrained optimization problem analytically and determined the optimal weighting vector. In 2003 using the Karush-Kuhn-Tucker second-order sufficiency conditions for optimality, Full´er and Majlender [155] computed the exact minimal variability weighting vector for any level of orness. In this Chapter we give a short survey of some later works that extend and develop these models.
4.1
Obtaining OWA Operator Weights
One important issue in the theory of OWA operators is the determination of the associated weights. One of the first approaches, suggested by O’Hagan, determines a special class of OWA operators having maximal entropy of the OWA weights for a given level of orness; algorithmically it is based on the solution of a constrained optimization problem. Another consideration that may be of interest to a decision maker involves the variability associated with a weighting vector. In particular, a decision maker may desire low variability associated with a chosen weighting vector. It is clear that the actual type of aggregation performed by an OWA operator depends upon the form of the weighting vector [344]. A number of approaches have been suggested for obtaining the associated weights, i.e., quantifier guided aggregation [338, 339], exponential smoothing [144] and learning [346]. Nettleton and Torra [264] suggested a genetic algorithm (GA). O’Hagan [267] determined a special class of OWA operators having maximal entropy of the OWA weights for a given level of orness. His approach is based on the solution of he following mathematical programming problem, C. Carlsson and R. Full´er: Possibility for Decision, STUDFUZZ 270, pp. 77–101. springerlink.com © Springer-Verlag Berlin Heidelberg 2011
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4 OWA Operators in Multiple Criteria Decisions n
disp(W ) = − ∑ wi ln wi
maximize
i=1
n−i subject to orness(W ) = ∑ · wi = α , 0 ≤ α ≤ 1 i=1 n − 1 n
(4.1)
w1 + · · · + wn = 1, 0 ≤ wi , i = 1, . . . , n. In 2001, using the method of Lagrange multipliers, Full´er and Majlender [153] transformed constrained optimization problem (4.1) into a polynomial equation which is then was solved to determine the maximal entropy OWA operator weights. By their method, the associated weighting vector is easily obtained by ln w j =
j−1 n− j ln wn + ln w1 =⇒ w j = n−1 n−1
and wn = then
n−1
n− j
j−1
w1 wn
((n − 1)α − n)w1 + 1 (n − 1)α + 1 − nw1
w1 [(n − 1)α + 1 − nw1]n = ((n − 1)α )n−1 [((n − 1)α − n)w1 + 1]
where n ≥ 3. For n = 2 then from orness(w1 , w2 ) = α the optimal weights are uniquely defined as w∗1 = α and w∗2 = 1 − α . Furthemore, if α = 0 or α = 1 then the associated weighting vectors are uniquely defined as (0, 0, . . . , 0, 1)T and (1, 0, . . . , 0, 0)T , respectively. An interesting question is to determine the minimal variability weighting vector under given level of orness [342]. The variance of a given weighting vector is computed as follows n 2 n 1 1 n 1 1 n 1 D2 (W ) = ∑ (wi − E(W ))2 = ∑ w2i − wi = ∑ w2i − 2 . ∑ n i=1 n i=1 n i=1 n i=1 n where E(W ) = (w1 + · · · + wn )/n = 1/n stands for the arithmetic mean of weights. In 2003 Full´er and Majlender [155] suggested a minimum variance method to obtain the minimal variability OWA operator weights. A set of OWA operator weights with minimal variability could then be generated. Their approach requires the solution of the following mathematical programming problem: minimize
1 n 1 D2 (W ) = · ∑ w2i − 2 n i=1 n n−i · wi = α , 0 ≤ α ≤ 1, n i=1 − 1 n
subject to orness(w) = ∑
w1 + · · · + wn = 1, 0 ≤ wi , i = 1, . . . , n.
(4.2)
4.1 Obtaining OWA Operator Weights
79
Using the Karush-Kuhn-Tucker second-order sufficiency conditions for optimality, Full´er and Majlender [155] computed the exact minimal variability weighting vector for any level of orness. The following disjunctive partition, originally introduced in [155], of (0, 1) is crucial in finding an optimal solution to problem (4.2): (0, 1) =
n−1
Jr,n ∪ J1,n ∪
r=2
n−1
J1,s .
(4.3)
s=2
where 1 2n + r − 2 1 2n + r − 3 Jr,n = 1 − · , 1− · , r = 2, . . . , n − 1, 3 n−1 3 n−1 1 2n − 1 1 n−2 J1,n = 1 − · , 1− · , 3 n−1 3 n−1 1 s−1 1 s−2 J1,s = 1 − · , 1− · , s = 2, . . . , n − 1. 3 n−1 3 n−1 Then the minimal variability weighting vector is computed by W ∗ = (0, . . . , 0, w∗r , . . . , w∗s , 0 . . . , 0)T ,
(4.4)
where w∗j = 0, if j ∈ / I{r,s} , 2(2s + r − 2) − 6(n − 1)(1 − α ) , (s − r + 1)(s − r + 2) 6(n − 1)(1 − α ) − 2(s + 2r − 4) w∗s = , (s − r + 1)(s − r + 2) s− j j−r w∗j = · wr + · ws , if j ∈ I{r+1,s−1}. s−r s−r w∗r =
(4.5)
and where I{r,s} = {r, . . . , s} and I{r+1,s−1} = {r + 1, . . . , s − 1}. We note that if r = 1 and s = n then we have 1 2n − 1 1 n−2 α ∈ J1,n = 1 − · , 1− · , 3 n−1 3 n−1
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4 OWA Operators in Multiple Criteria Decisions
and W ∗ = (w∗1 , . . . , w∗n )T , where 2(2n − 1) − 6(n − 1)(1 − α ) , n(n + 1) 6(n − 1)(1 − α ) − 2(n − 2) w∗n = , n(n + 1) n− j j−1 w∗j = · w1 + · wn , if j ∈ {2, . . . , n − 1}. n−1 n−1 w∗1 =
It can be shown (see [155]) that W ∗ , defined by (4.4), satisfies the Karush-KuhnTucker second-order sufficiency conditions for optimality ([98], page 58). Following [155] we will determine the minimal variability five-dimensional weighting vector under orness levels α = 0.1, . . . , 0.9. First, we construct the corresponding partition as (0, 1) =
4
r=2
where
Jr,5 =
Jr,5 ∪ J1,5 ∪
4
J1,s .
s=2
1 5−r−1 1 5−r 4−r 5−r · , · = , , 3 5−1 3 5−1 12 12
for r = 2, 3, 4 and J1,5 = and
1 5 − 2 1 10 − 1 3 9 · , · = , , 3 5−1 3 5−1 12 12
1 s−1 1 s−2 13 − s 14 − s J1,s = 1 − · , 1− · = , , 3 5−1 3 5−1 12 12
for s = 2, 3, 4, and, therefore we get, 1 1 2 2 3 3 9 (0, 1) = 0, ∪ , ∪ , ∪ , 12 12 12 12 12 12 12 9 10 10 11 11 12 ∪ , ∪ , ∪ , . 12 12 12 12 12 12 Without loss of generality we can assume that α < 0.5, because if a weighting vector W is optimal for problem (4.2) under some given degree of orness, α < 0.5, then its reverse, denoted by W R , and defined as wRi = wn−i+1 , is also optimal for problem (4.2) under degree of orness (1 − α ). Really, as was shown by Yager [339], we find that D2 (W R ) = D2 (W ) and orness(W R ) = 1 − orness(W ). Therefore, for any α > 0.5, we can solve problem (4.2) by solving it for level of orness (1 − α ) and then taking the reverse of that solution. Then we obtain the optimal weights from (4.5) as follows
4.1 Obtaining OWA Operator Weights
• if α = 0.1 then
81
α ∈ J3,5 =
1 2 , , 12 12
and the associated minimal variability weights are w∗1 (0.1) = 0, w∗2 (0.1) = 0, 2(10 + 3 − 2) − 6(5 − 1)(1 − 0.1) 0.4 = = 0.0333, (5 − 3 + 1)(5 − 3 + 2) 12 2 w∗5 (0.1) = − w∗3 (0.1) = 0.6334, 5−3+1 1 1 w∗4 (0.1) = · w∗3 (0.1) + · w∗5 (0.1) = 0.3333, 2 2 w∗3 (0.1) =
So, W ∗ (α ) = W ∗ (0.1) = (0, 0, 0.033, 0.333, 0.633)T , and, consequently, W ∗ (0.9) = (W ∗ (0.1))R = (0.633, 0.333, 0.033, 0, 0)T . with variance D2 (W ∗ (0.1)) = 0.0625. • if α = 0.2 then
α ∈ J2,5 =
2 3 , 12 12
and in a similar manner we find that the associated minimal variability weighting vector is W ∗ (0.2) = (0.0, 0.04, 0.18, 0.32, 0.46)T , and, therefore,
W ∗ (0.8) = (0.46, 0.32, 0.18, 0.04, 0.0)T ,
with variance D2 (W ∗ (0.2)) = 0.0296. • if α = 0.3 then
α ∈ J1,5 =
3 9 , 12 12
and in a similar manner we find that the associated minimal variability weighting vector is W ∗ (0.3) = (0.04, 0.12, 0.20, 0.28, 0.36)T , and, therefore, W ∗ (0.7) = (0.36, 0.28, 0.20, 0.12, 0.04)T , with variance D2 (W ∗ (0.3)) = 0.0128.
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4 OWA Operators in Multiple Criteria Decisions
• if α = 0.4 then
α ∈ J1,5 =
3 9 , 12 12
and in a similar manner we find that the associated minimal variability weighting vector is W ∗ (0.4) = (0.12, 0.16, 0.20, 0.24, 0.28)T , and, therefore, W ∗ (0.6) = (0.28, 0.24, 0.20, 0.16, 0.12)T , with variance D2 (W ∗ (0.4)) = 0.0032. • if α = 0.5 then W ∗ (0.5) = (0.2, 0.2, 0.2, 0.2, 0.2)T . with variance D2 (W ∗ (0.5)) = 0.
4.2
Constrained OWA Aggregations
Yager [341] considered the problem of maximizing an OWA aggregation of a group of variables that are interrelated and constrained by a collection of linear inequalities and he showed how this problem can be modeled as a mixed integer linear programming problem. The constrained OWA aggregation problem [341] can be expressed as the following mathematical programming problem max F(x1 , . . . , xn ) subject to Ax ≤ b, x ≥ 0, where F(x1 , . . . , xn ) = wT y = w1 y1 + · · · + wn yn and y j denotes the jth largest element of the bag < x1 , . . . , xn >. As an illustration of the general constrained OWA aggregation problem, Yager [341] considered problem (4.6) for n = 3 and showed how it can be modeled as a mixed integer linear programming problem. Then he used the Storm software to solve it. In this Section, following Carlsson, Full´er and Majlender [62] we will present a simple algorithm for exact computation of optimal solutions to a constrained OWA aggregation problem with a single constraint on the sum of all decision variables. Consider the following (nonlinear) constrained OWA aggregation problem max wT y subject to {x1 + · · · + xn ≤ 1, x ≥ 0}.
(4.6)
To find an optimal solution to (4.6) we should proceed as follows (see [62] for details): select the maximal element of the set
4.2 Constrained OWA Aggregations
83
2-nd n-th 1-st ! w1 + · · · + wn w1 + w2 H = w1 , ,..., , 2 n and, if its maximizing element is the k-th one then choose the k-th element element from the set 1-st k-th {( 1/k , . . . 1/k , 0, . . . , 0)T |k = 1, . . . , n} (4.7) for the unique solution of constrained OWA aggregation problem (4.6). As an example, consider the following 3-dimensional constrained OWA aggregation problem max F(x1 , x2 , x3 ) subject to {x1 + x2 + x3 ≤ 1, x ≥ 0}.
(4.8)
Then the set of all conceivable optimal values is constructed as
" w1 + w2 w1 + w2 + w3 H = w1 , , 2 3 and, the correspending optimal solutions are 1. If maxH = w1 then an optimal solution to problem (4.8) will be x∗1 = 1, x∗2 = x∗3 = 0 with
F(x∗ ) = w1 .
2. If maxH = (w1 + w2 )/2 an optimal solution to problem (4.8) will be x∗1 = x∗2 = 1/2, x∗3 = 0 with
F(x∗ ) = (w1 + w2 )/2.
3. If maxH = (w1 + w2 + w3 )/3 an optimal solution to problem (4.8) will be x∗1 = x∗2 = x∗3 = 1/3 with
F(x∗ ) = (w1 + w2 + w3 )/3.
From the commutativity of OWA operators it follows that all permutations of the coordinates of an optimal solution are also optimal solutions to constrained OWA aggregation problems.
84
4.3
4 OWA Operators in Multiple Criteria Decisions
Recent Advances
In this Section we will give a short chronological survey of some later works that extend and develop the maximal entropy and the minimal variability OWA operator weights models. We will mention only those works in which the authors extended, improved or used the findings of our original papers [153, 154]. In 2004 Liu and Chen [227] introduced the concept of parametric geometric OWA operator (PGOWA) and a parametric maximum entropy OWA operator (PMEOWA) and showed the equivalence of parametric geometric OWA operator and parametric maximum entropy OWA operator weights. In 2005 Wang and Parkan [327] presented a minimax disparity approach, which minimizes the maximum disparity between two adjacent weights under a given level of orness. Their approach was formulated as minimize
max
i=1,2,...,n−1
| wi − wi+1 |
n−i wi = α , 0 ≤ α ≤ 1, n i=1 − 1 n
subject to orness(w) = ∑
w1 + · · · + wn = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n. Majlender [245] developed a maximal R´enyi entropy method for generating a parametric class of OWA operators and the maximal R´enyi entropy OWA weights. His approach was formulated as maximize Hβ (w) =
n 1 β log2 ∑ wi 1−β i=1
n−i wi = α , 0 ≤ α ≤ 1, i=1 n − 1 n
subject to orness(w) = ∑
w1 + · · · + wn = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n. where β ∈ R and H1 (w) = − ∑ni=1 wi log2 wi . Liu [229] extended the the properties of OWA operator to the RIM (regular increasing monotone) quantifier which is represented with a monotone function instead of the OWA weighting vector. He also introduced a class of parameterized equidifferent RIM quantifier which has minimum variance generating function. This equidifferent RIM quantifier is consistent with its orness level for any aggregated elements, which can be used to represent the decision maker’s preference. Troiano and Yager [315] pointed out that OWA weighting vector and the fuzzy quantifiers are strongly related. An intuitive way for shaping a monotonic quantifier, is by means of the threshold that makes a separation between the regions of what is satisfactory and what is not. Therefore, the characteristics of a threshold can be directly related to the OWA weighting vector and to its metrics: the attitudinal character and the entropy. Usually these two metrics are supposed to be independent, although some limitations in their value come when they are considered jointly. They argued that these two metrics are strongly related by
4.3 Recent Advances
85
the definition of quantifier threshold, and they showed how they can be used jointly to verify and validate a quantifier and its threshold. In 2006 Xu [336] investigated the dependent OWA operators, and developed a new argument-dependent approach to determining the OWA weights, which can relieve the influence of unfair arguments on the aggregated results. Zadrozny and Kacprzyk [357] discussed the use of the Yager’s OWA operators within a flexible querying interface. Their key issue is the adaptation of an OWA operator to the specifics of a user’s query. They considered some well-known approaches to the manipulation of the weights vector and proposed a new one that is simple and efficient. They discussed the tuning (selection of weights) of the OWA operators, and proposed an algorithm that is effective and efficient in the context of their FQUERY for Access package [208, 209]. Wang, Chang and Cheng [328] developed the query system of practical hemodialysis database for a regional hospital in Taiwan, which can help the doctors to make more accurate decision in hemodialysis. They built the fuzzy membership function of hemodialysis indices based on experts’ interviews. They proposed a fuzzy OWA query method, and let the decision makers (doctors) just need to change the weights of attributes dynamical, then the proposed method can revise the weight of each attributes based on aggregation situation and the system will provide synthetic suggestions to the decision makers. Chang et al [100] proposed a dynamic fuzzy OWA model to deal with problems of group multiple criteria decision making. Their proposed model can help users to solve MCDM problems under the situation of fuzzy or incomplete information. Amin and Emrouznejad [10] introduced an extended minimax disparity model to determine the OWA operator weights as follows, minimize δ n−i wi = α , 0 ≤ α ≤ 1, i=1 n − 1 n
subject to orness(w) = ∑
w j − wi + δ ≥ 0, i = 1, . . . , n − 1, j = i + 1, . . ., n wi − w j + δ ≥ 0, i = 1, . . . , n − 1, j = i + 1, . . ., n w1 + · · · + wn = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n. In this model it is assumed that the deviation |wi − w j | is always equal to δ , i = j. In 2007 Liu [230] proved that the solutions of the minimum variance OWA operator problem under given orness level and the minimax disparity problem for OWA operator are equivalent, both of them have the same form of maximum spread equidifferent OWA operator. He also introduced the concept of maximum spread equidifferent OWA operator and proved its equivalence to the minimum variance OWA operator. Llamazares [236] proposed determining OWA operator weights regarding the class of majority rule that one should want to obtain when individuals do not grade their preferences between the alternatives. Wang, Luo and Liu [329] introduced two models determining as equally important OWA operator weights as possible for a given orness degree. Their models can be written as
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4 OWA Operators in Multiple Criteria Decisions
minimize J1 =
n−1
∑ (wi − wi+1)2
i=1
n−i wi = α , 0 ≤ α ≤ 1, n i=1 − 1 n
subject to orness(w) = ∑
w1 + · · · + wn = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n. and minimize J2 =
n−1
∑
i=1
wi wi+1 − wi+1 wi
2
n−i wi = α , 0 ≤ α ≤ 1, n i=1 − 1 n
subject to orness(w) = ∑
w1 + · · · + wn = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n. Yager [348] used stress functions to obtain OWA operator weights. With this stress function, a user can ”stress” which argument values they want to give more weight in the aggregation. An important feature of this stress function is that it is only required to be nonnegative function on the unit interval. This allows a user to completely focus on the issue of where to put the stress in the aggregation without having to consider satisfaction of any other requirements. In 2008 Liu [231] proposed a general optimization model with strictly convex objective function to obtain the OWA operator under given orness level, n
minimize
∑ F(wi )
i=1
n−i wi = α , 0 ≤ α ≤ 1, n i=1 − 1 n
subject to orness(w) = ∑
w1 + · · · + wn = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n. and where F is a strictly convex function on [0, 1], and it is at least two order differentiable. His approach includes the maximum entropy (for F(x) = x ln x) and the minimum variance (for F(x) = x2 problems as special cases. More generally, when F(x) = xα , α > 0 it becomes the OWA problem of R´enyi entropy [245], which includes the maximum entropy and the minimum variance OWA problem as special cases. Liu also included into this general model the solution methods and the properties of maximum entropy and minimum variance problems that were studied separately earlier. The consistent property that the aggregation value for any aggregated set monotonically increases with the given orness value is still kept, which gives more alternatives to represent the preference information in the aggregation of decision making. Then, with the conclusion that the RIM quantifier can be seen as the continuous case of OWA operator with infinite dimension, Liu [231] further suggested a general RIM quantifier determination model, and analytically solved it with
4.3 Recent Advances
87
the optimal control technique. Ahn [4] developed some new quantifier functions for aiding the quantifier-guided aggregation. They are related to the weighting functions that show properties such that the weights are strictly ranked and that a value of orness is constant independently of the number of criteria considered. These new quantifiers show the same properties that the weighting functions do and they can be used for the quantifier-guided aggregation of a multiple-criteria input. The proposed RIM and regular decreasing monotone (RDM) quantifiers produce the same orness as the weighting functions from which each quantifier function originates. the quantifier orness rapidly converges into the value of orness of the weighting functions having a constant value of orness. This result indicates that a quantifierguided OWA aggregation will result in a similar aggregate in case the number of criteria is not too small. In 2009 Wu et al [334] used a linear programming model for determining ordered weighted averaging operator weights with maximal Yager’s entropy [340]. By analyzing the desirable properties with this measure of entropy, they proposed a novel approach to determine the weights of the OWA operator. Ahn [5] showed that a closed form of weights, obtained by the least-squared OWA (LSOWA) method, is equivalent to the minimax disparity approach solution when a condition ensuring all positive weights is added into the formulation of minimax disparity approach. Liu [233] presented some methods of OWA determination with different dimension instantiations, that is to get an OWA operator series that can be used to the different dimensional application cases of the same type. He also showed some OWA determination methods that can make the elements distributed in monotonic, symmetric or any function shape cases with different dimensions. Using Yager’s stress function method [348] he managed to extend an OWA operator to another dimensional case with the same aggregation properties. In 2010 Ahn [6] presented a general method for obtaining OWA operator weights via an extreme point approach. The extreme points are identified by the intersection of an attitudinal character constraint and a fundamental ordered weight simplex that is defined as K = {w ∈ Rn | w1 + w2 + · · · + wn = 1, w j ≥ 0, j = 1, . . . , n}. The parameterized OWA operator weights, which are located in a convex hull of the identified extreme points, can then be specifically determined by selecting an appropriate parameter. Vergara and Xia [322] proposed a new method to find the weights of an OWA for uncertain information sources. Given a set of uncertainty data, the proposed method finds the combination of weights that reduces aggregated uncertainty for a predetermined orness level. Their approach assures best information quality and precision by reducing uncertainty. Yager [349] introduced a measure of diversity related to the problem of selecting of selecting n objects from a pool of candidates lying in q categories. In 2011 Liu [234] summarizing the main OWA determination methods (the optimization criteria methods, the sample learning methods, the function based methods, the argument dependent methods and the preference methods) showed some
88
4 OWA Operators in Multiple Criteria Decisions
relationships between the methods in the same kind and the relationships between different kinds. Hon [193] proved the extended minimax disparity OWA problem. Note 4.1. One of the main problems in group decision making - as pointed out by Carlsson, Fedrizzi and Full´er in 2003 [63] - is how to define a fusion method which considers the majority opinions from the individual opinions. Liu and Han [232] introduced a method for calculating OWA operator to consider priori preference of alternatives. Using the principle of ’most preferred OWA operator’ Emrouznejad [131] introduced an alternative OWA operator to consider majority opinion using experts choices and preference of alternatives. Wang and Parkan [330] proposed a preemptive goal programming method (PGPM) for group decision making problems to generate a set of optimal OWA operator weights. They stated the following preemptive goal programming model, m
minimize P1 ∑ hk |εk | + P2δ k=1
n−i subject to orness(w) = ∑ wi − αk = εk , 0 ≤ αk ≤ 1, k = 1, . . . , m, i=1 n − 1 n
wi − wi+1 − δ ≤ 0, i = 1, . . . , n − 1 wi − wi+1 + δ ≥ 0, i = 1, . . . , n − 1, w1 + · · · + wn = 1, 0 ≤ wi ≤ 1, i = 1, . . . , n. where P1 and P2 are preemptive priority factors, and hk is the relative importance of the k-th weight, k = 1, . . . , m. The proposed PGPM is an extension of their minimax disparity approach [327] to group decision making.
4.4
OCA Functions for Group Settlement Modeling
The challenge of group decision is deciding what action a group should take. Decision makers are invited to express their opinions/preferences on a set of alternatives. Their preferences can be elicited either by asking them to pairwise compare alternatives using suitable semantic scales [269, 290, 311], or by means of priority vectors, namely vectors whose components are scores of alternatives. Priority vectors are cardinal rankings and, to our aim, we do not need to set any constraint about their interpretation and their admissible values. Therefore, given a rating, say w = (w1 , w2 , . . . , wk ), on a set of k alternatives, the scale of admissible values can be either unipolar or bipolar, but also ratio, interval, absolute and ordinal as we only require that a real number representing a score or an utility level is associated to each alternative, i.e. w ∈ Rk . Note that, in the case of an ordinal scale, scores can be associated to alternatives, for example, by using the Borda count method [27]. An important step in obtaining a collective representation of preferences of a group of decision makers is the aggregation process. A large number of papers has been devoted to discuss this issue and various proposals have been made. Normally,
4.4 OCA Functions for Group Settlement Modeling
89
preference aggregation can be performed in two different ways: (i) by aggregating individual judgments (AIJ) and (ii) by aggregating individual preferences (AIP) [146]. The difference between them, is that the first one includes all those models performing the aggregation of preferences expressed by means of pairwise comparison matrices whereas the second one includes all those methods that apply priority vectors. In this Section, following Brunelli, Full´er and Mezei [41] we will describe AIP models. Our choice can be justified by the fact that, if the decision maker is perfectly consistent, then a pairwise comparison matrix and its associated priority vector are just two different ways for expressing the same opinions over a set of alternatives and also when the decision maker is not fully consistent but reasonably close to being such, the priority vector is assumed to be a sufficiently coherent estimation of his/her preferences. Last but not least, the handling of the ratings is easier and computationally less demanding than performing similar operations on pairwise comparison matrices. It might be useful to divide preference aggregation models into two families, with respect to the nature of the underlying model. The first one is to employ aggregation functions like the weighted arithmetic and geometric means as suggested in [283, 146] or the centroid, or center of mass of the preferences as suggested in [201]. To this class belongs also [17] where the authors suggested the middlemost value instead of the average values. The second family consists of all those methods where the collective representation of preferences is obtained by solving a usually nonlinear optimization problem (see [256] as examples). As a rule, this optimization method is used to minimize the total level of disagreement and an optimal – or consensual [254] solution – can be said to be the less disagreed one. Whenever such a consensual solution has to be found, a single valued, nonnegative cost function is assigned to each decision maker and then the overall cost is minimized. These cost functions are called opinion changing aversion (OCA) functions [255]. Several different cost functions have been proposed in the literature, e.g. Hamming distance, Euclidean distance, sigmoid functions. In this Section we will focus on quadratic OCA functions and show that the group decision (or settlement) will be nothing else but the center of gravity of the opinions of the decision makers. In 1993 Mich, Fedrizzi and Gaio [255] introduced a new measure of consensus depending on a function estimated for each expert according to her/his aversion to opinion changing. These opinion changing aversion (OCA) functions have been further developed in [28, 136, 138, 151]. In many cases, the group decision is heavily influenced by the degree of importance of participants. For example, the opinions of executives should be more reflected in the final conclusion of group decision. Yet, it could be also reasonable to assume that the importance of decision makers should be somehow related with their degrees of consistency [139, 140]. Therefore, we are now considering the relative importance weight of each expert. Let the degree of importance of i-th expert be βi , then his/her relative degree of importance is
βi . β1 + β2 + · · · + βn
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4 OWA Operators in Multiple Criteria Decisions
Let (ai , bi , ci ) denote the best possible settlement (or ideal solution) for the j-th expert. For simplicity, we will consider only three alternatives/criteria, but our results can easily be derived for the general case. We will use the following family of polynomial OCA functions, Cik (x, y, z) =
βi (x − ai )2k + (y − bi)2k + (z − ci)2k × , β 1 + β2 + · · · + βn M
(4.9)
where k ∈ N and M is defined by, M = max{(ai − a j )2k + (bi − b j )2k + (ci − c j )2k }. i, j
It is clear that Cik satisfies the conditions of an OCA function since Cik (ai , bi , ci ) = 0 and the directional derivative of Cik is positive in any direction at point (ai , bi , ci ) . However, for computational purposes we shall consider only the simplest polynomial OCA function, the quadratic one, defined by Ci (x, y, z) =
βi (x − ai )2 + (y − bi)2 + (z − ci )2 × , β1 + β2 + · · · + βn M
(4.10)
where M is defined by, M = max{(ai − a j )2 + (bi − b j )2 + (ci − c j )2 }. i, j
Here Ci (x, y, z) measures the aversion of the i-th expert to changing her/his opinion from (ai , bi , ci ) to (x, y, z), for i = 1, . . . , n.
4.4.1
Optimal Unconditional Settlement
The optimal strategy minimizes the overall aversion to opinion changing, and it can be computed by solving the following optimization problem, n
C (x, y, z) = ∑ Ci (x, y, z) → min; subject to x, y, z ∈ R
(4.11)
i=1
If we use quadratic OCA functions of the form (4.10) then the minimization problem can be stated as, n
βi
∑ β1 + β2 + · · · + β n ×
i=1
(x − ai )2 + (y − bi)2 + (z − ci)2 → min, M
4.4 OCA Functions for Group Settlement Modeling
91
which can be written in the form, n # $ 1 × ∑ βi (x − ai )2 + (y − bi)2 + (z − ci)2 → min M(β1 + β2 + · · · + βn) i=1
Following Brunelli, Full´er and Mezei [41] the unique optimal solution to minimization problem (4.11) can be written as ⎛ ⎞ a1 β 1 + a2 β 2 + · · · + an β n ⎜ β1 + β2 + · · · + βn ⎟ ⎜ ⎟ ⎟ ⎛ ∗⎞ ⎜ ⎜ ⎟ x ⎜ ⎟ ⎝y∗ ⎠ = ⎜ b1 β1 + b2 β2 + · · · + bn βn ⎟ ⎜ ⎟ β1 + β2 + · · · + βn ⎜ ⎟ z∗ ⎜ ⎟ ⎜ ⎟ ⎝ c1 β1 + c2 β2 + · · · + cn βn ⎠ β1 + β2 + · · · + βn which is nothing else but the center of gravity of the experts’ initial opinions. We show now some simple special cases. Example 4.1. [41] Suppose that we have a four-expert two-issue problem in which the ideal solutions and, consequently, the quadratic OCA functions for the experts are defined by First expert: The ideal solution is (a1 , b1 ) = (1, 1) and C1 (x, y) =
β1 (x − 1)2 + (y − 1)2 × . β 1 + β2 + β3 + β 4 2
Second expert: The ideal solution is (a2 , b2 ) = (1, 0) and C2 (x, y) =
β2 (x − 1)2 + y2 × . β 1 + β2 + β3 + β 4 2
Third expert: The ideal solution is (a3 , b3 ) = (0, 1) and C3 (x, y) =
β3 x2 + (y − 1)2 × β1 + β 2 + β3 + β4 2
Fourth expert: The ideal solution is (a4 , b4 ) = (0, 0) and C4 (x, y) =
β4 x2 + y2 × . β1 + β2 + β3 + β4 2
Then problem (4.11) collapses into, 4
C (x, y) = ∑ Ci (x, y) → min; subject to x, y ∈ [0, 1] i=1
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4 OWA Operators in Multiple Criteria Decisions
and we can easily compute that,
∂ C (x, y) β1 + β 2 = 2(β1 + β2 )(x − 1) + 2(β3 + β4 )x = 0 ⇒ x∗ = ∂x β1 + β2 + β 3 + β4 ∂ C (x, y) β1 + β 3 = 2(β1 + β3)(y − 1) + 2(β2 + β4 )y = 0 ⇒ y∗ = ∂y β1 + β2 + β 3 + β4 and the optimal cost is, 1 (β3 + β4 )(β1 + β2 )2 + (β1 + β2)(β3 + β4 )2 2(β1 + β2 + β3 + β4 )3 + (β2 + β4 )(β1 + β3 )2 + (β1 + β3)(β2 + β4 )2
C∗ =
Example 4.2. [41] Suppose that we have an eight-expert three-issue problem in which the ideal solutions and, consequently, the quadratic OCA functions for the experts are defined by (1,1,1): C1 (x, y, z) =
β1 (x − 1)2 + (y − 1)2 + (z − 1)2 3 ∑i βi
(1,1,0): C2 (x, y, z) =
β2 (x − 1)2 + (y − 1)2 + z2 3 ∑i βi
C3 (x, y, z) =
β3 (x − 1)2 + y2 + (z − 1)2 3 ∑i βi
C4 (x, y, z) =
β4 x2 + (y − 1)2 + (z − 1)2 3 ∑i βi
(1,0,1):
(0,1,1):
(0,0,1): C5 (x, y, z) =
β5 x2 + y2 + (z − 1)2 3 ∑i βi
C6 (x, y, z) =
β6 x2 + (y − 1)2 + z2 3 ∑i βi
C7 (x, y, z) =
β7 (x − 1)2 + y2 + z2 3 ∑i βi
(0,1,0):
(1,0,0):
(0,0,0): C8 (x, y, z) =
β8 x 2 + y 2 + z2 3 ∑i βi
4.4 OCA Functions for Group Settlement Modeling
93
Then problem (4.11) collapses into, 8
C (x, y, z) = ∑ Ci (x, y, z) → min; subject to x, y, z ∈ [0, 1] i=1
and we can easily compute that,
∂ C (x, y, z) = 2(β1 + β2 + β3 + β7 )(x − 1) + 2(β4 + β5 + β6 + β8 )x = 0, ∂x x∗ =
β1 + β2 + β3 + β7 , ∑i βi
and,
∂ C (x, y, z) = 2(β1 + β2 + β4 + β6)(y − 1) + 2(β3 + β5 + β7 + β8 )y = 0 ∂y y∗ =
β1 + β2 + β4 + β6 , ∑i βi
and
∂ C (x, y, z) = 2(β1 + β3 + β4 + β5 )(z − 1) + 2(β2 + β6 + β7 + β8 )z = 0 ∂z z∗ =
β1 + β3 + β4 + β5 . ∑ i βi
and the optimal cost is, C∗ =
1 × [(β1 + β2 + β3 + β7 )(β4 + β5 + β6 + β8 )2 + 3(β1 + β2 + · · · + β8 )3
(β1 + β2 + β3 + β7 )2 (β4 + β5 + β6 + β8 ) + (β1 + β2 + β4 + β6 )(β3 + β5 + β7 + β8 )2 + (β1 + β2 + β4 + β6 )2 (β3 + β5 + β7 + β8 ) + (β1 + β3 + β4 + β5 )(β2 + β6 + β7 + β8 )2 + (β1 + β3 + β4 + β5 )2 (β2 + β6 + β7 + β8 )].
4.4.2
Optimal Settlement under Budget Constraints
If there exist some budget constraints then the optimal strategy minimizes the overall aversion to opinion changing under these constraints, and it can be computed by solving the following optimization problem, n
C (x, y, z) = ∑ Ci (x, y, z) → min; subject to W (x, y, z)T ≤ q i=1
(4.12)
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4 OWA Operators in Multiple Criteria Decisions
where W ∈ Rm×3 and q ∈ Rm . Then problem (4.12) can be solved using the method of Lagrange multipliers. We will show a simple example. Example 4.3. Suppose that we have a four-expert two-issue problem in which the ideal solutions and, consequently, the quadratic OCA functions for the experts are defined in the same manner as in Example 4.1. Suppose further that we have the following budget constraint 1/4 ≤ x + y ≤ 3/4. Then problem (4.12) collapses into, 4
C (x, y) = ∑ Ci (x, y) → min i=1
subject to
x + y ≤ 3/4 x + y ≥ 1/4
and its Lagrangian function is defined by L (x, y, λ1 , λ2 ) = C (x, y) + λ1 (x + y − 3/4) + λ2(1/4 − x − y). Then the Karush-Kuhn-Tucker conditions can be written as ∂ L (x, y, λ1 , λ2 ) = 2(β1 + β2 )(x − 1) + 2(β3 + β4 )x + (β1 + β2 + β3 + β4 )(λ1 − λ2 ) = 0 ∂x ∂ L (x, y, λ1 , λ2 ) = 2(β1 + β3 )(y − 1) + 2(β2 + β4 )y + (β1 + β2 + β3 + β4 )(λ1 − λ2 ) = 0 ∂y
λ1 (x + y − 3/4) = 0 λ2 (1/4 − x − y) = 0 Then the optimal solution to problem (4.12) is (1/8, 1/8) if β1 − β2 − β3 − 3β4 > 0 and (3/8, 3/8) if β1 − β2 − β3 − 3β4 < 0. If β1 − β2 − β3 − 3β4 = 0, then (1/8, 1/8) and (3/8, 3/8) are both optimal solutions, and the optimal cost is, C∗ =
4.4.3
1 25 25 49 β 1 + β2 + β3 + β4 . 32 32 32 32
Optimal Settlement under Flexible Budget Constraints
Sometimes, in real-world decision making processes, e.g. project management, constraints cannot be defined precisely or, even if they are precise they are not required to be a totally tight restriction for the region of feasible solutions. Let us consider the optimal settlement problem with quadratic opinion changing aversion functions when budget constraints are flexible (fuzzy) and the group has a target level for the overall cost, denoted by q0 > 0. For other approaches the reader can consult [134, 135, 136, 138]. A symmetrical triangular fuzzy number with center a ∈ R and tolerance level d will be denoted by the pair (a, d).
4.4 OCA Functions for Group Settlement Modeling
95
If the budget constraints are fuzzy and the group has a target level for the overall cost then we can state the following fuzzy mathematical programming (FMP) problem [363]: Find (x∗ , y∗ , z∗ ) ∈ R3 such that it satisfies the following inequalities as much as possible C (x, y, z) ≤ (q0 , d0 ) w11 x + w12 y + w13z ≤ (q1 , d1 ) .. . wm1 x + wm2 y + wm3 z ≤ (qm , dm )
(4.13)
where q0 is the target level for the overall cost, (qi , di ) are fuzzy numbers of symmetric triangular form with center qi and tolerance level di > 0, for i = 0, 1, . . . , m, and the inequalities are understood in a possibilistic sense. That is, the degree of satisfaction of the ith constraint by a point (x, y, z) ∈ R3 , denoted μi (x, y, z), is defined by ⎧ 1 ⎪ ⎪ ⎨ w x + wi2 y + wi3 z − qi μi (x, y, z) = 1 − i1 ⎪ di ⎪ ⎩ 0
if wi1 x + wi2 y + wi3 z ≤ qi , if qi < wi1 x + wi2 y + wi3 z ≤ qi + di if wi1 x + wi2 y + wi3 z > qi + di
for i = 1, . . . , m. The degree of satisfaction of the fuzzy goal (q0 , d0 ) by a point (x, y, z) ∈ R3 is defined by ⎧ 1 ⎪ ⎪ ⎨ C (x, y, z) − q0 μ0 (x, y, z) = 1 − ⎪ d0 ⎪ ⎩ 0
if C (x, y, z) ≤ q0 , if q0 < C (x, y, z) ≤ q0 + d0 if C (x, y, z) > q0 + d0
If for a vector (x, y, z) ∈ R3 the value of wi1 x + wi2 y + wi3 z is less or equal than qi then (x, y, z) satisfies the ith budget constraint with the maximal conceivable degree: one. If qi < wi1 x + wi2 y + wi3 z < qi + di then (x, y, z) is not feasible in classical sense, but the group can still tolerate the violation of the crisp budget constraint, and accept (x, y, z) as a solution with a positive degree, however, the bigger the violation the less is the degree of acceptance. Finally, if wi1 x + wi2 y + wi3 z > qi + di then the violation of the ith constraint is intolerable by the group, that is, μi (x, y, z) = 0. Furthermore, if for a vector (x, y, z) ∈ R3 the value of overall cost, C (x, y, z), is less or equal to q0 then (x, y, z) satisfies the target level of overall cost with the maximal conceivable degree: one. If q0 < C (x, y, z) < q0 + d0 then (x, y, z) is not feasible in classical sense, but the group can still tolerate the exceeded overall cost, and accept (x, y, z) as a solution with a positive degree, however, the bigger the overstepping the less is the degree of acceptance. Finally, if C (x, y, z) > qi + di then the exceed of the target level is intolerable by the group, that is, μ0 (x, y, z) = 0.
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4 OWA Operators in Multiple Criteria Decisions
Then the (fuzzy) solution of the FMP problem (4.13) is defined as a fuzzy set on R3 whose membership function is given by
μ (x, y, z) = min{ μ0 (x, y, z), μ1 (x, y, z), . . . , μm (x, y, z)}, In this setup μ (x, y, z) denotes the degree to which all inequalities are satisfied at point (x, y, z) ∈ R3 . To maximize μ on R3 we have to solve the following crisp quadratic programming problem max λ λ d0 + C (x, y, z) ≤ q0 + d0 ,
λ d1 + w11 x + w12y + w13 z ≤ q1 + d1 , ··· λ dm + wm1 x + wm2 y + wm3 z ≤ qm + dm , 0 ≤ λ ≤ 1, x, y, z ∈ R. that is, max λ
βi × β + β + 2 · · · + βn i=1 1 n
λ d0 + ∑
(x − ai)2 + (y − bi)2 + (z − ci)2 M
≤ q0 + d0 ,
λ d1 + w11 x + w12 y + w13z ≤ q1 + d1, ··· λ dm + wm1 x + wm2 y + wm3 z ≤ qm + dm , 0 ≤ λ ≤ 1, x, y, z ∈ R. where M is defined by M = maxi, j {(ai − a j )2 + (bi − b j )2 + (ci − c j )2 }.
4.5
Olympic OWA Operators for Modeling Group Decisions
In this Section following Brunelli, Full´er and Mezei [40] we will show how to use olympic OWA operators for modeling group decisions. Mich, Fedrizzi and Gaio [255] introduced a new measure of consensus depending on a function estimated for each expert according to her/his aversion to opinion changing. These functions have become known in the literature as opinion changing aversion (OCA) functions, and have been further developed in [136, 138]. Fedrizzi et al. [137] used fuzzy linguistic quantifiers and OWA operators to define a fuzzy majority and to derive a degree of consensus under fuzzy preferences. Suppose that a certain amount of money exists and it has to be assigned to 3 different projects (alternatives) according to their importance. Moreover, let us also assume that n experts have been asked to evaluate these projects and they express their preferences under the form of normalized priority vectors, i.e. their components (portions) sum up to one [291]. The preferences
4.5 Olympic OWA Operators for Modeling Group Decisions
97
of the ith expert can be modeled by a triple (ai , bi , ci ), where ai denotes how much portion of the money should be assigned to the first project, and so on. For simplicity, we will now consider only three alternatives/criteria, but our results can easily be derived for the general case. We recall that an OWA operator [338] of dimension n is a mapping F : Rn → R, that has an associated n vector W = (w1 , w2 , . . . , wn )T such as wi ∈ [0, 1], 1 ≤ i ≤ n, w1 + w2 + · · · + wn = 1 Furthermore F(a1 , . . . , an ) = ∑nj=1 w j b j , where b j is the j-th largest element of the bag < a1 , . . . , an >. Olympic type OWA operators of dimension n, denoted by Fn , were introduced by [339] and defined by the weighting vector w = (0, 1/(n − 2), · · · , 1/(n − 2), 0), where n ≥ 3. That is, an Olympic OWA operator computes the arithmetic average of all aggregates except the smallest and the largest ones. Consider the portions the experts want to allocate for the first project a = (a1 , a2 , . . . , an ). In our model we will assume that the fair value of this distribution is computed by Yager’s OWA operator, Fn (a) = Fn (a1 , a2 , . . . , an ) =
∑ni=1 ai − mini ai − maxi ai n−2
(4.14)
That is, the fair value is defined as the arithmetic mean of the opinions without the two extreme opinions. We will call ⎛
⎞ ∑ni=1 ai − mini ai − maxi ai ⎛ ⎞ ⎜ ⎟ n−2 Fn (a) ⎜ n ⎟ b − min b − max b ⎜ ⎟ ∑ i i i i i ⎝Fn (b)⎠ = ⎜ i=1 ⎟ ⎜ ⎟ n−2 Fn (c) ⎝ n c − min ⎠ ∑i=1 i i ci − maxi ci n−2 as an Olympic-ideal solution the group assignment problem. To find a minimaldistance solution from the Olympic-ideal group assignment problem we have to solve the following nonlinear optimization problem (x − Fn(a))2 + (y − Fn(b))2 + (z − Fn(c))2 → min; subject to x, y, z ∈ X where X is a set of budget constraints. Example 4.4. [40] Suppose that a certain amount of money exists and it has to be assigned to 6 different projects (alternatives) according to their importance. Moreover, let us also assume that 6 experts have been asked to evaluate these projects and they express their preferences under the form of normalized priority vectors [291], i.e. their components (portions) sum up to one. Their preferences can be summarized in 6 weighting vectors in the following form (i)
(i)
(i)
w(i) = (w1 , w2 , . . . , w6 )
98
4 OWA Operators in Multiple Criteria Decisions (i)
where w j denotes how much portion of the money should be assigned to the j-th projects according to the ith expert. The 6 vectors of portions are actually given by w(1) = (0.2, 0.3, 0.1, 0.05, 0.15, 0.2) w(2) = (0.1, 0.1, 0.15, 0.2, 0.25, 0.2) w(3) = (0.4, 0.1, 0.05, 0.2, 0.3, 0.05) w(4) = (0.15, 0.1, 0.05, 0.2, 0.25, 0.25) w(5) = (0.2, 0.1, 0.2, 0.1, 0.3, 0.1) w(6) = (0.05, 0.05, 0.1, 0.4, 0.25, 0.15) The olympic-ideal solution is computed by (4.14) as, ⎞ ⎛ 0.1625 ⎜ 0.1 ⎟ ⎟ ⎜ ⎜ 0.1 ⎟ ⎟ ⎜ ⎜ 0.175 ⎟ . ⎟ ⎜ ⎝ 0.2625 ⎠ 0.1625 Let x j denote the portion of money to be assigned to project j. Then we have the natural constraint x1 + · · · + x6 = 1. Let us further assume that there exists a constraint for each of the portions, x j ≥ 1/8, for j = 1, . . . , 6. To find a minimal-distance solution from the Olympic-ideal group assignment problem we have to solve the following nonlinear optimization problem (x1 − 0.1625)2 + (x2 − 0.1)2 + (x3 − 0.1)2 + (x4 − 0.175)2+ (x5 − 0.2625)2 + (x6 − 0.1625)2 → min subject to x j ≥ 1/8, j = 1, . . . , 6 ,
6
∑ xj = 1
j=1
The unique solution of this six-expert six-alternatives problem is ⎞ ⎛ ∗⎞ ⎛ x1 0.159375 ⎜x∗ ⎟ ⎜ 0.125 ⎟ ⎟ ⎜ 2∗ ⎟ ⎜ ⎜x ⎟ ⎜ 0.125 ⎟ ⎟ ⎜ 3∗ ⎟ = ⎜ ⎜x ⎟ ⎜0.171875⎟ ⎟ ⎜ 4∗ ⎟ ⎜ ⎝x ⎠ ⎝0.259375⎠ 5 0.159375 x∗6 and the optimal distance is 0.00128906.
4.5 Olympic OWA Operators for Modeling Group Decisions
4.5.1
99
Group Decisions under Fuzzy Budget Constraints
Let us consider the optimal settlement problem with quadratic opinion changing aversion functions when budget constraints are flexible (fuzzy) and the group has a target level for the maximal Euclidean distance from the Olympic-ideal solution, denoted by q0 > 0. Definition 4.1. A fuzzy set of the real line given by the membership function ⎧ |a − t| ⎨ 1− if |a − t| ≤ d, A(t) = d ⎩ 0 otherwise, where d > 0 will be called a symmetrical triangular fuzzy number with center a ∈ R and tolerance level d and we shall refer to it by the pair (a, d). If the budget constraints are fuzzy and the group has a target level for the maximal Euclidean distance from the Olympic-ideal solution then we can state the following fuzzy mathematical programming (FMP) problem [363]: Find (x∗ , y∗ , z∗ ) ∈ R3 such that it satisfies the following inequalities as much as possible
τn (a, b, c) ≤ (q0 , d0 ) w11 x + w12 y + w13z ≤ (q1 , d1 ) .. . wm1 x + wm2 y + wm3 z ≤ (qm , dm )
(4.15)
where
τn (a, b, c) = (x − Fn(a))2 + (y − Fn(b))2 + (z − Fn(c))2 q0 is the target level for the maximal Euclidean distance from the Olympic-ideal solution, (qi , di ) are fuzzy numbers of symmetric triangular form with center qi and tolerance level di > 0, for i = 0, 1, . . . , m, and the inequalities are understood in a possibilistic sense. That is, the degree of satisfaction of the ith constraint by a point (x, y, z) ∈ R3 , denoted μi (x, y, z), is defined by ⎧ 1 ⎪ ⎪ ⎨ w x + wi2 y + wi3 z − qi μi (x, y, z) = 1 − i1 ⎪ di ⎪ ⎩ 0
if wi1 x + wi2 y + wi3 z ≤ qi , if qi < wi1 x + wi2 y + wi3 z ≤ qi + di if wi1 x + wi2 y + wi3 z > qi + di
for i = 1, . . . , m. The degree of satisfaction of the fuzzy goal (q0 , d0 ) by a point (x, y, z) ∈ R3 is defined by
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4 OWA Operators in Multiple Criteria Decisions
⎧ 1 ⎪ ⎪ ⎨ τ (a, b, c) − q0 μ0 (x, y, z) = 1 − n . ⎪ d0 ⎪ ⎩ 0
if τn (a, b, c) ≤ q0 , if q0 < τn (a, b, c) ≤ q0 + d0 if τn (a, b, c) > q0 + d0
If for a vector (x, y, z) ∈ R3 the value of wi1 x + wi2 y + wi3 z is less or equal than qi then (x, y, z) satisfies the ith budget constraint with the maximal conceivable degree: one. If qi < wi1 x + wi2 y + wi3 z < qi + di then (x, y, z) is not feasible in classical sense, but the group can still tolerate the violation of the crisp budget constraint, and accept (x, y, z) as a solution with a positive degree, however, the bigger the violation the less is the degree of acceptance. Finally, if wi1 x + wi2 y + wi3 z > qi + di then the violation of the ith constraint is untolerable by the group, that is, μi (x, y, z) = 0. Furthermore, if for a vector (x, y, z) ∈ R3 the value of Euclidean distance, τn (a, b, c), is less or equal to q0 then (x, y, z) satisfies the target level for the Euclidean distance with the maximal conceivable degree: one. If q0 < τn (a, b, c) < q0 + d0 then (x, y, z) is not feasible in classical sense, but the group can still tolerate the exceeded target level, and accept (x, y, z) as a solution with a positive degree, however, the bigger the overstepping the less is the degree of acceptance. Finally, if τn (a, b, c) > qi + di then the exceed of the target level is untolerable by the group, that is, μ0 (x, y, z) = 0. Then the (fuzzy) solution of the FMP problem (4.15) is defined as a fuzzy set on R3 whose membership function is given by
μ (x, y, z) = min{ μ0 (x, y, z), μ1 (x, y, z), . . . , μm (x, y, z)}, In this setup μ (x, y, z) denotes the degree to which all inequalities are satisfied at point (x, y, z) ∈ R3 . To maximize μ on R3 we have to solve the following crisp quadratic programming problem max λ
λ d0 + τn (a, b, c) ≤ q0 + d0 , λ d1 + w11 x + w12y + w13 z ≤ q1 + d1 , ··· λ dm + wm1 x + wm2 y + wm3 z ≤ qm + dm , 0 ≤ λ ≤ 1, x, y, z ∈ R. that is, max λ
λ d0 + (x − Fn(a)) + (y − Fn(b)) + (z − Fn (c))2 ≤ q0 + d0 , λ d1 + w11 x + w12 y + w13z ≤ q1 + d1 , ··· λ dm + wm1 x + wm2 y + wm3 z ≤ qm + dm, 0 ≤ λ ≤ 1, x, y, z ∈ R. 2
2
Chapter 5
Fuzzy Real Options for Strategic Decisions
In traditional investment planning investment decisions are usually taken to be nowor-never, which the firm can either enter into right now or abandon forever. The decision on to close/not close a production plant has been understood to be a similar now-or-never decision for two reasons: (i) to close a plant is a hard decision and senior management can make it only when the facts are irrefutable; (ii) there is no future evaluation of what-if scenarios after the plant is closed. Nevertheless, as we will show, it could make sense to work a bit with what-if scenarios as closing the plant will cut off all future options for the plant. Making hard decisions is the macho thing and new CEOs often want to earn their first spurs by closing production plants; they are quite often rewarded by the shareholders who think that decisive actions is the mark of a manager who is going to build good shareholder value. Nevertheless, the exact outcomes in terms of shareholder value of the decision are uncertain as a consequence of changing markets, changes in raw material and energy costs, changes in the technology roadmap, changes in the economic climate, etc. In order to support and motivate tough decisions a number of valuation methods have been developed and the standard approach is to use NPV or discounted cash flow (DCF) methods with a number of assumptions about the future development of key cost and profitability drivers. Only very few decisions are of the type now-or-never; it is often possible to postpone, modify or split up a complex decision in strategic components, which can generate important learning effects and therefore essentially reduce uncertainty. If we close a plant we lose all alternative development paths which could be possible under changing conditions; on the other hand, senior management may have a difficult time with shareholders if they continue operating a production plant in conditions which cut into its profitability as their actions are evaluated and judged every quarter. In these cases we can utilize the idea of real options. The new rule, derived from option pricing theory, is that we should only close the plant now if the net present value of this action is high enough to compensate for giving up the value of the option to wait. Because the value of the option to wait vanishes right after we C. Carlsson and R. Full´er: Possibility for Decision, STUDFUZZ 270, pp. 101–132. springerlink.com © Springer-Verlag Berlin Heidelberg 2011
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5 Fuzzy Real Options for Strategic Decisions
irreversibly decide to close the plant, this loss in value is actually the opportunity cost of our decision.
5.1
A Probabilistic Model
Real options in option thinking are based on the same principals as financial options. In real options, the options involve ”real” assets as opposed to financial ones [9]. To have a ”real option” means to have the possibility for a certain period to either choose for or against something, without binding oneself up front. The value of a real option is computed by [223] ROV = S0 e−δ T N(d1 ) − Xe−rT N(d2 ) where ln(S0 /X ) + (r − δ + σ 2 /2)T √ , σ T √ d2 = d1 − σ T , d1 =
and where S0 is the present value of expected cash flows, N(d) denotes the probability that a random draw from a standard normal distribution will be less than d, X is the (nominal) value of fixed costs, r is the annualized continuously compounded rate on a safe asset, T is the time to maturity of option (in years), σ is the uncertainty of expected cash flows, and finally δ is the value lost over the duration of the option. Furthermore, the function N(d) gives the probability that a random draw from a standard normal distribution will be less than d, i.e. 1 N(d) = √ 2π
d −∞
x2
e− 2 dx.
Facing a deferrable decision, the main question that a company primarily needs to answer is the following: How long should we postpone the decision - up to T time periods - before (if at all) making it? From the idea of real option valuation we can develop the following natural decision rule for an optimal decision strategy [20]. Let us assume that we have a deferrable decision opportunity P of length L years with expected cash flows (cf0 , cf1 , . . . , cfL ), where cfi is the cash inflows that the plant is expected to generate at year i, i = 0, 1, . . . , L. Where the maximum deferral time is T , make the investment (exercise the option) at time t , 0 ≤ t ≤ T , for which the option, ROVt , is positive and attends its maximum value,
5.1 A Probabilistic Model
103
ROVt = max ROVt = max {Vt e−δ t N(d1 ) − Xe−rt N(d2 )}, t=0,1,...,T
t=0,1,...,T
(5.1)
where Vt = PV(cf0 , . . . , cfL , βP ) − PV(cf0 , . . . , cft−1 , βP ) = PV(cft , . . . , cfL , βP ), is the present value of the aggregate cash flows generated by the decision, which we postpone t years before undertaking. Hence, L
t L cf j cf j cf j − cf − = , 0 ∑ ∑ j j j j=t (1 + βP) j=1 (1 + βP ) j=1 (1 + βP )
Vt = cf0 + ∑
where βP stands for the risk-adjusted discount rate of the decision (cf. [61] for details). Note 5.1. Obviously, in the case we obtain or learn some new information about the decision alternatives, their associated NPV table and the cash flows cfi which may change. Thus, we have to reapply this decision rule every time when new information arrives during the deferral period to see how the optimal decision strategy might change in light of the new information. Note 5.2. If we make the decision now without waiting, then V0 = PV(cf0 , . . . , cfL , βP ) =
L
cf j
∑ (1 + βP) j
j=0
and since we can formally write, lim d1 = lim d2 = +∞,
T →0
T →0
lim N(d1 ) = lim N(d2 ) = 1,
T →0
T →0
we obtain, ROV0 = V0 − X =
L
cf j
∑ (1 + βP) j − X,
j=0
That is, this decision rule also incorporates the net present valuation of the assumed cash flows. Real options are used as strategic instruments, where the degrees of freedom of some actions are limited by the capabilities of the company. In particular, this is the case when the consequences of a decision are significant and will have an impact on the market and competitive positions of the company. In general, real options should preferably be viewed in a larger context of the company, where management does have the degree of freedom to modify (and even overrule) the pure stochastic real option evaluation of decisions and de-investment opportunities.
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5.2
5 Fuzzy Real Options for Strategic Decisions
A Fuzzy Model
Let us now assume that the expected cash flows of the close/not close decision cannot be characterized with single numbers (which should be the case in serious decision making). With the help of possibility theory we can estimate the expected incoming cash flows at each year of the project by using a trapezoidal possibility distribution of the form cfi = (sLi , sRi , αi , βi ), i.e. the most possible values of the present value of expected cash flows lie in the interval [sLi , sRi ] (which is the core of the trapezoidal fuzzy number cfi ), and (sRi + βi ) is the upward potential and (sLi − αi ) is the downward potential for the present value of expected cash flows. In a similar manner one can estimate the expected costs by using a trapezoidal possibility distribution of the form X˜ = (xL , xR , α , β ), i.e. the most possible values of expected cost lie in the interval [xL , xR ] (which is the core of the trapezoidal fuzzy number X˜ ), and (xR + β ) is the upward potential and (xL − α ) is the downward potential for expected costs. Note 5.3. The possibility distribution of expected costs and the present value of expected cash flows could also be represented by nonlinear (e.g. Gaussian) membership functions. However, from a computational point of view it is easier to use linear membership functions and, more importantly, our experience shows that senior managers prefer trapezoidal fuzzy numbers to Gaussian ones when they estimate the uncertainties associated with future cash inflows and outflows. Let P be a deferrable decision opportunity with incoming cash flows and costs that are characterized by the trapezoidal possibility distributions given above. Furthermore, let us assume that the maximum deferral time of the decision is T , and the required rate of return on this project is βP . In these circumstances, we shall make the decision (exercise the real option) at time t , 0 < t < T , for which the value of the option, FROVt is positive and reaches its maximum value [61]. That is, FROVt = max FROVt = t=0,1,...,T
˜ −rt N(d2 (t))}, max {V˜t e−δ t N(d1 (t)) − Xe
t=0,1,...,T
where, ˜ 0 , . . . , cf ˜ L , βP ) − PV(cf ˜ 0 , . . . , cf ˜ t−1 , βP ) Vt = PV(cf L ˜j cf ˜ t+1 , . . . , cf ˜ L , βP ) = ∑ = PV(cf j j=t (1 + βP ) and where, d1 (t) =
ln(E(V˜t )/E(X˜ )) + (r − δ + σ 2 /2)T √ , σ T
√ d2 (t) = d1 − σ T ,
(5.2)
5.3 A Binomial Model
105
˜ stands for the the and where, E denotes the possibilistic mean value operator, E(X) possibilistic mean value of expected costs and
σ :=
σ (V˜t ) E(V˜t )
is the annualized possibilistic variance of the aggregate expected cash flows relative to its possibilistic mean (and therefore represented as a percentage value). However, to find a maximizing element from the set {FROV0 , FROV1 , . . . , FROVT }, is not an easy task because it involves ranking of trapezoidal fuzzy numbers. In our computerized implementation we have employed the following value function to order fuzzy real option values, FROVt = (ctL , ctR , αt , βt ), of trapezoidal form: v(FROVt ) =
ctL + ctR βt − αt + rA · , 2 6
where rA ≥ 0 denotes the degree of the investor’s risk aversion. If rA = 1 then the (risk neutral) manager compares trapezoidal fuzzy numbers by comparing their possibilistic expected values, i.e. he does not care about their downward and upward potentials. If rA > 1 then the manager is a risk-taker, and if rA < 1 then he is riskaverse.
5.3
A Binomial Model
For practical purposes and when working with senior management the binomial version of the real options model is easier to use and easier to explain in terms of the available data. For our case the basic binomial setting is presented as a setting of two lattices, the underlying asset lattice and the option valuation lattice; for adding insight we can also include a decision rule lattice. In Figure 5.1 the weights u and d describe the geometric movement (Brownian motion) of the cash flows V over time, q stands for a movement up and 1−q movement down, respectively. The value of the underlying asset develops in time according to probabilities attached to movements q and 1 − q, and weights u and d, as described in Figure 5.1. The input values for the lattice are approximated with the following set of formulae (see [111] for details), u = eσ d=e
√ Δt
(movement up)
√ −σ Δ t
1 1 α q= + × 2 2
(movement down) − 1/2 × σ 2√
σ
t
(probability of movement up)
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5 Fuzzy Real Options for Strategic Decisions
Asset value
q q
u 1-q
S 1-q
q
ud
d 1-q
0
u2
1
d2 2 Time
Fig. 5.1 Underlying asset lattice of two periods
The option valuation lattice is composed of the intrinsic values I of the latest time to invest retrieved as the maximum of present value and zero, the option values O generated as the maxi- mum of the intrinsic or option values of the next period (and their probabilities q and 1 − q) discounted, and the present value S − F of the period in question. This formulation describes two binomial lattices that capture the present values of movements up and down from the previous state of time PV and the incremental values I directly contributing to option value O. The relation of geometric movements up and down is captured by the ratio d = 1/u. The binomial model is a discrete time model and its accuracy improves as the number of time steps increases. Note 5.4. In summary, the benefit of using fuzzy numbers and the fuzzy real options model - both in the Black-Scholes and in the binomial version of the real options model - is that we can represent genuine uncertainty in the estimates of future costs and cash flows and take these factors into consideration when we make the decision to either close the plant now or to postpone the decision by t years (or some other reasonable unit of time). The simpler, classical representation does not adequately show the uncertainty.
5.4
Closing Production Plants - When and Where
The forest industry, and especially the paper making companies, has experienced a radical change of market since the change of the millennium. Especially in Europe the stagnating growth in paper sales and the resulting overcapacity have led to decreasing paper prices, which have been hard to raise even to compensate for increasing costs. Other drivers to contribute to the misery of European paper producers have been steadily growing energy costs, growing costs of raw material and
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the Euro/USD exchange rate which is unfavorable for an industry which is invoicing its customers in USD and pays its costs in Euro. The result has been a number of restructuring measures, such as closedowns of individual paper machines and production units. Additionally, a number of macroeconomic and other trends have changed the competitive and productive environment of paper making. The current industrial logic of reacting to the cyclical demand and price dynamics with operational flexibility is losing edge because of shrinking profit margins. Simultaneously, new growth potential is found in the emerging markets of Asia, especially in China, which more and more attracts the capital invested in paper production. This imbalance between the current production capacity in Europe and the better expected return on capital invested in the emerging markets has set the paper makers in front of new challenges and uncertainties that are different from the ones found in the traditional paper company management paradigm. In a global business environment both challenges and uncertainties vary from market to market, and it is important to find new ways of managing them in the current dynamic business environment.
Fig. 5.2 Population and paper consumption/capita worldwide [source: Finnish Forest Industries Federation]
In Figure 5.2 is shown the population and the paper consumption in different parts of the world in 2005 (the numbers are indicative for our purposes, the trends have developed for the worse by 2008); the problem for the paper producing companies is that the production capacity is concentrated in countries where the paper
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5 Fuzzy Real Options for Strategic Decisions
consumption/capita has reached saturation levels, not where the potential growth of consumption is highest.
5.4.1
Production, Exports, Jobs and Investments
The Finnish Forest Industries Federation continuously updates material on the forest industry on its website (cf. www.forestindustries.fi); from this material we can find the following key observations (cf. Figure 5.3).
Fig. 5.3 Value of the production, exports and imports of the forest industry in 2006
In 2006, the gross value of the forest industry’s production in Finland was about 21 billion, a third of which was accounted for by the wood products industry and two thirds by the pulp and paper industries. In 2006, the forest sector employed a total of 60,000, some 30,000 of whom worked for the paper industry and about 30,000 for the wood products industry. Finnish Forest Industries Federation member companies employed 50,000 people. In addition to their domestic functions, Finnish forest industry companies employed about 70,000 people abroad. The total investments of Finnish forest industry came to some e 2.2 billion in 2006, e 1.4 billion of which were invested abroad. World production of paper and paper board totals some 370 million tons. Growth is the most rapid in Asia, thanks mainly to the quick expansion of industry in China. Asia already accounts for well over a third of total world paper and paperboard production. In North America, by contrast, production is contracting; a number of Canadian mills have had to shut down because of weak competitiveness. Per capita consumption of paper and paperboard varies significantly from country to country and regionally. On average, one person uses about 55 kilos of paper a year; the extremes are 300 kilos for each US resident and some seven kilos for each African. Only around 35 kilos of paper per person is consumed in the populous area of Asia. This means that Asian consumption will continue to grow strongly in the coming years if developments there follow the precedent of the West. In Finland, per capita consumption of paper and paperboard is 205 kilos. Rapid growth in Asian paper production in recent years has increased the region’s self-sufficiency, narrowing the export opportunities available to both Europeans and Americans. Additionally, Asian paper has started to enter Western markets - from China in particular. Global competition has intensified noticeably as the new entrants’ cost level is significantly lower than in competing Western countries. The European industry has been dismantling overcapacity by shutting down unprofitable
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109
mills. In total, over five percent of the production volume in Europe has been closed down in the last couple of years. Globally speaking, the products of the forest industry are primarily consumed in their production country, so it can be considered a domestic-market industry. Globally speaking, the profitability of forest industry companies has been weak in recent years. Overcapacity has led to falling prices and this, coupled with rising production costs, is gnawing at the sector’s profitability. The Finnish forest industry has earlier enjoyed a productivity lead over its competitors. The lead is primarily based on a high rate of investment and the application of the most advanced technologies. Investments and growth are now curtailed by the long distance separating Finland from the large, growing markets as well as the availability and price of raw materials. Additionally, the competitiveness of Finnish companies has suffered because costs here have risen at a faster rate than in competing countries. Finnish energy policy has a major impact on the competitiveness of the forest industry. The availability and price of energy, emissions trading and whether wood raw material is produced for manufacturing or energy use will affect the future success of the forest industry. If sufficient energy is available, basic industry can invest in Finland. In decisions on how to use existing resources the challenges of changing markets become a reality when senior management has to decide how to allocate capital to production, logistics and marketing networks, and has to worry about the return on capital employed. The networks are interdependent as the demand for and the prices of fine paper products are defined by the efficiency of the customer production processes and how well suited they are to market demand; the production should be cost effective and adaptive to cyclic (and sometimes random) changes in market demand; the logistics and marketing networks should be able to react in a timely fashion to market fluctuations and to offer some buffers for the production processes. Closing or not closing a production plant is often regarded as an isolated decision, without working out the possibilities and requirements of the interdependent networks. Profitability analysis has usually had an important role as the threshold phase and the key process when a decision should be made on closing or not closing a production plant. Economic feasibility is of course an important consideration but - as pointed out - more issues are at stake. There is also the question of what kind of profitability analysis should be used and what results we can get by using different methods. Senior management worries - and should worry - about making the best possible decisions on the close/not close situations as their decisions will be scrutinized and questioned regardless of what that decision is going to be. The shareholders will react negatively if they find out that share value will decrease (closing a profitable plant, closing a plant which may turn profitable, or not closing a plant which is not profitable, or which may turn unprofitable) and the trade unions, local and regional politicians, the press etc. will always react negatively to a decision to close a plant almost regardless of the reasons.
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The idea of optimality of decisions comes from normative decision theory. The decisions made at various levels of uncertainty can be modeled so that the ranking of various alternatives can be readily achieved, either with certainty or with well-understood and non-conflicting measures of uncertainty. However, the real life complexity, both in a static and dynamic sense, makes the optimal decisions hard to find many times. What is often helpful is to relax the decision model from the optimality criteria and to use sufficiency criteria instead. Modern profitability plans are usually built with methods that originate in neoclassical finance theory. These models are by nature normative and may support decisions that in the long run may be proved to be optimal but may not be too helpful for real life decisions in a real industry setting as conditions tend to be not so well structured as shown in theory and - above all - they are not repetitive (a production plant is closed and this cannot be repeated under new conditions to get experimental data). In practice and in general terms, for profitability planning a good enough solution is many times both efficient, in the sense of smooth management processes, and effective, in the sense of finding the best way to act, as compared to theoretically optimal outcomes. Moreover, the availability of precise data for a theoretically adequate profitability analysis is often limited and subject to individual preferences and expert opinions. Especially, when cash flow estimates are worked out with one number and a risk-adjusted discount factor, various uncertain and dynamic features may be lost. The case for good enough solutions is made in fuzzy set theory [70]: at some point there will be a trade-off between precision and relevance, in the sense that increased precision can be gained only through loss of relevance and increased relevance only through the loss of precision. In a practical sense, many theoretically optimal profitability models are restricted to a set of assumptions that hinder their practical application in many real world situations. Let us consider the traditional Net Present Value (NPV) model - the assumption is that both the microeconomic productivity measures (cash flows) and the macroeconomic financial factors (discount factors) can be readily estimated several years ahead, and that the outcome of the project, such as a paper machine with an expected economic lifetime of 20-25 years, is tradable in the market of production assets without friction. In other words, the model has features that are unrealistic in a real world situation. The idea of the NPV is based on a fixed coupon bond that generates a fixed stream of cash flows during a pre-defined lifetime. For real investments with long economic life-times that are subject to intense competition, technological deterioration and radically changing context factors (currency exchange rates, energy costs, raw material costs, etc.) the NPV gives rather a simplistic picture of real life profitability. In reality, the decision makers have to face a complex set of interdependencies that change dynamically and are uncertain, and uncertain in their uncertainty. Having now set the scene, the problem we will address is the decision to close - or not to close - a production plant in the forest products industry sector. The plant we will use as a context is producing fine paper products, it is rather aged, the paper machines were built a while ago, the raw material is not available close by, energy costs are reasonable but are increasing in the near future, key markets are
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111
close by and other markets (with better sales prices) will require improvements in the logistics network. The intuitive conclusion is, of course, that we have a sunset case and senior management should make a simple, macho decision and close the plant. On the other hand we have the trade unions, which are strong, and we have pension funds commitments until 2013 which are very strict, and we have long-term energy contracts which are expensive to get out of. Finally, by closing the plant we will invite competitors to fight us in markets we have served for more than 50 years and which we cannot serve from other plants at any reasonable cost. We have shown in Section 5.2 that real options models will support decision making in which senior managers search for the best way to act and the best time to act. The key elements of the closing/not closing decision may be known only partially and/or only in imprecise terms, which are why we show that meaningful support, can be given with a fuzzy real options model. Following Heikkil¨a and Carlsson [181] and Carlsson et al [89] the real world case will be introduced in Subsection 5.4.2 where we show the dilemma(s) senior management had to deal with and the (low) level of precision in the data to be used for making a decision. In Subsection 5.4.3 we will show the models we worked with and the results we were able to get with fuzzy real options models. Subsection 5.4.7, finally, summarizes some discussion points and offers some conclusions.
5.4.2
The Production Plant and Future Scenarios
The production plant we are going to describe is a real case, the numbers we show are realistic (but modified for reasons of confidentiality) and the decision process is as close to the real process as we can make it. We worked the case with the fuzzy real options model in order to help senior management decide if the plant should (i) be closed as soon as possible, (ii) not closed, or (iii) closed at some later point of time (and then at what point of time). The background for the decisions can be found in the following general development of the profitability of the Finnish forest products companies (cf. Figure 5.4, the Finnish Forest Industries Confederation). The main reasons for the unsatisfactory development of the profitability are: (i) fine paper prices have been going down for six years, (ii) costs are going up (raw material, energy, chemicals), (iii) demand is growing slowly, (iv) production capacity cannot be used optimally, and (v) the e /USD exchange rate is unfavorable (sales invoiced in USD, costs paid in e ). The standard solution is to try to close the old, small and least cost-effective production plants. The analysis carried out for the production plant started from a comparison of the present production and production lines with four new production scenarios with different production line setups. In the analysis each production scenario is analyzed with respect to one sales scenario assuming a match between performed sales analysis and consequent resource allocation on production. Since there is considerable uncertainty involved in both sales quantities and sales prices the resource allocation decision is contingent to a number of production options that the management has to consider, but which we have simplified here in order to get to the core of the case.
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5 Fuzzy Real Options for Strategic Decisions
Fig. 5.4 Profit before taxes and ROCE, Finnish forest products companies
There were a number of conditions which were more or less predefined. The first one was that no capital could/should be invested as the plant was regarded as a sunset plant. The second condition was that we should in fact consider five scenarios: the current production setup with only maintenance of current resources and four options to switch to setups that save costs and have an effect on production capacity used. The third condition is that the plant together with another unit has to carry considerable administrative costs of the sales organization in the country. The fourth condition is that there is a pension scheme that needs to be financed until 2013. The fifth condition is the power contract of the unit which is running until 2013. These specific conditions have consequences on the cost structure and the risks that various scenarios involve.
Fig. 5.5 Production plant scenarios
Each scenario assumes a match between sales and production, which is a simplification; in reality there are significant, stochastic variations in sales which cannot be matched by the production. Since no capital investment is assumed there will be no costs in switching between the scenarios (which is another simplification). The
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113
possibilities to switch in the future were worked out as (real) options for senior management; the opportunity to switch to another scenario is a call option. The option values are based on the estimates of future cash flows, which are the basis for the upward/downward potentials. In discussions with senior management they (reluctantly) adopted the view that options can exist and that there is a not-to-decide-today possibility for the close/not close decision. The motives to include options into the decision process were reasoned through with the following logic: • New information changes the decision situation (Good or Bad News in Figure 5.6) • Consequently, new information has a value and it increases the flexibility of the management decisions • The value of the new information can be analyzed to enable the management to make better informed decisions
Fig. 5.6 Committing now vs. having options
In the discussions we were able to show that companies fail to invest in valuable projects because the options embedded in a project are overlooked and left out of the profitability analysis. The real options approach shows the importance of timing as the real option value is the opportunity cost of the decision to wait in contrast with the decision to act immediately. We also worked out the use of decision trees as a way to work with the binomial form of the real options model (cf. Figure 5.6). We were then able to give the following practical description of how the option value is formed: Option value = Discounted cash flow ∗ Value of uncertainty (usually standard deviation) - Investment ∗ Risk free interest If we compare this sketch of the actual work with the decision to close/not close the production plant with the theoretical models we introduced in Section 5.2, we cannot avoid the conclusion that things are much simplified. There are two reasons
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5 Fuzzy Real Options for Strategic Decisions
for this: (i) the data available is scarce and imprecise as the scenarios are more or less ad hoc constructs; (ii) senior management will distrust results of an analysis they cannot evaluate and verify with numbers they recognize or can verify as ”about right”.
5.4.3
Closing/Not Closing a Plant: A Case Study
The capital investment in a new paper machine - the type of project normally analyzed with the real options models - is a project of several hundred million Euros. As a capital investment such a project is a long-time venture of 10-15 years of operational lifetime, which means that the productivity and profitability of the machine should be worked out over this period of time. Productivity is largely defined by the technological deterioration rate. Generally, the longer the plant stays in the technological race for productivity, the longer it is able to compete profitably. The conventional wisdom in the paper making industry is to build a paper machine with the most advanced features of technology development, so that high profits can be retrieved during the early years to pay back the capital invested. The story is a bit different when we are nearing the end of the economic life time of a paper machine. Closing a paper mill is usually understood as a decision at the end of the operational life-time of the real asset. In the aging unit considered here the two paper machines were producing three paper qualities with different price and quality characteristics. The newer Machine 2 had a production capacity of 150 000 tons of paper per year; the older Machine 1 produced about 50 000 tons. The three products were, • Product 1, an old product with declining, shrinking prices, • Product 2, a product at the middle-cycle of its lifetime, • Product 3, a new innovative product with large valued added potential, As background information a scenario analysis had been made with market and price forecasts, competitor analyses and the assessment of paper machine efficiency. Our analysis was based on the assumptions of this analysis with five alternative scenarios to be used as a basis for the profitability analysis (cf. Figure 5.4). After a preliminary screening (a simplifying operation to save time) two of the scenarios, one requiring sales growth and another with unchanged sales volume were chosen for a closer profitability assessment. The first one, Scenario 1 (sales volume 200,000 tons) included two sub options, first 1A with the current production setup and 1B with a product specialization for the two paper machines. The 1B would offer possibilities for a close-down of a paper coating unit, which will result in savings of over e 700,000. Note 5.5. We have chosen Scenario 1A for analysis and for illustrations. Scenario 2 starts from an assumption of a smaller sales volume (150,000 tons) and allows a closedown of the smaller Machine 1, with savings of over e 3.5 M. In
5.4 Closing Production Plants - When and Where
115
addition to operational costs a number of additional cost items needed to be considered by the management. There is a pension scheme agreement which would cause extra costs for the company if Machine 1 is closed down. Additionally, the long term energy contracts would cause extra cost if the company wants to close them before the end term. The scenarios are summarized here as production and product setup options, and are modeled as options to switch a production setup. They differ from typical options - such as options to expand or postpone - in that they do not include major capital commitments; they differ from the option to abandon as the opportunity cost is not calculated to the abandonment, but to the continuation of the current operations.
5.4.4
A Binomial Analysis
Cash flow estimates for the binomial analysis were estimated for each of the scenarios from the sales scenarios of the three products and by considering changes in the fixed costs caused by the production scenarios. Each of the products had their own price forecast that was utilized as a trend factor. For the estimation of the cash flow volatility there were two alternative methods of analysis. Starting from the volatility of sales price estimates one can retrieve the volatility of cash flow estimates by simulation (the Monte Carlo method) or by applying expert opinions directly to the added value estimates. In order to illustrate the latter method the volatility is here calculated from added value estimates (AVE) (with fuzzy estimates: a: AVE ∗ -10%, b: AVE ∗ 10%, α : AVE ∗ 10%, β : AVE ∗ 10%) (cf. Fig. 5.7).
Fig. 5.7 Added value estimates, trapezoidal fuzzy interval estimates and retrieved volatilities (STDEV)
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5 Fuzzy Real Options for Strategic Decisions
The annual cash flows in the option valuation were calculated as the cash flow of postponing the switch of production subtracted with the cash flows of switching now. The resulting cash flow statement of switching immediately is shown below (Figure 5.8). The cash flows were transformed from nominal to risk-adjusted in order to allow risk-neutral valuation. Year Fixed Cost Total, Scenario 1 A Added Value Total, Scenario 1A EBDIT, Scenario 1A Risk-neutral valuation parameter EBDIT NPV, no delay
0 0 0 0 1,000 0 7 174 624
1 -5 620 750 6 465 000 844 250 0,955 805 875 8 148 015
2 -5 757 269 7 358 000 1 600 731 0,911 1 458 518
3 -5 899 200 7 913 000 2 013 800 0,870 1 751 484
4 -6 056 180 8 881 000 2 824 820 0,830 2 345 185
5 -6 257 835 8 902 000 2 644 165 0,792 2 095 423
Fig. 5.8 Incremental cash flows and NPV with no delay in the switch to Scenario 1A
The switch immediately to Scenario 1A seems to be profitable. In the following option value calculation the binomial process results are applied in the row ”EBDIT, from binomial EBDIT lattice”. The calculation shows that when given volatilities are applied to all the products and the retrieved Added Value lattices are applied to EBDIT, the resulting EBDIT lattice returns cash flow estimates for the option to switch, adding 24 million of managerial flexibility (cf. Fig. 5.9).
Year Fixed Cost Total, Scenario 1A Added Value Total, Scenario 1A EBDIT, Scenario 1A Risk-neutral valuation parameter EBDIT NPV, no delay NPV at year 2006 NPV,delay: 1 year(s) EBDIT, from binomial EBDIT lattice Option to switch, value at year 2006 Option to switch Flexibility
0 0 0 0 1,000 0 7 174 624
1 -5 620 750 6 465 000 844 250 0,955 805 875 8 148 015 7 777 651 603 027
2 -5 757 269 7 358 000 1 600 731 0,911 1 458 518
3 -5 899 200 7 913 000 2 013 800 0,870 1 751 484
4 -6 056 180 8 881 000 2 824 820 0,830 2 345 185
5 -6 257 835 8 902 000 2 644 165 0,792 2 095 423
6 -6 390 171 8 786 900 2 396 729 0,756 1 813 003
3 711 963 33 047 232 31 545 085 24 370 461
6 718 118
8 067 557
10 802 222
9 651 783
12 064 213
Fig. 5.9 Incremental cash flows, the NPV and Option value assessment when the switch to Scenario 1A is delayed by 1 year
The binomial process is applied to the Added Value Estimates (AVEs). The binomial process up and down parameters, u and d, are retrieved from the volatility (σ ) and time increment dt. The binomial process is illustrated in Figure 5.10 Figure 5.12.
5.4.5
A Fuzzy Interval Analysis
The fuzzy interval analysis allows management to make scenario based estimates of upward potential and downward risk separately. The volatility of cash flows is defined from a possibility distribution and can readily be manipulated if the potential
5.4 Closing Production Plants - When and Where
117
Binomial process, example
Up Down Trend 10% 1.108203497 0.902361346 0.99 10% 1.108203497 0.902361346 0.9974194 10% 1.108203497 0.902361346 0.99
Volatility of Product 1 Volatility of Product 2 Volatility of Product 3
Delay, years
2
0 1 2 3 4
Binomial value added process
Binomial lattice: Product 1 process with volatility of 10.2740233382816% 0 1 2 3 4 5 6 0 0 118 0 0 0 0 0 105 129 0 0 0 0 0 94 116 142 0 0 0 0 84 103 127 156 0 0 0 75 92 113 139 171
0 Sales revenue 1 2 process 3 Added value * Sales 4
8
0 0 0
0 0
0
129=118*up*trend
105=118*down*trend Product 1 sales 0 0 0 0 0 0
7
1 0 0 0 0 0
2 2596000 2319105 2071744 1850767 1653360
3 0 1553524 1387822 1239793 1107554
4 0 0 994236 888189 793452
5 0 0 0 1090798 974451
1 0 0 0 0 0
2 1535949 811524 164086 -414548 -931692
3 0 2540158 1721123 989150 334984
4 0 0 4087253 3119647 2254411
5 0 0 0 4700974 3670258
6
7
8
0 0 0 1196737
0 0 0
0 0
0
6
7
8
9
0 0 0 5289020
0 0 0
0 0
0
EBDIT
EBDIT process = Sales revenue – Fixed Costs
0 1 2 3 4
0 0 0 0 0 0
Fig. 5.10 Binomial value added process, and following steps
Binomial process, example EBDIT
EBDIT of year 1 extrapolated to the future with a trend factor
0 0 0 0 0 0
0 1 2 3 4
1 0 0 0 0 0
2 1535949 811524 164086 -414548 -931692
3 0 2540158 1721123 989150 334984
4 0 0 4087253 3119647 2254411
5 0 0 0 4700974 3670258
6
7
8
9
0 0 0 5289020
0 0 0
0 0
0
Calculation rolls down from up (to the future)
1,535,949 1,537,138 1,538,327 1,539,518 1,540,709
0 1 2 3 4
1,948,444 1,035,607 2,345,683 346,506 1,335,483 2,785,291 0 497,294 1,700,238 3,257,473 0 0 713,701 2,129,549 3,748,311
Calculation rolls up from down (from the future)
713701 =Max((2254411-1540709),0)
Fig. 5.11 Binomial process, final node assessment
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5 Fuzzy Real Options for Strategic Decisions
Binomial process, example EBDIT 0 1 2 3 4
0 0 0 0 0 0
0 1 2 3 4
For NPV calculation with options cash flow of 1948444 is used instead of 1535949
1 0 0 0 0 0
2 1535949 811524 164086 -414548 -931692
3 0 2540158 1721123 989150 334984
4 0 0 4087253 3119647 2254411
5 0 0 0 4700974 3670258
6
7
8
9
0 0 0 5289020
0 0 0
0 0
0
1,948,444 1,035,607 2,345,683 346,506 1,335,483 2,785,291 0 497,294 1,700,238 3,257,473 0 0 713,701 2,129,549 3,748,311
497294 0
1700238 713701
497294 = 0* (1-0.696781795) + 713701 * 0.696781795
Risk-adjusted probability: 0.696781795 = (erf - down)/(up-down)
Fig. 5.12 Binomial process, node-wise comparison
and risk profiles of the project change. Assuming that the volatilities of the three product-wise AVEs were different from the ones presented in Figure 5.7 to reflect a higher potential of Product 3 and a lower potential of Product 1, the following volatilities could be retrieved (Figure 5.13). Note that the expected value with products 1 and 3 now differs from the AVEs.
Fig. 5.13 Fuzzy Added Value intervals and volatilites
5.4 Closing Production Plants - When and Where
119
The fuzzy cash flow based profitability assessment allows a more profound analysis of the sources of a scenario value. In real option analysis such an asymmetric risk/potential assessment is realized by the fuzzy ROV (cf. Section 5.2). Added values can now be presented as fuzzy added value intervals instead of single (crisp) numbers. The intervals are then run through the whole cash flow table with fuzzy arithmetic operators. The fuzzy intervals described in this way are called trapezoidal fuzzy numbers (cf. Fig. 5.14).
Sales volume Product 1, incremental Sales volume Product 2, incremental Sales volume Product 3, incremental Sales volume total, incremental Added Value Product 1, Crisp Added Value Product 1, Support up Added Value Product 1, Core up Added Value Product 1, Core down Added Value Product 1, Support down Added Value Product 1, Fuzzy EV Added Value Product 1, St. Dev. Added Value Product 1, St. Dev. %
-1,0% 10,0% 5,0% -10,0% -20,0%
0 0 0 0 115 126,50 120,75 103,50 92,00 111,17 9,80 8,8%
37000 3000 10000 50000 114 125,40 119,70 102,60 91,20 110,20 9,71 8,8%
22000 3000 25000 50000 113 124,30 118,65 101,70 90,40 109,23 9,63 8,8%
12000 3000 35000 50000 112 123,20 117,60 100,80 89,60 108,27 9,54 8,8%
7000 8000 40000 55000 111 122,10 116,55 99,90 88,80 107,30 9,46 8,8%
7000 3000 45000 55000 109 119,90 114,45 98,10 87,20 105,37 9,29 8,8%
Fig. 5.14 Fuzzy interval assessment, applying interval assumptions to Added Value
In case of the risk-neutral valuation the discount factor is a single number. In our analysis the discounting is done with the fuzzy EBDIT based cash flow estimates by discounting each component of the fuzzy number separately. The expected value (EV) and the standard deviation (St.Dev) are shown in Figure 5.15 (see also Section 5.2).
Risk-neutral valuation parameter EBDIT, risk neutral EBDIT, risk neutral , Support up EBDIT, risk neutral , Core up EBDIT, risk neutral , Core down EBDIT, risk neutral , Support down EBDIT, risk neutral , Fuzzy EV EBDIT, risk neutral , St. Dev. EBDIT, risk neutral , St. Dev. %
0.955 805875 2040102 1422989 188761 -428352 805875 634024 78.7%
0.911 1458518 2799376 2128947 788088 117659 1458518 688801 47.2%
0.870 1751484 3127935 2439710 1063258 375032 1751484 707085 40.4%
Fig. 5.15 Fuzzy interval assessment, discounting a fuzzy number
As a result from the analysis a NPV calculation now supplies results of the NPV and fuzzy ROV as fuzzy numbers. Also flexibility is shown as a fuzzy number (Figure 5.16).
5.4.6
A Comparative Analysis
For illustrative reasons this comparative analysis is made by applying a standard volatility (10.3%) for each product, scenario and option valuation method.
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2004 Present value at delay Present value at delay, Support up Present value at delay, Core up Present value at delay, Core down Present value at delay, Support down Present value at delay, Fuzzy EV Present value at delay, St. Dev. Present value at delay, St. Dev. % NPV at present year, 2005 Delay value without flexibility Delay value with flexibility, Support Up Delay value with flexibility, Core Up Delay value with flexibility, Core Down Delay value with flexibility, Support Down Delay value with flexibility, Fuzzy EV Delay value with flexibility, St. Dev. Delay value with flexibility, St. Dev. % Delay
Flexibility -1,283,804 3,667,612 3,172,855 2,183,343 1,688,587 2,925,477 508,314 17.4%
2005 7,174,624 9,834,912 7,552,125 2,986,552 703,765 6,410,732 2,345,340 36.6%
2006
7,174,624 9,834,912 7,552,125 2,986,552 703,765 6,410,732 2,345,340 36.6%
2
2007 6,494,629 14,886,532 11,824,291 5,699,809 2,637,568 10,293,171 3,146,154 30.6% 5,890,820 13,502,524 10,724,981 5,169,895 2,392,352 9,336,209 2,853,654 30.6%
Fig. 5.16 Fuzzy interval assessment, NPV and Fuzzy Real Option Value (FROV)
Fig. 5.17 Comparing the results from fuzzy interval method and binomial process graphically, the option to Switch to Scenario 1A at 2006
Figure 5.18 summarizes the results from binomial process and cash flow interval analysis (the analogous analysis of the switch to Scenario 2 has not been shown). The analysis shows that there are viable alternatives to the ones that result in closing the paper mill and that there are several options for continuing with the current operations. The uncertainties in the Added Value processes, which we have modeled in two different ways, show significantly different results when, on the one hand, both risk and potential are aggregated to one single (crisp) number in the
5.4 Closing Production Plants - When and Where
121
Fig. 5.18 Results comparison
binomial process and, on the other hand, there is a fuzzy number that allows the treatment of the downside and the upside differently. In this case study management is faced with poor profitability and needs to assess alternative routes for the final stages of the plant with almost no real residual value. The specific costs of closedown (the pension scheme and the energy contracts) are a large opportunity cost for an immediate closedown (the actual cost is still confidential). The developed model allows for screening alternative paths of action as options. The binomial assessment, based on the assumptions of the real asset tradability, overestimates the real option value, and gives the management flexibilities that actually are not there. On the other hand, the fuzzy cash flow interval approach allows an interactive treatment of the uncertainties on the (annual) cash flow level and in that sense gives the management powerful decision support. With the close/not close decision, the fuzzy cash flow interval method offers both rigor and relevance as we get a normative profitability analysis with readily available uncertainty and sensitivity assessments. Here we showed one scenario analysis in detail and sketched a comparison with a second analysis. For the real case we worked out all scenario alternatives and found out that it makes sense to postpone closing the paper mill with several years. Note 5.6. The paper mill was closed on January 31, 2007 at a significant cost.
5.4.7
Discussion and Conclusions
In decisions on how to use existing resources the challenges of changing markets become a reality when senior management has to decide how to allocate capital to production, logistics and marketing networks, and has to worry about the return on capital employed. The networks are interdependent as the demand for and the prices of forest industry products are defined by the efficiency of the customer production processes and how well suited they are to market demand; the production
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Fig. 5.19 Results from the binomial option valuation
should be cost effective and adaptive to cyclic (and sometimes random) changes in market demand; the logistics and marketing networks should be able to react in a timely fashion to market fluctuations and to offer some buffers for the production processes. Closing or not closing a production plant is often regarded as an isolated decision, without working out the possibilities and requirements of the interdependent networks. The problem we have addressed is the decision to close - or not to close - a production plant in the forest products industry sector. The plant was producing fine paper products, it was rather aged, the paper machines were built a few decades ago, the raw material is not available close by, energy costs are reasonable but are increasing in the near future, key markets are close by and other markets (with better sales prices) will require improvements in the logistics network. The intuitive conclusion was, of course, that we have a sunset case and senior management should make a simple, macho decision and close the plant.
5.4 Closing Production Plants - When and Where
123
Fig. 5.20 Results from the Fuzzy Real Option Valuation
On the other hand we have the trade unions, which are strong, and we have pension funds commitments until 2013 which are very strict, and we have long-term energy contracts which are expensive to get out of. Finally, by closing the plant we will invite competitors to fight us in markets we have served for more than 50 years and which we cannot serve from other plants at any reasonable cost. We showed that real options models will support decision making in which senior managers search for the best way to act and the best time to act. The key elements of the closing/not closing decision may be known only partially and/or only in imprecise terms; then meaningful support can be given with a fuzzy real options model. We found the benefit of using fuzzy numbers and the fuzzy real options model both in the Black-Scholes and in the binomial version of the real options model to be that we can represent genuine uncertainty in the estimates of future costs and cash flows and use these factors when we make the decision to either close the plant now or to postpone the decision by t years (or some other reasonable unit of time).
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5 Fuzzy Real Options for Strategic Decisions
We used a real world case to show the dilemma(s) senior management had to deal with and the (low) level of precision in the data to be used for making a decision. We worked the case with fuzzy real options models and were able to find ways to work out the consequences of closing or not closing the plant. Then, management made their own decision.
5.5
A Fuzzy Pay-Off Method for Real Option Valuation
In this Section, following Collan, Full´er and Mezei [107] we will describe a fuzzy pay-off method for real options valuation. In 2007 a novel approach to real option valuation was presented in [117, 250], where the real option value is calculated from a pay-off distribution, derived from a probability distribution of the NPV for a project that is generated with a (Monte-Carlo) simulation. The authors show that the results from the method converge to the results from the analytical Black-Scholes method. The method presented greatly simplifies the calculation of the real option value, making it more transparent and brings real option valuation as a method a big leap closer to practitioners. The most positive issue in this method is that it does not suffer from the problems associated with the assumptions connected to the market processes connected to the Black-Scholes and the binomial option valuation methods. The method utilizes cash-flow scenario based estimation of the future outcomes to derive the future pay-off distribution - this is highly compatible with the way cash-flow based profitability analysis is commonly done in companies. These models and methods use probability theory in their treatment of uncertainty, there are however, other ways than probability to treat uncertainty, or imprecision in future estimates, namely fuzzy logic and fuzzy sets. In classical set theory an element either (fully) belongs to a set or does not belong to a set at all. This type of bi-value, or true/false, logic is commonly used in financial applications (and is a basic assumption of probability theory). Bi-value logic, however, presents a problem, because financial decisions are generally made under uncertainty. Uncertainty in the financial investment context means that it is in practice impossible, ex-ante to give absolutely correct precise estimates of, e.g., future cash-flows. There may be a number of reasons for this, see, e.g., [218], however, the at the end of the day we our estimations are less than fully accurate. On the other hand, fuzzy sets can be used to formalize inaccuracy that exists in human decision making and as a representation of vague, uncertain or imprecise knowledge, e.g., future cash-flow estimation, which human reasoning is especially adaptive to. Two recent papers [250, 117] present a practical probability theory based method for the calculation of real option value (ROV) and show that the method and results from the method are mathematically equivalent to the Black-Scholes formula [24]. The method is based on simulation generated probability distributions for the NPV of future project outcomes. The method implies that: the real-option value can be understood simply as the average net profit appropriately discounted to Year 0, the date of the initial R&D investment decision, contingent on terminating the project if a loss is forecast at the future launch decision date. The project outcome probability distributions are used to generate a pay-off distribution, where the negative
5.5 A Fuzzy Pay-Off Method for Real Option Valuation
125
outcomes (subject to terminating the project) are truncated into one chunk that will cause a zero pay-off, and where the probability weighted average value of the resulting pay-off distribution is the real option value. Collan, Full´er and Mezei [107] use fuzzy numbers in representing the expected future distribution of possible project costs and revenues, and hence also the profitability (NPV) outcomes. When using fuzzy numbers the fuzzy NPV itself is the pay-off distribution from the project. The method presented in [117] implies that the weighted average of the positive outcomes of the pay-off distribution is the real option value; in the case with fuzzy numbers the weighted average is the fuzzy mean value of the positive NPV outcomes (which is nothing more than the possibility weighted average). Derivation of the fuzzy mean value is presented in [54].
"
! ! (
#!' !
$!'
(
Fig. 5.21 A triangular fuzzy number A, defined by three points {a, α , β } describing the NPV of a prospective project; (percentages 20% and 80% are for illustration purposes only). [107]
This means that calculating the real option value (ROV) from a fuzzy NPV (distribution) is straightforward, it is the fuzzy mean of the possibility distribution with values below zero counted as zero, i.e., the area weighted average of the fuzzy mean of the positive values of the distribution and zero (for negative values). In 2009 Collan, Full´er and Mezei [107] introduced a new fuzzy pay-off method for real options valuation. Definition 5.1. [107] We calculate the real option value from the fuzzy NPV as follows ∞
A(x)dx × E(A+) −∞ A(x)dx
ROV = ∞0
(5.3)
where A stands for the fuzzy NPV, E(A+ ) denotes the fuzzy mean value of the ∞ positive side of the NPV, and −∞ A(x)dx computes the are below the whole fuzzy ∞ number A, and 0 A(x)dx computes the area below the positive part of A (see Fig. 5.5.)
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5 Fuzzy Real Options for Strategic Decisions
Fig. 5.22 Three NPV scenarios for the duration of the synergies that are used to generate (triangular) fuzzy NPV. [107]
It is easy to see that when the whole fuzzy number is above zero then ROV is the fuzzy mean of the fuzzy number, and when the whole fuzzy number is below zero the ROV is zero. The components of this method are simply the observation that real option value is the probability weighted average of the positive values of a pay-off distribution of a project, which is nothing more than the fuzzy NPV of the project, and that for fuzzy numbers the probability weighted average of the positive values of the pay-off distribution is nothing more than the weighted fuzzy mean of the positive values of the fuzzy NPV, when we use fuzzy numbers. Following Collan, Full´er and Mezei [108] we show now how to use credibility measure and the credibilistic expected (or mean) value for fuzzy real option evaluation. To compare the results with the possibilistic mean value, we will use the same examples from [107]. In case of credibilistic expected value, the calculation of the mean of the positive part is defined by the formula, EC [A+ ] =
∞ 0
Cr{A ≥ r}dr.
There are five cases when calculating the credibilistic mean value of the positive area of a triangular fuzzy pay-off, • Case 1: 0 < a − α . In this case we have EC (A+ ) = EC (A) = a +
β −α . 4
We note here that the possibilistic mean value of a triangular fuzzy number is E p (A) = a +
β −α . 6
5.5 A Fuzzy Pay-Off Method for Real Option Valuation
127
Comparing this value to the result above, we can observe that E p (A) − a ≤ |EC (A) − a|. Also important to note, that E p (A) ≤ EC (A) if and only if the left width, α , is smaller than the right width, β . • Case 2: a − α < 0 < a. Then the credibilistic expected value has the following form: EC [A+ ] =
∞ 0
Cr{A ≥ r}dr = =
a 1 0
a−r ( + )dr + 2 2α
a+β 1 a
r−a ( − )dr 2 2β
a a2 β + + 2 4α 4
• Case 3: a < 0 < a + β . In this case EC [A+ ] =
∞ 0
Cr{A ≥ r}dr =
a+β 1 0
r−a a a2 β ( − )dr = + + . 2 2β 2 4β 4
• Case 4: a + β < 0. Then it is easy to see that E(A+ ) = 0 For computing the real option value from an NPV (pay-off) distribution of a trapezoidal form we must consider a trapezoidal fuzzy pay-off distribution A defined by ⎧ u a1 − α ⎪ ⎪ − if a1 − α ≤ u ≤ a1 ⎪ ⎪ α α ⎪ ⎨ 1 if a1 ≤ u ≤ a2 A(u) = u a + β 2 ⎪ ⎪ + if a2 ≤ u ≤ a2 + β ⎪ ⎪ β ⎪ ⎩ −β 0 otherwise where the γ -level of A is defined by [A]γ = [γα + a1 − α , −γβ + a2 + β ]. In trapezoidal case the credibility has the following form: ⎧ 0 if r ≤ a1 − α ⎪ ⎪ ⎪ ⎪ ⎪ 1 a−x ⎪ ⎪ ⎪ − if a1 − α ≤ x ≤ a1 ⎪ ⎪ 2 2α ⎪ ⎪ ⎨1 Cr{A ≤ r} = if a1 ≤ x ≤ a2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ 1 x−a ⎪ ⎪ ⎪ + if a2 ≤ x ≤ a2 + β ⎪ ⎪ 2 2β ⎪ ⎪ ⎩ 0 if a2 + β ≤ x Then to calculate the credibilistic expected value for the positive part, we need to consider the following five cases:
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5 Fuzzy Real Options for Strategic Decisions
• Case 1: 0 < a1 − α . In this case we have EC (A+ ) = EC (A). EC (A+ ) = EC (A) =
a 1 + a2 β − α + . 2 4
Note 5.7. The possibilistic mean value of a trapezoidal fuzzy number is E p (A) =
a1 + a2 β − α + . 2 6
Comparing this value to the result above, we can observe that E p (A) − a ≤ |EC (A) − a|. Also important to note, that E p (A) ≤ EC (A) if and only if the left width, α , is smaller than the right width, β . • Case 2: a1 − α < 0 < a1 . Then the credibilistic expected value can be calculated as: a1 a2 a2 +β 1 a1 − r 1 1 r − a2 EC [A+ ] = + dr + dr + − dr 2 2α 2 2β 0 a1 2 a2 a2 a21 β + + . 2 4α 4 • Case 3: a1 < 0 < a2 . In this case we have, a2 a2 + β 1 1 r − a2 a2 β EC [A+ ] = dr + − dr = + . 2 2 2 β 2 4 0 a2 =
• Case 4: a2 < 0 < a2 + β . In this case we have a 2 +β 1 r − a2 a2 a22 β EC [A+ ] = − dr = + + . 2 2β 2 4β 4 0 • Case 5: a2 + β < z. Then it is easy to see that E(A|z) = 0.
5.6
Optimal R&D Project Portfolios
A major advance in development of project selection tools came with the application of options reasoning to R&D. The options approach to project valuation seeks to correct the deficiencies of traditional methods of valuation through the recognition that managerial flexibility can bring significant value to a project. The main concern is how to deal with non-statistical imprecision we encounter when judging or estimating future cash flows. In this Section we develop a model for valuing options on R&D projects, when future cash flows and expected costs are estimated by trapezoidal fuzzy numbers.
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129
With a real option we understand the opportunity to invest or not invest in a project which involves the acquisition or building of real assets. By working out schemes for phasing and scheduling of projects, every step in the project opens or closes the possibility for further options. By defining phases and actively scheduling activities we can collect information to decide and to analyze/reanalyze if we want to go ahead with the investment or not. The real options models (or actually real options valuation methods) were first tried and implemented as tools for handling very large investments, so-called giga-investments, as there was some fear that capital invested in very large projects, with an expected life cycle of more than a decade is not very productive and that the overall activity around giga-investments is not very profitable (The Waeno project; Tekes 40470/00). Giga-investments compete for major portions of the risk-taking capital, and as their life is long, compromises are made on their short-term productivity. The shortterm productivity may not be high, as the life-long return of the investment may be calculated as very good. Another way of motivating a giga-investment is to point to strategic advantages, which would not be possible without the investment and thus will offer some indirect returns. Giga-investments made in the pulp and paper industry, in the heavy metal industry and in other base industries, today face scenarios of slow (or even negative) growth (2-3% p.a.) in their key markets and a growing overcapacity in Europe. The energy sector faces growing competition with lower prices and cyclic variations of demand. There is also some statistics, which shows that productivity improvements in these industries have slowed down to 1-2% p.a., which opens the way for effective competitors to gain footholds in their main markets. There are other issues. Global financial markets make sure that capital cannot be used non-productively, as its owners are offered other opportunities and the capital will move (often quite fast) to capture these opportunities. The capital market has learned ”the American way”, i.e. there is a shareholder dominance among the actors, which has brought (often quite short-term) shareholder return to the forefront as a key indicator of success, profitability and productivity. There are lessons learned from the Japanese industry, which point to the importance of immaterial investments. These lessons show that investments in buildings, production technology and supporting technology will be enhanced with immaterial investments, and that these are even more important for re-investments and for gradually growing maintenance investments. The core products and services produced by giga-investments are enhanced with lifetime service, with gradually more advanced maintenance and financial add-on services. These make it difficult to actually assess the productivity and profitability of the original giga-investment, especially if the products and services are repositioned to serve other or emerging markets. New technology and enhanced technological innovations will change the life cycle of a giga-investment. The challenge is to find the right time and the right innovation to modify the life cycle in an optimal way. Technology providers are involved throughout the life cycle of a gigainvestment, which should change the way in which we assess the profitability and the productivity of an investment.
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5 Fuzzy Real Options for Strategic Decisions
Now, rather surprisingly, the same type of arguments can be found when senior management ponders portfolios of R&D projects even if the funds to be invested are quite limited when compared to the giga-investments. R&D projects - and more specifically portfolios of R&D projects - may generate commitments, which are (i) showing long life-cycles, (ii) uncertain (sometimes vague, overly optimistic) future cash flow estimates, (iii) uncertain (sometimes questionable) profitability estimates, (iv) quite imprecise assessments of future effects on productivity, market positions, competitive advantages, shareholder value, etc. and (v) generating series of further investments. Jensen and Warren [205] propose to use options theory to value R&D in the telecom service sector. The reasons are rather similar to those we identified above: research managers are under pressure to explain the value of R&D projects to senior management and at the same time they need to evaluate individual projects to make management decisions on their own R&D portfolio, or simply put - to get and defend an R&D budget in negotiations with senior management and then to allocate this budget to individual projects so that the future value of the portfolio is optimized. The research in real options theory has evolved from general presentations of flexibility in investment and industrial cases, to more theoretical contributions and the application of real options to the valuation of both industrial, and research and development projects. The term real option was introduced in 1984 by Kester [214] and Myers [260]. The option to postpone an investment is discussed in McDonald and Siegel [253], and Pakes [272] looks at patents as options. Siegel, Smith, and Paddock [300] discuss the option valuation of offshore oil properties. Majd and Pindyck [242] look at the optimal time of building and the option value in investment decisions. Trigeorgis’ [314] book on managerial flexibility and strategy in resource allocation presents a theory of real options. Abel, Dixit, Eberly, and Pindyck [1] discuss a theory of option valuation of real capital and investments. Faulkner [141] discusses the application of real options to the valuation of research and development projects at Kodak. Kulatilaka, Balasubramanian and Storck [220] discuss a capability based real options approach to managing information technology investments. The use of fuzzy sets to work on real options is a new approach, which has not been attempted too much. One of the first papers to use fuzzy mathematics in finance was published by Buckley [42], in which he works out how to use fuzzy sets to represent fuzzy future value, fuzzy present value, and the fuzzy internal rate of return. The instruments were used to work out ways for the ranking of fuzzy investment alternatives. Buckley returns to the discussion about comparing mutually exclusive investment alternatives with internal rate of return in Buckley [43], and proposes a new definition of fuzzy internal rate of return. Carlsson and Full´er [50] also dealt with the fuzzy internal rate of return in another context (the investment decisions to control several paper mills), and Carlsson and Full´er [51] developed a method for handling capital budgeting problems with fuzzy cash flows. As was pointed out by Carlsson, Full´er, Heikkil¨a and Majlender [79] R&D management has several common features with strategic management. It actively aims at utilizing possibilities supplied by new technologies and innovations in business
5.6 Optimal R&D Project Portfolios Fit with market strategy - FROV, EV (1000 EUR) ; (Bubble size: Market uncertainty) 3000 2500 FROV, EV (1000 EUR)
Fig. 5.23 Bubble diagram of four strategic R&D options in terms of profitability by real option valuation, market strategy and market uncertainty [79]
131
Radical Delay 1 year
2000 Incremental 1 Delay 3 years
1500 1000
Incremental 2 Delay 1 year
Explorative Delay 1 year
500 0 1
2
3
4
5
-500 Fit with market strategy (1 = Current declining market, 2 = Current important market, 3 = Future key market, 4 = Outside strategic focus) Incremental 1
Explorative
Incremental 2
Radical
operations. Similarly to strategic management, R&D management also has to define objectives for the R&D operations. Following the basic R&D management approach, they have created a support tool for evaluating R&D opportunities - the Extended Project Portfolio Tool (XPT) - for the following purposes: (1) to detect shadow options that are not yet measurable in terms of cash flows, and (2) to include such options into the decision making with R&D portfolios. The approach they applied in XPT balances the portfolio selection process with respect to the various modes of strategic change, including (i) the incremental continuous strategy; (ii) the radical changing strategy; (iii) the explorative pilot strategy. Furthermore, instead of the mean-variance-based risk-return criteria, the management has to deal with the following three criteria: (1) return, (2) uncertainty and (3) strategic fit (see Figs. 5.23 and 5.24). The choice is subject to the particular mechanism applied by the management in practice when determining the budgets that allocate capital and other resources. It is also important to know how exogenous strategic knowledge and endogenous knowledge about expected project contribution can change the budgetary limits.
Fit with technology strategy - FROV, EV (1000 EUR); (Bubble size: Technological uncertainty) 3000
FROV, EV (1000 EUR)
Fig. 5.24 Fit with technology strategy (1 = Basic technology, 2 = Current key technology, 3 = Future key technology, 4 = Outside strategic focus) [79]
Radical Delay 1 year 2000 Incremental 1 Delay 3 years 1000
Incremental 2 Delay 1 year
Explorative Delay 1 year
0 1
2
3
4
5
-1000 Fit with technology strategy (1 = Basictechnology, 2 = Current key technology, 3 = Future key technology, 4 = Outside strategic focus) Incremental 1
Explorative
Incremental 2
Radical
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5 Fuzzy Real Options for Strategic Decisions
The usage of uncertainty instead of risk as a criterion may require some more explanation. This choice addresses the situation, where the decision maker has to evaluate economic benefits in lack of some well-functioning efficient market mechanism which could integrate the expected financial benefits and perceived financial risks into a single risk-return framework. For example a technological expert may state that if the project were to start now, then it would be possible to apply for a patent after a few years to protect a new production method. However, the expert cannot give any detail (e.g. probability distribution) about the likelihood that the patent will be granted or that the technological foresight will eventually come true. In such a case the patent is a pre R&D option that becomes a real R&D option’ when the uncertainty about the underlying technological foresight is determined. It would not make much sense if the technological expert presented some subjective probabilities to characterize outcomes that are unknown (not only for him or her but also for the whole community of technological experts). Thus, an expectation involving this kind of uncertainty cannot be represented as a probability of some known future state. However, it can be considered as a possibility, that is, a foresight characterizing the currently unknown state. Formulating from this point of view, we seek to correct the deficiencies of traditional investment valuation methods by incorporating the managerial flexibility that can (and usually does) bring significant value to projects. From our experience, we found that the main issue in the options approach to strategic project valuation is the correct characterization of the non-statistical imprecision that we encounter when judging or estimating future cash flows. Then the basic optimal R&D project portfolio selection problem can be formulated as a fuzzy mixed integer programming problem (see Carlsson, Full´er, Heikkil¨a and Majlender [79] for details). There are now a growing number of papers in the intersection of these two disciplines: real options and possibility theory. In one of the first papers on developing the fuzzy Black-Scholes model, Carlsson and Full´er [52] present a fuzzy real option valuation method, and in Carlsson and Full´er [53] show how to carry out real option valuation in a fuzzy environment. Muzzioli and Torricelli [258] use fuzzy sets to frame the binomial option pricing model, and Carlsson and Full´er [55] discuss the optimal timing of investments with fuzzy real options. Muzzioli and Torricelli [259] present a model for fuzzy binomial option pricing. Carlsson, Full´er, and Majlender [56] develop and test a method for project selection with fuzzy real options, and Carlsson, Full´er [61] work out a fuzzy approach to real options valuation. Majlender [244] presented a comprehensive framework of the development of investment valuation methods in a possibilistic environment.
Chapter 6
Portfolio Selection with Imprecise Future Data
In 1952 Markowitz [247] published his pioneering work which laid the foundation of modern portfolio analysis. The mean-variance methodology for the portfolio selection problem has played an important role in the development of modern portfolio selection theory. It combines probability with optimization techniques to model the behavior investment under uncertainty. The basic principle of the meanvariance model is to use the expected return of a portfolio as the investment return and to use the variance of the expected returns of the portfolio as the investment risk. Most of existing portfolio selection models are based on probability theory. Though probability theory is one of the main techniques used for analyzing uncertainty in finance, the financial market is also affected by several non-probabilistic factors such as vagueness and ambiguity. In many important cases it might also be easier to estimate the possibility distributions of rates of return on securities rather than the corresponding probability distributions. Decision makers are commonly provided with information which is characterized by linguistic descriptions such as high risk, low profit, high interest rate, etc. [298] With the introduction of fuzzy set theory by Zadeh [351], it was realized that imperfect knowledge of the returns on the assets and the uncertainty involved in the behavior of financial markets may be captured by means of fuzzy quantities and/or fuzzy constraints. Since the early seventies fuzzy set theory has been widely used to solve many problems including financial risk management. By using fuzzy approaches, the experts’ knowledge and the investors’ subjective opinions can be better integrated into a portfolio selection model.
6.1
Possibilistic Choice of Portfolios with Highest Utility Score
The mean-variance methodology for the portfolio selection problem, originally proposed by Markowitz [247], has been one of the most important research fields in modern finance theory [335]. Following [26] we will assume that each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of those portfolios. The utility score may be viewed as a C. Carlsson and R. Full´er: Possibility for Decision, STUDFUZZ 270, pp. 133–144. springerlink.com © Springer-Verlag Berlin Heidelberg 2011
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means of ranking portfolios. Higher utility values are assigned to portfolios with more attractive risk-return profiles. One reasonable function that is commonly employed by financial theorists assigns a risky portfolio P with a risky rate of return rP , an expected rate of return E(rP ) and a variance of the rate of return σ 2 (rP ) the following utility score [26]: U(P) = E(rP ) − 0.005 × A × σ 2(rP ),
(6.1)
where A is an index of the investor’s risk aversion (A ≈ 2.46 in US). The factor of 0.005 is a scaling convention that allows us to express the expected return and standard deviation in equation (6.1) as percentages rather than decimals. Equation (6.1) is consistent with the notion that utility is enhanced by high expected returns and diminished by high risk. Because we can compare utility values to the rate offered on risk-free investments when choosing between a risky portfolio and a safe one, we may interpret a portfolio’s utility value as its certainty equivalent rate of return to an investor. That is, the certainty equivalent rate of a portfolio is the rate that risk-free investments would need to offer with certainty to be considered as equally attractive as the risky portfolio. Now we can say that a portfolio is desirable only if its certainty equivalent return exceeds that of the risk-free alternative. In the mean-variance context, an optimal portfolio selection can be formulated as the following quadratic mathematical programming problem n n n 2 (6.2) U ∑ ri xi = E ∑ ri xi − 0.005 × A × σ ∑ ri xi → max i=1
i=1
subject to
i=1
! ∑ xi = 1, xi ≥ 0, i = 1, . . . , n , n
i=1
where n is the number of available securities, xi is the proportion invested in security (or asset) i, and ri denotes the risky rate of return on security i, i = 1, . . . , n. Denoting the rate of return on the risk-free asset by r f , a portfolio is desirable for the investor if and only if n U ∑ ri xi > r f . i=1
we will assume that the rates of return on securities are modeled by possibility distributions rather than probability distributions. That is, the rate of return on security iwill be represented by a fuzzy number ri , and ri (t), t ∈ R, will be interpreted as the degree of possibility of the statement that ’t will be the rate of return on security i’. we will consider only trapezoidal possibility distributions, but our method can easily be generalized to the case of possibility distributions of type LR. In standard portfolio models uncertainty is equated with randomness, which actually combines both objectively observable and testable random events with subjective judgments of the decision maker into probability assessments. A purist on
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theory would accept the use of probability theory to deal with observable random events, but would frown upon the transformation of subjective judgments to probabilities. The use of probabilities has another major drawback: the probabilities give an image of precision which is unmerited - we have found cases where the assignment of probabilities is based on very rough, subjective estimates and then the subsequent calculations are carried out with a precision of two decimal points. This shows that the routine use of probabilities is not a good choice. The actual meaning of the results of an analysis may be totally unclear - or results with serious errors may be accepted at face value. In standard portfolio theory the decision maker assigns utility values to consequences, which are the results of combinations of actions and random events. The choice of utility theory, which builds on a decision maker’s relative preferences for artificial lotteries, is a way to anchor portfolio choices in the von NeumannMorgenstern axiomatic utility theory. In practical applications the use of utility theory has proved to be problematic (which should be more serious than having axiomatic problems): (i) utility measures cannot be validated inter-subjectively, (ii) the consistency of utility measures cannot be validated across events or contexts for the same subject, (iii) utility measures show discontinuities in empirical tests (as shown by Tversky (cf. [317])), which should not happen with rational decision makers if the axiomatic foundation is correct, and (iv) utility measures are artificial and thus hard to use on an intuitive basis. As the combination of probability assessments with utility theory has these wellknown limitations we have explored the use of possibility theory as a substituting conceptual framework. Recall that if A ∈ F is a fuzzy number with [A]γ = [a1 (γ ), a2 (γ )], γ ∈ [0, 1], then the (crisp) possibilistic mean (or expected) value and variance of A is defined as [54] E(A) =
σ 2 (A) =
1
1 2
0
γ (a1 (γ ) + a2 (γ ))d γ ,
1 0
2 γ a2 (γ ) − a1 (γ ) d γ .
It is easy to see that if A = (a, b, α , β ) is a trapezoidal fuzzy number then E(A) =
a+b β −α + . 2 6
and (b − a)2 (b − a)(α + β ) (α + β )2 + + 4 6 24 b − a α + β 2 (α + β )2 = + + . 2 6 72
σ 2 (A) =
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Watada [332] proposed a fuzzy portfolio selection model where he used fuzzy numbers to represent the decision maker’s aspiration levels for the expected rate of return and a certain degree of risk. Inuiguchi and Tanino [198] introduced a novel possibilistic programming approach to the portfolio selection problem: their approach, which prefers a distributive investment solution, is based on the minimax regret criterion (the regret which the decision maker is ready to undertake). In many important cases it might be easier to estimate the possibility distributions of rates of return on securities rather than the corresponding probability distributions. In 2002 Carlsson, Full´er and Majlender [57] stated the portfolio selection problem with possibility distributions n n n 2 U ∑ ri xi = E ∑ ri xi − 0.005 × A × σ ∑ ri xi → max (6.3) i=1
i=1
i=1
subject to {x1 + · · · + xn = 1, xi ≥ 0, i = 1, . . . , n}. where ri = (ai , bi , αi , βi ), i = 1, . . . , n are fuzzy numbers of trapezoidal form. It is easy to compute that n n 1 1 E ∑ ri xi = ∑ ai + bi + (βi − αi ) xi , 3 i=1 i=1 2 and
σ2
n
∑ ri xi
i=1
=
2 2 1 1 1 n b − a + ( α + β ) x + ( α + β )x ∑ i i 3 i i i ∑ i i i . 72 i=1 i=1 2 n
Introducing the notations 1 1 ui = ai + bi + (βi − αi ) , 2 3 √ 0.005A 1 vi = bi − ai + (αi + βi ) , 2 3 √ 0.005A wi = √ (αi + βi ), 72 they represent the ith asset by a triplet (vi , wi , ui ), where ui denotes its possibilistic expected value, and v2i + w2i is its possibilistic variance multiplied by the constant 0.005 × A. It is also assumed that there are at least three distinguishable assets, with the meaning that if two assets have the same expected value and variance then they are considered indistinguishable. That is, we assume that ui = u j or v2i + w2i = v2j + w2j for i = j. Then Carlsson, Full´er and Majlender [57] stated the possibilistic portfolio selection problem (6.3) as
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u, x − v, x2 − w, x2 → max;
(6.4)
subject to {x1 + · · · + xn = 1, xi ≥ 0, i = 1, . . . , n}. The convex hull of {(vi , wi , ui ) : i = 1, . . . , n}, denoted by T , and defined by T = conv{(vi , wi , ui ) : i = 1, . . . , n} ! n n n n = ∑ vi xi , ∑ wi xi , ∑ ui xi : ∑ xi = 1, xi ≥ 0, i = 1, . . . , n . i=1
i=1
i=1
i=1
is a convex polytope in R3 . Then (6.4) turns into the following three-dimensional nonlinear programming problem −(v20 + w20 − u0 ) → max subject to (v0 , w0 , u0 ) ∈ T, or, equivalently, f (v0 , w0 , u0 ) := v20 + w20 − u0 → min; subject to (v0 , w0 , u0 ) ∈ T,
(6.5)
where T is a compact and convex subset of R3 , and the implicit function gc (v0 , w0 ) := v20 + w20 − c, is strictly convex for any c ∈ R. This means that any optimal solution to (6.5) must be on the boundary of T . Consider three assets (vi , wi , ui ), i = 1, 2, 3, which are not colinear: (α1 , α2 , α3 ) ∈ R3 , (α1 , α2 , α3 ) = 0, such that ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ v3 v1 v2 α1 ⎣ w1 ⎦ + α2 ⎣ w2 ⎦ − (α1 + α2 ) ⎣ w3 ⎦ = 0. u1 u2 u3 Then the 3-asset optimal portfolio selection problem with not-necessarily nonnegative weights reads (v1 x1 + v2 x2 + v3 x3 )2 + (w1 x1 + w2 x2 + w3 x3 )2 − (u1 x1 + u2 x2 + u3 x3 ) → min (6.6) subject to x1 + x2 + x3 = 1. Let us denote L(x, λ ) = (v1 x1 + v2 x2 + v3 x3 )2 + (w1 x1 + w2 x2 + w3 x3 )2 −(u1 x1 + u2x2 + u3 x3 ) + λ (x1 + x2 + x3 − 1),
(6.7)
the Lagrange function of the constrained optimization problem (6.6). The KarushKuhn-Tucker necessity conditions are
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2v1 (v1 x1 + v2 x2 + v3 x3 ) + 2w1 (w1 x1 + w2 x2 + w3 x3 ) − u1 + λ = 0, 2v2 (v1 x1 + v2 x2 + v3 x3 ) + 2w2 (w1 x1 + w2 x2 + w3 x3 ) − u2 + λ = 0, 2v3 (v1 x1 + v2 x2 + v3 x3 ) + 2w3 (w1 x1 + w2 x2 + w3 x3 ) − u3 + λ = 0, x1 + x2 + x3 = 1, which lead us to the following linear equality system 2 q1 + r12 q1 q2 + r1 r2 x1 1/2(u1 − u3) − q1 v3 − r1 w3 = , x2 1/2(u2 − u3) − q2 v3 − r2 w3 q1 q2 + r1 r2 q22 + r22
(6.8)
where we used the notations q1 = v1 − v3 , q2 = v2 − v3 , r1 = w1 − w3 and r2 = w2 − w3 . It can be shown (see [57]) that if (vi , wi , ui ), i = 1, 2, 3, are not colinear then equation (6.8) has a unique solution, and, furthermore, x∗ = (x∗1 , x∗2 , 1 − x∗1 − x∗2 ) satisfies the Karush-Kuhn-Tucker sufficiency condition, i.e. L (x, λ ) is matrix at x = x∗ in the subset defined by a positive definite 3 y = (y1 , y2 , y3 ) ∈ R : y1 + y2 + y3 = 0 . Consider now a 2-asset problem with (v1 , w1 , u1 ) and (v2 , w2 , u2 ), such that (v1 , w1 , u1 ) = (v2 , w2 , u2 ): (v1 x1 + v2 x2 )2 + (w1 x1 + w2 x2 )2 − (u1x1 + u2 x2 ) → min; s.t. x1 + x2 = 1.
(6.9)
Let us denote L(x, λ ) = (v1 x1 + v2 x2 )2 + (w1 x1 +w2 x2 )2 −(u1 x1 + u2 x2 )+ λ (x1 +x2 − 1), (6.10) the Lagrange function of the constrained optimization problem (6.6). The KarushKuhn-Tucker necessity conditions leads us to the following linear equation 1 2 2 (v1 − v2 ) + (w1 − w2 ) x1 = (u1 − u2 ) − (v1 − v2 )v2 − (w1 − w2 )w2 . (6.11) 2 If (v1 − v2 )2 + (w1 − w2 )2 = 0 then we find that x∗ = (x∗1 , 1 − x∗1), where x∗1
1 1 = (u1 − u2 ) − (v1 − v2 )v2 − (w1 − w2 )w2 , (6.12) (v1 − v2 )2 + (w1 − w2 )2 2
is the unique solution to equation (6.11). If v1 = v2 and w1 = w2 then from (6.11) we find u1 = u2 , which would contradict the initial assumption that the two assets are not identical. It can L (x∗ , λ ) is a positive definite matrix in easily be seen that 2 the subset defined by y = (y1 , y2 ) ∈ R : y1 + y2 = 0 . So, x∗ is the unique optimal solution to (6.9), and if x∗ > 0 then x∗ is an optimal solution to (6.4) with n = 2. We will illustrate the above algorithm by a simple example from Carlsson, Full´er and Majlender [57]. Let us consider a 3-asset problem with A = 2.46 and with possibility distributions of trapezoidal form
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139
r1 = (−10.5, 70.0, 4.0, 100.0), r2 = (−8.1, 35.0, 4.4, 54.0), r3 = (−5.0, 28.0, 11.0, 85.0), and, therefore, (v1 , w1 , u1 ) = (6.386, 1.359, 45.750), (v2 , w2 , u2 ) = (3.469, 0.763, 21.717), (v3 , w3 , u3 ) = (3.604, 1.255, 23.833). It should be noted that the first asset may yield negative rates of return with degree of possibility one. First consider the 3-asset problem with (v1 , w1 , u1 ), (v2 , w2 , u2 ) and (v3 , w3 , u3 ). Since q r 2.782 0.105 det 1 1 = det = −1.352 = 0, q2 r 2 −0.135 −0.491 we get
and, since,
1 x∗1 0.259 0.427 0.800 0.124 = × = , x∗2 0.044 0.373 −1.3522 0.427 7.751 [x∗1 , x∗2 , x∗3 ] = [0.124, 0.373, 0.503] > 0.
we get U∗ = −9.386 and x∗ = [0.124, 0.373, 0.503]. Thus [0.124, 0.373, 0.503] is a qualified candidate for an optimal solution to (6.3). Let us consider all conceivable 2-asset problems (1, 2), (1, 3) and (2, 3), where the numbers stand for the corresponding assets chosen from the bag {(v1 , w1 , u1 ), (v2 , w2 , u2 ), (v3 , w3 , u3 )}. Here we are searching for optimal solutions on the edges of the triangle generated by the assets. Select (1,2).
Since (v1 − v2 )2 + (w1 − w2 )2 = 8.864 = 0, we get U∗ := −9.336 and [x∗1 , x∗2 ] = [0.163, 0.837] > 0.
Thus [0.163, 0.837, 0] is a qualified candidate for an optimal solution to (6.3). Select (1,3). Since (v1 − v3 )2 + (w1 − w3 )2 = 7.751 = 0, we get U∗ := −9.352 and [x∗1 , x∗3 ] = [0.103, 0.897] > 0. Thus [0.103, 0, 0.897] is a qualified candidate for an optimal solution to (6.3). Select (2,3). Since (v2 − v3 )2 + (w2 − w3 )2 = 0.259 = 0, we get U∗ := −9.277 and [x∗2 , x∗3 ] = [0.171, 0.829] > 0.
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Thus [0, 0.171, 0.829] is a qualified candidate for an optimal solution to (6.3). Finally, we compute the utility values of all the three vertexes of the triangle generated by the three assets: v21 + w21 − u1 = −3.122, and [1, 0, 0] is the corresponding feasible solution to (6.3). v22 + w22 − u2 = −9.101, and [0, 1, 0] is the corresponding feasible solution to (6.3). v23 + w23 − u3 = −9.269, and [0, 0, 1] is the corresponding feasible solution to (6.3). Comparing the utility values of all feasible solutions we find that the only solution to the 3-asset problem is x∗ = [0.124, 0.373, 0.503] with a utility value of 9.386. The optimal risky portfolio will be preferred to the risk-free investment (by an investor whose degree of risk-aversion is equal to 2.46) if r f < 9.386%.
6.2
Recent Advances
In this Section we will give a short chronological survey of some later works on fuzzy portfolio selection models. We will mention only those works in which the authors extended, improved or used the findings of our original paper [57]. In 2005 Zmeskal [364] described an approach to modeling uncertainty of the international index portfolio by the value at risk (VAR) methodology under soft conditions by fuzzy-stochastic methodology. The generalized term uncertainty is understood to have two aspects: risk modeled by probability (stochastic methodology) and vagueness sometimes called impreciseness, ambiguity, softness is modeled by fuzzy methodology. Thus, hybrid model is called fuzzy-stochastic model. Input data for a stochastic model are unique distribution functions and crisp (real) data. Input data for fuzzy model are fuzzy numbers and crisp (real) data. Input data for hybrid model are fuzzy probability distribution functions, unique distribution functions, and crisp (real) data. Risk is modeled by stochastic methodology on the VAR basis and vagueness is modeled through the fuzzy numbers. The analytical delta normal VAR methodology for international index portfolio under soft conditions is described. In 2006 Huang [194] considered two types of credibility-based portfolio selection model are provided according to two types of chance criteria. By one chance criterion, the objective is to maximize the investor’s return at a given threshold confidence level; by another chance criterion, the objective is to maximize the credibility of achieving a specified return level subject to the constraints. Let xi denote the investment proportions in securities i, ξi the fuzzy returns for the ith
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141
securities, i = 1, 2, . . . , n, respectively, α the predetermined confidence level the investor accepts. Then Huang’s first model is maximize f¯ subject to Cr[x1 ξ1 + x2 ξ2 + · · · xn ξn ] ≥ α x1 + · · · + xn = 1, xi ≥ 0, i = 1, . . . , n where max f¯ is the α -return defined as max{ f¯ | Cr[x1 ξ1 + x2 ξ2 + · · · xn ξn ] ≥ f¯} ≥ α }, which means the maximal investment return the investor can obtain at confidence level α . It is obvious that the combination of securities that can obtain f¯ is the optimal portfolio the investor should select. Huang’s second model is maximize Cr[x1 ξ1 + x2 ξ2 + · · · xn ξn ] ≥ a subject to x1 + · · · + xn = 1, xi ≥ 0, i = 1, . . . , n where a is the predetermined investment return level the investor feels satisfactory. Huang also designed a hybrid intelligent algorithm integrating fuzzy simulation and genetic algorithm to solve these problems. In 2007 Zhang [359] discussed the portfolio selection problem for bounded assets based on the lower and upper possibilistic means and variances of fuzzy numbers. If the possibility distribution of rates of return on the ith security is given by a trapezoidal fuzzy number ri = (ai , bi , αi , βi ) for i = 1, . . . , n, then the lower possibilistic mean-standard deviation model of portfolio selection is stated as n αi maximize ∑ ai − 3 i=1 √ 6 n subject to ∑ αi xi ≤ σ 2 i=1 x1 + · · · + xn ≤ 1, li ≤ xi ≤ ui , i = 1, . . . , n. and the upper possibilistic mean-standard deviation model of portfolio selection is stated as n βi maximize ∑ bi + 3 i=1 √ 6 n subject to ∑ βi x i ≤ σ 2 i=1 x1 + · · · + xn ≤ 1, li ≤ xi ≤ ui , i = 1, . . . , n.
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where 0 ≤ li ≤ ui ≤, i = 1, . . . , n and ∑ni=1 li < 1. Zhang [359] pointed out that the lower possibilistic efficient portfolios construct the lower possibilistic efficient frontier. All the upper possibilistic efficient portfolios construct the upper possibilistic efficient frontier. Solving the above problems for all tolerated risk level σ , the lower and upper possibilistic efficient frontiers are derived explicitly. Smimou et al [304] considered the derivation of portfolio modeling under a fuzzy situation using probability theory, and provides various other (non-probabilistic) scenarios with their utility in risk modeling. They also proposed a simple method for identification of mean-entropic frontier. In 2008 Gupta et al [172] incorporated fuzzy set theory into a semi-absolute deviation portfolio selection model five criteria: short term return, long term return, dividend, risk and liquidity. In their proposed model, for a given return level, the investor penalizes the negative semi-absolute deviation that is dened as a risk. From computational point of view, the semi-absolute deviation halves the number of required constraints with respect to the absolute deviation. Huang [195] presented two mean-semivariance models for fuzzy model portfolio selection. Suppose that xi denotes the investment proportions in the ith security, and ξi stands for the fuzzy return on the ith security, where p + d i − p i ξi = i , pi for i = 1, 2, . . . , n, and where pi is the estimated closing price of the ith security in the following year, pi is the closing price of the ith security now, and di is the estimated dividends of the ith security during the coming year. In the first model the investor is in the position to give a maximal level of risk, denoted by γ , he can still tolerate, and wants to maximize the expected return at the given level of risk. Then the fuzzy mean-semivariance model can be formulated as maximize E[x1 ξ1 + x2 ξ2 + · · · xn ξn ] subject to S[x1 ξ1 + x2 ξ2 + · · · xn ξn ] ≤ γ x1 + · · · + xn = 1, xi ≥ 0, i = 1, . . . , n where E denotes the expected value operator [225], and S denotes the semivariance [195] of the fuzzy variables. In the second model the investor presets an expected return level, denoted by α , that he feels satisfactory, and wants to minimize the risk for this given level of return, maximize E[x1 ξ1 + x2 ξ2 + · · · xn ξn ] subject to S[x1 ξ1 + x2 ξ2 + · · ·xn ξn ] ≥ α x1 + · · · + xn = 1, xi ≥ 0, i = 1, . . . , n Huang [195] also provided a fuzzy simulation based genetic algorithm to solve these optimization problems. Vercher [321] considered a capital market with n risky
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143
assets and a risk-free asset with a fixed rate of return. It is assumed that that each investor can assign a preference level to competing investment portfolios based on the expected return and risk of those portfolios. Furthermore, x j denotes the proportion of the total investment fund devoted to j-th asset, the uncertainty on its return is modeled by fuzzy numbers of trapezoidal form, R˜ j , R f denotes the rate of return on the risk-free asset, and the P(x) = {x1 , . . . , xn } denotes the portfolio built up from assets x1 , . . . , xn . Different definitions of the average of a fuzzy number can be used to evaluate both the expected return and the risk of the portfolio P(x). He used a fuzzy downside risk function [221] to measure the risk of the investment, which evaluates the mean absolute semi-deviation with respect to the total return, wE (P(x)) = E(max{0, E(R˜ P (x)) − R˜ P (x)}). Then he formulates the fuzzy portfolio selection problem as, minimize wE (P(x)) n
subject to
∑ x j R˜ j ≥ R f
j=1
x1 + · · · + xn = 1, l j ≤ x j ≤ u j , j = 1, . . . , n. where l j (respectively, u j ) represents the minimum (maximum) amount of fund which can be invested in j-th asset. It is assumed that short selling is not allowed, therefore l j is a non-negative number. Moreover, for a given j, at most one bound can be active at each feasible point, then l j < u j , and in order to ensure feasibility the next inequality holds ∑nj=1 u j ≥ 1. Then he derives the optimal portfolio using semi-infinite programming in a soft framework. In 2009 Hasuike and Ishii [179] proposed several mathematical models with respect to portfolio selection problems, particularly using the scenario model including the ambiguous factors. These mathematical programming problems with probabilities and possibilities are called to stochastic programming problem and fuzzy programming problem, and it is difficult to find the global optimal solution for those problems. They also managed to develop an efficient solution method to find the global optimal solution of such a nonlinear programming problem. Zhang et al [360] proposed a new portfolio selection model with the maximum utility based on the interval-valued possibilistic mean and possibilistic variance, which is a twoparameter quadratic programming problem. They also presented a sequential minimal optimization (SMO) algorithm to obtain the optimal portfolio. The remarkable feature of their algorithm is that it is extremely easy to implement, and it can be extended to any size of portfolio selection problems for finding an exact optimal solution. Chen [104] developed two weighted possibilistic portfolio selection models with bounded constraint, which can be transformed to linear programming problems under the assumption that the returns of assets are trapezoidal fuzzy numbers. Dia [120] presented a four-step methodology based on Data Envelopment Analysis for
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portfolio selection of decision-making units (DMUs) which can be stocks or other financial assets. Along the steps of the methodology, DMUs efficiency ratios are first computed, and then, the generation of a portfolio is carried out by a mathematical model which optimizes the weighted sum of the DMUs’ efficiency ratios included in this portfolio, which is optimal for the Decision Maker’s system of preferences. Huang [196] showed some credibilistic portfolio selection models, including meanrisk model, mean-variance model, mean-semivariance model, credibility maximization model, α -return maximization model, entropy optimization model and game models. In 2010 Zhang et al [361] introduced a possibilistic risk tolerance model for the portfolio adjusting problem based on possibility moments theory. An SMO-type decomposition method is developed for finding exact optimal portfolio policy without extra matrix storage. Furthermore, they presented a simple method to estimate the possibility distributions for the returns of assets. Chen [105] considered securities with fuzzy rate of return and developed a possibilistic mean-variance safety-first portfolio model. Using the possibilistic means and variances, he transformed the possibilistic programming problem into a linear optimal problem with an additional quadratic constraint and proposed a cutting plane algorithm to solve it. Petreska and Kolemisevska-Gugulovska [276] introduced a methodology that can be useful in the management of assets against certain given liability and risk estimation of different portfolio structures. In 2011 Huang [197] discussed the uncertain portfolio selection problem when security returns cannot be well reflected by historical data. He proposed that uncertain variable should be used to reflect the experts’ subjective estimation of security returns. Regarding the security returns as uncertain variables, he introduced a risk curve and developed a mean-risk model. Zhang et al [362], based on possibilistic mean and variance theory, dealt with the portfolio adjusting problem for an existing portfolio under the assumption that the returns of risky assets are fuzzy numbers and there exist transaction costs in portfolio adjusting precess. They proposed a portfolio optimization model with V-shaped transaction cost which is associated with a shift from the current portfolio to an adjusted one.
Chapter 7
Risk Assessment in Grid Computing
There is an increasing demand for computing power in scientific and engineering applications which has motivated the deployment of high performance computing (HPC) systems that deliver tera-scale performance. Current and future HPC systems that are capable of running large-scale parallel applications may span hundreds of thousands of nodes. In 2006 the highest processor count was 131K nodes according to top500.org [282]. For parallel programs, the failure probability of nodes and computing tasks assigned to the nodes has been shown to increase significantly with the increase in number of nodes. Large-scale computing environments, such as the current grids CERN LCG, NorduGrid, TeraGrid and Grid’5000 gather (tens of) thousands of resources for the use of an ever-growing scientific community. Many of these Grids offer computing resources grouped in clusters, whose owners may share them only for limited periods of time and Grids often have the problems of any large-scale computing environment to which is added that their middleware is still relatively immature, which contributes to making Grids relatively unreliable computing platforms. Long et al. [237] collected a dataset on node failures over 11 months from 1139 workstations on the Internet to determine their uptime intervals. Plank and Elwasif [277] collected a dataset on failure information for a collection of 16 DEC Alpha work-stations at Princeton University; the size of this network is smaller and is a typical local cluster of homogeneous processors; the failure data was collected for 7 months and shows similar characteristics as for the larger clusters. Schroeder and Gibson [294] analyze failure data collected over 9 years at Los Alamos National Laboratory (LANL) and which includes 23000 failures recorded on more than 20 different systems, mostly large clusters of SMP and NUMA nodes. Their study includes root cause of failures, the mean time between failures, and the mean time to repair. They found that average failure rates differ wildly across systems, ranging from 20-1000 failures per year, mean repair time varies from less than an hour to more than a day. Most applications (about 70%) running in the LANL are short duration computing tasks of < 1 hour, but there are also large-scale, longrunning 3D scientific simulations. These applications perform long periods (often months) of CPU computation, interrupted every few hours by a few minutes of I/O C. Carlsson and R. Full´er: Possibility for Decision, STUDFUZZ 270, pp. 145–165. springerlink.com © Springer-Verlag Berlin Heidelberg 2011
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for check-pointing. When node failures occur in the LANL, hardware was found to be the single largest cause (30-60%); software is the second largest contributor (with 5-24%), but in most systems the root cause remained undetermined for 2030% of the failures [294]. They also found that the yearly failure rate varies widely across systems, ranging from 17 to an average of 1159 failures per year for several systems. The main reason for the differences was that the systems vary widely in size and that the nodes run different workloads. Iosup et al [200] fit statistical distributions to the Grid’5000 data using maximum likelihood estimation (MLE) to find a best fit for each of the model parameters. They found that the best fits for the inter-arrival time between failures, the duration of a failure, and the number of nodes affected by a failure, are the Weibull, Log-Normal, and Weibull distributions, respectively. The results for inter-arrival time between consecutive failures indicate an increasing hazard rate function, i.e. the longer a computing node stays in the system, the higher the probability for the node to fail, which will prevent long jobs to finish. Iosup et al [200] also wanted to find out if they can decide where (on which nodes or in which cluster) a new failure could/should occur. Since the sites are located and administered separately, and the network between them has numerous redundant paths, they found no evidence for any other assumption than that there is no correlation between the occurrence of failures at different sites. For the LANL dataset Schroeder and Gibson [294] studied the sequence of failure events and the time between failures as stochastic processes. This includes two different views of the failure process: (i) the view as seen by an individual node; (ii) the view as seen by the whole system. They found that the distribution between failures for individual nodes is well modeled by a Weibull or a Gamma distribution; both distributions create an equally good visual fit and the same negative log-likelihood. For the system wide view of the failures the basic trend is similar to the per node view during the same time. The Weibull and Gamma distributions provide the best fit, while the lognormal and exponential fits are significantly worse. A significant amount of the literature on grid computing addresses the problem of resource allocation on the grid [37, 113, 116, 227, 241]. The presence of disparate resources that are required to work in concert within a grid computing framework increases the complexity of the resource allocation problem. Jobs are assigned either through scavenging, where idle machines are identified and put to work, or through reservation in which jobs are matched and pre-assigned with the most appropriate systems in the grid for efficient workflow. In grid computing a resource provider [RP] offers resources and services to other Grid users based on agreed service level agreements [SLAs]. The research problem we have addressed is formulated as follows: • the RP is running a risk to be in violation of his SLA if one or more of the resources [nodes in a cluster or a Grid] he is offering to prospective customers will fail when carrying out the tasks • the RP needs to work out methods for a systematic risk assessment [RA] in order to judge if he should offer the SLA or not if he wants to work with some acceptable risk profile.
7.1 Risk Assessment
147
In the context we are going to consider (a generic grid computing environment) resource providers are of various types which mean that the resources they manage and the risks they have to deal with are also different; we have dealt with the following RP scenarios (but we will report only on extracts due to space): • RP1 manages a cluster of n1 nodes (where n1 < 10) and handles a few (< 5) computing tasks for a T ; • RP2 manages a cluster of n2 nodes (where n2 < 150) and handles numerous (≈ 100) computing tasks for a T ; RP2 typically uses risk models building on stochastic processes (Poisson-Gamma) and Bayes modeling to be able to assess the risks involved in offering SLAs; • RP3 manages a cluster of n3 nodes (where n3 < 10) and handles numerous (≈ 100) computing tasks for a T ; if the computing tasks are of short duration and/or the requests are handled online RP3 could use possibility models that will offer robust approximations for the risk assessments; • RP4 manages a cluster of n4 nodes (where n4 < 150) and handles numerous (≈ 100) computing tasks for a T ; typically RP4 could use risk models building on stochastic processes (Poisson-Gamma) and Bayes modeling to assess the risks involved in offering SLAs; if the computing tasks are of short duration and/or the requests are handled online hybrid models which combine stochastic processes and Bayes modeling with possibility models could provide tools for handling this type of cases. During the execution of a computing task the fulfillment of the SLA has the highest priority, which is why an RP often is using resource allocation models to safeguard against expected node failures. When spare resources at the RP’s own site are not available outsourcing will be an adequate solution for avoiding SLA violations. The risk assessment modeling for an SLA violation builds on the development of predictive probabilities and possibilities for possible node failures and combined with the availability of spare resources. In Section 7.1 we will work out the basic conceptual framework for risk assessment, in Sections 7.2-7.4 we will introduce the Bayesian predictive probabilities as they apply to the SLAs for RPs in grid computing, in Sections 7.6-7.7 we will work out the corresponding predictive possibilities and show the results of the validation work we carried out for some RP scenarios; we will work out the mathematical basis and the models in detail because this type of tools have not previously been used to assess the risks related to Grid computing services; in Section 7.5 we compute a lower limit for the probability of success of computing tasks in a grid.
7.1
Risk Assessment
There is no universally accepted definition of business risk but in the RP context we will understand risk to be a potential problem which can be avoided or mitigated [86]. The potential problem for an RP is that he has accepted an SLA and may not
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be able to deliver the necessary computing resources in order to fulfill a computing task within an accepted time frame T . Risk assessment is the process through which a resource provider tries to estimate the probability for the problem to occur within T and risk management the process through which a resource provider tries to avoid or mitigate the problem. In classical decision theory risk is connected with the probability of an undesired event; usually the probability of the event and its expected harm is outlined with a scenario which covers the set of risk, regret and reward probabilities in an expected value for the outcome. The typical statistical model has the following structure, R(θ , δ (x)) =
L(θ , δ (x)) f (x|θ )dx
(7.1)
where L is a loss function of some kind, x is an observable event (which may not have been observed) and δ (x) is an estimator for a parameter θ which has some influence on the occurrence of x. The risk is the expected value of the loss function. The statistical models are used frequently because of the very useful tools that have been developed to work with large datasets. The statistical model is influenced by the modern capital markets theory where risk is seen as a probability measure related to the variance of returns. Markowitz [247] initiated the modern portfolio theory stating that investors should focus on selecting portfolios based on portfolios’ risk-reward characteristics instead of compiling portfolios of assets that each individually has attractive risk-reward characteristics. The risk is classified in systematic (”market”) risk and idiosyncratic (”company” or ”individual”) risk. The analogy would be that an RP – handling a large number of nodes and a large number of computing tasks – will reach a steady state in his operations so that there will be a stable systematic risk (”market risk’) for defaulting on an SLA which he can build on as his basic assumption and then a (”small’) idiosyncratic risk which is situation specific and which he should estimate with some statistical models. We developed a hybrid probabilistic and possibilistic model to assess the success of computing tasks in a Grid. The model first gives simple predictive estimates of node failures in the next planning period when the underlying logic is the Bayesian probabilistic model for observations on node failures. When we apply the possibilistic model to a dataset we start by selecting a sample of k observations on node failures. Then we find out how many of these observations are different and denote this number by l; we want to use the two datasets to predict what the (k + 1)th observation on node failures is going to be. The possibility model is used to find out if that number is going to be 0, 1, 2, 3, . . . etc.; for this estimate the possibility model uses the ”most usual’ numbers in the larger dataset and makes an estimate which is ”as close as possible’ to this number. The estimate we use is a triangular fuzzy number, i.e. an interval with a possibility distribution. The possibility model turned out to be a faster and more robust estimate of the (k + 1)th observation and to be useful for online and real-time risk assessments with relatively small samples of data.
7.3 The Predictive Nature of Bayesian Inference
7.2
149
Predictive Probabilities
In the following we will use node failures in a cluster (or a Grid) as the focus, i.e. we will work out a model to predict the probabilities that n nodes will fail in a period covered by an SLA (n = 0, 1, 2, 3, . . .). In the interest of space we have to do this by sketches as we deal with standard Bayesian theory and modeling [86]. The first step is to determine a probability distribution for the number of node failures for a time interval (t1 ,t2 ] by starting from some basic property of the process we need to describe. Typically we assume that the node failures represent a Poisson process which is non-homogenous in time and has a rate function λ (t),t > 0. The second step is to determine a distribution for λ (t) given a number of observations on node failures from r comparable segments in the interval (t1 ,t2 ]. This count normally follows a Gamma(α , β ) distribution and the posterior distribution p(λt1 ,t2 ), given the count of node failures, is also a Gamma distribution according to the Bayesian theory. Then, as we have been able to determine λt1 ,t2 we can calculate the predictive distribution for the number of node failures in the next time segment; Bayes’ theory shows that this will be a Poisson-Gamma distribution. The third step is to realize that a computing task can be carried out successfully on a cluster (or a Grid) if all the needed nodes are available for the scheduled duration of the task. This has three components: (i) a predictive distribution on the number of nodes needed for a computing task covered by an SLA; (ii) a distribution showing the number of nodes available when an assigned set of nodes is reduced with the predicted number of node failures and an available number of reserve nodes is added (the number of reserve nodes is determined by the resource allocation policy of the RP); (iii) a probability distribution for the duration of the task. The fourth step is to determine the probability of an SLA failure. We need to use a multinomial distribution to work out the combinations. Consider a Grid of k clusters, each of which contains nc nodes, leading to the total number of nodes n = ∑ nc , where c = 1, . . . , k, in the Grid. Let in the following λ (t),t > 0, denote generally a time non-homogeneous rate function for a Poisson process N(t). We will assume that we have the RP4 scenario as our context, i.e. we will have to deal with hundreds of nodes and hundreds of computing tasks with widely varying computational requirements over the planning period for which we are carrying out the risk assessment.
7.3
The Predictive Nature of Bayesian Inference
Bayesian reasoning regarding uncertainty has recently gained momentum in many seemingly unrelated fields of applied sciences, ranging from genetics [19] and developmental psychology [296] , to a bulk of applications in technology and engineering, such as aircraft navigation, computer vision, digital communication [124]. However, a vast majority of putative applications of this line of reasoning are surely yet unexplored, as scientists are successively becoming aware of the potential residing in the approach, where the cornerstone is the Bayes’ theorem [23, 203]. Even
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though the Bayes’ theorem itself has been present in standard textbooks on probability calculus for decades, the field of statistics and uncertainty reasoning was much unaffected by its existence for the most of the 20th century. The result, which in fact is a simple consequence of the basic modern axioms of probability, was earlier usually overlooked either by the fact that its applicability was considered very limited apart from some esoteric situations, or due to the subjective a priori probability judgement included in the theorem. The very fact that traditional frequentist statistical approaches based on hypothesis testing and parameter estimation do not usually provide directly the sought answers, in particular in typical technological applications, have laid the ground for the emergence of alternative, more appropriate methods for handling uncertainty. Therefore, when the awareness of powerful Monte Carlo simulation methods to solving complex issues in Bayesian computation increased in the early 1990’s, the time was ripe for adopting the Bayesian approach to handling uncertainty in many challenging applications [124, 287]. Additionally, the general awareness of the methodology has also sparked an interest in reconsidering fairly standard statistical problems, where advanced computational methods are not necessarily needed. To illustrate the predictive nature of Bayesian characterizations of uncertainty, we use a simple example model and contrast the strategy with the maximum likelihood method. Other similar approaches relying on point estimates of model parameters in predictive tasks could equally be considered. Proofs of the results from standard probability calculus and Bayesian inference can be found, e.g. in [23]. Consider a data set comprising information from six exchangeable sampling units, each of which can give raise to a number Xi of events, such as errors in coding, typing, transmission etc, for i = 1, . . . , 6. When Xi = 0, it is concluded that no errors occurred within the ith sampling unit. If it is anticipated that the number of realized errors is small in relation to the number of potentially possible errors, a Poisson distribution /indexPoisson distribution is one common characterization of the uncertainty related to the intensity at which the errors occur among comparable sampling units. Using this distribution as the likelihood component for each of the observations, we obtain the joint conditional distribution of the data as, p(x1 , ..., x6 |λ ) ∝ exp(−6λ )λ ∑i=1 xi , 6
where λ is the unknown Poisson intensity parameter. In the Bayesian framework uncertainty is handled by attempting to describe all unknown quantities of interest or use, by assigning them probability distributions, which encode existing knowledge. Then, depending on the purpose of the modeling task, various characterizations of uncertainty may be derived using the laws of probability. In our example we have a single unknown quantity, the Poisson parameter λ , for which the uncertainty is quantified prior to the arrival of the observations x1 , . . . , x6 . The standard prior used for this purpose is the Gamma(α , β ) family of distributions, which has the following density representation,
7.3 The Predictive Nature of Bayesian Inference
p(λ |α , β ) =
151
β α α −1 λ exp(−β λ ), Γ (α )
where α and β have the following relation to the distribution moments E(λ ) =
α α , E(λ − E(λ ))2 = 2 . β β
The posterior distribution of the intensity parameter λ , i.e. the conditional distribution of λ given the data x =(x1 , . . . , x6 ), is then also a Gamma distribution and its density function equals, p(λ |x) =
(β + 6)α +z α +z−1 λ exp(−(β + 6)λ ), Γ (α + z)
where z = ∑6i=1 xi is the value of the sufficient statistic for the data under the Poisson model. Under a squared error loss, the Bayesian estimate of λ equals the posterior mean, which can be written as (α + z)/(β + 6). The asymptotic behavior of this estimate as a function of the number of observations, say n, is easily characterized, because the posterior mean approaches the maximum likelihood estimate λˆ = z/n, provided that α and β are bounded, i.e. n→∞ (α + z)/(β + n) → λˆ .
In many technological applications it is not of primary interest to estimate model parameters, but rather use a statistical model for handling uncertainty in a given situation. For instance, in our example it might be of interest to provide a probability statement regarding whether a future number of events Xn+1 would exceed a certain threshold, deemed crucial for the acceptability of the error rate in the system. The Bayesian answer to this question is the predictive distribution of the anticipated future observations, given the knowledge accumulated so far. Formally, this distribution is defined as p(x7 |x1 , ..., x6 ) =
∞ 0
p(x7 |λ )p(λ |x1 , ..., x6 )d λ ,
(7.2)
where p(λ |x1 , ..., x6 ) is the posterior distribution α
p(λ |x1 , ..., x6 ) ∝ ∞ 0
∝ ∞ 0
exp(−6λ )λ y Γβ(α ) λ α −1 exp(−β λ ) α
exp(−6λ )λ y Γβ(α ) λ α −1 exp(−β λ )d λ
(7.3)
exp(−6λ )λ y λ α −1 exp(−β λ ) , exp(−6λ )λ y λ α −1 exp(−β λ )d λ
which is recognized as the Gamma(α + z, β + 6) distribution. By evaluating the integral (7.2) with respect to the above Gamma distribution, we obtain a special case of a Poisson-Gamma distribution, which is a mixture of Poisson distributions.
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The Poisson-Gamma distribution has in this case the probability mass function, p(x7 |x1 , ..., x6 ) =
(β + 6)α +z Γ (α + z + x 7 ) . Γ (α + z) x7 !(β + 6 + 1)α +z+x7
It is important to notice that the Poisson-Gamma mixture has higher variance than the Poisson distribution, which reflects the uncertainty remaining about the parameter λ after observing the data x. This can be interpreted as an honest quantification of the predictive uncertainty given the axioms of probability, and the judgement that Poisson model provides an appropriate description of the event probabilities. Also, it shows that the amount of uncertainty regarding future events is underestimated, if they are quantified by using the ordinary Poisson distribution, either with a Bayesian or maximum likelihood estimate of λ . However, when the length of the data vector x increases towards infinity, the Poisson-Gamma mixture tends to the ordinary Poisson distribution, which illustrates how the additional uncertainty stemming from the unknown parameter λ vanishes as the amount of data available for statistical learning increases. Assuming the observed error rates equal to x = (3, 4, 2, 1, 2, 3) and β = .001, α = 1/2, in our Poisson example, a concrete comparison can be made between the Bayesian predictive distribution and the distribution obtained by replacing λ with the maximum likelihood estimate 15/6 = 2.5.
7.4
Predictive Probabilities in Grid Computing Management
Dynamically gathered observations of events within a grid computing system enable updating of the knowledge concerning the system reliability and resource availability. It is expected that resource management should optimally be based on the characteristics learned from the particular grid under consideration. Moreover, as a grid may undergo substantial technological and operational changes during its life cycle, the actual management procedures should be allowed to continuously adapt to the eventual changes in system. By monitoring grid behavior and feeding the appropriate information to a statistical model, it is possible to better anticipate future behavior and manage risks associated with component failures. It is worth noticing that such a dynamic perspective on information processing is in a perfect harmony with the Bayesian predictive framework, which updates knowledge about probabilities related to future events every time relevant new information is made available. The evident stochasticity of a grid, concerning the availability of a particular resource at some point in time, suggests that probability-based decisions are most appropriate in a grid environment. Using the predictive probabilistic approach, our aim here is to develop a framework for resource management in grid computing. The works cited earlier illustrate the substantial level of variation that exists over different grids with respect to the degree of exploitation, maintenance policies and system reliability. Several factors are of importance when the uncertainty about the expected amount of resources available at a given time point is considered. In the current work we consider explicitly the effects of computing task characteristics in terms of execution time and the
7.4 Predictive Probabilities in Grid Computing Management
153
amount of resources required, as well as failure rate and maintenance for individual system components. A basic predictive model with a modular structure is derived, such that several ramifications with respect to the distributional assumptions can be made when deemed necessary for any particular grid environment. Consider a grid of k clusters, each of which contains nc operative units termed nodes, with the total number of nodes in the grid denoted by n = ∑kc=1 nc . For simplicity of notation, we will below treat the clusters as exchangeable units concerning predictive statistical learning. However, if such an assumption is deemed implausible for a particular environment, the derived model can be applied separately to the individual clusters for statistical learning. To introduce a stochastic model for tendencies of node failures, let λ (t) > 0, t ≥ 0, denote generally the rate function for a Poisson process N(t). The rate function specifies the expected number of failure events in any given time interval (t1 ,t2 ] according to,
λt1 ,t2 =
t2
t1
λ (t)dt,
and the probability distribution for the number of events X = N(t2 ) − N(t1 ) is given by the Poisson distribution, p(X = x) =
e−λt1 ,t2 (λt1 ,t2 )x , x = 0, 1, . . . x!
Under many circumstances it is feasible to simplify the Poisson model by assuming that the rate function is constant over time, or over certain time segments. In the Poisson process it assumed that the waiting times between successive events are exponentially distributed with the rate parameter λ governing the expected waiting times. Consider a time interval (t1 ,t2 ] with the corresponding rate parameter λt1 ,t2 , such that x events have been observed during (t1 ,t2 ]. Then, the likelihood function for the rate parameter is given by, p(x|λt1 ,t2 ) =
e−λt1 ,t2 (λt1 ,t2 )x . x!
By collecting data (counts of events) x1 , . . . , xr from r comparable time segments, we obtain the joint likelihood function p(x1 , ..., xr |λt1 ,t2 ) ∝ e−rλt1 ,t2 (λt1 ,t2 )∑i=1 xi . r
(7.4)
In accordance with the predictive example considered in the previous section, using the Gamma prior distribution for λt1 ,t2 we then obtain the predictive Poisson-Gamma probability of having y events in future on a comparable time interval, which equals (β + r)α +z Γ (α + z + y) , Γ (α + z) y!(β + r + 1)α +z+y where z = ∑ri=1 xi .
(7.5)
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When a computing task is slated for execution in a grid environment, its successful completion will require a certain number of nodes to be available over a given period of time. To assess the uncertainty about the resource availability, we need to model both the distribution of the number of nodes and the length of the task required by the tasks. The most adaptable choice of a distribution for the number of nodes required, say M, is the multinomial distribution p(M = m) = pm , m = 1, ..., u,
(7.6)
where u is an a priori specified upper limit for the number of nodes. Such a vector of probabilities will be denoted by p in the sequel. Here u could, e.g., equal the total number of nodes in the system, however, such a choice would lead to an inefficient estimation of the probabilities, and therefore, the upper bound value should be carefully assessed using empirical evidence. An advantage of the use of the multinomial distribution in this context is its ability to represent any types of multimodal distributions for M, in contrast to the standard parametric families, such as the Geometric, Negative Binomial and Poisson distributions. For instance, if there are two major classes of computing tasks or maintenance events, such that one class is associated with relatively small numbers of required nodes, and the other with relatively large numbers, the system behavior in this respect is well representable by a multinomial distribution. On the other hand, standard parametric families of distributions would not enable an appropriate representation, unless some form of a mixture distribution were utilized. Such a choice would complicate the inference about the underlying parameters due to the fact that the number of mixture components would be unknown a priori. In general, estimation of parameters and calculation of predictive probabilities under a such mixture model requires the use of a Monte Carlo simulation technique, e.g. the EMor Gibbs sampler algorithm [287]. A disadvantage of the multinomial distribution is that it contains a large number of parameters when u is large. However, this difficulty is less severe when the Bayesian approach to the parameter estimation is adopted. Given observed data on the number of nodes required by computing tasks, the posterior distribution of the probabilities p is available in an analytical form under a Dirichlet prior, and its density function can be written as p(p|w) =
Γ (∑um=1 αm + wm ) u αm +wm −1 , ∏ pm ∏um=1 Γ (αm + wm ) m=1
where wm corresponds to the number of observed tasks utilizing m nodes, αm is the a priori relative weight of the mth component in the vector p, and w is the vector (wm )um=1 . The corresponding predictive distribution of the number of nodes required by a generic computing task in the future equals the Multinomial-Dirichlet distribution, which is obtained by integrating out the uncertainty about the multinomial parameters with respect to the posterior distribution. The Multinomial-Dirichlet distribution is in our notation defined as
7.4 Predictive Probabilities in Grid Computing Management
Γ (∑um=1 αm + wm ) ∏um=1 Γ (αm + wm + I(m = m∗ )) Γ (1 + ∑um=1 αm + wm ) ∏um=1 Γ (αm + wm ) u Γ (∑m=1 αm + wm ) Γ (αm∗ + wm∗ + 1) = . Γ (αm∗ + wm∗ ) Γ (1 + ∑um=1 αm + wm )
p(M = m∗ |w) =
155
(7.7)
To simplify the inference about the length of a task affecting a number of nodes we assume that the length follows a Gaussian distribution with expected value μ and variance σ 2 . Let the data t1 , ...,tb represent the lengths (say, in minutes) of b tasks. This leads to the sample mean t¯ = ∑bi=1 ti and variance s2 = b−1 ∑bi=1 (ti − t¯)2 . Assuming the standard reference prior [23] for the parameters, we obtain the predictive distribution for the length of a future task, say T , which has the T-distribution with parameters t¯, ((b −1)/(b +1))s2 , b − 1, i.e. the probability density of the distribution equals
p(t|t¯, ((b − 1)/(b + 1))s2, b − 1) =
1 2 Γ (b/2) 1 × 2 b−1 Γ ( 2 )Γ (1/2) (b + 1)s − b 2 1 2 1+ (t − t¯) . 2 b + 1)s
(7.8)
The probability that a task lasts longer than any given time t equals P(T > t) = 1 − P(T ≤ t), where P(T ≤ t) is the cumulative distribution function (CDF) of the T-distribution. The value of the CDF can be calculated numerically using functions existing in most computing environments. However, it should also be noted that for a moderate to large b, the predictive distribution is well approximated by the Gaussian distribution with the mean t¯ and the variance s2 (b+1) (b−3) . Consequently, if the Gaussian approximation is used, the probability P(T ≤ t) can be calculated using the cumulative density function (CDF) of the Gaussian distribution. We now consider an approximation to the probability that a particular computing task is successful. This happens if there will always be at least a single idle node available in the system in the case of a node failure. Let S = 1 denote the event that the task is successful, and S = 0 the opposite event. We formulate the probability of the success as the sum of the probabilities P(”none of the nodes allocated to the task max fail”) + ∑m m=1 P(”m of the nodes allocated to the task fail & at least m idle nodes are available as reserves”). Here mmax is an upper limit for the number of failures considered. The value can be chosen by judging the size of the contribution of each event, determined by the corresponding probability. Thus, the sum can be simplified by considering only those events that do not have vanishingly small probabilities. A conservative bound for the success probability can be derived by assuming that the m failures take place simultaneously, which leads to
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P(S = 1) = 1 − P(S = 0) = 1− = 1− ≥ 1−
(7.9)
mmax
∑ P(m failures occur & less than m free nodes available)
m=1 mmax
∑ P(m failures occur)P(less than m free nodes available)
m=1 mmax
∑ P(m failures occur)P(less than m free nodes at any time point)
m=1
Fig. 7.1 Prediction of node failures [Probability of Failure: 0.1509012] with 136 time slots of LANL cluster data for computing tasks running for 9 days, 0 hours
The probability P(m failures occur) is directly determined by the failure rate model discussed above. The other term, the probability P(less than m free nodes at any time point), on the other hand, is dependent both on the level of workload in the system and the eventual need of reserve nodes by the other tasks running simultaneously. Thus, the failure rate model can be used to calculate the probability distribution of the number of reserve nodes that will be jointly needed by the other tasks (that are using a certain total number of nodes) during the computation time that has the distribution specified above for a single node.
7.5
A Hybrid Approach to Computing a Measure of Success
Grid computing is a form of distributed computing where clusters of networked, loosely-coupled computers, are acting in concert to perform very large and/or a large number of tasks [148]. However, running applications on the Grid environment poses significant challenges due to the diverse failures encountered during execution. If one computing node fails during the job execution, the whole job fails and has to be restarted. To handle resource failures and avoid restarting the job from the initial state, the fault-tolerance mechanisms checkpointing and migration have
7.5 A Hybrid Approach to Computing a Measure of Success
157
Fig. 7.2 Probability that less than m nodes will be available for computing tasks requiring < 10 nodes; the number of nodes randomly selected; 75 tasks simulated over 236 time slots; LANL
been developed. The Resource Management System makes a snapshot of the job’s execution state, transfers the snapshot to another computing node, and resumes the job execution after all nodes failed have been replaced with nodes that are working [324]. Users negotiate for resource usage through a Grid resource broker which queries resource providers on their behalf to find suitable resources. They require a job execution with a desired level of priority and quality [121]. Failure intensity usually increases with age for mechanical equipment. Power law and loglinear Poisson processes are often used to model failure intensity. The distinguishing feature of Poisson processes is that the previous history of failure times t1 , . . . , tn does not affect failure intensity [16]. Let us suppose that the number of all nodes in a grid is equal to Z. Let N be an upper limit for the number of failures for all classes and at any time during the simulations (this number is estimated by the broker). We will suppose that N = mmax . Let M be the most possible value for the number of failures. Let S∗ = 1 denote the event that the task is successful. Let Q be the possibility distribution for maximal number of failures. That is, Q(m) is interpreted as the degree of possibility of the statement that the maximal number of failures is equal to m.
Fig. 7.3 A linear possibility distribution for maximal number of failures
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7 Risk Assessment in Grid Computing
We use the notation
Π (Q = m) = Q(m). That is, Q(m) denotes the degree of possibility to which m is considered to be the maximal number of failures. Let Ω be a finite set. We recall that a function Π : Ω → [0, 1] is said to be a possibility measure on Ω if Π (∅) = 0, Π (Ω ) = 1, Π (U ∪V ) = max{Π (U ), Π (V )} for any U,V ⊂ Ω [18]. It follows that the possibility measure on finite set is determined by its behavior on singletons:
Π (U ) = max Π ({ω }). ω ∈U
Now we compute the probability of success (7.9) taking into consideration the possibility distribution for maximal number of failures derived from the broker’s observations, P(S∗ = 1) = N
∑ Π (Q = m)P(m failures occur & less than m free nodes available)
= 1− = 1− ≥ 1− = 1−
m=1 N
∑ Π (Q = m)P(m failures occur)P(less-than-m-anytime)
m=1 N
∑ Π (Q = m)P(m failures occur)P(less-than-m-anytime)
m=1 M
∑ P(m failures occur)P(less-than-m-anytime)
m=1
−
N
∑
Π (Q = m)P(m failures occur)P(less-than-m-anytime)
m=M+1
Here less-than-m-anytime stands for the event less than m free nodes available at any time point. Here, for simplicity we used the same notation P for the hybrid measure for success. For example, if Q is linear ⎧ 1 if m ≤ M ⎪ ⎪ ⎪ ⎨ m−M Qlinear (m) = 1 − if M ≤ m ≤ N ⎪ N−M ⎪ ⎪ ⎩ 0 if N ≤ m ≤ Z then using the equality, Qlinear (m) = 1 −
m−M N −m = N −M N −M
7.5 A Hybrid Approach to Computing a Measure of Success
159
we get M
P(S∗ = 1) = 1 −
∑ P(m failures occur)P(less-than-m-anytime)
m=1
N −m − ∑ × P(m failures occur)P(less-than-m-anytime) m=M+1 N − M N
Since N
P(S = 1) = 1 −
∑ P(m failures occur)P(less-than-m-anytime)
m=1
we get P(S∗ = 1) ≥ P(S = 1). Fig. 7.4 A quadratic possibility distribution for maximal number of failures
Depending on the observations, the broker could also use a quadratic function for the possibility distribution of maximal number of failures.
Qquadratic (m) =
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
1 N−m N −M 0
2
if m ≤ M if M ≤ m ≤ N if N ≤ m ≤ Z
In this case, the we get P(S∗ = 1) = 1 − −
N
∑
m=M+1
M
∑ P(m failures occur)P(less-than-m-anytime)
m=1
N −m N −M
2 × P(m failures occur)P(less-than-m-anytime)
The probability of success with a quadratic possibility distribution is bigger than in the linear case. It follows from the relationship Qquadratic (m) < Qlinear (m) if m > M. That is, the quadratic possibility, Qquadratic (m), that m is the maximal number of failures diminishing more quickly than the linear possibility, Qlinear (m), that that m is the maximal number of failures for any m > M.
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Note 7.1. We presented a hybrid probabilistic and possibilistic technique for assessing the success of a computing task in a grid environment. The probability of success in a hybrid environment is bigger than in the pure probabilistic environment since the hybrid approach takes into consideration the possibility distribution for maximal number of failures derived from the broker’s observations.
7.6
Predictive Possibilities
In this Section we will introduce a possibilistic method for simple predictive estimates of node failures in the next planning period. We will interpret the new model as an alternative for predicting the number of node failures. In this way we will have a possibilistic estimates of the expected number of node failures. An RP may use either one estimate for his risk assessment or use a combination of both. In the AssessGrid project we have implemented both models as one software solution to give the user two alternative routes. We will now sketch the possibilistic regression model which is a simplified version of the fuzzy nonparametric regression model with crisp input and fuzzy number output introduced by [331]. This is essentially a standard regression model with parameters represented by triangular fuzzy numbers - typically this means that the parameters are intervals and represent the fact that the information we have is imprecise and/or uncertain. Recall that a fuzzy number A is a fuzzy set of the real line R with a normal, fuzzy convex and continuous membership function of bounded support. Fuzzy numbers can be considered as possibility distributions. If A is a fuzzy number and x ∈ R then A(x) can be interpreted as the degree of possibility of the statement ”x is A”. A fuzzy number A is said to be a triangular fuzzy number, denoted A = (a, α , β ), with center a, left width a − α > 0 and right width β − a > 0 if its membership function is defined by A(t) = (t − α )/(a − α ) if α ≤ t ≤ a, A(t) = (t − β )/(a − β ) if a < t ≤ β , and A(t) = 0 otherwise. If A is symmetrical, α = β , then we use the notation A = (a, α ). A triangular fuzzy number with center a may be seen as a fuzzy quantity ”x is approximately equal to a”. Let F denote the set of all triangular fuzzy numbers. We consider the following univariate fuzzy nonparametric regression model, Y = F(x) + ε = (a(x), α (x), β (x)) + ε .
(7.10)
where x is a crisp independent variable (input) with domain D, where D ⊂ R. The output Y ∈ F is a triangular fuzzy variable, F : D → F is an unknown fuzzy regression function with its center, lower and upper limit a(x), α (x), β (x), and ε may also be considered a fuzzy error or a hybrid error containing both fuzzy and random components. We will take a sample of a dataset (in our case, a sample from the LANL dataset) which covers inputs and fuzzy outputs according to the regression model. Let xi ,Yi be a sample of the observed crisp inputs and fuzzy outputs of model (7.1). The main goal of fuzzy nonparametric regression is to estimate F(x) at any x ∈ D from (xi ,Yi ),
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i = 1, 2, . . . , n. And the membership function of an estimated fuzzy output should be as close as possible to that of the corresponding observed fuzzy number [331]. From this point of view, we shall estimate a(x), α (x), β (x) for each x ∈ D in the sense of best fit with respect to some distance that can measure the closeness between the membership functions of the estimated fuzzy output and the corresponding observed one. There are actually a few such distances available. We use the distance proposed by [123] as a measure of the fit because it is simple to use and, more importantly, can result in an explicit expression for the estimated fuzzy regression function when the local linear smoothing technique is used. Let A = (a, α1 , β1 ), B = (b, α2 , β2 ) be two triangular fuzzy numbers. Then the squared distance between A and B is defined by [123], d 2 (A, B) = (α1 − α2 )2 + (a − b)2 + (β1 − β2 )2 . Using this distance we will extend the local linear smoothing technique in statistics to fit the fuzzy nonparametric model (7.10). Let us now assume that the observed (fuzzy) output is Yi = (a, αi , βi ), then with the Diamond distance measure and a local linear smoothing technique we need to solve a locally weighted least-squares problem in order to estimate F(x0 ), for a given kernel K and a smoothing parameter h, where |xi − x0 | Kh (|xi − x0 |) = K , (7.11) h The kernel is a sequence of weights at x0 to make sure that data that is close to x0 will contribute more when we estimate the parameters at x0 than those that are farther away, i.e. are relatively more distant in terms of the parameter h. Let Fˆ(i) (xi , h) = (aˆ(i) (x0 , h), αˆ (i) (x0 , h), βˆ(i) (x0 , h)).
(7.12)
be the predicted fuzzy regression function at input xi . Compute Fˆ(i) (xi , h) for each xi and let CV (h) =
1 l 2 ∑ d (Yi , Fˆ(i) (xi , h)). l i=1
(7.13)
We should select h0 so that it is optimal in the following expression CV (h0 ) = min CV (h). h>0
(7.14)
By solving this minimalization problem, we get the following estimate of F(x) at x0 (see Wang et al. [331] for details) , ˆ 0 ) = (a(x F(x ˆ 0 ), αˆ (x0 ), βˆ (x0 )) = (eT1 H(x0 , h)aY , eT1 H(x0 , h)αY , eT1 H(x0 , h)βY ), (7.15) where H(x0 , h) = (XT (x0 )W)T (x0 , h)X(x0 ))−1 XT (x0 )W(x0 , h),
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⎛
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 x1 − x0 a1 α1 β1 ⎜1 x2 − x0 ⎟ ⎜ a2 ⎟ ⎜α2 ⎟ ⎜β2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜. ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ aY = ⎜ . ⎟ αY = ⎜ . ⎟ βY = ⎜ . ⎟ X(x0 ) = ⎜ ⎜. ⎟ ⎜.⎟ ⎜.⎟ ⎜.⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝. ⎠ ⎝.⎠ ⎝.⎠ ⎝.⎠ 1 xn − x0 an αn βn and W(x0 , h) = Diag(Kh (|x1 − x0 |), . . . , Kh (|xn − x0|)) and e1 = (1, 0)T . Note 7.2. If the observed inputs are symmetric, in a way that, ai − αi = βi − ai , then ˆ 0 ) is also a symmetric triangular fuzzy number. In fact, if the ith symmetric fuzzy F(x output is denoted by Yi = (ai , ci ), where ci = ai − αi = βi − ai denotes the spread of Yi , then the spread vector of the n observed fuzzy outputs can be expressed as cY = aY − αY = βY − aY . According to (7.15) we get, a(x ˆ 0 ) − αˆ (x0 ) = βˆ (x0 ) − a(x ˆ 0) = ˆ 0 ) = (eT1 H(x0 , h)aY , eT1 H(x0 , h)cY ) is a symmetric eT1 H(x0 , h)cY and, therefore, F(x triangular fuzzy number.
7.7
Predictive Possibilities in Grid Computing Management
Suppose there are n observations, {x1 , . . . , xn }, where xi denotes the number of node failures in the ith planning period, and l of them are different. Without loss of generality we will use the notation {x1 , . . . , xl } for the l different observations. Let Xn+1 denote the (unknown) possibility distribution of the number of node failures in the next planning period. Then Xn+1 is defined over the set of non-negative integers {0, 1, 2, 3, . . .}. It is clear that from the first n observations we can not predict the exact value of the (n+1)th observation, but using a fuzzy nonparametric regression technique we can estimate the possibility of the statement ”the number of node failures in the (n+1)th planning period will be x0 ” where x0 is a non-negative integer. Let us introduce a notation vi = | { j : the jth observation is equal to xi } |. In our case the center of the symmetric triangular fuzzy number will be ai = xi , and its lower and upper limits are, vi vi αi = xi − , βi = xi + . n n Since we work with the symmetrical case, we shall use the notation ci = vi /n. We should choose a kernel function and h. Let h = 1/n and let 1 x2 K(x) = √ exp − , 2 2π the Gaussian kernel. Let ki = Kh (|xn − x0 |)). We calculate now the matrix eT1 H(x0 , h) by, ∑li=1 ki ∑li=1 ki (xi − x0 ) (XT (x0 )W)T (x0 , h)X(x0 ) = =⇒ ∑li=1 ki (xi − x0 ) ∑li=1 ki (xi − x0 )2
7.7 Predictive Possibilities in Grid Computing Management
(XT (x0 )W)T (x0 , h)X(x0 ))−1 =
163
1
×
(∑li=1 ki )(∑li=1 ki (xi − x0 )2 ) − (∑li=1 ki (xi − x0 ))2 l ∑i=1 ki (xi − x0 )2 − ∑li=1 ki (xi − x0 ) − ∑li=1 ki (xi − x0 ) ∑li=1 ki
Let s = ∑li=1 ki , t = ∑li=1 ki (xi − x0 ) and v = ∑li=1 ki (xi − x0 )2 . Then, v − t(x1 − x0 ) . . . v − t(xl − x0 ) (XT (x0 )W)T (x0 , h)X(x0 ))−1 XT (x0 ) = s(x1 − x0 ) − t . . . s(xl − x0 ) − t and H(x0 , h) = (XT (x0 )W)T (x0 , h)X(x0 ))−1 XT (x0 )W(x0 , h) k1 (v − t(x1 − x0)) . . . kl (v − t(xl − x0 )) = , k1 (s(x1 − x0 ) − t) . . . kl (s(xl − x0 ) − t) and eT1 H(x0 , h) = k1 (v − t(x1 − x0 )) . . . kl (v − t(xl − x0 )) , and finally, l
l
eT1 H(x0 , h)aY = ∑ xi ki (v − t(xi − x0 )),
eT1 H(x0 , h)cY = ∑ ci ki (v − t(xi − x0 )).
i=1
i=1
So we can write our estimation as, ˆ 0 ) = (eT1 H(x0 , h)aY , eT1 H(x0 , h)cY ) = F(x
l
l
i=1
i=1
∑ xi ki (v − t(xi − x0 )), ∑ ci ki (v − t(xi − x0 ))
,
which is the predicted possibility distribution of symmetric triangular form of node failures. So, the estimate of the possibility that ”the number of node failures in (n + 1)th planning period will be x0 ”, denoted by Pos(Xn+1 = x0 ), is computed by, Pos(Xn+1 = x0 ) = ⎧ l l ⎪ ⎨ 1 − | ∑i=1 xi ki (v − t(xi − x0 )) − x0 |, if | ∑i=1 xi ki (v − t(xi − x0 )) − x0 | ≤ 1 l ∑i=1 ci ki (v − t(xi − x0 )) ∑li=1 ci ki (v − t(xi − x0 )) ⎪ ⎩ 0 otherwise. The possibilistic model is a faster and more robust estimate (that is the possibility of a node failure always exceeds the probability of a node failure) and will therefore be useful for online and real-time risk assessments with relatively small samples of data. Now we can use the new model as an alternative for predicting the number of node failures and use it as part of the Bayesian model for predictive probabilities. In this way we will have hybrid estimates of the expected number of node failures - both probabilistic and possibilistic estimates. An RP may use either one estimate for his risk assessment or use a combination of both. We carried out a number of validation tests in order to find out (i) how well the predictive possibilistic models
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can be fitted to the LANL dataset, (ii) what differences can be found between the probabilistic and possibilistic predictions and (iii) if these differences can be given reasonable explanations. The testing was structured as follows: − 5 time frames for the possibilistic predictive model with a smoothing parameter from the smoothing function: h = 382.54 × Nr of timeslots−0.5325 ; − 5 feasibility adjustments from the hybrid possibilistic adjustment model to the probabilistic predictive model. In the testing we worked with short and long duration computing tasks scheduled on a varying number of nodes and the SLA probabilities of failure estimates remained reasonable throughout the testing. The smoothing parameter h for the
Fig. 7.5 Possibilistic and probabilistic prediction of n node failures for a computing task with a duration of 8 days on 10 nodes; 153 time slots simulated
Fig. 7.6 Comparison of probabilistic and possibilistic success for an SLA for computing tasks on a 6 node cluster with two spare nodes; simulated for 60-19500 minutes
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possibilistic models should be determined for the actual cluster of nodes, and in such a way that we get a good fit between the probabilistic and the possibilistic models. The approach we used for the testing was to experiment with combinations of h and then to fit a distribution to the results; the distribution could then be used for interpolation. We run the tests for all the RP1-RP4 scenarios [86] but here we are showing only results for the RP1 scenario (cf. fig. 7.5-7.6) in the interest of space. In Fig 7.5 we show the differences between probabilistic and possibilistic predictions of having n node failures for a computing task of 8 days on 10 nodes. The reason for the different results is that LANL data covers relatively few computing tasks running for 8 days which means that there are few observations of this type of computing tasks for the regression models. Fig. 7.6 shows the probability and the possibility of SLA success with 1-2 reserve nodes for computing tasks requiring 6 nodes.
Chapter 8
Knowledge Mobilization
Fuzzy ontologies have been proposed as a solution for addressing semantic meaning in an uncertain and inconsistent world. As with fuzzy logic, reasoning is approximate rather than precise. The aim is to avoid the theoretic pitfalls of monolithic ontologies, facilitate interoperability between different and independent ontologies [112], and provide flexible information retrieval capabilities. The Knowledge Mobilization project (KNOWMOBILE) has been a joint effort by Institute for Advanced ˚ Akademi University and VTT Technical ReManagement Systems Research, Abo search Centre of Finland. Its goal was to better ”mobilize” knowledge stored in heterogeneous databases for users with various backgrounds, geographical locations and situations. The working hypothesis of the project was that fuzzy mathematics combined with domain-specific data models, in other words, fuzzy ontologies, would help manage the uncertainty in finding information that matches the user’s needs. In this way, KNOWMOBILE places itself in the domain of knowledge management. In this Chapter, drawing heavily on Tommila, Hirvonen and Pakonen [313]; Hirvonen et al. [185]; and Carlsson, Full´er and Fedrizzi [95] we will describe an industrial demonstration of fuzzy ontologies in information retrieval in the paper industry where problem solving reports are annotated with keywords and then stored in a database for later use. Furthermore, using Bellmann-Zadeh’s principle to fuzzy decision-making we will show a method for identifying keyword dependencies in the keyword taxonomic tree. In the KNOWMOBILE project, we developed a concept of a tool for searching plant knowledge with a search engine based on a fuzzy ontology. The usage scenario for the tool was that a process expert, dealing with a problem in the process chemistry of a paper machine, wishes to find past problem solving cases of a similar setting in order to find possible solutions to a current issue. This setting is a universal one: pieces of knowledge, called ”nuggets”, are written and stored by companies on different domains in the form of incident reports. C. Carlsson and R. Full´er: Possibility for Decision, STUDFUZZ 270, pp. 167–183. springerlink.com © Springer-Verlag Berlin Heidelberg 2011
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8 Knowledge Mobilization
Ontologies
Ontology is a conceptualization of a domain into a human understandable, machinereadable format consisting of entities, attributes, relationships, and axioms [171]. An ontology can be considered as a formal representation of a set of concepts within a domain and the relationships between those concepts. Ontologies are used in artificial intelligence, the Semantic Web, software engineering, biomedical informatics, library science, and information architecture as a form of knowledge representation. The most important components of ontologies are, • Classes - concepts that are also called type, sort, category, and kind - can be defined as an extension or an intension. According to an extensional definition, they are abstract groups, sets, or collections of objects. According to an intensional definition, they are abstract objects that are defined by values of aspects that are constraints for being member of the class. The first definition of class results in ontologies in which a class is a subclass of collection. The second definition of class results in ontologies in which collections and classes are more fundamentally different. Classes may classify individuals, other classes, or a combination of both. • Individuals are the basic - ground level - components of an ontology. The individuals in an ontology may include concrete objects such as people, animals, tables, automobiles, molecules, and planets, as well as abstract individuals such as numbers and words (although there are differences of opinion as to whether numbers and words are classes or individuals). Strictly speaking, an ontology need not include any individuals, but one of the general purposes of an ontology is to provide a means of classifying individuals, even if those individuals are not explicitly part of the ontology. • Attributes - aspects, properties, features, characteristics, or parameters that objects (and classes) can have. Objects in an ontology can be described by relating them to other things, typically aspects or parts. These related things are often called attributes, although they may be independent things. Each attribute can be a class or an individual. The kind of object and the kind of attribute determine the kind of relation between them. A relation between an object and an attribute express a fact that is specific to the object to which it is related. • Relations between objects in an ontology specify how objects are related to other objects. Typically a relation is of a particular type that specifies in what sense the object is related to the other object in the ontology. Much of the power of ontologies comes from the ability to describe relations. Together, the set of relations describes the semantics of the domain. The set of used relation types (classes of relations) and their subsumption hierarchy describe the expression power of the language in which the ontology is expressed. The most important type of relation is the subsumption relation – is-a-superclass-of, – the converse of is-a,
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– is-a-subtype-of – is-a-subclass-of The addition of the is-a-subclass-of relationships creates a hierarchical taxonomy; a tree-like structure (or, more generally, a partially ordered set) that clearly depicts how objects relate to one another. In such a structure, each object is the ’child’ of a ’parent class’. As well as the standard is-a-subclass-of and is-by-definition-a-part-of-a relations, ontologies often include additional types of relations that further refine the semantics they model, for example, – – – – –
relation types for relations between classes relation types for relations between individuals relation types for relations between an individual and a class relation types for relations between a single object and a collection relation types for relations between collections
• Function terms: complex structures formed from certain relations that can be used in place of an individual term in a statement. • Restrictions: formally stated descriptions of what must be true in order for some assertion to be accepted as input • Rules: statements in the form of an if-then (antecedent-consequent) sentence that describe the logical inferences that can be drawn from an assertion in a particular form. • Axioms: assertions (including rules) in a logical form that together comprise the overall theory that the ontology describes in its domain of application. It should be noted that this definition differs from that of ”axioms” in generative grammar and formal logic. In these disciplines, axioms include only statements asserted as a priori knowledge. Axioms also include the theory derived from axiomatic statements. • Events: the changing of attributes or relations. Ontologies are commonly encoded using ontology languages. Ontology languages are formal languages used to construct ontologies. They allow the encoding of knowledge about specific domains and often include reasoning rules that support the processing of that knowledge. The Web Ontology Language (OWL) is a family of knowledge representation languages for authoring ontologies, and is endorsed by the World Wide Web Consortium. An OWL ontology consists of a set of axioms which place constraints on sets of individuals (called ”classes”) and the types of relationships permitted between them. These axioms provide semantics by allowing systems to infer additional information based on the data explicitly provided.
8.2
Description Logics
Description Logics (DLs) are the basis of several ontology languages. The current standard for ontology representation is OWL (Web Ontology Language), which
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comprises three sublanguages (OWL Lite, OWL DL and OWL Full). OWL 2 is a recent extension which is currently being considered for standardization. The logical counterparts of OWL Lite, OWL DL and OWL 2 are the DLs SHIF(D), SHOIN(D), and SROIQ(D), respectively [25]. That is, description logics are a family of logics that are decidable fragments of first-order logic. The semantics for OWL is given through translation to a particular DL. Therefore OWL is both a syntax for describing and exchanging ontologies, and has a formally defined semantics that gives the meaning. For example, OWL DL corresponds to the SHOIN(D) description logic, while OWL 2 corresponds to the SROIQ(D) logic. In addition, sound, complete, and terminating reasoners (i.e. systems which are guaranteed to derive every consequence of the knowledge in an ontology) exist for many DLs including those underlying OWL. Description logic has become a cornerstone of the Semantic Web for its use in the design of ontologies. In description logics, concept names are regarded as atomic concepts, role names are regarded as atomic roles. Fig. 8.1 Syntax for DL: Concepts [307]
The syntax of a member of the description logic family is characterized by its recursive definition, in which the constructors that can be used to form concept terms are stated. Some common constructors include logical constructors in first-order logic such as intersection or conjunction of concepts, union or disjunction of concepts, negation or complement of concepts, value restriction (universal restriction), existential restriction, etc. Other constructors may also include restrictions on roles which are usual for binary relations, for example, inverse, transitivity, functionality, etc. Especially for intersection and union, description logics use the symbols and to distinguish them from the first-order logic and and or. The semantics of description logics is defined by interpreting concepts as sets of individuals and roles
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Fig. 8.2 Syntax for DL: Ontology [307]
Fig. 8.3 Syntax for DL: Roles [307]
as sets of pairs of individuals. Fuzzy description logic combines fuzzy logic with Description Logics. Since many concepts that are needed for intelligent systems lack well defined boundaries, or precisely defined criteria of membership, we need fuzzy logic to deal with notions of vagueness and imprecision. This offers a motivation for a generalization of description logics towards dealing with imprecise and vague concepts.
8.3
The Knowledge Mobilization Project
Fuzzy ontologies have been proposed as a solution for addressing semantic meaning in an uncertain and inconsistent world. As with fuzzy logic, reasoning is approximate rather than precise. The aim is to avoid the theoretic pitfalls of monolithic ontologies, facilitate interoperability between different and independent ontologies [112], and provide flexible information retrieval capabilities. The Knowledge Mobilization project (KNOWMOBILE) has been a joint effort by Institute
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˚ Akademi University and VTT for Advanced Management Systems Research, Abo Technical Research Centre of Finland. Its goal was to better ”mobilize” knowledge stored in heterogeneous databases for users with various backgrounds, geographical locations and situations. The working hypothesis of the project was that fuzzy mathematics combined with domain-specific data models, in other words, fuzzy ontologies, would help manage the uncertainty in finding information that matches the user’s needs. In this way, KNOWMOBILE places itself in the domain of knowledge management. Drawing heavily on Tommila, Hirvonen and Pakonen [313]; Hirvonen et al. [185]; and Carlsson, Full´er and Fedrizzi [95] we will describe an industrial demonstration of fuzzy ontologies in information retrieval in the paper industry where problem solving reports are annotated with keywords and then stored in a database for later use. Furthermore, using Bellmann-Zadeh’s principle to fuzzy decision-making we will show a method for identifying keyword dependencies in the keyword taxonomic tree. In the KNOWMOBILE project, we developed a concept of a tool for searching plant knowledge with a search engine based on a fuzzy ontology. The usage scenario for the tool was that a process expert, dealing with a problem in the process chemistry of a paper machine, wishes to find past problem solving cases of a similar setting in order to find possible solutions to a current issue. This setting is a universal one: pieces of knowledge, called ”nuggets”, are written and stored by companies on different domains in the form of incident reports.
Fig. 8.4 System concept-a knowledge base of event reports [185]
In the KNOWMOBILE project we have focused on the chemistry of the ”wet end” in order to limit the work effort needed to construct the domain ontology and concentrate on a subject on which domain expertise and actual data were available. Nuggets are documents than can contain all kinds of raw data or multimedia extracted from different information systems. An expert author annotates the nuggets with suitable keywords, and it is these keywords that the search is then
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based on. In addition to providing exact results to queries, the tool uses a fuzzy domain ontology to extend the query to related keywords (see Figure 8.4). As a result, the search results include nuggets that may not necessary deal with exactly the same process equipment, variable, function or chemical, but nuggets that may still provide valuable insight to solving the problem at hand. The terms defined in the ontology should be familiar to the users of the system, e.g. for plant operators, process developers and chemical suppliers. On the other hand, the terminology should be based on sound principles of conceptual modeling and ontology development. Furthermore, the goal should be to have common concepts and plant models with equipment manufacturers and engineering contractors that provide the initial information and often participate in modifications and upgrade projects. This creates a link to the ongoing development of engineering data models for various industrial areas. Fuzzy keyword ontologies should thus make use of relevant product and plant modeling standards and more general upper ontologies. The key research question of the KNOWMOBILE project is [185]: How to build fuzzy ontologies for the process industry domain to enhance knowledge retrieval?
8.3.1
Fuzzy Ontology for Process Industry
Our demonstration works with both engineering and operational knowledge of an industrial plant. Therefore, the fuzzy ontology should not be developed in separation from existing engineering tools and knowledge repositories, but existing terminologies, taxonomies and data models should be used if possible. This leads to a taxonomic system consisting of several layers, • Top layer: general concepts (i.e. based on international standards) that apply to several industries. • Middle layer: vocabulary defined and shared by business partners (within a certain industry, again based on standards) to share knowledge of, e.g. the type and structure of process equipment. This layer extends the top layer with domainspecific keywords. • Bottom layer: custom, company-specific concepts, e.g. specific products and component types, or even individual process plants. In order to speak about an ontology, our system of keywords should represent concepts, properties, relationships, axioms, and reasoning schemes relevant for the application area. On the basis of various upper ontologies and industrial data models we identified that the following keyword categories are needed to characterize event reports: • Systems: types of real-world components of a process plant, e.g. machines, buildings, software and people. • Functions: phenomena and activities carried out at an industrial plant in order to fulfill its purpose.
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• Variables: properties and state variables of various entities, e.g. temperature. • Events: types of interesting periods of plant life described in event reports, e.g. test runs or equipment failures. • Materials: raw materials, products, consumables etc. handled in a process plant. Our basic approach to conceptualize our application is shown in the informal UML class diagram below (Figure 8.5).
Fig. 8.5 Overall domain concepts [313]
Event reports describe events that are related to various entities of a process plant, e.g. to equipment, processing functions and materials. Nothing is assumed about the internal structure of event reports. Instead, they are characterized by an expert with keywords selected from a fuzzy ontology. The expert can select the keywords from five categories: event, system, function, material and variable. All keywords represent an entity type and can have subtypes and smaller parts. In the KNOWMOBILE demonstration tool keywords are used to characterize event reports and other nuggets stored in a database. Therefore, keywords can be understood as representatives of populations of realworld entities that overlap and are related in many ways. For example, the keyword ”paper machine” (cf. Figure 8.6) might represent the set of all paper machines in the world. Classification (is-a) and decomposition (part-of) can be found in most ontologies and data models. They are important in t he industrial context as well. So, the keywords in each category are linked by is-a and part-of relationships as illustrated in Figure 8.6. Furthermore, the ontology should model functional and other kinds of dependencies between keywords in various keyword categories. As an example,
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Fig. 8.6 Simplified event classification matrix [313]
systems can be or are used for some purposes, i.e. they play various roles in carrying out one or more functions. This creates a link between the keywords ”wire section (a part of paper machine)” and ”formation (a quality measure of the produced paper)”. Modeling classifications, decompositions and various dependencies leads to a situation where we have a taxonomy tree for each keyword category and a set of partonomy (part-of relationships) trees describing the decomposition to various domain entities (Figure 8.7). In addition, there are dependency relationships linking keywords to each other.
8.3.2
The Keyword Ontology
For developing a software tool the ontology should be expressed and stored in a more formal way. The basic approach for representing a fuzzy ontology is illustrated in Figure 8.9 with a combination of UML class and instance diagrams.
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Fig. 8.7 The fuzzy ontology defines classification, decomposition and miscellaneous dependency relationships between keywords. [313].
From all kinds of nuggets, the demonstration only looks at event reports that usually describe problematic situations. Each report, like the specific instance ”Report #1” in the diagram, is related to types of systems, plant functions, events, etc. Only the reference ”Report #1” to the keyword ”Holes” is shown in Figure 8.9 . As indicated by the ”instance of” associations, all keywords seen by the users are individuals, .i.e. instances of a subclass of ”Keyword category”. Fuzzy dependencies between keyword instances are described by a few fundamental relationship types like is-a (specialization), part-of and, as an option, instantiation. In the first version we focus on ”specializations”, i.e. fuzzy classification of keywords. The degree of overlapping (or inclusion) of the sets represented by the keywords is described by linguistic labels, i.e. natural language words like ”moderate” or ”significant”. So, the instance named ”Specialization #1” in Figure 8.9 tells us that ”Holes” is ”to a large extent” understood as a subclass of ”Quality problem but only represents a minor part of its scope”. In addition, the keyword ”Holes” may also specialize other problem types. In the KNOWMOBILE project we have modeled dependencies with one single dependency relationship. For simplicity it is symmetric and has a strength value between zero (independence) and one (full dependence).
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Fig. 8.8 A fragment of the event type fuzzy taxonomy [313]
Fig. 8.9 Representing fuzzy keyword ontology with object classes and their instances [313]
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8.3.3
8 Knowledge Mobilization
A Minimax Approach to Assess Keyword Dependencies
Now we show a method for approximating keyword dependencies in the keyword ontology. This method uses Bellman-Zadeh’s principle to fuzzy decision making [22]. An event type fuzzy taxonomy is shown in Figure 8.8. For example, consider the second column of event classification matrix ”Problem”. All ”Technical problems” are ”Problems” and they represent around 80% of all possible problems. That is ”Technical problem” covers ”Problem” with degree 0.8. Similarly, ”Human errors” are ”Problems ” and they represent around 30% of all possible problems. That is ”Human error” covers ”Problem” with degree 0.3. Following Bordogna and Pasi ([29], page 804) we will assume that if A and B are two keywords in the keyword taxonomic tree then coverage(A, B) = fuzzy inclusion(A, B). For example, ”System fault = {Device fault, Design flaw}” that is, ”System fault” is a union of these two events. Furthermore, ”Function failure = {Design flaw, Drift, Oscillation}” that is ”Function failure” is the union of these three events. ”Design flaw” covers ”System fault” with degree 0.6 and at the same time ”Design flaw” covers ”Function failure” with degree 0.4 (see Figure 8.8). Moreover, ”System fault” and ”Function failure” do not have any more component in common. We compute the degree of dependency between ”System fault” (SF) and ”Function failure” (FF) as their joint coverage by ”Design flaw” (DF) dependency(SF, FF) = min{coverage(DF, SF), coverage(DF, FF)}, that is, dependency(SF, FF) = min{0.6, 0.4} = 0.4 It is easy to see in Figure 8.8 that keywords ”Function failure” and ”Fire” are independent since they do not have any component in common. In this case we have, dependency(”Function failure”, ”Fire”) = 0. Zero means independence, one means full dependence, and values between zero and one denote intermediate degrees of dependency between keywords. It can happen that two keywords have more than one joint component. Then we apply BellmanZadeh’s principle (max-min approach) to fuzzy decision making to measure their dependency. For example, suppose that ”System fault” and ”Function failure” were to have two joint components, where the first one is ”Design flaw” and the second one ”Fluctuation” that has a coverage values 0.7 and 0.5, respectively. Then we measure the degree of dependency between ”System fault” and ”Function failure” according to Bellman-Zadeh’s principle to fuzzy decision making as dependency{SF, FF} = max{min{0.6, 0.4}, min{0.7, 0.5}} = max{0.4, 0.5} = 0.5.
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Supposing that all the coverage degrees are given by experts. Then we can summarize our algorithm as follows: Compute the degrees of dependency between keywords on the immediate upper level using the max-min approach. Then repeat this procedure until the top layer. For example, consider keywords ”Technical problem” (TP) and ”Operational problem” (OP). Then we find (Figure 8.8), dependency{T P, OP} = min{coverage(FF, T P), coverage(FF, OP)} = 0.5. One can further improve this model by introducing degrees of inclusion and coverage between concepts as suggested by Holi and Hyv¨onen [188], and Holi [189].
8.3.4
Demo Architecture and Implementation
The component-based demo architecture has been implemented by VTT Technical Research Centre of Finland, using the Prot´eg´e ontology editor to maintain the fuzzy ontology in OWL format (as shown in Figure 8.10). The GUI component (Graphical User Interface) guides the user in specifying the information query, and presents the results. Tools for browsing and evaluating the fuzzy reasoner component directly were also provided. A database adapter is used to access report data, which in this case was stored locally in XML files. Similarly, an ontology adapter is used to provide access to the fuzzy ontology, in this case stored in OWL files.
Fig. 8.10 Component-based demo application architecture [313]
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The adapters help hide the different interfaces and protocols of different data sources (e.g. SQL, HTTP) and provide transparent access via an agreed interface. The fuzzy ontology reasoner component is used to process ontology-based information. Its main function in the demo is to extend a list of query keywords to a list of their closest neighbors in terms of fuzzy ontology relationships. For maintenance and evaluation purposes, the component interface also provides methods for directly accessing the ontology concepts and relationships. Finally, the application logic component binds all the functionality together by taking the query, using the reasoner component to extend it, passing the extended query to the report database and then combining and ordering the results for the GUI. The fuzzy ontology with fuzzy concepts, relations, and instances was defined using Prot´eg´e version 3.4. The developed ontology was exported for the reasoning software as a standard OWL file. Prot´eg´e a widely used tool for developing ontologies. As such it provides some advantages, for example a forms-based interface for editing the basic classes and adding the individual keywords, reports, as well as their relationships. Prot´eg´e does not have a built-in support for modeling uncertainties, however in late 2010 Straccia [308] introduced new plug-in, called SoftFacts, which is an ontology mediated top-k information retrieval system over relational databases.
8.4
An Approximate Reasoning Approach to Rank the Results of Fuzzy Queries
In 2010 Carlsson, Full´er and Mezei [90], suggested the use of a context-dependent fuzzy aggregation method to rank the results of fuzzy queries over fuzzy ontologies. In their approach the fuzzy aggregation rules are provided by the experts, the coefficients of the consequence part of the rules are derived from the linguistic values used in the conditional part of the rules and the rank of a search result is determined by the Takagi-Sugeno fuzzy reasoning scheme. They present a fuzzy case-based reasoning approach to refine the results of fuzzy queries over fuzzy ontologies. It consists of two stages: The first stage is to retrieve the results from a fuzzy ontology by using a fuzzy (keyword) search. It can be done, for example, by using the FQUERY for Access system designed by [209, 358]. In the second stage they rank the results of fuzzy queries by using fuzzy aggregation rules and a Takagi-Sugeno fuzzy reasoning scheme. Now we will describe the approximate reasoning approach suggested in [90]. Without loss of generality we will suppose that we have (fuzzy) three keywords and a fuzzy query gives us an output with degrees of similarity (or match) x1 , x2 and x3 , usually in percentage terms. We should combine these degrees of similarities into an overall degree of similarity. Then we can rank the results of a fuzzy query according to their overall degree of match. One can use a fixed aggregation operator to find the overall match to the query. In many situations, however, this simple aggregation operator may not bring us the correct answer. More often than not we should use a context-dependent aggregation process instead. This context-dependent
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aggregation can be implemented by using a set of fuzzy if-then rules supplied by experts. Suppose the experts can provide us with some fuzzy aggregation rules of type, if x1 is A1 and x2 is B1 and x3 is C1 then o1 = a1 x1 + b1x2 + c1x3 where A1 , B1 and C1 are fuzzy numbers, and a1 , b1 and c1 are crisp numbers depending on A1 , B1 and C1 . For example, if x1 is SMALL and x2 is SMALL and x3 is MEDIUM then o1 = 1/5x1 + 1/5x2 + 3/5x3 which should be interpreted as, if the degree of match to the first keyword, x1 , is small and the degree of match to the second keyword x2 is small and the degree of match to the third keyword x3 is medium then the aggregated value should be o1 = 1/5x1 + 1/5x2 + 3/5x3. Suppose we have the following fuzzy aggregation rules, if x1 is A1 and x2 is B1 and x3 is C1 then o1 = a1 x1 + b1 x2 + c1 x3 ... if x1 is Am and x2 is Bm and x3 is Cm then om = am x1 + bm x2 + cmx3 where Ai , Bi and Ci are fuzzy numbers, i = 1, . . . , m. The procedure for obtaining the fuzzy output of such a knowledge base consists of the following three steps: • Find the firing level of each of the rules. • Find the output of each of the rules. • Aggregate the individual rule outputs to obtain the overall system output. Sugeno and Takagi use the following architecture [309]. The firing levels of the rules are computed by
αi = t-norm(Ai (s1 ), Bi (s2 ),Ci (s3 )) where t-norm is a triangular norm, and s1 , s2 and s3 are the crisp observations. The individual rule outputs are computed by, oi = ai s1 + bis2 + ci s3 , i = 1, . . . , m, and then the crisp output is obtained by, o=
α1 o1 + · · · + αm om α1 + · · · + αm
(8.1)
Suppose further that we can assign some context-dependent weights to the fuzzy numbers in the conditional part of the rules, denoted by w(Ai ), w(Bi ) and w(Ci ) and let us derive the coefficients of the consequence part of the rules from the linguistic values of the conditional part as oi =
w(Ai ) × s1 + w(Bi ) × s2 + w(Ci ) × s3 w(Ai ) + w(Bi ) + w(Ci )
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It is clear that 0 ≤ oi ≤ 1 for all i = 1, . . . , m. The tricky part here is the determination of the weights from the logical combination of the keywords. Suppose we have three fuzzy keywords, K1 , K2 and K3 , and we search for f (K1 , K2 , K3 ), where f denotes any fuzzy logic-based function that may connect the keywords. For example, if f (K1 , K2 , K3 ) = K1 ∩ K2 ∩ K3 then the weights of the linguistic values can be obtained from their natural ordering. Suppose we have three fuzzy keywords, K1 = {Very High Brightness} K2 = {High Consistency} K3 = {Low Air content} of equal importance and we search for {Very High Brightness} and {High Consistency} and {Low Air content} that is, we use a simple fuzzy connective ’and’. Suppose further that we have five linguistic values {VERY SMALL, SMALL, MEDIUM, BIG, VERY BIG} for fuzzy numbers in the conditional part of the rules. From the definition of logical ’and’ operation it follows that we can assign the following weights to linguistic values w(VERY SMALL) = 1, w(SMALL) = 2, w(MEDIUM) = 3, w(BIG) = 4, w(VERY BIG) = 5, and suppose the experts can provide us with the following three fuzzy aggregation rules, if x1 is SMALL and x2 is SMALL and x3 is MEDIUM then o1 = 1/5x1 + 1/5x2 + 3/5x3 if x1 is BIG and x2 is SMALL and x3 is MEDIUM then o2 = 4/9x1 + 2/9x2 + 3/9x3 if x1 is SMALL and x2 is MEDIUM and x3 is MEDIUM then o3 = 2/8x1 + 3/8x2 + 3/8x3 Then a search result with similarity degrees (s1 , s2 , s3 ) is propagated through this fuzzy inference system and its overall degree of similarity is computed by (8.1). For example, let the universe of discourse be the unit interval and let we define the following semantics (membership functions) for linguistic labels, VERY SMALL(u) = (1 − u)2, SMALL(u) = 1 − u, MEDIUM(u) = max{0, (1 − 2|1/2 − u|)}, BIG(u) = u VERY BIG(u) = u2
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and let us have a result with degrees of similarity (s1 , s2 , s3 ) = (0.3, 0.4, 0.8). Then the firing levels of the rules are computed by
α1 = min{SMALL(0.3), SMALL(0.4), MEDIUM(0.8)} = min{0.7, 0.6, 0.4} = 0.4 α2 = min{BIG(0.3), SMALL(0.4), MEDIUM(0.8)} = min{0.3, 0.6, 0.4} = 0.3 α3 = min{SMALL(0.3), MEDIUM(0.4), MEDIUM(0.8)} = min{0.7, 0.8, 0.4} = 0.4 and the individual rule outputs are, o1 = 1/5 × 0.3 + 1/5 × 0.4 + 3/5 × 0.8 = 0.420, o2 = 4/9 × 0.3 + 2/9 × 0.4 + 3/9 × 0.8 = 0.488, o3 = 2/8 × 0.3 + 3/8 × 0.4 + 3/8 × 0.8 = 0.525, Then the overall score (degree of match) is
α1 o1 + α2 o2 + α3 o3 0.4 × 0.420 + 0.3 × 0.488 + 0.4 × 0.525 = α1 + α2 + α3 1.1 0.1680 + 0.144 + 0.21 = = 0.47. 1.1
o=
Chapter 9
Mobile Value Services
This chapter may appear to deviate from the central theme of the book. Nevertheless, we will show - briefly - that fuzzy logic and fuzzy ontology may have a key role to play in the design of the next generation of mobile value services. Mobile technology is one of the most important technologies in the history of humankind with now more than 5 billion mobile phone subscriptions in the world. It is also described as one of the ”disruptive technologies” as it is changing the daily routines of ordinary people, the information and communication systems in business and industry and even the communication infrastructure of whole countries. Mobile technology is a platform for mobile services through which the expected changes will take place. It is widely assumed that the future of mobile telephony will rely on mobile services and that the use of mobile services will be an integral part of the revenues that were expected to be generated by third generation mobile telephony. The adoption of new mobile services contradicts this proposition as it has been much slower than expected. Several reasons have been suggested for the slow uptake of mobile services, ranging from cultural differences to the business models of the actors, lacking innovation capabilities of content and service providers, and national mobile network operator (MNO) strategies. If we look at some of the scenarios offered in the multitude of commercial market studies carried out each year there is now said to be an evolution of mobile telephony from voice and text communication to the use of value added services. This change is notable in three ways, which have precedents in the history of communication and media technology: (i) it changes the nature of mobile telephony; (ii) it is a challenge to the continuity of mobile telephony, making it a financially risky step to take; and (iii) we cannot yet say how and for what purposes the services will be used in the future [32]. We have carried out adoption studies with large consumer surveys in Finland 2003-2009; a major finding is that different groups of mobile services are developed and adopted at a different pace despite having the same technology base. Accordingly, one of the generalizations is that the adoption of mobile services takes place asynchronously with the development of mobile technology (see Carlsson et al. [73, 74]). These results are consistent with the previous research by Orlikowski and Iacono [270]. C. Carlsson and R. Full´er: Possibility for Decision, STUDFUZZ 270, pp. 185–246. springerlink.com © Springer-Verlag Berlin Heidelberg 2011
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This raises the question if the mobile services are designed in such a way that they fit into the day-to-day routines of the intended consumers, as well as offer viable and feasible business models to the actors in the mobile eco-system of mobile service producers, -distributors and -users. Neither adoption nor acceptance studies help designers and developers of new mobile services to understand the potential use of mobile services. Mobile service providers don’t follow formal procedures. Design teams collaborate only during the development of a specific service, but return to day-to-day operation when a project is finished. In practice, learning effects and reuse of design methods are limited. Experience with accumulation of knowhow on business cases is limited. When we used an open questionnaire in a recent research project (see Bouwman et al [33]) the response on which design methods or methodologies were used the answers were limited, and if there was a response the focus was on human computer interaction, user segmentation, usability tests, SWOT analysis and business case analyses. New Product Design (NPD) was mentioned twice; however no reference to service design methods was made, and no reference at all to understanding the value of mobile service in everyday life (usage in context). Therefore we want to introduce the metaphor of mobile value services, and focus on the value that a mobile service can offer to customers when we understand their day-to-day routines, as well as the value capturing by the providers of the services. A metaphor is ”a rhetorical figure of speech that describes a first object as being (or being equal to) a second object in some way; it is an implied comparison” (cf. Wikipedia). In saying that mobile value services is a better metaphor for mobile services we state that mobile services could (or should) be redesigned and implemented as mobile value services, i.e. mobile service designs should be developed starting from the current day-to-day practise of the intended user. But also the value for the provider should be understood by focussing on the underlying business model. Therefore the value of a service is formed by the way in which the service is contributing to day-to-day routines and the way in which it is ultimately changing user behavior. We build on (i) existing design approaches, and (ii) design science guided work focussed on understanding day-to-day routines, to illustrate our case - that we need a better metaphor for mobile services. We will use different research approaches in order to tackle different aspects of mobile services - service design, understanding daily routines, use context - in order to motivate our mobile value service design as systematically as possible. We will combine the results we have from the empirical work - a series of seven consumer studies and work with building a mobile service prototype - with a key observation from service marketing (Gr¨onroos [170], page 310): The firm cannot create value for customers. Its role is, first of all, to serve as value facilitator. By providing customers with value-facilitating goods and services as input resources into customers’ self-service value generating processes, the firm is indirectly involved in the customers’ value creation.
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We will use this insight as a basis when we discuss the design and implementation of mobile value services and as a way to explain what kind of business models would be useful for developing and implementing them. In Section 9.1 we will position our research in design science and new service design approaches; in Section 9.2 we will discuss relevant theories with regard to barriers and benefits, adoption, acceptance and empirical research in order to point out the first differences between mobile services and mobile value services; in Section 9.3 we work through the design and implementation of mobile value services with the Bomarsund Mobile Guide as an example of the design and implementation work; Section 9.4 is an attempt at generalizing mobile service design based on the experience we gained from the work with the Bomarsund; in Section 9.5 we will summarize and conclude and then offer some guidelines for building a viable business model for mobile value services; in Section 9.6 we will discuss two prototypes, MobiFish and Travel MoCo that were developed and implemented. MobiFish is a complete system for managing, selling and inspecting fishing permits for the ˚ registered fishing areas in the Aland archipelago, while Travel MoCo is a mobile community-based service; in Section 9.7 we will present some evidence of the development of mobile TV in Finland and other countries. The results of this Section show that the mobile TV, as a new mobile service to consumers, is far from being accepted and used in people’s daily life. The dilemma has to be studied and solutions have to be provided both through academic research and business development; and finally in Section 9.8 we show how fuzzy ontology can be used for new design program for mobile value services,
9.1
Design Science and Mobile Service Design
Design science did draw a lot of attention in IS research since the seminal paper of Hevner et al. [183] in 2004. In this Section two paradigms i.e. the perspective of behavioral science and the paradigm of design science are more or less presented as two paradigms that have conflicting objectives, e.g. developing and verifying (we assume that Hevner et al. [183] mean falsifying) theories, and creating new and innovative artefacts. We hold the opinion that a theory on design is needed. There has been limited focus on the embedding of the design of artefacts in generic knowledge on designs, design processes, phasing, and design methods. Although some work has been done in this area (Herder and Stikkelman [182]) a clear conceptual basis or a theory on design is largely missing, as is rightfully argued in a number of recent publications (e.g. Vaishnavi and Kuechler [318]). Mobile service design needs to become a real tool based in a working theory because we have found the need for it to become more and more urgent every year. The facts are simple: by the end 2011 there will be close to 4.9 billion mobile phones in use worldwide; more and more people start to depend on mobile phones - and the mobile services they make possible and support - as part of their everyday lives;
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market studies we and others have carried out - we will show results from studies carried out 2003-2009 in Section 9.2 - show that demand for mobile services is not increasing; this indicates that users do not make use of the enhanced capabilities of their mobile phones; the issues at hand are obviously in part design issues - the mobile services do not have the proper design to fit the actual needs of users in their everyday life routines. The conceptual basis for design science is rooted both in theories on design as well as in kernel theories that are relevant to the design artefact. In the case of this work the design artefact are mobile value services, with a focus on the value that mobile services create both for the user as well for the providers. The kernel theories are related to adoption and diffusion of innovation literature, acceptance models, domestication and value generation. Kernel theories play a role in the design of applications. Therefore we will come back to those theories in the next section. Although we will discuss a single mobile application in this Section, design science is not about one single artefact, but about generic knowledge, instantiations, and solutions to be applied to a class of similar design artefacts, i.e. mobile value services in general. In the domain of service innovation and development extensive literature on new service development is emerging that is relevant to researchers in the mobile service domain. Stage Gate Models by Cooper [110] have been developed in order to assess the viability of technical and market aspects of the service innovation in different phases of development, e.g. ideation (first screen), preliminary investigation (second screen), detailed evaluation of business case, development; testing and validation of artefact, and pre- and post implementation review. Moreover, stakeholder analysis can be combined with the new service design (NSD) model (Smith et al [303]). The process of service design is similar to the design cycle as used in design science. In the mobile service domain Bouwman et al [32] (see also Faber and Vos, [132]), for a practical manual) developed an approach for mobile services design and the underlying business models. In this approach there is a clear focus on user requirement research as well as stakeholder analysis, both focussed on finding an appealing service definition. Based on this service definition the enabling technologies as well as other resources and capabilities are organized. Relevant partners in offering the services are selected and managed, and a business case is developed. The model discusses the critical design issues and success factors for four domains, i.e. service, technology, organization and finance (STOF). In the service domain: a description of the value proposition (added value of a service offering), the market segment at which the offering is targeted, branding and customer retention are critical design issues. In the technology domain the focus is on the description of the technical functionality required to realize the service offering. Critical design issues are security, quality of service, system integration, accessibility to core enabling technologies, and management of user profiles. In the organization domain: a description of the structure of the multi-actor value network required to create and distribute the service offering, and to describe the focal firm’s
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position within this value network is given. Critical issues are partner selection, network openness, complexity and governance. In the last domain, finance, a description of the way a value network intends to generate revenues from a particular service offering and of the way risks, investments and revenues are divided across the different actors in a value network is analyzed. Understanding the role of critical design issues contributes to design theory (design science). Although the model as briefly discussed deals with multiple domains, we will focus on the customer value that is created. Therefore a closer look into the kernel theories is necessary. For mobile value service design the STOF model works in two ways: (i) the service and the technology are core constructs, (ii) the organization and finance are enabling constructs.
9.2
Mobile Services and Mobile Value Services
A basic understanding of what forms mobile value services is needed: mobile services become mobile value services when they become part of everyday routines (Keen and Mackintosh [212]). The user of a service will change his/her everyday routines, which start to rely on the service and he/she will be reluctant to change as long as the service is available. This has proved to be a both simple and effective insight for identifying mobile value services. To study customer value the focus can be directed towards socio-demographic variables that actually explain little about the adoption of mobile services. Instead, a user’s relationship to technology seems important, but the meaning of this relationship is not straightforward. Those who enjoy technology are early adopters, but so are some of those who do not. It appears that services are adopted by those who are not interested in technology once the services appear to be useful. This gives a slightly broader picture than diffusion theory typically does (cf. De Marez and Verleye [119]) where categorized consumer groups bear differences in characteristics. The result makes sense keeping in mind that mobile phones have become tools for everybody on an almost global scale. Barriers and benefits were found to predict little of the use of mobile services (Carlsson et al [74]; Bouwman et al [32]) when studied with large random samples (1000 consumers representative of the Finnish population) that were repeated for every year in 2003-2009. This is an unexpected result because barriers and benefits typically receive much attention when new markets are evolving. Correspondingly, diffusion theory does not give any particular guidance on how the adoption of mobile services is going to proceed. It represents a macro level approach to a subject that might require the incorporation of micro elements. The adoption of mobile services is initiated by users, and mobile services constitute a much diversified supply. IS acceptance research has been influenced by intention-based models rooted in cognitive psychology, such as the theory of reasoned action and the theory of planned behavior (L´opez-Nicols et al [238]). The adoption of technological products and services has been predominantly explained using these two theories and their extensions, such as the Technology Acceptance Model (Davis [118])). We have
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repeatedly found evidence (with the same large random samples described above) that the Technology Acceptance Model (TAM) does not offer any good explanations for the adoption of mobile services (Carlsson et al [78]; Bouwman et al [32]) or that it explains the value that customers perceive. Customer value can be defined in several ways and it can be expected that not every mobile service will fulfill customer needs - we actually found out that a large majority of the sample of 30 mobile services that we have studied did not fulfill customer needs. There are four interrelated concepts of value: intended and delivered value on the part of the service provider, and expected and perceived value on the part of the service user. In their study Bouwman et al [32] noted that users redefine the value and the way they use technology and/or services in a way that fits their preferences and their behavior. In many cases the intended value is not the value that will be delivered so the perceived customer value, that made the customer decide to try the service in the first place, has often little to do with the expected value (Bouwman et al [32, 34]; Chen and Dubinsky [102]). Perceived value is what matters to the customers. This conceptual background has served as a basis for seven market studies carried out in 2003-2009 with random samples of 1000 Finnish consumers (in 2009 with 1300 consumers). The market studies have been published in a number of papers (Bouwman et al [31, 32, 35]; Carlsson et al. [73, 78]; Walden et al [325]), which is why we here have compiled the results as an overview with a simplified, longitudinal model (see Fig. 9.1 discussing actual use of services and Fig. 9.2 future use of services) as this will be sufficient for our purposes . The studies were instrumental in the sense that they would help us to understand user behavior and demand for mobile services, as well as the value that a user would attribute to mobile services. The studies shows that the Finnish use of mobile services did not change much in 2003-2009 and was limited to mainly three: SMS, ordering ring tones and icons, and search services. This is despite the efforts made by the mobile network operators (MNOs) to launch additional 20-30 mobile services. The studies also show that the market is changing slowly and that the introduction of more advanced phones in 2005-2006 did not drive the use of new mobile services (as could be expected). We should also note that new advanced smart phones require that the daily routines in uploading new software and applications has to fit user behavior, and has to become routine. The reasons we gave previously that service producers (often MNOs) regard better technology as a service driver, but consumers as a service enabler, and the observations on the asynchronous adoption of mobile technology and mobile services - all point to a simple fact: the normal mobile phone users have not seen how the mobile services will benefit them in their everyday routines significantly enough to replace daily routines in such a way that there would be no fall back to the old routines. In a rather recent empirical study of Finnish mobile services (Viestint¨avirasto [323]) we note that: (i) 64% do not find any use for the developed mobile services; (ii) 18% cannot use the services (too complicated); (iii) 16% found the services too expensive/of no interest; (iv) 3% are willing to try out new mobile services given the existing pricing, but (v) 59% were not under any circumstances willing to try new mobile services. This study supports the conclusions we made
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from our empirical studies. As a consequence mobile services are not in demand as expected. When we compare the consumers’ actual use of mobile services with their proposed future use of mobile services (cf. Fig. 9.1 and Fig. 9.2) we notice that the patterns are strikingly different. The actual use shows very conservative behavior with only three services (cf. above SMS, icons and ringtones, search services) being part of everyday routines but that an additional 27 mobile services have not moved beyond the ”I have tried” + ”I have never used” group despite serious marketing efforts by the MNOs. The future use of mobile services shows a very different pattern - the consumers find (beyond the previously mentioned three services) that also multimedia and video services, mobile banking, lotto information, family search, event data and information, and mobile travel support would be mobile value services when they think about their future use of mobile services. Most of the participants had tried several of the services they proposed as future mobile value services. As we carried out the studies each year from 2003 to 2009 an obvious idea was that the proposed future use of mobile services would show up as an increased actual use one, two (or even several) years later. Despite very careful statistical analysis (cf. Bouwman et al [35, 36]) there was no indication of any correlation between the results on the future use of mobile services in 2003, 2004, . . . , 2008 and the actual use of mobile services in 2004, 2005, . . . , 2009; he pattern was the same with a time lag of 2 years, and also with 3 years, 4 years and so on. Therefore our first proposal is that consumers do not use mobile services they think they would (or even should) be using because they have no need for them. Overall it is fair to state that ordinary mobile phone users in Finland are more concerned with the mobile services than the technology and that the basic form of the services may well be sufficient for the everyday needs of a user. The introduction of higher speed networks, multimedia data services and the parallel use of multiple services seem not to be value-adding for the user. There is a supply-demand mismatch for mobile services in Finland. Even in Japan and Korea, which are considered to be forerunners in the adoption of mobile services, rather basic services (messaging and ring tones) have been most successful (Funk [166] Kim et al. [215]; Srivastava [305]). The same holds true in several European countries - basic services have been the most popular (Mylonopoulos and Doukidis [261]) but more advanced services have not yet found their way into the everyday lives of consumers. Thus the penetration of mobile services will probably be better understood if we get deeper insights in user requirements, user day-to-day lives and the way mobile services fit their behavior. Current service designs as initiated by MNOs appear to be driven by ad hoc situations and by service concepts that are based on ”more of the same”. Implying that if a service is successful variations of the service will emerge. Kernel theories like TAM, Diffusion of Innovation and classical empirical research methods focussed on (i) why mobile services have not been adopted on a much larger scale and (ii) why we do not have a thriving and growing mobile service industry, proved to be insufficient. The simplest answer (however, nothing is this simple as we have found out) is that mobile services have not been designed to give value to users in their everyday routines, i.e. where mobile services would matter most.
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Fig. 9.1 Mobile service markets in Finland 2003-2009
Fig. 9.2 Future mobile service markets in Finland 2003-2009
9 Mobile Value Services
9.3 Design and Implementation of Mobile Value Services
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Therefore our second proposal is that there are flaws in the mobile service designs which explain why the demand for mobile services is not growing.
9.3
Design and Implementation of Mobile Value Services
Hence we will shift to a design science research methodology and find out how to design and implement mobile value services in a quest to find some principles and design models. This follows our time-honoured belief that the methodology should be adapted to the research problems we try to solve instead of adapting research problems to methodologies we happen to master. In 2006-2008 one of our research groups was involved in an EU-funded project and designed, developed (and actually built the software from Open Source software modules) and implemented a dozen ˚ mobile service prototypes for tourists visiting the Aland Islands. Also for offering mobile services to tourists, although not day-to-day routines in general, we still have to understand the routines of tourists and how the mobile service fits in. Here we will work through one of the prototypes, a mobile guide service. The Bomarsund mobile guide is part of typical mobile technology developments in many EC countries as tourism is one of the major industries in Europe. This is evident from the growing interest to support tourists with web- and mobile technologies. Tourists are supported with digital information throughout their travel and when they want to share information with fellow travelers (Carlsson et al. [83, 84]). Developed web-services (e.g. for instance http://en.wikipedia.org/wiki/Bomarsund), advanced mobile networks (e.g. 3G, Wi-Fi, Bluetooth etc.), mobile handsets and smart phones, and new generations of data oriented mobile services are designed and developed to offer information and entertainment to tourists. Mobile phones are personal and user-adaptive, and smart phones will be ideal for giving tourists a continuous access to interactive and personalized travel information and services. This is in line with Rasinger et al [284] who found that mobile tourist guides show potential for supporting tourists when they are in places which are new and unknown for them. Compared to the Internet, mobile technologies promise to offer tourists a new level of freedom to explore various sites when they are in place, which offers new experiences. The basic idea for the Bomarsund mobile guide is to provide personalized, contextadapted guide information (cf. Carlsson and Walden [94]) for a detailed discussion of the design and development work that went into the mobile guide). Similar proposals have been worked on in a number of recent research and development projects, which have been well published (Schwinger et al [293]). There are two mobile guide systems which have features close to those we were striving for: (i) the Time Treks combines a story-based game with cultural and historical information to create a mobile tour; (ii) the m-ToGuide offers city travelers location-specific (with GPS) multimedia information about places of interest on a digital map and supports virtual tours of the city; an innovative feature is that it also supports the recording of experiences with text and snapshots for after-tour personal diaries. There are also other approaches: the IMAGE project (Edwards et al [130]) created a Mobility
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Agent to mimic a traditional travel agent to assist a user with travel decisions for short-distance land transport, to purchase travel and accommodation, and to inform about possible activities en-route or at the destination; mobile positioning technologies (GPS) are used to give a tourist locations and site-connected information in several applications (GUIDE (Cheverst et al [101]), CRUMPET (Poslad et al [278]) and COMPASS (Van Setten et al [297]) and LoL@ (Pospischil et al [279]). The objective for the Bomarsund was to design and develop a mobile service for tourists which offers more than (fragmentary) mobile services and official tourist services that are already available. The methodology for designing and developing the mobile guide was an iterative, design science approach (Hevner et al [183]), which is normally used for fast prototyping when the users are involved in the development process. First, an animated prototype was constructed to simulate the key features and user interfaces of the mobile guide. Then an assessment team (consisting of an art designer, a technical developer and two users) evaluated the prototype in terms of screen layout, task design and technical feasibility. Based on the findings from the evaluation, a second prototype was built to work on a mobile phone but with limited interactive functionality; this was used as an instrument for the second iteration and to be tested by a larger group of users. The testing covered all aspects of the usability of the prototype which also included a set of tests carried out in a usability laboratory. Then again, feedback from the second round of tests was used for the development of the final prototype, which was carefully tested and evaluated by the future owners of the mobile guide service (and a number of minor details were corrected and improved). The Bomarsund mobile guide was built to serve as a very insightful local guide, which offers a possibility for the tourist to get context-adapted material (Fig. 9.3) which tells the story about the Bomarsund fortress from different user perspectives. The guide (for instance) presents material in English for British visitors about the British fleet and the victory won at Bomarsund from an English perspective and the fact that the first Victoria Cross was awarded at Bomarsund. For French tourist the material is available in French and tells the French part of the story to the point why there is an inscription about Bomarsund at the Arc de Triomphe in Paris. Also Swedish and Finnish versions are provided. The mobile guide is built around movie clips, pictures, 3d graphics, speech and sound effects, in order to illustrate the building process, what the area used to look like and how the battles were fought (not unlike the Time Treks). The mobile guide has five main parts, each linked to specific places at Bomarsund (the numbers 1-5 in Fig. 9.3). The first part contains information about the planning and politics involved; the second part focuses on the building of the fortress and the community which was growing around the fortress; the third part looks at the battle at Bomarsund from the perspective of the Russian defenders, while the fourth part looks at the same battle from the perspective of the French-British attackers; the fifth part is focused on the following peace and the history and development of ˚ Aland and Finland. There is a sixth part (the number 6 in Fig. 9.3) with separate presentations of the ruins of the three towers (only three of the planned six were actually built). The material of the multimedia presentations was collected from
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Fig. 9.3 The Bomarsund Mobile Guide
historical sources and from local historians to ensure that the facts are correct and the narrative local and genuine; the material is very rich and can be adapted to and enhanced to support different groups with varying interests in the history (i.e. the guide adapts to different purposes as an experienced and knowledgeable guide will do). The development environment was the Nokia N73 smart phones (with good multimedia support) but most of the material can be accessed and used also with phones of less functionality, based on premium SMS. The fortress of Bomarsund cannot be seen anymore as it was completely destroyed during the Crimean War in 1854. The mobile guide ”makes the fortress live again” as it tells the story of the fortress and its fate from different angles as the tourists walk through the area and visit different parts of the area and the ruins. In this sense when compared with what now exists - just the silent ruins - the mobile guide offers added value to a tourist visiting the fortress. Professional guides are available at a cost but they are mostly not available for ad hoc visitors. The mobile guide is a mobile value service in the way we have envisioned it. The implementation proved to be a different story: designing, building and testing prototypes is rather straight-forward and it was possible to show that the design science approach actually works, but the Bomarsund should be made operational in the tourist market in order to turn it into a mobile value service. There are a number of principles available to guide this process (e.g. the STOF-model by Bouwman et al [32]) but so far the implementation process has not been carried out. The starting point for any service business model should be the value of the service that an individual company (or even a network of companies) has to offer and which will satisfy customer needs. Careful research in user behavior directed to understanding daily routines either by co-involvement of users, participatory observation, focus groups,
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and repeated user tests (for instance) with conjoint analysis (see for instance Bouwman and van de Wijngaert [34]), helps to understand the perceived values of mobile services within daily routine behavior. Only when the user value becomes apparent the service and the underlying business model can be implemented. As discussed before the design approach of Bouwman et al [32] discusses both the service, i.e. the way value is created for customers and the underlying business model i.e. the business logic of a service and the way providers can turn created value to revenue. Service, technology, organization and finance are the domains that interact with each other and describe the context of the business models (Bouwman et al. [32]). Translated to the Bomarsund mobile guide service the implementing (business, public) organization needs to be able to continuously work with the tourists to find out and understand their needs and then to adapt the service as needed. The present approach with a dominating MNO which is able to charge up to 40% for network services seems to prevent any viable business with the mobile guide. The increasing availability of smart phones, supported by mobile web services developed by application providers, will enable users to run applications on the phone instead of via the network of mobile operators. The shift is towards open platforms that will enable business developers to implement new services independently from MNOs. Open innovation networks with a keen focus on user value are more likely to realize advanced mobile services. Note 9.1. This far we have now worked out (i) that consumers do not use mobile services they think they would (or even should) be using because they have no need for them; (ii) that there are flaws in the mobile service designs which explain why the demand for mobile services is not growing. The first result is based on facts from a series of seven market studies with random samples in the years 2003-2009. The second result is a proposal we have explored and tested by designing, developing and implementing a mobile guide service which we worked out as a mobile value service. It turned out that the service will obviously produce value for tourists and visitors but that it (so far) has not been a business success, which shows that we also need to work on a business model. Then our final step will be two-pronged - we will work out a formal design process for mobile value services and we will summarize some ideas on how to design a viable business model.
9.4
Designing Mobile Value Services
There are number of lessons that can be drawn from the more than 30 mobile service design projects we have been involved in or carried out during the last 4-5 years. We will summarize these lessons as they have lead to our approach of designing mobile value services (see also [32] for a detailed discussion) which follows the general design science process of building and evaluating artefacts (cf. Hevner et al [183]). First of all, the traditional view is that value has to be created both for the organizations involved in designing the service as well as for customers. The interest of service providers has to be balanced with the interest of the users. From the provider side value addition is more complicated, as the provider will not in most cases be
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able to provide the service by themselves and is obliged to work together with and create value for other parties. In many of the projects we have seen that relevant parties are left out in the early phases of designing a service (as the development process needs to proceed quickly, there is no time to teach users design methods or concepts, a particular software that is used in the process, etc.) with the result that a prototype is made, but that parties that are expected to implement the prototype are not interested, as they have not been involved from the start and the assumed value creation on their behalf is not clear to them. So it is clear that partner selection is an important and critical design issue. We have seen that involving users can be done in a number of ways. One of the key issues for understanding user behavior is that the context of the behavior is understood. Thus observing user behavior, participatory observation, discussions within focus groups and individual talks with expected users are all means to specify use cases that are realistic in nature. Information gathered in this way can be a starting point for conjoint analysis in which users are confronted with use case scenarios and get an opportunity to choose from a number of alternatives to facilitate their actions. The conjoint analysis is a tool for finding out what users say they will use to support their behavior in a day-to-day context, and what in the end they actually will do. In one of our design cases police officers told us that they had a high interest in using smart phones, but when confronted with use-case scenarios discussing specific situations and tasks to be executed, they actually never opt for the smart phone solution (Bouwman and Van de Wijngaert [34]). Moreover the research showed that kernel theories like TAM don’t help to predict the actual decision to use a specific mobile service (Bouwman and Van de Wijngaert [34]) . The added value has to be very explicit, fitting tasks and behavior in everyday situations. Next to users testing mobile service concepts it is smart to include experts and representatives of co-providers to review concepts, and to discuss technical solutions. We have learned that the teams, that are invited to review concepts and to discuss the critical design issues (like target group specification, pricing levels, trust/branding, customer retention opportunities and personalization), have to be selected from all relevant disciplines, i.e. marketing, operations, software, telecom and accounting (to assess risk and investment consequences). As a concept of a mobile service is defined (the first artefact), and validated by users and providers, the next step is to derive the technical specifications and to look for the available technical solutions. The example of Bomarsund makes it clear that sometimes high end and low tech solutions have to be developed in parallel. The critical choices with regard to the interface-design, security, level of technological integration, quality of service, management of user profiles are being made. Based on the specification a first prototype, as was the case with the Bomarsund Mobile Guide, can be built. This second artefact can be tested by potential and expected users. Preferably alternative mock ups are available, and via pseudo experiments the developers can get insights in what the expected users would prefer. The degree to which the services can be defined in reusable building blocks (think about mobile web services) the easier the experiments with alternative solutions will become. Then it will be possible - with limited input of time and resources - to build and
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test extensions, or even to carry out an expansion to other, alternative mobile value services (Carlsson et al. [84]). Parallel with the building of the prototype, the selection of (technology) partners takes places. However, we have found out that the selection of partners, and the setting up of collaboration needs to be carried out in cooperation with the developers of the mobile service. It has turned out that in engineering driven projects most of the times decisions are made that in the long run lead to problems when one wants to roll out the service as partners that have a key role in the operational phases (for instance insurance companies in the case of mobile accident management systems) are not included in time. Thus besides dealing with technical issues it is important to work with technology and potential other partners, to discuss their strategic motivations, and to find out about their interest in creating and generating value from the mobile services. In trying to get an effective collaboration established the network size (number of partners), network openness, and the governance of the network of involved partners are critical issues to consider. Next to the value offered the consumer, the bottom-line of the project has to be considered. Price setting, costs, investments, risk assessment and management all need to get attention. In a number of our mobile service projects we have found it necessary to have a clear balance between the involved partners in terms of the tangible and intangible benefits, i.e. the strategic gains and the revenues, at the one hand, and the costs and risks at the other. If there is an imbalance in terms of cost, time and/or monetary value among the actors - one actor carries the cost while the revenue goes to another - the service will never come about. We have found this balancing activity to be at the core of the development efforts to turn mobile services into viable business. Moreover, we have found that in practice this process is not linear, which means that the parties are involved in different phases of design, roll out and market development in different ways and that their use of resources will differ over time. Design science research is relevant for the whole process from concept design to a marketable service but should also deal with technology, business models and economic and financial issues. More focused topics would be the re-use of service components and the development of generic mobile services that could be quickly turned into families of mobile value services by adding to or changing components. We have found that the design, development and implementation of mobile services will contribute to design science research by adding the specific mobile service domain knowledge and by adding elements from research results in the services, technology, organization and finance domains. Finally, we have seen that within the whole process from concept to a mobile value service that is being accepted and used by consumers, many different design tools and methods can be used. The evaluation of the effectiveness of these tools, as well as building knowledge on the design artefacts, the methods and tools that have been used, requires further conceptualization and research. A meta-theory on the design of mobile value services is largely missing and here we will offer a first step. We can easily verify that the experiences we have summarized are well in line with
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the Guidelines for Design Science in IS Research that Hevner et al. [183] proposed a few years ago (here summarized): we should build an innovative purposeful artefact (cf. Bomarsund), for a specific purpose (cf. testing if a mobile guide can be a mobile value service); we should thoroughly evaluate the artefact (cf. test programs with service providers, intended users, service organizations, etc.); the artefact must be innovative, solving unsolved problems or known problems in a more effective or efficient way (cf. tourists need to get information on and an understanding of some ruins that once were a major fortress); the artefact must be rigorously defined, formally represented, coherent and internally consistent (cf. developing second, third and progressively more advanced and specified prototypes with better and better validated software); the process of creating an artefact should be similar to a search process carried out in a problem space (cf. successive prototypes represent better and more advanced problem solving); a design artefact is complete and effective when it satisfies the requirements and constraints of the problem it was meant to solve (cf. tourists will use the ”Bomarsund” regularly and would be deeply dissatisfied if it was not available). Finally, Hevner et al. [183] recommend that the design science research is communicated effectively - as there are now close to 4.9 billion mobile phones in use worldwide there is no doubt that any approach used to build successful mobile value services will be communicated rather effectively. Research in service logic and service management (Gr¨onroos [170]) suggests that the mobile service development work has been on the wrong track: ”the firm cannot create value for customers; its role is, first of all, to serve as value facilitator; by providing customers with value-facilitating goods and services as input resources into customers’ self-service value-generating processes the firm is indirectly involved in the customers’ value creation” (Gr¨onroos [170], page 310). Thus for mobile value services to be developed we should aim at building the services with the users, not for the users. This is rather a different process from the traditional mobile service design as practised by MNOs. The customer service logic is created interactively with individual users who design artefacts that will add pleasurable features to their everyday life routines through audio or graphic or knowledge functions added to mobile phones. From our own experience we note that although the focus is not only on customers, we also have to take potential other stakeholders into account, such as actors that will be responsible for the implementation and operational processes, and the commercial exploitation. The provider service logic is found through interactive design research with individual users and works out hardware and software solutions that fit the designs by building, modifying, adapting and fitting standard modules. The design and implementation process as we have discussed above, offers new challenges for design science research. The joint design research processes, combining customer service logic and provider service logic, will work out over web- or mobile platforms through rapid evaluation and rapid rejection/scaling
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up. It will be possible to adopt the customer/provider service logic to the design, development and implementation of mobile value services.
9.5
A Business Model
The longitudinal material from the 2003-2009 market studies in Finland shows that mobile services beyond the basic 2-3 services are not in demand; it also shows that intentions to start using mobile services in the future did not show up as actual use after even 2-3 years. Understanding the behavior of users in adopting and using mobile services requires a clear understanding of how mobile services fit into people’s everyday lives. Existing kernel theories, like TAM and Diffusion of Innovation are not very helpful in the design process. This notion has implications for the design research focus in the IS domain when we want to develop mobile services to mobile value services. Instead of pursuing the barren road of yet another TAM study, the focus should be redirected to small scale studies in user behavior and daily routines of users that are directing the design of mobile value services. Hence our proposal that there are flaws in the mobile service designs which explain why the demand for mobile services is not growing is gaining support. Through recent research in critical design issues as proposed by Bouwman et al [32], and by focussing on service logic and service management we suspect that the mobile service development work should aim at building the services with the users, not for the users, and understand user behavior in context. The final step in our analyses is to sketch the outline of a business model for mobile value services. The context is the tourist mobile services we introduced with the Bomarsund in Section 9.3. Now we focus on Company A, which is an (mobile value) service (MVS) provider in that context and which works as part of a tourist service network including tourist service providers (offer independent or package tours) and tourist tour organizers (local or national or multinational; offer independent or package tours). Company A provides software solutions (and hardware modules if needed) for mobile phones. The software supports the building of (i) mobile value services with modules for (ii) localization, (iii) context adaptive and (iv) personalized services, but the company is not developing all the software by itself - there are out-tasking and in-sourcing partners (a Partner B building solutions that Company A has developed and for which it has IPR is an out-tasking partner, or, a Partner C, selling their own solutions to which they have IPR as part of Company A’s mobile value service is an in-sourcing partner). The other key parts of A’s value chain are the interactive design processes with customers to work out a customer service logic and then to work this out for MVS compositions to get specifications for the software development work. The final stage - maintenance and development - can be expected to be a true cash flow generator if the MVS design and implementation are successful. This business model remains to be tested - after which a number of companies have been persuaded to try it out - and this will now be undertaken in a couple of new research projects.
˚ 9.6 The New Interactive Media (NIM) ALand Project
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Fig. 9.4 The Mobile Services Business Model
9.6
˚ The New Interactive Media (NIM) ALand Project
˚ Although most of the people of the Aland islands are Swedish speaking, the islands ˚ are demilitarized and are a sovereign part of Finland. Aland is an EU member, but is not part of the EU tax union. It has its own government, and a legislative assembly, the Lagtinget, which, together with the local government, makes their own laws . The population is mainly concentrated in the region’s only city Mariehamn (40%. The main island encompasses about 70% of the area and contains 90% of the total population. Less than 10% of the population lives on islands outside mainland ˚ Aland. The economic environment is described as follows by the local government: ˚ Aland is a wealthy, modern and technology-oriented society characterized by an entrepreneurial spirit and creativity. Unemployment has been consistently low (2%) since 2003. Although shipping, trade, (ecological) farming and fishing are important sectors, the service sector and the tourist industry dominate the economy. When we look at the five characteristics for a RIS, i.e. public administration policy towards innovations, the role of firms, the R&D public system, the environment and innovation support infrastructures, the following picture emerges. The public ˚ policy of the Aland Autonomy Government towards innovations is supportive and there are a number of political initiatives aimed at promoting new industrial and service projects to use innovations in the ICT sector. In building the I Tiden technology centre, an incubator, the government has concentrated its resources and created an environment where potential entrepreneurs can get help and support for developing their ideas, financial support for their first business venture and eventually - as the business gets more mature - obtain support for relocating in the archipelago. However, the actual number of companies is less than 10, which means that the impact on the island economy has thus far been rather limited. Although several IT-companies
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with an international business network are located on the islands, most of them are spin-offs of the major regional bank and gambling association and focus mainly on banking applications, innovative gaming and gambling applications. A typical innovative company is the Chips Inc, which builds on the very good quality potatoes grown on the islands, uses advanced technology to produce branded potato products and has acquired a very strong market position. The company was sold to the Norwegian Orcla Foods a few years ago, which is typical in the sense that mid-sized ˚ companies have more or less disappeared from Aland due to the high costs for manufacturing companies on the islands. This is one of the reasons why the local government stimulates service innovation. The public R&D system is mainly helping local companies and organizations obtain EU funding for their R&D projects and ˚ giving advice on the use of public R&D funding. Aland has its own polytechnic, but there are also close relations to mainland universities in Sweden and Finland. The innovation support services are mainly provide by the mainland universities and the science park (with an incubator organization, venture capital, patent organizations, international cooperative networks, etc.) in the Turku-region in Finland, the closest mainland community. ˚ The economic, institutional and physical setting (archipelago) makes the Aland islands an excellent test bed for understanding the development and testing of mobile value services within the context of a knowledge transfer and regional develop˚ ment program funded by the European Regional Development Fund and the Aland ˚ ˚ Autonomy Government. The main mobile operator in Aland is Alands Mobiltelefon (GSMWorld, 2007). All the major mobile operators are listed as roaming partners ˚ and the network coverage for Aland is quite extensive. We can conclude that the conditions for mobile services are positive. The NIM project was carried out in the period 2006-2008, with the aim of improving the service level in the private and public sectors through innovative, advanced mobile services. One of the means to achieve this objective is to transfer knowledge on interactive mo-bile technologies from mainland universities to the local community, and to develop some mobile applications to demonstrate how the services could be used and contribute to local economy. Knowledge transfer was stimulated by seminars and an international conference organized in Mariehamn. A number of applications were proposed, prototyped, selected and developed, ranging from simple communication aides in the form of interactive calendars to fully fledged information portals with mobile extensions. An typical example of a such an application is a mobile service focused on obtaining and paying via a mobile phone for a fishing permit (MobiFish). Other applications focus on creating timetables for public (ferries) transport, as well as providing real-time information, for instance on ice conditions that may affect the time tables, on the information that is available, on hotel and restaurant booking via a mobile handset, including web 2.0 applications enabling travelers to share their experiences, and on a mobile guide directing visits to a historical site, to name a few. The NIM project had to address an existing problem in an innovative way and be feasible, and local organizations had to be able to implement and operate them. This resulted in a number of more or less conflicting requirements. For example, the applications should be used with current
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state of the art mobile phones (for instance by making use of MMS and/or premium SMS), while on the other hand they should support map- or location-based functionalities (requiring web-based and JAVA-applet solutions). Moreover, the mobile services had to be cost-effective and not require additional efforts from the organi˚ zations involved. For instance, the fishing experts from the Tourist Board of Aland made it explicit that the system should not include any additional work, but only improve the cash flow. Prototypes were developed in close collaboration with relevant parties, local software and consultancy firms and potential users. In the first stages of the project, the applications were designed in collaboration ˚ with local experts in Aland. The project started with a review of current systems and solutions, and moved on to a systematic effort to process re-engineer the routines that had been the drivers of the existing solutions The applications were developed using the principles of agile soft-ware development. The basic idea behind iterative enhancement is to develop a software system incrementally, allowing the developer to take advantage of lessons learned from the development of earlier versions of the system. Although all the applications being developed were different, there were some common elements. It was decided to develop different front-end for the mobile part of the system based on the same principles as the mobile version and to use the same back-end, but to use recent Internet technology to provide increased usability and more options to the user. For most services, a low-end as well as a high-end application was developed. After evaluating the different possible payments options, it was decided the focus should be on SMS-based payments. To make the system flexible and not dependent on GPRS/3G network coverage, it was decided that the high-end Java version should also use SMS to order, pay for and receive information. To maintain a high level of reliability, the system was designed in such a way as to allow for updates and maintenance of the database without any downtime. Because most systems focus on tourists and travelers, multi-language support was one of the initial requirements. It was decided that the prototype should support Swedish, Finnish, English and German. Now we will briefly discuss two prototypes that were developed (and implemented), the first of which is called MobiFish, a complete system for managing, ˚ selling and inspecting fishing permits for the registered fishing areas in the Aland archipelago. MobiFish was designed to allow fishermen to buy a fishing permit anytime and anywhere, to make it easy for the owners of the fishing waters to sell permits to their fishing areas and to offer inspectors a quick and easy way to check fishing permits. As far as consumers are concerned, MobiFish has three different interfaces designed for different mobile phones. Inspectors can check the validity of the permits offline and online. For the owners of the fishing waters, the system provides online statistics on the sales of permits for the relevant fishing areas. The second prototype is a mobile community-based service called Travel MoCo (Fig. 9.6). Because social interaction plays an essential role in providing tourists with information, we developed a service that allows users to look for travel-related information through social interaction with other people and to share experiences with each other anytime and anywhere via mobile phone. Our research focus also builds on a social perspective on mobile tourism services that do not directly provide
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Fig. 9.5 Screenshots of the MobiFish service
travel information, but enable tourists to obtain the information through social interactions that are mediated by the service. The service offers 24/7 access to tourist information, even when official or public information channels are not available.
Fig. 9.6 Screenshots from the Tourist service
Although throughout the project serious efforts were made to involve local business partners who would deliver the core resources and capabilities necessary for the applications to operate, a structural partner developing and exploiting the application was lacking, which ultimately led to a search for businesses that may be interested in exploiting some of the applications.
˚ 9.6 The New Interactive Media (NIM) ALand Project
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Stakeholder and User Involvement
Stakeholder and user requirements were discussed in meetings with the stakeholders, including potential users. Stakeholders that focused on technological resources, for instance SGNET (a telecom infrastructure provider), the local mobile operator and Quedro AB (a billing and SMS platform provider), as well as investors, were involved in the development of the prototypes. Although, potentially, they would stand to benefit the most, organizations that were involved in streamlining the operational processes, including restaurants, hotels, tourist organizations, local advertisers, landowners, inspectors and other supporting facility providers, were - reluctant to become deeply involved in the development of the innovative solutions. These stakeholders are also the local users, and in a sense co-creators of the service by being providers as well as consumers of services. Consequently, the focus was also on persuading the local community to adopt and use the mobile technologies. Usability tests were carried out in a usability laboratory, with the aim of detecting and solving any software-related as well as hardware-related problems. The results were very satisfactory and the system was delivered for actual use to some companies that had expressed an interest. To make sure the application would be adopted, much attention was paid to the service support element, both with regard to local business and local users. In addition, extensive survey research was carried out, to collect data on user acceptance as well as to create awareness of the advanced mobile services. We define advanced mobile services as data services that have the look and feel of Internet pages and are accessible via mobile or handheld devices, and via mobile telecommunication networks (2.5 and 3G+ networks) [238].
9.6.2
User Acceptance Studies
Acceptance of advanced mobile services by the local community depends on a number of variables. In recent decades, perceived ease of use and perceived usefulness have been considered important in determining the people’s acceptance and use of advanced technologies, and their adoption depends on a various other factors, such as the technological characteristics, the target users, and the context. Although more and more research information is becoming available, market analysis indicates that ˚ consumers (in our case in Finland and more specifically in Aland) are still reluctant to use their mobile phones for more advanced mobile services. Practitioners and academics [265] are struggling to explain and predict the use of mobile services and the reasons why consumers decide whether or not to adopt these services. There are several models that can be used to examine the acceptance of advanced mo-bile services. To date, Information System (IS) acceptance research has been predominantly influenced by intention-based models rooted in cognitive psychology, i.e. the theory of reasoned action (TRA) by Fishbein and Ajzen [145], the theory of planned behavior (TPB) by Ajzen [8] and the technology acceptance model (TAM) by Davis [118]. TAM [118] predicts people’s intentions to use a technology based on their perceptions of its ease of use and usefulness. As far as the enablers and barriers ˚ in the adoption of advanced mobiles services on the Aland islands are concerned,
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the TPB offers the more appropriate approach, as we look at internal and external constraints and facilitating conditions, such as supplier support, ease of use, service flexibility, and costs. Cost barrier. Potential adopters of advanced mobile services are consumers who will consider prices and evaluate the services based on their benefits and costs [216]. The cost barrier is defined as the extent to which people believe that the costs of using advanced mobile services will prevent them from adopting them [12]. Monetary costs include the price of the advanced mobile service (or bundle), whereas nonmonetary costs usually include time and effort, as well as other factors [216]. The monetary and non-monetary costs of the advanced mobile service are considered among the highest barriers in the adoption of these new services [31]. Perceived customer value must at least equal, and preferably exceed, the price or cost of the ˚ service [32]. In the context of the Aland islands, tourists and local people adopt and use advanced mobile services for personal purposes, and they also foot the bill. From the outset, it was clear that costs would be the core issue on which the advanced mobile services would be judged, even if it was obvious that the services would simplify daily routines and thus save time and money. One aspect that requires further research is the fact that people have a ”sunk cost” invested in their established routines and are reluctant to change their established patterns. Consequently, we posit the following: H1. The cost barrier reduces the future use of advanced mobile services. Perceived ease of use (PEOU). Ease of use has been widely considered from a predominantly technological point of view and as a potential threshold in the adoption of advanced mobile services [216]. It is defined as the degree to which an individual believes that using a particular system would be free of physical and mental effort [118]. In terms of design, functionality and navigation, ease of use is often seen as one of the most important factors influencing the adoption of advanced mobile services [249]. Perceived ease of use has a positive effect on behavioral intention (BI) to adopt a new technology. We posit the following: H2. Perceived ease of use increases the future use of advanced mobile services. Perceived flexibility. According to Bouwman et al. [31], the benefits of mobile services are related to the nomadic value of mobile services, as reflected in concepts like anytime and anywhere. These benefits are highly relevant in a geographical environment were physical constraints require users to be flexible in time and space. Mobile devices and services offer people the opportunity to move around while maintaining access to relevant services and staying (socially) connected [204]. Flex˚ ibility may be specially valued by consumers in an isolated region like the Aland islands. Accordingly, we propose: H3. Perceived flexibility increases the future use of advanced mobile services. Service support. Some studies indicate that mobile device size, access procedures, ease of use, mobile interface, and training and support are significant factors
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in the use of new applications [167]. Service support includes various user-related factors, such as technical consultants, related training programs, appropriate and sufficient resources from either internal or external organizations. In this Section, service support is defined as assistance and help provided by mobile operators and service providers to solve problems related to the use of mobile services. The NIM project was focused specifically on providing support to the local community and local users of the advanced mobile services. Valuable support will efficiently enhance individual capabilities and their perceptions. Hasan [178] reports that technological support and training increases the adoption of mobile systems. Consequently, we posit the following: H4. Service support increases PEOU of advanced mobile services. In the mobile market, customer services are seen as a ”hygiene factor”: if they are is poor, they will create dissatisfaction, but if they is good, they will not improve customer satisfaction [168]. As far as mobile services are concerned, the cognitive effort associated with overcoming the complexity of a service may also be perceived as a cost factor [216]. Support and assistance may reduce cost barriers related to ease of use and have a positive impact on the perceived flexibility and adoption of advanced mobile services, which is why we posit the following: H5. Service support reduces cost barrier of advanced mobile services; H6. Service support increases perceived flexibility of advanced mobile services.
Fig. 9.7 Research model, summarizing the hypotheses
We argue that the costs related to adopting advanced mobile services can be reduced by enhancing perceived ease of use. Thus, we posit the following hypothesis: H7. PEOU reduces cost barrier of advanced mobile services. Constantinides [109] has argued that the process of transferring e-commerce to mobile commerce involves a variety of costs, which makes mobile commerce more
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expensive then wired e-commerce. This implies that the perceived cost barrier of mobile services is higher when compared to fixed systems. However, within the ˚ context of Aland, where there are more mobile connections than fixed ones, we think transferring to ubiquitous systems may reduce cost barriers. Thus, we posit the following: H8. Perceived flexibility reduces cost barriers of advanced mobile services. These 8 hypotheses are summarized in Fig. 9.7.
9.6.3
Method
• Sample and procedure. The information used in this study was provided by ˚ a self-administered cross-sectional survey on the Aland islands, collected over a four year period (2004-2007), i.e. before and during the NIM project. The survey is thus representative of the prevalent attitudes regarding mobile services on the islands within the relevant time frame. The questionnaire was mailed to a sample of ˚ the Aland population between the ages 16 and 64, whose mother tongue was either ˚ Finnish or Swedish and who resided on the Aland Islands. The effective response rates were 52.3% in 2004, 56.6% in 2005, 53.6% in 2006 and 48.6% in 2007. The results indicated adequate levels of sample representativeness. The combined responses over the four year period (N= 634) were used for the analysis. • Questionnaire and measurement development. The questionnaire consists of three parts. The first part contains questions about devices and subscriptions. In the second, part items are presented that have to do with barriers, benefits and attitudes towards mobile devices, services and innovations. The third part contains questions about the actual and future use of services. The multi-item scales were drawn from previous studies (Bouwman et al. [31]). The items used in the analysis are presented in the appendix. • Scales properties. The measures were refined by assessing their unidimensionality and reliability. First, an initial exploration of uni-dimensionality was conducted using principal component factor analyses. In each analysis, eigenvalues were greater than 1, lending preliminary support to a claim of uni-dimensionality in the constructs. Next, confirmatory factor analysis (CFA) and alpha reliability analysis were used to establish the required convergent validity, discriminant validity and reliability. As can be observed in Table 9.1, the results of the eight factor model provide an acceptable fit ( χ 2 (44)=77.82; CFI=.99, NNFI=.98, RMSEA=.04). All item-construct loadings are high and significant, providing evidence of adequate convergent validity. Because the research contains several multi-item reflective scales, the psychometric properties of the measures described above were analyzed via the scale composite reliability index [15] and the average variance extracted index (Fornell and Larrcker [147]). Both indexes exceeded the recommended benchmark of .60 and .50 respectively. Further evidence of discriminant validity among the dimensions was provided by two different procedures recommended in literature, as follows:
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Table 9.1 Reliability and convergent validity (n=477); a Scale composite reliability, Average variance extracted, χ 2 (44)=77.82, CFI=.99, NNFI=.98, RMSEA=.04 Mean Service support 3.06 Perceived ease of use 2.94 Cost of the service 2.64 Flexibility benefits of the service 4.08 Future use advanced mobile services 2.80
SD 1.16 1.0 1.27 1.03 1.08
Cronbach’s alpha .88 .80 .86 .92 .84
Eigenvalue 1.78 1.64 1.73 1.87 2.76
SCRa .89 .80 .86 .93 .85
AVEb .80 .65 .75 .87 .60
1) the 95% confidence interval constructed around the correlation estimate between two latent variables never includes value 1 (Anderson and Gerbing [11]); and 2 the square of the correlation of two constructs was less than the average variance extracted estimates of the two constructs (Fornell and Larcker [147]). These findings provide evidence of discriminant validity among the components and constructs. Overall, the results obtained provided strong evidence of scales reliability.
9.6.4
Results
After checking the reliability of all the measures, we developed a structural model using Lisrel 8.8 to test the hypotheses. As is shown in Fig. 9.8, based on the Lisrel analysis, most of the hypotheses were supported. The overall adjustment indexes also fall within the limits recommended in literature for each of the services (χ 2 (46)= 82.47; CFI=.99, NNFI=.98, RMSEA=.04). It is generally agreed, however, that researchers should compare rival models and not just test the performance of a proposed model (Bagozzi and Yi [15]), which is why the proposed model was compared to another model that also estimates the relationship between service support and BI with regard to the use of advanced mobile services. In 1988 Anderson and Gerbing [11] recommend this procedure and suggest the use of a chi-square difference test (CDT) to test the null hypothesis: TM-AM=0. Accordingly, a nonsignificant CDT would lead to the acceptance of the more parsimonious of the two models. Based on the non-significant change in chi-square between the TM and the AM, it can be concluded that TM offers a better specification. As expected, the model shows that costs play a crucial role when it comes to the ˚ adoption of advanced mobile services on the Aland Islands. Although services support helps reduce the costs (note that the correlation is negative and the formulation of the items is also negative), and have a positive effect on the expected ease of use as well as the expected flexibility, costs are the basic inhibitor. The model clearly illustrates that financial considerations, i.e. costs are the main barrier for future use of advanced mobile services.
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Fig. 9.8 Final model
9.6.5
Discussion, Limitations and Conclusions
This Section contributes to regional economic studies by focusing on a serviceoriented industry - tourism - in which concepts for advanced mobile value services, rather than patents and traditional R&D, are the research focus. The research indicates that focusing mainly on concepts based in regional economics, regional innovation systems, and learning regions, is too limited an approach if we are to understand the regional impact of a knowledge transfer project focused on service innovation and design. Based on the RIS, we can conclude that many criteria were met within the NIM project . However, focusing on services rather than on traditional R&D and industry focused spill-over led to major complications. Because services are produced locally by providers and consumers, it is important to understand local demand and supply and therefore how and why services are accepted by the local community. To answer these questions, we studied the regional innovation systems, as well as the demand for and the acceptance of advanced mobile value services, based on approaches that are common in Service Design and Information Systems research. The aim of this Section r was to show how knowledge transfer focused on developing mobile value service concepts and applications has to be combined with service design, and Information Systems acceptance research in order to overcome and to exploit specific natural, institutional, cultural and economic constraints. In most cases, bringing innovation to a specific economic region is motivated by an assessment of the potential benefits of a certain technology for the regional economy. The people who have to use the technology are rarely involved in the decisionmaking process. From the outset, the researchers and practitioners involved in the ˚ NIM Aland project understood that the development of and knowledge transfer with regard to know-how on new mobile services was not enough. For the project to be
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a success, it was important to focus on the behavior of the potential users as well. Service innovation is different from traditional product innovation because of the nature of services. The value of a service is created in an interaction between producer and consumer. While products are highly dependent on patents, services depend primarily on the co-creation of services by local providers and customers, which means that detailed knowledge is needed regarding user acceptance and the behavior of providers and users alike. Providing information on local demand, attitudes and perceptions of characteristics of mobile services is highly relevant. The project had to solve existing problems in an innovative way and be feasible. In addition, local organizations had to be able to implement and operate the services, which resulted in a number of more or less conflicting requirements. Because the mobile services were supposed to be cost-effective and not to require additional efforts from the organizations involved, the outset information was gathered regarding people’s attitudes and preferences based on the actual use of a set of more or less established and potential innovative mobile services that would be beneficially to the regional economy. Based on the results of this study, we may conclude that ordinary mobile phone ˚ users in Aland are more concerned with the costs of the mobile services than they are with the technology, and that the basic form of the services may well be sufficient to meet their everyday needs. It would appear that the introduction of higher speed networks, multimedia data services and the parallel use of multiple services adds little by way of perceived user value , which is supported in a number of empirical studies [73, 78] that indicate that there is a supply-demand mismatch when it comes to mobile services. More advanced services have not yet found their way into the everyday lives of consumers. ˚ The longitudinal description of the 2004-2007 studies in Aland shows that there is little demand for mobile services, with the exception of three or four basic services. In addition, people’s stated intentions to use mobile services in the future failed to translate into actual use, even after two to three years. Although most hypotheses in this study are supported and the importance of service support, ease of use and the value of flexibility, was well understood by the practitioners and researchers involved in the project and confirmed by the empirical analysis, the empirical data ˚ also show that cost is a major barrier. Although Aland belongs to the 20th wealthiest regions of Europe, people are very careful when spending money. For the local population, the costs-benefit discussion is crucial, although they are aware of the relevance of flexibility in a region where communication and logistics are heavily influenced by the geographical nature of the archipelago. The study also has some limitations. First of all, from a regional studies perspective, the study is a single case, and the results have a limited external validity. Moreover, the focus of this study has been services rather than product-oriented R&D ˚ spill-over. Finally, in light of its geographical location, the Aland archipelago is an a-typical region from an institutional and economic perspective. At a theoretical level, the research is limited due to the fact that there is no-coherent framework that
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combines insights from regional economics, service in-novation and Information Systems research. In our view, a multi-theory, multi-method approach as presented in this Section is always limited and subject to criticism, due to the fact that the research approach is multi-disciplinary in nature, and assumptions behind theories and concepts may be conflicting. However, in our view, a multi-theory and multimethod approach increases the richness to the data and to the interpretation of the empirical results from different disciplines. ˚ The users, tourists and local inhabitants of the Aland islands were consulted in the development of the mobile applications at various stages of the project. However, the survey was aimed exclusively at the local inhabitants, although in some cases they may not have been the primary target group, for instance in the case of tourist services. However, tourist services are potentially relevant as well, because inhabitants are involved in offering the tourist services and the information provided by tourist via the blog-functions may be relevant to them as well. In addition, they also use the services when traveling or looking for a place to eat. An important limitation is the fact that the assessment of the prototypes of the advanced mobile services could not be included in the questionnaire, mainly because we developed of the prototypes and the surveys in parallel. However, the prototypes have since been subject of focus group discussions and results of these focus groups were taken in account in the of the questionnaire The aim of the project was to improve the service level in private and public sectors, by using innovative, advanced mobile ICT services for economic development. A number of prototypes were developed and implemented. The project had to address existing problems in innovative ways and find feasible solutions that could be implemented and operated by local organizations, which resulted in more or less conflicting requirements. Further limitations resulted from the insistence that the mobile services should be cost-effective and not require additional efforts from the organizations involved. Stakeholder and user requirements were discussed in meetings with various stakeholders, including potential users. Stakeholders focused on technological resources and investors were involved in the development of the prototypes. Some organizations were more involved in the streamlining of operational processes, including restaurants, hotels, tourist organizations, local advertisers, landowners, inspectors and supporting facility providers. Although these stakeholders would potentially benefit the most from the services, they proved reluctant to take any business risks with the innovative solutions. The results of the survey indicated that, while developing applications that may be relevant from a provider point of view, it is important to take the attitudes, opinions and behavior of the inhabitants of the specific region into account, using theories and concepts that are relevant from the perspective of Information Systems acceptance theories. In literature on regional development, the relevance of the match of regional demand and supply is often emphasized at a macro-level. Our research clearly shows that using a micro-level approach can shed light on potential problem areas.
9.7 Will Mobile TV Be a Value Service for Consumers?
9.7
213
Will Mobile TV Be a Value Service for Consumers?
Mobile television (TV) is a new mobile service that goes beyond simple voice and data connections. It can be narrowly defined as ”real-time broadcast transmission of content to mobile devices” (Jarvenpaa and Loebbecke [202] ). Or more broadly described as ”any video played on a mobile device”. Mobile TV is often described as a natural and evolutionary next step as mobile phones get more multimedia features [81]. In the last few years, we have also seen advances in the infrastructure technology for Mobile TV. Different standards are developed worldwide for technology that allows television signals to be received on mobile devices, for example, Digital Video Broadcasting to Handheld (DVB-H) in Europe, Digital Multimedia Broadcasting (DMB) in South Korea, Integrated Service Digital Broadcasting-Territorial mobile multimedia (ISDM-Tmm) in Japan, China Mobile Multimedia Broadcasting (CMMB) in China, and Forward Link Only (FLO) in the USA. Research and development is taking place for a multi-standard-multi-platform technology that will make it possible for device manufactures to bring mobile TV to all portable devices (http://www.Axel.fi). The multi-standard-multi-platform technology would, when in place, support all the key mobile TV standards and it is seen as a way to turn mobile TV to a mass-market service. The development of mobile TV technology for mobile devices and infrastructures is mov-ing the development of mobile services to a new phase, in which we are able to get more knowledge of the consumers’ behaviors regardless of their location. Technically, in broad-casting networks the same signal is transmitted to all users. This means that we are not able to get any detailed information about the number of users, their location or the quality of the signal they experience. In the case of cable TV services information can be found and used about the fixed location of the audience. When it comes to cellular (mobile) networks there exist a return channel which means that we can know who the consumers are and how many they are, their movements and the quality of service they experience. Another important thing to observe is that in point-to-point systems different signals and content is transmitted to all users. This is one of the significant differences between broadcasting and cellular networks [184]. Mobile TV which is a broadcasting service for the service provider but a cellular service for the user is promoted as a nice and useful service for the consumers and a large and growing market for the service providers. So far this has not actually worked out in this way. The advances and readiness of the mobile TV technology have triggered expectations for mobile TV from consumers. It has been predicted that mobile TV will attract as many as 500 million consumers worldwide and generate revenue of up to 20 Billion Euros by 2011. Jupiter Research on the other hand predicts that by 2013 some 330 million consumers will have mobile TV enabled phones. On the contrary to the positive forecasting, there is a big gap between the prediction and the actual usage of the service by consumers. There is insufficient evidence that mobile TV will be truly used by consumers, and there is no certainty that it will return the investments of the shareholders [115]. In the media discussion in Finland mobile TV has for a couple of years been widely promoted as an
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attractive service for users; several pilot studies and surveys have been conducted among potential users. The argument has been that as Finland is at the forefront of advanced mobile technology, the adoption and use of mobile TV would create an exciting example for the global industry [81]. But in March 2009 the service was closed down due to the fact that there is very low demand from the consumers. In this Section, we present some empirical results that show the gap between the promoted uptake of mobile TV services and the actual use of the services in the daily lives of Finnish consumers. We first discuss the Braudel Rule which is fundamental for creating mobile value service. We then review the mobile TV development in some countries and the results of a number of earlier user studies. Then we present the results on consumers’ use and intention to use mobile TV based on a longitudinal summary of consumer surveys which have been carried out in Finland each year 2005 - 2008.
9.7.1
The Braudel Rule
Fernand Braudel, one of the most influential historians of the twentieth century, has identified the core of progress in civilization and capitalism as expanding the limits of the possible in the structures of everyday life of all people [38]. Braudel [38] studied several technological developments, or in his term ”techniques”, in the areas of the military, transport and agriculture. He concludes that people start trying a new technology when they meet ”limits” within the structures of their everyday lives, and the existing technology is not able to help them cope with the ”limits”. If the new technology expands the ”limits” of what is possible for people, enabling them to achieve what was believed to be impossible, then the technology will be adopted and used successfully by members of the society. On the other hand, as long as everyday life continues without problem, within the structures inherited from the past, people will not have any economic motivation to adopt a new technology. A recent view of Braudel’s thinking on the impact of technology on everyday life has been presented by Keen and Mackintosh [212]. They formed the Braudel Rule and described it in the following way: ”freedom becomes value when it changes the limits of the possible within the structures of everyday life” (ibid. p. 31). They have then applied this Rule to analyze the links between technology and what they call ”the freedom economy”, another term they introduced. They assert that it is the best base we have for planning in the unpredictable world of mobile commerce and wireless innovation. They classify the impact of technology on everyday life in three different categories: ”freedom”, ”convenience” and ”features”. Freedom is the main driver of the adoption and diffusion of a mobile innovation, it changes/expands what is possible in the structures of everyday life; conveniences and features do not fit the Braudel Rule, because they are neat ideas and solutions to problems that no one may care about. As an illustration of the Braudel Rule, they argue that interactive TV and Web TV are not killer applications that build on new technologies; they are business killers (ibid. p. 36-37). They show that interactive TV may bring consumers convenience and more choices or features when casting votes or when watching a pay-per-view movie; the technology does not change the structures of people’s
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everyday lives. As mobile TV in some sense is a further development of interactive TV and Internet TV Keen’s and Mackintosh’s arguments apply to mobile TV as well. Watching TV is one of most people’s daily routines. It has become embedded in the structures of our daily lives since TV was first introduced to consumers in the 1930s. Mobile TV has the potential to expand the current home-oriented TV routines to other time slots and places, but two conditions have to be met to trigger people’s adoption of mobile TV. The first is that there are ”limits” to our routines to watch TV that we are aware of and annoyed about, and the second is that existing TV technology cannot help consumers to overcome these ”limits”. The promoters of mobile TV claim that we should be able to watch TV anytime and any-place whenever there is something being broadcast that may be of interest to us. Studies show, however, that this may not be true for consumers in general - we will show some of the studies in the following - even if the mobile TV promoters bring up a number of study or peer groups that have found the mobile TV experience delightful. Using the Braudel Rule we could argue that one of the reasons why it is hard to find a growing demand for mobile TV is that people in general find the traditional routine of watching TV in the peace of their homes and in t heir favorite recliner a superior routine to any other mode of TV watching.
9.7.2
A Brief Review: The Mobile TV Development and User Studies
In recent years, there have been numerous trials worldwide parallel with launching mobile TV. Different technologies have been adopted and deployed, and various users’ experiences have been reported. In the following, we briefly review the development of mobile TV and relevant user studies in the leading countries, e.g. South Korea, Japan, EU, China and USA. A summary of the review is shown in Table 9.2 at the end of this section. South Korea is at the forefront of the development of mobile TV services both in Asia and in the world. The country, together with Japan, jointly launched the MBSAT satellite in March 2004, which provides the world’s first satellite based multimedia mobile broadcasting services for mobile users [310]. In May 2005, South Korea officially introduced the world’s first mobile TV services on its own standard Satellite-Digital Multimedia Broadcasting (S-DMB) to consumers [295]. After four months, a qualitative study of eight young early adopters [114] showed that on average, users viewed mobile TV six hours per week and the radio con-tents were the favorite. The value of novelty was the main motivation for adopting the services [114] . The usage had typically taken place when users were involved in evening commuting, at home, or at school. Cui et al. [114] claimed that mobile TV should be considered as personal TV, since the value of mobile TV is in its personal use rather than mobility. Another study also reported that users’ experience of flow and contents are the key factors in influencing the adoption of mobile TV in South Korea [206].
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Integrated Service Digital Broadcasting-Territorial mobile multimedia (ISDBTmm) is a Japanese standard for mobile TV. The services called ”One-Seg” was launched on April 1, 2006, and by July 2007, more than 10 million One-Seg mobile handsets had been shipped in the Japanese domestic market. Japanese consumers’ behavior was studied by Miyauchi et.al. in Tokyo, August 2007 [257]. Differences were found between the users who were using mobile TV and those using mobile video contents during their commuting time. The results suggested that mobile TV users wanted to be relaxed after work with as little as possible effort to set up the services. By contrast, mobile video users are more tolerant in content preparation due to the fact that they will study the contents for enhancing their working skills. The MediaFLO (Media Forward Link Only) which is developed by Qualcomm has been tested in November, 2007 in Tokyo. It has become a strong competitor of ISDB-Tmm in the Japanese market [286]. The EU has implemented a three-pillar commission strategy for rolling out mobile TV services across Europe in 2008. The adoption of the DVB-H standard for mobile broadcasting has been processed in the 27 member states. To date, DVB-H has been commercially launched in Austria on 30 May, 2008, Finland on 1 December, 2006, Italy on 5 June, 2006, the Netherlands on 19 August 2008, and in two European countries outside the EU, i.e. in Albania on 20 December, 2006, and Switzerland on June, 2008, with trials in several other European countries. In most pilot trials, consumers were reported to be in favor of the services provided. Consumers have experienced difficulties with some of the mobile devices, which were regarded as too heavy and clumsy to use when watching mobile TV. Consumers also complained that content providers did not provide many channels, which made their experience of use rather limited. The unclear pricing model also made consumers reluctant to use the services. In general, the uptake of mobile TV in Europe is relatively slower than it is in South Korea and Japan. Italy, which is the most successful country in the field in Europe, has attracted 1% of its mobile consumers to use mobile TV services. China is the largest mobile market in the world with up to 600 million mobile subscribers by the mid-2008, according to the figures presented by ITU. Several mobile TV pilot trials have been launched in China since 2005 based on different mobile broadcasting standards, e.g., DVB-H, DMB and MediaFLO (Forward Link Only) . China’s own standard CMMB (China Mobile Multimedia Broadcasting), as a showcase for the Beijing Olympic 2008, covered 37 major cities and gained a substantial market share. Users were attracted by the services and showed a high degree of satisfaction. The full coverage of the Olympic Games also promoted usage nationally. CMMB is going to flourish in the coming years in China and will probably change people’s TV watching experiences. Although several trials of DVB and DMB mobile TV services have been conducted, Medi-aFLO is likely to become the most popular standard in the US market. Verizon Wireless and AT&T have launched the services-FLO TV - to their subscribers in 2007 and 2008 respectively. In trials with test consumers, Qualcomm,
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the developer of FLO, has found that consumers were willing to spend long periods watching newscasts, sporting events, and even full-length feature films on their mobile devices. The company is also confident that FLO TV will emulate the TV in the living room. At the Mobile World Congress in Barcelona in February 2009, the best mobile TV service was awarded to a company named MobiTV. The company launched its first television service for mobile phones in 2003 together with Sprint. Today there are more than 5 million subscribers in North and South America. The service offered for consumers is made for mobile and video on demand content from major broadcasters. The mobile users of the service are provided personalized and targeted advertising and the MobiTV company is providing its broadcast and operator partners with a profile of the preferences and general behavior of the end user. Here we can summarize that there are different broadcasting standards available for launching mobile TV. The EU has decided that the standard is DVB-H, in the US the most likely standard will be MediaFLO, South Korea have their own standard S-DMB and Japan has ISDB-Tmm but is also testing MediaFLO. China has tested MediaFLO and DVB-H in addition to their own standard CMMB. The early Mobile TV trials reveal that early users watch mobile TV basically while traveling, commuting and at home. But we have to be note that the consumers’ early experience of using the service does not necessarily lead to increased use at a later stage, and it also does not show that more and more consumers will start using the service if they will have to pay for the service.
Table 9.2 Mobile development Country /region
Standard
Launch time
South Korea
S-DMB
May 2006
Japan
ISDB-Tmm FLO trial
1 April,2006 November 2007
EU
DVB-H
China
CMMB Other standards trails
USA
FLO Others trials
Mobile TV development User experience • Used it 6 hours/week • Used it while commuting • Users experience of flow and contents are key factors for adoption
5 June, 2006 Italy 1 Dec. 2006 Finland 20 Dec. 2006 Albania 30 May 2008 Austria June 2008 Switzerland 19 August 2008 the Netherland August 2008 Pilot trials since 2005 Trials in 2007 & 2008 by AT & T MobiTV service in 2003
• •
Used it while commuting Different behaviors are found if viewing mobile TV services or video on demand contents
• • • •
Difficulties with some of the mobile TV devices No much contents provided Unclear pricing model Uptake is general slower than that in South Korea and Japan
• •
High satisfaction with the service of broadcasting the Olympic Games Likely to change people’ TV viewing experience
• • •
Willing to spend more time on mobile TV Will emulate the TV in the living room MobiTV provides more services
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9.7.3
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Early Experience from Finland
Finland, as one of the pioneers of providing mobile TV services with the digital video broadcasting-handheld (DVB-H) standard, has accumulated some national knowledge from two pilot studies carried out in years 2005 and 2007 in Helsinki [81, 207] and from the Podcast-ing project (2005-2007) [268]. In 2006, the Finnish government offered a 20-year license for mobile television broadcast and service, based on DVB-H mobile TV. The license was acquired by Digita Oy. The results from the pilot studies show that Finland has been leading the mobile TV development in the world. One of the early results shows that content is a key driver for the development of mobile TV [81] but it appears that the service providers were not aware of this fact or that they simply did not have the technical resources nor the time to produce content tailor-made for mobile TV. Broadcasting, Internet content, video-on-demand, Podcasting, and various forms of interactivity will merge in order to deliver sufficient content that is wanted by consumers. Technical challenges, including the convergences of different delivery channels or networks and compatible with the consumers’ mobile TV devices have to be solved in order to make mass marketing a possibility. On of the pilot studies claims that consumers want to watch TV programs on their mobile phones and that they prefer a fixed price model; the pilot study participants spent about 20 minutes per day watching mobile TV and the usual context was while traveling with public transport to and from work; the most common reason for watching TV on a mobile phone was in order to relax or to keep up to date with the latest news [81] .
9.7.4
The Use of Mobile TV in Finland - The Consumer Survey
A number of consumer surveys were carried out in Finland in 2003-2008 to study the consumers’ demand for and actual use of mobile services. Each year a random sample of 1000 Finns was selected, the data was collected with a questionnaire mailed to the consumers and a lottery on a top-level mobile phone contributed to a very good answering rate, close to 50% each year which is rather remarkable for a consumer survey. Statistical tests each year showed that the sample was representative for the Finnish population. Among the mobile services included in the study, mobile TV became of interest for the first time in 2005. In the survey, both the consumers’ current use of and future intention to use mobile TV were studied in the years 2005-2008. Table 9.3 summarizes the questions and the scales used in the surveys. In 2005 there were not too many Finns who had a mobile phone with mobile TV watching capabilities. In 2005-2007 it was commonly assumed that if mobile TV would be launched in Finland it would require a separate license for watching. There were several heated discussions about copyright issues and whether mobile TV was a separate channel (and thus would require a separate license). The situation changed in spring 2007 when the copyright organizations and the TV-reporters found a joint understanding
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Table 9.3 Consumer use of mobile TV in Finland
of how to handle the copyright fees for mobile TV. Thus the mobile TV license was no longer an issue in 2008.
9.7.5
Data Analysis and the Results
An aggregation algorithm [326] was used to compute a representative summary of consumer behavior regarding the actual use and the intention to use mobile TV. As the samples are representative for Finnish consumers this summary would give an indication for what can be expected of the future of mobile TV. In our model we used an aggregation method (see [326] for details) to find some representative consumer behavior for actual use and intention to use mobile TV. In particular, we computed an average score that measured the degree to which the consumers accepted mobile TV. Since the number of respondents was high (a 40%-46% response rates) in the yearly samples we assume that the behavior of the group we analyzed reflects (in a statistically significant way) all consumers’ behavior in the market. Thus, we can view the derived aver-age score as the degree to which consumers would accept using mobile TV. In the following, we show the results from the data sets that were collected on the consumers’ current and future intentions to use mobile TV. The scale used was defined as follows: a category of non users (scale value 1), indecisive users (scale value 2), and users (scale values 3, 4 and 5). On the other hand, when analyzing the answers we got on the consumers’ future intention to use mobile TV, we redefined scale values 1 and 2 (no intention to use), scale value 3 (neutral to use), and scale values 4 and 5 (intention to use). After transforming the scales in this way, the responses of each consumer on its current or future use of mobile TV could be characterized by the following three choices:
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− used or will use the service (acceptance); − did not use or will not use the service (rejection); − tried or will try mobile TV without having a decisive opinion about using or rejecting it (trial). We assigned specific values to each of these three choices to quantify the degree of acceptance of using mobile TV by a particular consumer. Let us represent (i) the acceptance by value 1, (ii) the rejection by value -1, and (iii) the trial by value 0. Hence, for every consumer we have a definite value that measures the degree to which it accepts mobile TV. Let ai be the degree that consumer i accepts the service, and let us assume that there are n respondents from which we received answers. Then, ai ∈ {−1, 0, 1}, for i = 1, 2, . . . , n, and we can compute the average score by r=
1 n ∑ ai . n i=1
We also aggregated the data from 2006-2007 to compute the average value of consumers’ behaviors towards mobile TV (with or without requiring a license) by r=
r(license) + r(nolicense) 2
where r(license) and r(nolicense) denotes the average score of all consumers in our sample who specified its usage towards mobile TV at years 2006 and 2007 with license and without license, respectively. The results are shown in Fig. 9.9 - consumers’ current usage of mobile TV and Fig. 9.10 - consumers’ future intention to use mobile TV. Fig. 9.9 Consumers’ current usage of mobile TV
The results show that the consumers are not yet ready for using mobile TV in their daily lives as the acceptance level of mobile TV in Finland is very low. The aggregate value of the representative consumer behavior towards mobile TV is far below the value 0 in the whole period of 2005-2008. Consumers have a bit more positive evaluation of mobile TV 2007-2008 which is shown by an increasing curve in the figure. Consumers appear to be more favorable to mobile TV for future
9.7 Will Mobile TV Be a Value Service for Consumers?
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Fig. 9.10 Consumers’ future intention to use mobile TV
use - they think that they may want to watch mobile TV. In studies of other mobile services with the data sets it appears that this ”thinking” does not get trans-formed to actual use, not even with time lags of 2-3 years (cf. [326] ). Mobile TV promoters could argue that mobile TV shows a positive trend that might result in consumer acceptance sometimes in the future. The surveys also collected information on what kind of mobile phones the Finnish consumers have, which is shown in Table 9.3. There is a clear trend that more and more consumers have 3G mobile phones that are capable of getting wireless broadcasting signals and would support mobile TV. In 2008, 23.5% of the consumers had 3G mobile phones. It indicates that consumers at least have the possibility to make mobile TV part of their daily lives. But as the results show in Figures 9.9 and 9.10, there is still some years to go before the adoption rate will go above the value 0, which means that mobile TV service would be widely used by the consumers. The results also prove that beyond the pilots, Finnish consumers have a rather low acceptance rate of mobile TV services.
9.7.6
Discussion
The results from the preliminary review of the leading countries of mobile TV development and the Finnish market show that mobile TV is now technologically feasible. Different standards have been launched in the countries which is leading the development of mobile TV, and trials has been implemented almost world wide. Mobile TV is likely to become a promising mobile service for customers. In the various pilot and user studies, the results have shown that consumers do use the service in some occasions, e.g. while they are commuting. More contents are desired, like special events (Olympic Games), video on demand contents for learning purpose, or news briefing etc. Service providers are very enthusiastic that mobile TV may replace the traditional TV in our living room. Negative user experiences are also found that hinder consumers acceptance of the service. The results from the pilot studies in Finland as well as our survey have indicated the similar insights with those from other countries. Consumers do have some good experiences during the pilot, but in the long term, they are reluctance to adopt the service in their daily life. The survey data have shown the increasing interests of consumers to use the service in the future. But it remains unclear whether it will become a success, and
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when it will come to the tipping point. The empirical results show that there is an obvious gap between the business hype of up-take mobile TV service and the consumers’ acceptance of it in their daily life. There is insufficient evidence that it will be truly adopted, and there is no certainty that it will return the investments of the shareholders [115]. So far, not much research has been carried out to describe and explain the adoption and diffusion processes of mobile TV. Therefore, the present research did not give us sufficient evidence to find out which factors facilitate or hinder consumers’ acceptance of mobile TV. Consumers’ TV watching experiences and requests for content are mostly embedded in their daily life behaviors. It is important that we can observe their TV viewing behaviors from their daily life, their needs, their values and the social circumstances. We found out that mobile TV could have the potential to expand the current home-oriented TV routines to other time slots and places. The Braudel Rule points out that there is a mechanism that needs to be activated for this to happen: first there should be ”limits” to our routines for watching TV that we are aware of and annoyed with, and second the existing TV technology should either hinder or be unhelpful to consumers if they want to overcome these ”limits”. The promoters of mobile TV claim that we should be able to watch TV anytime and any-place whenever there is something being broadcast that may be of interest to us (here is a ”limit”). The studies we have been discussing show, however, that this is not a genuine need for most consumers even if there are a number groups of technology promoters that have found the mobile TV experience delightful. We have argued that one of the reasons why it is hard to find a growing demand for mobile TV is that people like to watch TV in their favorite recliner and that this is a superior routine to trying to catch the action on the small screen of a mobile phone when traveling on public transport or when walking around in public places in the noise of urban traffic and/or in daytime sunshine. In this Section, we first presented some evidence of the development of mobile TV in Finland and other countries. The results show that the mobile TV, as a new mobile service to consumers, is far from being accepted and used in people’s daily life. The dilemma has to be studied and solutions have to be provided from both through academic research and business development. Without a growing demand from consumers and a growing daily use of the new media service, mobile TV will not get into any growth phase in the media market and will probably disappear as one more mobile technology that was possible but for which it was hard to find any actual need. This will of course have a negative impact on the technology development of mobile TV, and business investments in the service. This ”soon to die” mobile service may also disappoint consumers to the extent that they would be reluctant to try any similar new services that build on mobile technology. Therefore, some new approach to study when and how consumers will use mobile TV is necessary. We have here shown how the Braudel Rule could be used to guide future research on what drivers will be the key to the adoption of mobile TV and what factors will facilitate the diffusion process. We will then have a better basis for an understanding of how mobile TV services should be designed - when the next mobile TV technology generation appears in the next 2-3 years. Technology
9.8 Mobile Value Services: Conclusions and the Next Steps
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development will never be denied even if we as consumers try to resist and claim that we see no use for the technology.
9.8
Mobile Value Services: Conclusions and the Next Steps
Mobile service design needs to become a real tool based in a working theory because we have found the need for it to become more and more urgent every year as more and more people start to depend on their mobile phones - and the mobile services they make possible and support - as part of their everyday lives. On the other hand, market studies us and others have carried out - we discussed results from Finland in 2003-2009 - show that demand for mobile services is not increasing. This seems to prove that mobile phone users do not enhance their use of mobile services and do not make use of the new technical capabilities they get when they upgrade their mobile phones to new versions. We found and had to accept the fact that ordinary mobile phone users in Finland are more concerned with the mobile services than the technology, and that the basic form of the services may well be sufficient for the everyday needs of a user. Even in Japan and Korea, which are considered to be forerunners in the adoption of mobile services, rather basic services (messaging and ring tones) have been most successful. Then - not surprisingly - a number of studies in several European countries have found that basic services have been the most popular but more advanced services have not yet found their way into the everyday lives of consumers. We assumed that part of the issues at hand are design issues - the mobile services do not have the proper design to fit the actual needs of users in their everyday life routines, i.e. where mobile services would matter most. We worked out (i) that consumers do not use mobile services they think they would (or even should) be using because they have no need for them; (ii) that there are design flaws that should be understood and corrected. We have studied the design problems by working out a number of prototypes for mobile services that we actually designed as mobile value services. It turned out that the services will obviously produce value for the users (tourists and visitors) but that it (so far) has not been a business success, which showed that there was also a need for work on a business model; thus we worked out a generic business model that could be used for implementing mobile value ser˚ vices. In our case context - the Aland Islands - that was no success either, which have a number of other reasons than simply the design of mobile value services. We worked out a formal design process for mobile value services and offered it as a first step towards a meta-theory on the design that follows some of the guidelines for the use of design science in Information Systems research that were developed by Hevner et al [183]. They work with the design and implementation of an artifact that should be (i) innovative and purposeful; (ii) thoroughly evaluated; (iii) proved to solve previously unsolved problems or known problems in more effective or efficient ways; (iv) be rigorously defined, formally represented, coherent and internally consistent; (v) created through search processes in a problem space; (vi) a design that satisfies the requirements and constraints of the problem it was meant to solve.
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This design process points towards the possibilities to use some more formal and technology oriented approach. We have also built insight with the support of theory development in other fields. Research in service logic and service management (Gr¨onroos [170]) suggests that the mobile service development work has been on the wrong track: ”the firm cannot create value for customers; its role is, first of all, to serve as value facilitator; by providing customers with value-facilitating goods and services as input resources into customers’ self-service value-generating processes the firm is indirectly involved in the customers’ value creation” (Gr¨onroos [170], page 310). Thus for mobile value services to be developed we should aim at building the services with the users, not for the users. This is rather a different process from the traditional mobile service design as practiced by MNOs, in which they worked out what would be ”cool and nice” from an engineering viewpoint and then assumed that the service users would be as excited as them about the possibilities offered by the technical solutions. The idea that mobile services ”should be built with the users, not for the users” appears in a similar form in the so-called Braudel Rule: ”freedom becomes value when it changes the limits of the possible within the structures of everyday life”. We cannot find out what will ”change the limits of the possible” without working with the users (and potential users) themselves. We also worked on and tested the assumption that mobile TV could have the potential to expand the current home-oriented TV routines to other time slots and places. The Braudel Rule points out that there is a mechanism that needs to be activated for this to happen: first there should be ”limits” to our routines for watching TV that we are aware of and annoyed with, and second the existing TV technology should either hinder or be unhelpful to consumers if they want to overcome these ”limits”. The promoters of mobile TV claim that we should be able to watch TV anytime and any-place whenever there is something being broadcast that may be of interest to us (here is a ”limit”). We found out that this is not a genuine need for most consumers - the promoters assumed that this need would be genuine without bothering to actually test it. The Braudel Rule is actually quite helpful in finding out what would be the genuine needs and should be used in the testing programs for mobile value services. The work we have done on fuzzy ontology actually shows the way towards a new design program for mobile value services, which we will briefly outline in the following: • Let us assume that mobile services could be designed, built and offered just a moment before they are going to be used [this would eliminate the problem of finding the optimal portfolio of mobile services to offer a potential service user] • Let us further assume that each service will be adaptive to the context for which it is going to be used [this would eliminate the need for complex generic designs that could be used in any context] • Let us also assume that each service will be adaptive to the user’s cognitive profile, i.e. it is built in such a way that it will fit the knowledge and skill level of the user [this will reduce the need for time-consuming training and the use of extensive manuals]
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• Mobile value services are mobile services that are (i) adaptive to the cognitive profile of the user, (ii) adaptive to the context, and (iii) designed, built and paid for [at a price that is relevant to the context and the needs of the user] at the moment when they are going to be used • Mobile value services are built of entities which are generic service components; the sets of service components may be very large [typically hundreds of thousands or millions of component]; the sets of service components may be part open source and part proprietary, commercial; cloud computing resources would be typical platforms for storing, maintaining and accessing service components • Fuzzy ontology can be used as a meta-data framework for identifying relevant entities and for composing context adaptive services; the composition of services can be done with approximate reasoning schemes to find the most suitable sets of compositions for any context for which services are needed • Approximate reasoning schemes, OWA-operators and fuzzy algorithms can be used to adapt the sets of compositions to the cognitive profiles of the users (or potential users) • The composition of entities to mobile value services should be automatic, very fast and with a minimum input from the users; possible mobile technology platforms for the compositions could be Android, iOS and the new Windows 7; the Android offers middleware for building mobile clouds, which could be a possible way to build the resource base of sets of service components The actual design of mobile value services using fuzzy ontology and approximate reasoning is forthcoming in a series of research papers.
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Index
α -cut 8 ˚ Aland Islands 201, 203, 205, 223 Łukasiewicz t-conorm 11 Łukasiewicz t-norm 10 additive generator 11 andlike OWA operator 19 Android 225 approximate reasoning 225 Archimedean t-norm 11 arithmetic mean 17 averaging operator 16 bag 17 ball-shaped joint distribution 61 Bayes modeling 147 Bayesian probabilistic model 148 Bayesian reasoning 149 Bellman-Zadeh’s principle 19, 178 Black-Scholes formula 124 Bomarsund mobile guide viii, 187, 193 Braudel rule 224 Brownian motion 105 certainty equivalent 134 chance criterion 140 complement of a fuzzy set 10 conditional probability 59, 60, 63 convex fuzzy set 8 correlation coefficient 25 covariance 25 credibilistic expected value 126 credibilistic portfolio selection 144 credibility measure 23
cumulative density function 155 cutting plane algorithm 144 data envelopment analysis 143 deferrable decision opportunity 104 deferral time 102 degree of membership 7 degree of necessity 22 degree of possibility 134 degree of similarity 180 density function 24 description logic 169 design science 187 discounted cash flow 101 entropy 19 equality of fuzzy sets 10 event report 174 expected cash flows 104 expected costs 104 expected value 24 exponential random variable 70 extended addition 13, 73 extended subtraction 13 extension principle 12 Finnish forest industry 4, 108 flexible budget constraints 95 forest industry 3 fuzzy aggregation 180 fuzzy algorithms 225 fuzzy Black-Scholes model 132 fuzzy cash flow 119 fuzzy decision making 178
248 fuzzy downside risk function 143 fuzzy mathematical programming 95 fuzzy NPV 125 fuzzy number 8, 28 fuzzy number of type LR 9 fuzzy ontology 167, 180, 187 fuzzy pay-off method 125 fuzzy point 10 fuzzy quantity 7 fuzzy random variable 36 fuzzy real option 111 fuzzy set 7 fuzzy subsethood 10 fuzzy taxonomy 178 fuzzy-stochastic model 140 Gamma distribution 146 Gaussian distribution 155 generalized p-mean 17 geometric mean 17 Gibbs sampler algorithm 154 giga-investment 129 grid computing 146, 156 group assignment 97 group decision 88 Hamacher t-conorm 12 Hamacher t-norm 11 hard decisions 1 harmonic mean 17 high performance computing 145 identity quantifier 21 index of interactivity 69 interactive difference 76 interactive fuzzy numbers 73 interactive sum 75, 76 intersection of fuzzy sets 10 interval-valued probabilistic mean 31
Index lower modal value 31 lower possibilistic mean 31 lower probability mean 31 Mamdani t-norm 72 managerial flexibility 132 marginal distributions 37, 39, 44, 54, 55 Markowitz optimal portfolio 133 maximal entropy 77 maximum t-conorm 11 mean absolute semi-deviation 143 mean-risk model 144 mean-semivariance 142 measure of andness 19 measure of consensus 89, 96 measure of dispersion 19 measure of interactivity 57, 69 measure of orness 18 minimal variability 78 minimax disparity model 85 minimum t-norm 10 MobiFish 187, 202, 203 mobile service design 187 mobile service domain 188 mobile TV 213, 214, 218, 221 mobile value services 189, 225 Monte Carlo method 115 negotiation support systems 3 net present value 110 Nguyen’s theorem 14 NIM project 202, 210 non-interactive fuzzy numbers 31, 44 non-interactive sum 30 non-probabilistic factors 133 normal fuzzy set 7 now-or-never decision 101 nuggets 172
Karush-Kuhn-Tucker conditions 94, 138 keyword dependencies 178 keyword ontology 178 Knowledge Mobilization project 167
OCA function 89 Olympic OWA operator 97 ontology 168 ontology adapter 179 optimal R&D project portfolio 132 option valuation lattice 106 orlike OWA operator 19 orness level 77, 87 OWA operator 17
Lagrange multipliers 78 linguistic quantifiers 20
paper machine 114, 167 part-of relationships 175
joint possibility distribution 22, 27, 37, 54, 61, 73, 76 joint random variable 24
Index pay-off distribution 125 perfect correlation 45, 73 Poisson distribution 152 Poisson model 151 Poisson process 70, 149 Poisson-Gamma mixture 152 portfolio selection problem 136 possibilistic correlation coefficient 42, 72 possibilistic correlation ratio 56 possibilistic efficient portfolio 142 possibilistic efficient portfolios 142 possibilistic interactivity 64 possibilistic mean value 28, 66, 105 possibilistic mean-standard deviation 141 possibilistic portfolio selection 136 possibilistic variance 37 possibility distribution 27, 73, 116 predictive possibilities 164 predictive probabilities 163 preemptive priority factor 88 priority vector 88 probabilistic correlation ratio 55 probabilistic metric space 10 probabilistic t-conorm 11 product t-norm 10 product-sum 15 Prot´eg´e ontology editor 180 pulp and paper industry 129 quadratic OCA function 92 quantifier-guided aggregation 87 quasi fuzzy number 64, 67, 69 quasi-arithmetic mean 17 quasi-triangular fuzzy number 10 R&D budget 130 R&D investment decision 124 R&D project portfolio selection 132 random variable 56 real option 102 real option value 125 real options model 113 real R&D option 132 Resource Management System 157 resource provider 146 risk assessment 148 risk-adjusted discount rate 103 risk-neutral valuation 119
249 scenario analysis 114 security returns 144 semantic web 168 sequential minimal optimization 143 service innovation 188 service level agreement 146 service support 207 smart computing methods 3 smart phones 193 soft decision-making 3 standard normal distribution 102 sup-min convolution 76 support of fuzzy set 7 systematic risk assessment 146 t-conorm 11 t-conorm-based union 12 t-norm 10, 73 t-norm-based extension principle 23 t-norm-based intersection 11 Takagi-Sugeno reasoning scheme 180 trade-off 16 trapezoidal fuzzy number 9, 34, 36, 104, 119, 135, 143 trapezoidal possibility distribution 104 triangular fuzzy number 8, 29, 36 uncertain variable 144 uniform distribution 28, 58, 65 union of fuzzy sets 10 upper modal value 31 upper possibilistic mean 31 upper probability mean 31 utility theory 135 value at risk 140 variance 25 Waeno project 129 weak t-norm 11, 41, 54 web ontology language 169 weighted index of interactivity 69 weighted possibilistic covariance 37 weighted possibilistic mean 29, 30 weighted possibilistic variance 35 weighting function 27, 32, 55 weighting vector 80 window type OWA operator 18 Windows 7 225
