Production Planning in Production Networks: Models for Medium and Short-term Planning

Production Planning in Production Networks

Pierluigi Argoneto • Giovanni Perrone Paolo Renna • Giovanna Lo Nigro Manfredi Bruccoleri • Sergio Noto La Diega

Production Planning in Production Networks Models for Medium and Short-term Planning

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Pierluigi Argoneto, Dr. Ing. Dipartimento di Tecnologia Meccanica Produzione ed Ingegneria Gestionale (DTMPIG) Università degli Studi di Palermo Viale delle Scienze 90128 Palermo Italy

Giovanni Perrone, Prof. Ing. Dipartimento di Tecnologia Meccanica Produzione ed Ingegneria Gestionale (DTMPIG) Università degli Studi di Palermo Viale delle Scienze 90128 Palermo Italy

Paolo Renna, Dr. Ing. Dipartimento di Ingegneria e Fisica dell’Ambiente (DIFA) Università degli Studi della Basilicata Macchia Romana 85100 Potenza Italy

Giovanna Lo Nigro, Prof. Ing. Dipartimento di Tecnologia Meccanica Produzione ed Ingegneria Gestionale (DTMPIG) Università degli Studi di Palermo Viale delle Scienze 90128 Palermo Italy

Manfredi Bruccoleri, Dr. Ing. Dipartimento di Tecnologia Meccanica Produzione ed Ingegneria Gestionale (DTMPIG) Università degli Studi di Palermo Viale delle Scienze 90128 Palermo Italy

Sergio Noto La Diega, Prof. Ing. Dipartimento di Tecnologia Meccanica Produzione ed Ingegneria Gestionale (DTMPIG) Università degli Studi di Palermo Viale delle Scienze 90128 Palermo Italy

ISBN 978-1-84800-057-5

e-ISBN 978-1-84800-058-2

DOI 10.1007/978-1-84800-058-2 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2008921524 © 2008 Springer-Verlag London Limited Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copy-right Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: eStudio Calamar S.L., Girona, Spain Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Preface

Globalisation is pushing manufacturing companies toward a more distributed production approach. Indeed, corporate manufacturing firms are spreading their production all over the world in order to stay close to the customers, while medium manufacturing firms organise themselves in networks in order to scale their production to a global level. This tendency is putting a lot of stress on production planning. Indeed, the more distributed production facilities are, the more difficult and complex production planning becomes. Both multi-plant facility and manufacturing networks require to be coordinated in order to reach effectiveness and efficiency required by the competitive arena. Most of the production planning tools, such as Advanced Planning and Scheduling (APS) tools, are designed to manage a centralised production, i.e. a production that is accomplished in a single plan. When it comes to managing a network of plants or a manufacturing network, the application of APS tools becomes complex, since the complexity of the planning problem scales up and most of all, because the necessity to recollect data from different sites in a centralised planner causes several problems of data consistency and updating. This is the reason why, in managing production networks, decentralised production planning tools have been recommended by researchers and industrial managers. Indeed, decentralised production planning has several positive outcomes when it comes to managing production networks: a) data management is easier and more trustworthy; b) production planning systems are more robust, scalable, reliable; c) as a result production planning activities are easier and more reliable. However, distributed production planning has some drawbacks: a) production planning outcomes are considered less efficient than centralised production planning outcomes; b) coordination among the different entities involved in production planning activities needs to be properly designed; c) properly commercial tools are not available yet.

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This book concerns the above-mentioned issues. It faces the production planning problem in complex and very structured manufacturing firms such as those involved in the semiconductor industry. The book presents research work answering major issues dealing with decentralised production planning; in particular, for the considered research context it shows: ƒ How to structure and organise decentralised production planning for a complex multi-national corporate; ƒ How to organise business processes among the decentralised entities involved in the production planning process; ƒ Which kind of methodological tools can be used to obtain a reliable, effective and efficient cooperative production plan; ƒ Which kind of technology can be used to develop a distributed cooperative production planning system; ƒ A benchmark analysis showing how the proposed approach and methodologies allow obtaining realistic production plans in the considered research context. Very briefly (a more detailed outline of this book is given at the end of Chapter 1), the book is organised as follows: Chapter 1 introduces the research problem and the research context with reference to the state of the art; Chapters 2 and 3 provide respectively an overview of Game Theory and Negotiation Theory which are the methodological tools used to build cooperative production plans in a distributed environment; Chapter 4 presents an overview of the Agent Theory that is the technological tool suggested to develop a distributed production planning tool; Chapter 5 presents our approach for organising and structuring a distributed production planning system in a complex environment such as the semiconductor industry. Chapters 6 and 7 present the methodological approach suggested to reach a cooperative production plan in distributed networks such as described in Chapter 5. Specifically, Chapter 6 presents the methodological approach for planning production at medium-term level, and Chapter 7 at plant level. Chapter 8 presents the integration of the methodological approach presented in Chapters 6 and 7 in order to show how the proposed algorithms integrate with each other in order to provide a consistent production planning tool; finally, Chapter 9 presents the conclusions of the research developed in the book. I wish to thank all the researchers who have been involved in this project. Special thanks go to Dr. Pierluigi Argoneto who, with his Ph.D. work, has allowed us to develop a consistent and unitary body of methodological approaches for planning production in production networks. Palermo, 25 July 2007

Giovanni Perrone

Contents

1 Introduction and Literature Overview ...........................................................1 1.1 Introduction ................................................................................................1 1.2 Production Planning in High-tech, High-volume Industry.........................2 1.3 Strategic and Tactical Level .......................................................................2 1.4 Operational Models: Optimization and Decision Support..........................5 1.4.1 Mathematical Approach ....................................................................7 1.4.2 Queuing and Stochastic Approaches .................................................8 1.4.3 Heuristics and Simulation-based Approaches ...................................8 1.5 Motivation ..................................................................................................9 1.6 Book Outline ............................................................................................10 1.7 References ................................................................................................10 2 Game Theory: an Overview...........................................................................13 2.1 Introduction ..............................................................................................13 2.2 Game Setup ..............................................................................................14 2.3 Non-cooperative Static Games .................................................................15 2.4 Existence of Equilibrium..........................................................................16 2.5 Multiple Equilibria ...................................................................................17 2.6 Dynamic Games .......................................................................................17 2.7 Simultaneous Moves: Repeated and Stochastic Games ...........................18 2.8 Cooperative Games ..................................................................................18 2.9 N-Person Cooperative Games ..................................................................19 2.10 Characteristic Function and Imputation ...................................................20 2.11 Shapley Value ..........................................................................................21 2.12 The Bargaining Game Model ...................................................................22 2.13 References ................................................................................................23 3 Negotiation: an Overview...............................................................................25 3.1 Introduction ..............................................................................................25 3.2 Negotiation and Rational Self-interested Agents .....................................28 3.3 Negotiation Models ..................................................................................29

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3.4 3.5 3.6 3.7

Underlying Principle for Electronic Negotiation......................................30 Electronic Negotiation Protocols..............................................................31 Characteristics that Differentiate Negotiations Protocols.........................31 Modelling Approaches and Solution Concepts ........................................32 3.7.1 Decision Theory ..............................................................................33 3.7.2 Game Theory...................................................................................34 3.7.3 Negotiation Analysis .......................................................................34 3.8 Strategic Negotiation................................................................................35 3.9 Negotiation Strategies ..............................................................................35 3.10 References ................................................................................................36

4 Multiple-agent Systems: an Overview...........................................................41 4.1 Introduction ..............................................................................................41 4.2 Applications .............................................................................................42 4.3 Challenging Issues....................................................................................43 4.4 Individual Agent Reasoning .....................................................................44 4.5 Observable Worlds ...................................................................................45 4.6 Stochastic Transitions and Utilities ..........................................................45 4.7 Distributed Decision Making ...................................................................47 4.8 Recognising and Resolving Conflicts.......................................................48 4.9 Communicating Agents............................................................................48 4.10 References ................................................................................................49 5 Distributed Production Planning in Reconfigurable Production Networks .........................................................................................................51 5.1 Introduction ..............................................................................................51 5.2 Production Planning in DPS.....................................................................52 5.2.1 Context of the Semiconductor Industry...........................................52 5.2.2 PP in the Considered Industrial Case ..............................................54 5.2.3 IDEF0 Architecture .........................................................................55 5.2.4 Agent Architecture ..........................................................................55 5.3 Top PP Level............................................................................................57 5.4 High PP Level ..........................................................................................57 5.5 Medium PP Level.....................................................................................58 5.6 Low PP Level ...........................................................................................58 5.7 Shop-floor PP Level .................................................................................59 5.8 References ................................................................................................60 6 Distributed Models for Planning Capacity of Reconfigurable Production Networks at Medium Term............................................................................63 6.1 Introduction ..............................................................................................63 6.2 Initial State ...............................................................................................63 6.3 The Centralised Model .............................................................................64 6.4 The Negotiation Model ............................................................................65 6.5 The Game-theoretical Model....................................................................66 6.5.1 Case 1: Characteristic Function hij > 0.............................................67 6.5.2 Case 2: Characteristic Function hij < 0.............................................67

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6.5.3 The Bargaining Solution..................................................................70 6.6 Simulation Case Study .............................................................................71 6.6.1 The Simulation Environment...........................................................71 6.6.2 Simulation Case Study ....................................................................74 6.7 Results ......................................................................................................74 6.7.1 Two-way Analysis of Variance .......................................................75 6.7.2 Design of Experiment (DoE)...........................................................88 6.8 References...................................................................................................95 7 Distributed Models for Plant Capacity Allocation.......................................97 7.1 Introduction ..............................................................................................97 7.2 Initial State ...............................................................................................97 7.3 The Centralised Model .............................................................................98 7.4 The Negotiation Model ............................................................................99 7.4.1 Generative Function ........................................................................99 7.4.2 Reactive Function............................................................................99 7.5 The Game -theoretical Model.................................................................100 7.6 The Simulation .......................................................................................101 7.6.1 The Simulation Environment.........................................................101 7.6.2 The Simulation Case Study ...........................................................104 7.7 Results ....................................................................................................105 7.7.1 Efficiency Performance Analysis: Two-way ANOVA .................105 7.7.2 Efficiency Performance Analysis: DoE.........................................115 7.7.3 Distance Performance Analysis: Two-way ANOVA ....................121 7.7.4 Distance Performance Analysis: DoE ...........................................130 7.7.5 Number of Reconfigurations Performance Analysis: Two-way ANOVA .................................................................................................136 7.7.6 Number of Reconfigurations Performance Analysis: DoE............147 7.7.7 Absolute Residual Performance Analysis: Two-way ANOVA.....153 7.7.8 Absolute Residual Performance Analysis: DoE ............................164 8 Distributed Production Planning Models: an Integrated Approach........171 8.1 Introduction ............................................................................................171 8.2 The Simulation Case Study ....................................................................172 8.3 Results ....................................................................................................173 8.3.1 Efficiency Performance Analysis: Two-way ANOVA .................173 8.3.2 Efficiency Performance Analysis: DoE.........................................182 8.3.3 Distance Performance Analysis: Two-way ANOVA ....................192 8.3.4 Distance Performance Analysis: DoE ..........................................200 8.3.5 Absolute Residual Performance Analysis: Two-ways ANOVA ...209 8.3.6 Absolute Residual Performance Analysis: DoE ............................214 8.4 Conclusions ............................................................................................223 9 Conclusions....................................................................................................225 9.1 Summary ................................................................................................225 9.2 Major Scientific Contributions of This Book .........................................226 9.3 Directions for Future Work ....................................................................227

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Appendix A: Simulation Results Related to Chapter 6...................................229 Appendix B: Simulation Input Parameters and Results Related to Chapter 7 ..........................................................................................233 Appendix C: Simulation Input Parameters and Results Related to Chapter 8 ..........................................................................................247 Index ....................................................................................................................255

1 Introduction and Literature Overview

1.1 Introduction In recent years, manufacturing companies have entered a new era in which all manufacturing enterprises must compete in a global economy. Global competition increases customers’ purchasing power, which, in turn, drives frequent introduction of new products and causes large fluctuations in product demand. To stay competitive, companies must use production systems that not only produce their goods with high productivity, but also allow for rapid response to market changes and customers’ needs. A new manufacturing capability that allows for a quick production launch of new products, with production quantities that might unexpectedly vary, becomes a necessity. Reconfigurable manufacturing systems (RMS), offer this capability. If we look at the evolution steps of factories in the last twenty years, we basically find different factory concepts and production concepts in each decade depending on different goals and criteria. Stable and well-predicted markets led to a functional factory in which know-how was concentrated in departments in order to achieve optimisation with a high flexibility of resources. Later, the necessity of a more consistent orientation to markets and products led to segmented factories. These concepts led to an enormous increase of the efficiency of the total business through independently acting units oriented towards products and markets. The growing complexity within the value-added chain can no longer be steered merely by means of hierarchical leadership in competences. In the future, networks as well as increasing cooperation and decentralisation will be regarded as a main solution. These concepts are called decentralised production networks and are regarded as a possible answer to the new challenges of the global competition in order to achieve shorter response time and stronger orientation towards customer needs, through a high innovation ability. Prerequisites for the successful participation in production networks are changeable and reconfigurable production processes, resources, structures and layouts as well as their logistical and organisational concepts. This ability is necessary to withstand the permanent change and the turbulence of the surroundings of production companies and can be described as changeability.

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Production Planning in Production Networks

1.2 Production Planning in High-tech, High-volume Industry In high-tech, high-volume industries such as semiconductors, consumer electronics and telecommunications, but also in some pharmaceutical industries, the ability of a firm to manage production capacity is arguably the most critical factor for its long-term success. Even in a stable economy, the demand for high-tech products is volatile and difficult to forecast; the rapid rate of technology innovation causes short product lifecycles, low production yield and, often, long production lead time, all of which hamper the firm’s ability to respond to market changes. Uncertain economic times exacerbate these challenges. Whereas in an environment of sustained demand growth, firms might build inventories or hold excess capacity to buffer against demand variability, most are reluctant, or unable, to assume such financial risks in a downward market. Nevertheless, high-tech companies recognize that in order to sustain their customer base and to seize revenue opportunities, they must be able to manage successive technological innovations effectively, e.g., introducing high-margin innovative products at the right moment while maximising the return on investment for older, more mature products. To do so, firms must structure capacities in their supply chain so that over time it is possible to respond to demand surges from new product introduction and market upturn, and to absorb short-term decline due to technological migration and market downturn. The role of capacity management is even more important in industries in which capital equipment cost is high. For example, in the semiconductor industry, manufacturers are faced with exorbitant capacity costs, long capacity lead times, high obsolescence rates and high demand volatility. Moreover, the rapid technology innovation leads to short product lifecycles and thus to higher obsolescence rates and increased equipment usage costs. To make the situation worse, the demand variability during a particular quarter may peak above 80% of the average sales and the equipment procurement lead times are usually as long as 6–12 months. This means that the demand beyond the capacity lead times is highly uncertain. The above environment drives semiconductor manufacturers to adopt exceedingly conservative capacity expansion policies [1]. However, in a fastgrowing global market, the conservative capacity-expansion policy leads to severe shortfalls in service levels. In this chapter, scientific literature relevant to high-tech capacity planning and management will be reviewed: we examine the impact of capacity from strategic as well as tactical and, especially, operational perspectives. The final goal of the research presented in this book is building up a decision support model for a specific operational environment, as will be presented in the following.

1.3 Strategic and Tactical Level At the strategic level, capacity planning involves not only the firm’s own capacity investment, but also its supply chain partners’ investments. The capacity investment of one firm in the supply chain could have enormous impact on the performance of all upstream and downstream firms; thus, strategic interactions

Introduction

3

between two or more players need to be taken into account. In addition to monolithic models that employ tools such as expected utility theory and dynamic programming, the scientific literature increasingly considers settings that model independent multiple decision makers in the context of supply chain management. Research in this area utilises game-theoretical models focusing on issues such as contracting, coordination and risk-sharing mechanisms. At the tactical level, capacity planning focuses on capacity expansion tactics as related to the operational aspects of the firm. Demand uncertainty and the short product lifecycles of high-tech products are two key factors that influence capacity expansion models. High-tech manufacturers avoid carrying inventory due to high obsolescence rates. In fact, high-tech products are often treated as perishable goods. Inventory models developed in this context typically use news vendor networks settings with single-period and stochastic demand. These models consider capacity investment by a single or multiple independent decision makers in a stationary environment; once capacity is built it stays unchanged during the planning horizon. This literature stream employs aggregate planning for the acquisition and allocation of resources to satisfy customer demand over a specific time period. The short product lifecycles in high-tech make such an approach quite appropriate. In [2], it is observed that firms achieve better financial results by optimising their capacity and production/inventory decisions simultaneously. In [3] is proposed a four-way trade-off among capacity, production, subcontracting, and inventory levels over a finite horizon. In [4], two-stage supply chains have been considered and the impact of supply/demand uncertainty on capacity and outsourcing decisions has been analysed. They conclude that greater supply uncertainty encourages vertical integration. In contrast, outsourcing becomes more attractive as uncertainty in demand increases. [5] investigates capacity investment and pricing decisions for a manufacturer and a subcontractor with guaranteed availability. In [6] the interplay among capacity, inventory and pricing decisions are examined. The authors study the impact of the timing on these decisions and on the firm’s profitability; they examine different market settings such as monopoly, oligopoly and perfect competition. Other news-vendor like game-theoretical models study sizing [7], timing [8], and allocation [9, 10] of capacity under competitive settings. Few researchers tackle the capacity expansion problem with multiple products and multiple resources; however, there is growing interest in models that consider more general product/resource settings. Using the multidimensional news vendor approach, in [11] a two-product setting in which the firm has the option to invest in two product-dedicated resources or one flexible resource that can process both products has been studied. The paper examines the impact of price, cost, demand uncertainty and demand correlation on the investment decisions. Later, the research presented in [6] extends these models to news vendor networks that incorporate multiple products, multiple resources and multiple storage points. The authors observe that when demand is normally distributed, the optimal expected investment value is an increasing function of the demand vector and a decreasing function of any variance term. In [12], news vendor networks have also been used to investigate investment policies for product-flexible versus dedicated resources in a two-product setting with correlated demand.

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Production Planning in Production Networks

Capacity investment in high-tech industry typically involves substantial cash exposures, volatile market demand and changing supply (technological) specifications. In this environment, the return on investment is highly uncertain; thus, the variability in returns is at least as important as the expected return. Moreover, most investment expenditures are irreversible and can be considered sunk costs for the firm once invested. In general, capacity investment implicates considerable risk for high-tech firms. The traditional approach of maximising net present value (NPV) when analysing capacity investment is not sufficient to capture the managerial flexibility that exists throughout the capacity planning process [13]. A more accurate model should consider not only the purchasing and installation costs, but also the value of the options related to alternative investments [14]. Option theory provides a powerful tool to value risky investments through risk-neutral discounting and to incorporate risk without explicitly defined utility functions. Since the seminal paper by Black and Scholes [15], option pricing has become a popular topic in finance. The enormous attention in the literature on options is summarised in the comprehensive survey of [16]. Although option theory has been primarily studied in finance, the potential benefits of real options have been recognised by researchers in operations management and engineering economics in recent years [17]. Real options theory is particularly relevant to capacity planning because it focuses on the combined importance of uncertainty and managerial decisions and it offers a dynamic view of a firm’s investment and operational decisions. In [18] it is asserted that since operational decisions have the goal of maximising value, the framework of real options can be used to evaluate decisions under risk. Making the connection between the effect of capacity on the firm and the pricing of a call option, it is shown that risks can be incorporated into planning models through capacity adjustments. A model is proposed that integrates financial risk attitudes into a linear capacity investment problem. In general, options theory has broad appeal to a variety of application areas in capacity and production planning. Examples include options to expand, options to defer production [10], options to abandon a project [19], options to wait or temporarily shut down production [20], and options to switch [21]. Paper [22] reports that in recent years, high-tech companies have been making use of real options to determine their investment and operating strategies, more specifically, the timing and choice of capacity adjustments. There is a growing espousal of real options models for high-tech capacity planning in the operations management literature. A broad description and discussion of real options can be found in [23] and [24]. Each unit of capacity provides the firm options to produce a certain quantity of the product throughout its lifecycle; such options are referred to as the operating options. The investment in capacity is the premium for the option, while the production cost corresponds to the exercise price. On the other hand, the firm usually has options to add more capacity, known as growth options. In general, options are early investments associated with firms’ ability to expand in the future; the investment may be the acquisition of (or access to) land facility, technology, know-how or other resources. Following the legacy of option pricing, a majority of papers in this area employ geometric Brownian motion to model demand changes; as such, demands in future time periods can be modelled by a lognormal distribution. As pointed out earlier, high-tech products typically have a short

Introduction

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lifecycle. Therefore a straightforward adoption of the Black and Scholes formula in modelling capacity expansion may lead to inaccurate conclusions since the drift can change direction [25]. In [26], it is demonstrated that simple stochastic processes may not accurately represent capacity investment in many important manufacturing sectors, such as the semiconductors because they are characterised by well-defined product lifecycles with bell-shaped growth patterns. A generalisation of the real options valuation is proposed that explicitly incorporates stochastic product lifecycle into the firm’s capacity investment and production planning problem. The product lifecycle is represented using a regime-switching process in which the planning horizon starts with a growth regime (increasing demand) and then switches stochastically to a decay regime (decreasing demand). Both the fixed (irreversible) capacity and the flexible (reversible) capacity are considered. In the former case, the capacity must be fixed at the beginning of the project and cannot be adjusted throughout the lifecycle. The firm chooses the capacity investment that maximises the project’s NPV, given optimal production policies across periods, the cost of investment, the demand shift parameter and the discount factor. At the beginning of each period when it is possible to expand and contract capacity, the firm chooses the current capacity level based on the trade-off between the cost of adjusting the capacity and the change in expected future profits. To model the stochastic demand in the two-regime context, the author employs the Wiener process; however, he considers different drift values for the demand parameter in the growth and decay regimes. While the drift is assumed to be positive in the growth regime, negative drift is used to model the decay regime. In a given period, the probability of switching from growth to decay is defined by a cumulative normal distribution function of the time elapsed since the beginning of the project. The author examines the sensitivity of the project and option values to demand uncertainty and the project lifecycle through a numerical analysis. He shows that by ignoring the product lifecycle, traditional approaches may undervalue the contraction option, by underestimating the probability that the demand may fall at some point in the future, and may overvalue the expansion option, by implicitly assuming that demand is expected to grow indefinitely.

1.4 Operational Models: Optimization and Decision Support We now consider, in a more expansive way, operational models: the literature focuses on optimisation and decision support models, which can be categorised according to the level of detail they capture and the length of the planning horizon they consider. In [27] several fabrication installations (fabs) of different sizes that can achieve the same throughput levels measured in wafer starts per month have been considered. They compare multiple fabs of smaller capacity, a single fab of larger capacity and one or more fabs of expandable capacity, all of which provide the same total wafer output level. The trade-offs are that smaller and multiple plants do not achieve economies of scale as good as a single larger plant of the same total capacity; however, building small plants reduces the risk of underutilisation. However, long installation times for an individual plant (up to 18

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months) may cause lost business opportunities. The decisions are when and how much capacity to install under demand uncertainty, which is modelled as a continuous time Brownian motion. The objective is to identify the configuration that maximises the total expected profits, which is found by applying a stochastic control approach. This study shows that, in general, it is more costly to install a fab before demand is realised than to install it late. In that regard, large-capacity expandable fabs provide a good compromise by delaying the capacity increments and also achieving economies of scale better than low-capacity fabs. Paper [28] considers capacity planning across multiple fabs. Microelectronics technologies are aggregated and distinguished by their capacity consumption rate. Individual fab capacity is modelled as a single resource whose level is a random variable and measured in wafer output per period. This approach takes into account yield variability at an aggregate level. Uncertainty in demand for technologies and uncertainty in capacity levels are reconciled in the form of discrete scenarios in a multi-period, multi-stage, stochastic programming model. The objective is to minimise the expected mismatch between planned and actual capacity allocation as defined in the scenarios. The authors illustrate various methodologies for preparing demand scenarios as input to their optimisation model. In [29], based on previous work, the authors expand their model to include capacity expansion decisions and aiming at minimising the sum of capacity expansion and expected fab reconfiguration costs. An efficient solution approach, inspired by the decentralised decision-making environment is developed. The proposed approach captures the trade-off between the extent of the information shared between participating decision makers and the quality of capacity planning decisions. It also quantifies the value of information sharing. Another line of research tries to determine whether there is enough tool capacity to satisfy the anticipated demand through a medium-term planning horizon and if so, when and in what numbers are additional tools required. These are very important decisions because tools cost up to several million dollars each and have long delivery lead times (6 to 12 months). Further group research in these tactical planning problems comes under three categories differentiated by the level of detail the models capture and their respective solution approaches: mathematical programming, queuing networks and simulation and other heuristics. Mathematical programming approaches distinguish processes only by production rate which are otherwise treated as generic. Yield and cycle time variability is modelled as a constant yield factor and as such congestion effects are not modelled. The planning period is up to two years which reflects the delivery lead time for newly acquired tools. Some studies also incorporate the assignment of operations to tool groups that are capable of doing multiple operations at different rates into their optimisation models. These are also referred to as routing decisions. The batch-processing nature of semiconductor manufacturing lends itself well to modelling with queuing networks. This approach can compute the cycle time and throughput without running computationally expensive simulation models. Although this approach has been applied extensively in performance evaluation of manufacturing systems (and semiconductor manufacturing as well), we restrict our scope to studies that use this approach within a scheme to make tool capacity planning decisions. In such studies, a

Introduction

7

queuing network model guides the search for a minimum cost tool configuration that achieves a desired performance level for a wafer fab. One restrictive assumption of this approach in the literature observed so far is that the manufacturing flow, the assignment of operations to tools and the routing of batches is assumed to be predetermined. Detailed simulation models are also required to test the solutions obtained by optimisation methods or by heuristics. However, the substantial computational requirements of these models often prohibit their use for evaluating more than a handful of alternative configurations. These and some ad hoc approaches for capacity planning at tactical levels (such as spreadsheet applications) fall into the last category that we consider. 1.4.1 Mathematical Approach There are a few researchers who study the problem with a deterministic model. In [30], a linear programming (LP) based decision support system for tactical planning at an IBM wafer fab which is referred to as the CAPS system has been developed. The decision problem is to find the most profitable product mix for a given tool set and an expected demand profile. An important simplifying assumption is that there is no inventory or back-order between two consecutive planning periods. In [31], a deterministic integer model that relaxes this assumption has been developed. The model minimises the total cost of inventory, tool acquisition and tool operating costs such that the expected demand is satisfied. The authors develop a Lagrangian relaxation-based heuristic which is empirically shown to produce solutions with a small duality gap. However, demand volatility and long manufacturing lead times in the semiconductor industry make it important to integrate demand uncertainty into any decision model. Hence, most studies use a stochastic programming approach where the demand distribution is included as a set of discrete scenarios in the model. Swaminathan [32] studies a tool procurement problem in a single period under the simplifying assumption that operation-to-tool assignments are already known (rather than optimised by the model). Although the underlying model is simplified, the study illustrates the benefits of incorporating demand scenarios versus optimising with respect to expected demand only. Later, the same author, in [33], generalises his earlier work by extending it to a multi-period setting and including operation-tool assignments as decision variables. Heuristic solution methods are developed and their performance is investigated. The study reinforces the benefits of the demand scenario approach and applies the solution methods to an industry data set. Zhang et al. [34] focus on the large number of demand scenarios that are required to capture uncertainty accurately, since this aspect is one of the main reasons for the large size of the resulting model. Their model is developed under the simplifying assumptions that there is no inventory and there are no backorders between periods and that the allocation of products to tools follows simple rules rather than explicitly being optimised by the model. The multi-variate nature of the demand distribution (for all products combined) complicates the model considerably.

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1.4.2 Queuing and Stochastic Approaches Chen et al. [35] are among the first to use a queuing network model to study the performance of a wafer fab as an alternative to a detailed simulation study. The advantage of a queuing model is that it provides a closed-form solution that relates control policies and fab configuration to system performance (e.g., manufacturing cycle time, throughput, work in progress). Such an analytical model makes it easy to explore the performance consequences of different facility configurations and operating policies without heavy computational requirements. In [36], the authors make use of the closed-form solution of a queuing model to develop a set of nonlinear equations that represent the cycle time as a function of the total number of tools in the system. A non-linear mixed integer optimisation model is developed, which aims to minimise cycle time with respect to a given demand profile and a given budget available for tool purchases. The authors develop several heuristic solution approaches and validate them with a set of industry data. Hopp et al. [37] combine a queuing network model with simulation. The study attempts to find a minimum cost capacity configuration while the cycle times are within acceptable limits and throughput is as desired. The queuing model is used to guide the search of the optimisation model to achieve this feat Batch processes, machine changeovers and re-entrant flows are incorporated. In [38], the authors also use a queuing network model to derive cycle time and production costs as functions of the number of tools in tool groups and of the throughput for products. The model is used to determine the number and size of tool groups with the criteria being the trade-off between manufacturing costs and the product throughput. They use the model to analyse the impact of fab scale on cycle time and production costs. They show that the degree of versatility of the tool groups determines the required fab capacity to achieve a given performance: as the versatility increases the required fab scale decreases. Agile mini-fabs could therefore be achieved by installing versatile tools. 1.4.3 Heuristics and Simulation-based Approaches There are a large number of studies that develop rough-cut heuristic approaches for finding easy-to-understand practical capacity planning solutions. However, development complexity and high runtime requirements preclude simulation from being a mainstream application tool. In [39], the authors apply simulation for capacity planning in a development wafer fab and look at the work in process (WIP), cycle time, bottleneck operations and the impact of randomness on the performance. The model is based on company-wide tool availability, process requirement and demand data which are used in simple equations to turn first-cut aggregate results into capacity requirements. In [40], the authors develop a materials requirement planning model tailored to a twin wafer fab system. The model keeps track of WIP amounts and current capacity loads and also decides on wafer releases assuming unlimited capacity. Its main goal is to smooth loading across fabs and across planning periods. Kotcher and Chance [9] describe an easyto-apply methodology for assessing the sensitivity of tool capacity with respect to product mix. The methodology could be used in connection with a simulation

Introduction

9

model to identify bottleneck tool groups and make tool acquisition decisions. In [21], the authors evaluate tool and factory floor (shell space) capacity planning and expansion methodologies that are commonly practiced in industry and highlight their relative performance under different conditions. They make use of an optimum-seeking algorithm developed previously by the authors that finds the optimal tool purchase time and quantities such that the sum of expected lost demand and tool purchase costs are minimised. The results of this algorithm serves as a common comparison basis for the heuristic methods that are practised.

1.5 Motivation As has been shown in the previous sections, different approaches have been utilised to face the strategic/tactical/operational-level manufacturing decisions of the high-tech industries. Many of these studies aim to analyse and optimise, where possible, some performance indicators related to a specific decision level. No organic study on the capacity planning through all the levels constituting a hightech, high-volume company has been detected in the literature. This lack is the starting point of the research presented in this book: an innovative distributed and hierarchical approach for planning capacity at different levels of time horizon will be introduced. A general framework for distributed production planning of complex and multi-plant manufacturing companies in high-tech, high-volume industry will be analyzed using different mathematical tools. For each level bargaining objectives, actors, information and roles will be defined and simulated using a specific simulation environment based on the multiple-agent-system (MAS) paradigm. The focus pertains to the study of coordination policies in a distributed environment. Specifically, the goal is to evaluate their impact in a company, which we call reconfigurable enterprise (RE), which is a production network made of different and geographically dispersed plants which can be reconfigured in order to gather a specific production objective of the enterprise. The distributed and self-organising nature of REs allows responsiveness with regard to the unpredictable market changes: quite normally significant variations in the demand rate arise suddenly and in an unpredictable way. In this situation, the low degree of flexibility characterising production capacity leads almost inevitably to some criticalities; however, the RE’effectiveness and efficiency depends on the extent to which overall performance is obtained; in order to reach this objective REs require strong coordination tools. As shown in the literature overview of the previous sections, basically two approaches support coordination in a distributed organisation: centralised planning tools and decentralised ones. In a centralised planning perspective decisions are taken by a central planner: the most important drawback concerns the lack of flexibility of such a solution that leads this approach to be unusable in the real practice where the scale of the planning problem increases. The other possible solution is to use decentralised decision-making approaches. This means that each actor involved in the planning process takes its own planning decision, while common goals are reached through coordination mechanisms. This book will introduce a new approach concerning capacity allocation methodologies, where managerial and organisational tools are proposed

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with the aim of supporting human decision making, rather than substituting it. In particular, the authors combine techniques from cooperative game theory, negotiations models and multi-agent systems to produce highly reactive, lightweight, useful tools to be used in a real-time system environment in distributed enterprises.

1.6 Book Outline The book is organised as follows: In Chapter 2 game theory in its most relevant topics for the present research is introduced. An overview of the scientific literature concerning the applications of this theory in the capacity planning field is presented. In Chapter 3 negotiation and its relevant solutions concepts related to the conflicts among independent agents is discussed. An overview of the scientific literature concerning the applications of this theory in the capacity planning field is presented. In Chapter 4 the multi-agent systems (MAS) approach and its application to the simulation environment necessary to simulate the coordination approaches proposed in this is presented. In Chapter 5 an innovative distributed and hierarchical approach for planning capacity at different levels of time horizon is introduced. A general framework for distributed production planning of complex and multi-plant manufacturing companies in high-tech, high-volume industries is structured. For each level, bargaining objectives, actors, information and roles are defined. In Chapter 6 distributed models for capacity planning at medium term are analysed and simulated. Specifically, the mathematical formulations of the approaches are shown and then simulated with a specific simulation environment developed by using the MAS paradigm. A statistical analysis is conducted to evaluate differences existing between the proposed models and to evaluate the impact of the input parameters on the estimated performance measures. In Chapter 7 distributed model for plant allocation is analysed. Also in this case, the mathematical formulations of the approaches are shown and then simulated with a specific simulation environment and a statistical analysis is conducted to evaluate models and input parameters impact on the estimated performance measures. In Chapter 8 an innovative integrated approach of the high and medium levels of capacity planning is simulated. The impact of the input parameters and models on the estimated performance measures is evaluated using a statistical analysis. In Chapter 9 conclusions and future work are reported.

1.7 References [1]

Erkoc M, Wu SD (2004) Capacity reservation across multiple buyers. Working paper, Lehigh University, Bethlehem, PA

Introduction

[2] [3] [4] [5] [6] [7] [8]

[9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

[23] [24] [25]

11

Bradley JR, Arntzen BC (1997) The simultaneous planning of production, capacity and inventory in seasonal demand environments. Oper Res 47(6):795–806 Atamturk A, Hochbaum DS (2001) Capacity acquisition, subcontracting, and lot sizing. Manag Sci 47(8):1081–1100 Kouvelis P, Milner JM (2002) Supply chain capacity and outsourcing decisions: the dynamic interplay of demand and supply uncertainty. IIE Trans 34(8): 717–728 Tan B (2002) Subcontracting with availability guarantees: production control and capacity decisions,” IIE Trans 36(8):711–724 van Mieghem JA, Rudi N (2002) Newsvendor networks: inventory management and capacity investment with discretionary activities. M&SOM 4(4):313–335 Bernstein F, DeCroix GA (2004) Decentralized pricing and capacity decisions in a multitier system with modular assembly. Manag Sci 50(9):1293–1308 Ferguson ME, DeCroix GA, Zipkin PH (2002) When to commit in a multi-echelon supply chain with partial information updating. Working paper, Georgia Tech, Atlanta, GA Kotcher R, Chance F (1999) Capacity planning in the face of product-mix uncertainty. In: IEEE international symposium on semiconductor manufacturing conference proceedings, 11–13 October 1999, Santa Clara, CA, pp 73–76 Pindyck RS (1988) Irreversible investment, capacity choice, and the value of the firm. Am Econ Rev 78(5):969–985 van Mieghem JA (1998) Investment strategies for flexible resources. Manag Sci 44(8):1071–1078 Bish E, Wang Q (2004) Optimal investment strategies for flexible resources, considering pricing and correlated demands. Oper Res 52(6):954–964 Feinstein SP (2002) A better understanding of why npv undervalues managerial flexibility. Eng Econ 47(4):418 Dixit A, Pindyck RS (1994) Investment under uncertainty. Princeton University Press, Princeton, NJ Black F, Scholes M (1999) The pricing of options and corporate liabilities. J Polit Econ 81(3):637–654 Broadle M, Detemple JB (2004) Option pricing: valuation models and applications. Manag Sci 50(9):1145 Miller LT, Park CS (2002) Decision making under uncertainty—real options to the rescue?. Eng Econ 47(2): 105–150 Birge JR (2000) Option methods for incorporating risk into linear capacity planning models. Manuf Serv Oper Manag 2(1):19–31 Majd S, Myers S (1990) Abandonment value and project life. Adv Futures Options Res 4:1–21 McDonald R, Siegel D (1985) Investment and the valuation of firms when there is an option to shut down. Int Econ Rev 26:331–349 Cakanyildirim M, Roundy RO (2002) Optimal capacity, expansion and contraction under demand uncertain. Working paper, University of Texas, Dallas, TX Johnson B, Billington C (2003) Creating and leveraging options in the high technology supply chain. In: Harrison TP, Lee HL, Neale JJ (eds) The practice of supply chain management: where theory and application converge. International series in operations research and management science, Kluwer Trigeorgis L (1996) Real options: managerial flexibility and strategies in resource allocation. MIT Press, Cambridge Amram M, Kulatilaka N (1999) Real options: Managing strategic investment in an uncertain world. Harvard Business School Press, Boston Bowman EH, Moskowitz GT (2001) Real options analysis and strategic decision making. Organ Sci 12(6):772–777

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[26] Bollen NP (1999) Real options and product life cycles. Manag Sci 45(5):670 [27] Benavides DL, Duley JR, Johnson BE (1999) As good as it gets: optimal fab design and deployment. IEEE Trans Semicond Manuf 12(3):281–287 [28] Christie RME, Wu SD (2002) Semiconductor capacity planning: stochastic modelling and computational studies. IIE Trans 34(2):131–144 [29] Karabuk S, Wu SD (2003) Coordinating strategic capacity planning in the semiconductor industry. Oper Res 51(6):839–849 [30] Bermon S, Hood SJ (1999) Capacity optimization planning system [caps]. Interfaces 29(5):31–51 [31] Catay B, Erenguc SS, Vakharia AJ (2003) Tool capacity planning in semiconductor manufacturing. Comput Oper Res 30(9):1349–1366 [32] Swaminathan JM (2000) Tool capacity planning for semiconductor fabrication facilities under demand uncertainty. Eur J Oper Res 120(3):545–558 [33] Swaminathan JM (2002) Tool procurement planning for wafer fabrication facilities: a scenario-based approach. IIE Trans 34(2):145–155 [34] Zhang F, Roundy R, Cakanyildirim M, Huh WT (2004) Optimal capacity expansion for multi-product, multi-machine manufacturing systems with stochastic demand. IIE Trans 36(1):23–37 [35] Chen H, Harrison JM, Mandelbaum A, Van Ackere A, Vein LM, (1988) Empirical evaluation of a queuing network model for semiconductor wafer fabrication. Oper Res 36(2):202–215 [36] Bard JF, Srinivasan K, Tirupati D (1999) An optimization approach to capacity expansion in semiconductor manufacturing facilities. Int J Prod Res 37(15):3359– 3382 [37] Hopp, Wallace J, Spearman ML (1996) Factory Physics, Irwin/McGraw-Hill, Boston [38] Iwata Y, Taji K, Tamura H (2003) Multi-objective capacity planning for semiconductor manufacturing. Prod Plan Control 14(3):244–254 [39] Chou W, Everton J (1996) Capacity planning for development wafer fab expansion. In: Advanced semiconductor manufacturing conference and workshop proceedings, IEEE/SEMI, Cambridge, MA, pp 17–22 [40] Chen JC, Fan YC, Wang JY, Lin TK, Leea SK, Wu SC, Lan YJ (1999) Capacity planning for a twin fab. In: IEEE International symposium on semiconductor manufacturing conference proceedings, 11–13 October 1999, Santa Clara, CA, pp 317–320

2 Game Theory: an Overview

2.1 Introduction “Game theory is a branch of mathematics that is concerned with the actions of individuals who are conscious that their actions affect each other”. As such, game theory (hereafter GT) deals with interactive optimisation problems. While many economists in the past few centuries have worked on what can be considered game-theoretical (hereafter G-T) models, John von Neumann and Oskar Morgenstern are formally credited as the fathers of modern game theory. Their classic book Theory of Games and Economic Behavior [1] summarises the basic concepts existing at that time. GT has since enjoyed an explosion of developments, including the concept of equilibrium [2], games with imperfect information [3], cooperative games [4, 5], and auctions [6], to name just a few. Citing Shubik [7], “In the 50s ... game theory was looked upon as a curiosum not to be taken seriously by any behavioural scientist. By the late 1980s, game theory in the new industrial organisation has taken over: game theory has proved its success in many disciplines.” GT is divided into two branches, called the non-cooperative and cooperative branches. The two branches of GT differ in how they formalise interdependence among the players. In the non-cooperative theory, a game is a detailed model of all the moves available to the players. By contrast, the cooperative theory abstracts away from this level of detail, and describes only the outcomes that result when the players come together in different combinations. Though standard, the terms noncooperative and cooperative game theory are perhaps unfortunate. They might suggest that there is no place for cooperation in the former and no place for conflict, competition etc. in the latter. In fact, neither is the case. One part of the non-cooperative theory (the theory of repeated games) studies the possibility of cooperation in ongoing relationships. And the cooperative theory embodies not just cooperation among players, but also competition in a particularly strong, unfettered form. The non-cooperative theory might be better termed procedural game theory, the cooperative theory combinatorial game theory. This would indicate the real distinction between the two branches of the subject, namely that the first specifies

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various actions that are available to the players while the second describes the outcomes that result when the players come together in different combinations. The goal of this chapter is to give a brief overview about GT and, specifically, about G-T concepts and tools. Obviously, due to the need of short explanations, all proofs will be omitted, and we will only focus on the intuition behind the reported results.

2.2 Game Setup To break the ground for next section on non-cooperative games, basic GT notation will be introduced: the reader can refer to Friedman [8] and Fudenberg and Tirole [9] if a more deep knowledge is required. A game in the normal form consists of: players (indexed by i 1, 2,..., n) , a set of strategies (denoted by xi , i 1, 2,..., n) available to each player and payoffs ( S i ( x1 , x2 ,..., xn ) , i 1, 2,..., n) received by each player. Each strategy is defined on a set Xi , xi  Xi , so we call the Cartesian product X1 u X 2 u ... u Xn the strategy space (typically the strategy space is R n ). Each player may have a onedimensional strategy or a multi-dimensional strategy. However, in simultaneousmove games each player’s set of feasible strategies is independent from the strategies chosen by the other players, i.e., the strategy choice of one player does not limit the feasible strategies of another player. A player’s strategy can be thought of as the complete instruction for which actions to take in a game. For example, a player can give his or her strategy to a person that has absolutely no knowledge of the player’s payoff or preferences and that person should be able to use the instructions contained in the strategy to choose the actions the player desires. Because each player’s strategy is a complete guide to the actions that are to be taken, in the normal form the players choose their strategies simultaneously. Actions, which are adopted after strategies, are thus chosen and those actions correspond to the given strategies. The normal form can also be described as a static game, in contrast to the extensive form which is a dynamic game. If the strategy has no randomly determined choices, it is called a pure strategy; otherwise it is called a mixed strategy. There are situations in economics and marketing in which mixed strategies have been applied: e.g., search models [10] and promotion models [11]. In a non-cooperative game the players are unable to make binding commitments regarding which strategy they will choose before they actually choose their strategies. In a cooperative game players are able to make these binding commitments. Hence, in a cooperative game players can make sidepayments and form coalitions. The overview here reported starts with noncooperative static games.

Game Theory: an Overview

15

2.3 Non-cooperative Static Games In non-cooperative static games the players choose strategies simultaneously and are thereafter committed to their chosen strategies. The solution concept for these games was formally introduced by John Nash [2] although some instances of using similar concepts date back to a couple of centuries. The concept is best described through best response functions. Definition 1. Given the n-player game, player i’s best response (function) to the strategies x i of the other players is the strategy xi* that maximizes player i’s payoff S i ( xi , x i ) : x * ( x i ) arg max S i ( xi , x i ) . i

xi

If S i is quasi-concave in xi the best response is uniquely defined by the firstorder conditions. Clearly, given the decisions of other players, the best response is the one that the best player i can hope for. Naturally, an outcome in which all players choose their best responses is a candidate for the non-cooperative solution. Such an outcome is called a Nash equilibrium (hereafter NE) of the game. Definition 2. An outcome ( x1* , x2* ,..., xn* ) is a Nash equilibrium of the game if xi* is a best response to x* i for all i = 1, 2, ..., n. One way to think about an NE is as a fixed point of the best response mapping n R o R n . Indeed, according to the definition, the NE must satisfy the system of equations wS i wxi 0 , for all i. Recall that a fixed point x of mapping f(x), R n o R n is any x such that f ( x ) x . Define f i ( x1 , x2 ,..., xn ) wS i wxi  xi . By the definition of a fixed point, fi ( x1* , x2* ,..., xn* ) wS i ( x1* ,..., xn* ) wxi  xi* o wS i ( x1* ,..., xn* ) wxi 0 , all i.

Hence, x * solves the first-order conditions if and only if it is a fixed point of mapping f(x) defined above. The concept of the NE is intuitively appealing. Indeed, it is a self-fulfilling prophecy. To explain, suppose a player is able to guess the strategies of the other players. A guess would be consistent with payoff maximisation (and therefore reasonable) only if it presumes that strategies are chosen to maximise every player’s payoff given the chosen strategies. In other words, with any set of strategies that is not an NE there exists at least one player that is choosing a non payoff maximizing strategy. Moreover, the NE has a selfenforcing property: no player wants to unilaterally deviate from it since such behaviour would lead to lower payoffs. Hence the NE seems to be the necessary condition for the prediction of any rational behaviour by players. Although attractive, numerous criticisms of the NE concept exist. Two particularly vexing problems are the non-existence of equilibrium and the multiplicity of equilibria. Without the existence of an equilibrium, little can be said regarding the likely outcome of the game. If there are multiple equilibria, then it is not clear which one will be the outcome. Indeed, it is possible the outcome is not even an equilibrium because the players may choose strategies from different equilibria. In some situations it is possible to rationalise away some equilibria via a refinement of the NE concept: e.g., trembling hand perfect equilibrium [12], sequential equilibrium [13] and proper equilibria [14]. In fact, it may even be

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possible to use these refinements to the point that only a unique equilibrium remains. An interesting feature of the NE concept is that the system optimal solution (a solution that maximises the total payoff to all players) need not be an NE. In fact, an NE may not even be on the Pareto frontier: the set of strategies such that each player can be made better off only if some other player is made worse off. A set of strategies is Pareto optimal if they are on the Pareto frontier; otherwise a set of strategies is Pareto inferior. Hence, an NE can be Pareto inferior. The Prisoner’s Dilemma game is the classic example of this: only one pair of strategies is Pareto optimal (both “cooperate”), and the unique Nash equilibrium (both “defect”) is Pareto inferior.

2.4 Existence of Equilibrium An NE is a solution to a system of n equations (first-order conditions), so an equilibrium may not exist. Non-existence of an equilibrium is potentially a conceptual problem since in this case it is not clear what the outcome of the game will be. However, in many games an NE does exist and there are some reasonably simple ways to show that at least one NE exists. As already mentioned, an NE is a fixed point of the best response mapping. Hence fixed-point theorems can be used to establish the existence of an equilibrium. There are three key fixed point theorems, named after their creators: Brouwer, Kakutani and Tarski. (see [15] for details and references.) However, direct application of fixed-point theorems is somewhat inconvenient and hence generally not done (see [16] for existence proofs that are based on Brouwer’s fixed-point theorem). Alternative methods, derived from these fixed-point theorems, have been developed. The simplest (and the most widely used) technique for demonstrating the existence of an NE is through verifying concavity of the players’ payoffs, which implies continuous best response functions. Theorem 1. [17]. Suppose that for each player the strategy space is compact and convex and the payoff function is continuous and quasi-concave with respect to each player’s own strategy. Then there exists at least one pure strategy NE in the game. If the game is symmetric (i.e., if the players’ strategies and payoffs are identical), one would imagine that a symmetric solution should exist. This is indeed the case, as the next theorem ascertains. Theorem 2. Suppose that a game is symmetric and for each player the strategy space is compact and convex and the payoff function is continuous and quasiconcave with respect to each player’s own strategy. Then there exists at least one symmetric pure strategy NE in the game.

Game Theory: an Overview

17

2.5 Multiple Equilibria Many games are just not blessed with a unique equilibrium. The next best situation is to have a few equilibria. (The worst situation is either to have an infinite number of equilibria or no equilibrium at all.) The obvious problem with multiple equilibria is that the players may not know which equilibrium will prevail. Hence, it is entirely possible that a non-equilibrium outcome results because one player plays one equilibrium strategy while a second player chooses a strategy associated with another equilibrium. However, if a game is repeated, then it is possible that the players eventually find themselves in one particular equilibrium. Furthermore, that equilibrium may not be the most desirable one. If one does not want to acknowledge the possibility of multiple outcomes due to multiple equilibria, one could argue that one equilibrium is more reasonable than the others. For example, there may exist only one symmetric equilibrium and one may be willing to argue that a symmetric equilibrium is more focal than an asymmetric equilibrium. In addition, it is generally not too difficult to demonstrate the uniqueness of a symmetric equilibrium. If the players have one-dimensional strategies, then the system of n first-order conditions reduces to a single equation and one need only show that there is a unique solution to that equation to prove the symmetric equilibrium is unique. If the players have m-dimensional strategies, m > 1, then finding a symmetric equilibrium reduces to determining whether a system of m equations has a unique solution (easier than the original system, but still challenging).

2.6 Dynamic Games The simplest possible dynamic game was introduced by Stackelberg [18]. In a Stackelberg duopoly model, player 1 chooses a strategy first (the Stackelberg leader) and then player 2 observes this decision and makes his own strategy choice (the Stackelberg follower). To find an equilibrium of a Stackelberg game (often called the Stackelberg equilibrium) we need to solve a dynamic two-period problem via backwards induction: first find the solution x2*  x1 for the second player

as a response to any decision made by the first player: wS 2 ( x2 , x1 ) x ( x1 ) : 0. wx2 Next, find the solution for the first player anticipating the response by the second player: * dS 1 ( x1 , x2* ( x1 )) wS 1 ( x1 , x2* ) wS 1 ( x1 , x2 ) wx 2 0.  dx1 wx1 wx2 wx1 Intuitively, the first player chooses the best possible point on the second player’s best response function. Clearly, the first player can choose an NE, so the leader is always at least as well off as he would be in NE. Hence, if a player were allowed to choose between making moves simultaneously or being a leader in a game with complete information he would always prefer to be the leader. * 2

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2.7 Simultaneous Moves: Repeated and Stochastic Games A different type of dynamic game arises when both players take actions in multiple periods. Two major types of this game exist: without and with time dependence. In the multi-period game without time dependence the exact same game is played over and over again (hence the term repeated games). The strategy for each player is now a sequence of actions taken in all periods. Consider one repeated game version of the competing newsvendor game in which the newsvendor chooses a stocking quantity at the start of each period, demand is realised and then leftover inventory is salvaged. In this case, there are no links between successive periods other than the players’ memory about actions taken in all the previous periods. A fascinating feature of repeated games is that the set of equilibria is much larger than the set of equilibria in a static game and may include equilibria that are not possible in the static game. At first, one may assume that the equilibrium of the repeated game would be to play the same static NE strategy in each period. This is, indeed, an equilibrium but only one of many. Since in repeated games the players are able to condition their behaviour on the observed actions in the previous periods, they may employ so-called trigger strategies: the player will choose one strategy until the opponent changes his play, at which point the first player will change the strategy. This threat of reverting to a different strategy may even induce players to achieve the best possible outcome (i.e., the centralised solution) which is called an implicit collusion. Many such threats are, however, non-credible in the sense that once a part of the game has been played, such a strategy is not an equilibrium anymore for the reminder of the game. To separate out credible threats from non-credible, Selten [19] introduced the subgame, a portion of the game (that is a game in itself) starting from some time period and a related notion of subgameperfect equilibrium (this notion also applies in other types of games, not necessarily repeated), and equilibrium for every possible subgame (see Hall and Porteus [20] and van Mieghem and Dada [21] for solutions involving subgameperfect equilibria in dynamic games).

2.8 Cooperative Games The idea behind cooperative game theory has been expressed in this way: “Cooperative theory starts with a formalization of games that abstracts away altogether from procedures and concentrates, instead, on the possibilities for agreement. There are several reasons that explain why cooperative games came to be treated separately. One is that when one does build negotiation and enforcement procedures explicitly into the model, then the results of a non-cooperative analysis depend very strongly on the precise form of the procedures, on the order of making offers and counter-offers and so on. This may be appropriate in voting situations in which precise rules of parliamentary order prevail, where a good strategist can indeed carry the day. But problems of negotiation are usually more amorphous; it is difficult to pin down just what the procedures are. More fundamentally, there is a feeling that procedures are not really all that relevant; that it is the possibilities for coalition forming, promising and threatening that are decisive, rather than whose

Game Theory: an Overview

19

turn it is to speak. Detail distracts attention from essentials. Some things are seen better from a distance; the Roman camps around Metzada are indiscernible when one is in them, but easily visible from the top of the mountain” [22]. The subject of cooperative games first appeared in the seminal work of von Neumann and Morgenstern [1]. However, for a long time cooperative game theory did not enjoy as much attention in economics literature as non-cooperative GT. Cooperative GT involves a major shift in paradigms as compared to noncooperative GT: the former focuses on the outcome of the game in terms of the value created through cooperation of (a subset of) players but does not specify the actions that each player will take, while the latter is more concerned with the specific actions of the players. Hence, cooperative GT allows us to model outcomes of complex business processes that otherwise might be too difficult to describe (e.g., negotiations) and answers more general questions (e.g., how well is the firm positioned against competition). In what follows, we will cover transferable utility cooperative games (including two solution concepts: the core of the game and the Shapley value).

2.9 N-Person Cooperative Games Recall that the non-cooperative game consists of a set of players with their strategies and payoff functions. In contrast, in this case, although players are autonomous decision makers, they may have an interest in making binding agreements in order to have a bigger payoff at the end of the game. This agreement or partnership is the basic ingredient of the mathematical model of a cooperative game, and it is called a coalition. Mathematically, a coalition is a subset of the set of players N and we can denote it by S. To form a coalition S, it is required that agreements take place involving all players in the future coalition S. Whenever all players approve joining in a new entity called coalition, we can say that the new coalition is formed. Joining a coalition S also implies that there is no possible agreement between any member of S and any member not in S (set N\S). In short, the essential feature of a coalition is its foundational agreement that binds and reconstitutes the individuals as a coordinated entity. The grand coalition of all n players will be referred as coalition N (there is a total of 2n  1 possible coalitions); The empty coalition is a coalition made up of no members (the null set ‡ ). A coalition structure is a means of describing how the players divide themselves into mutually exclusive coalitions. Any exhaustive partition of the players can be described by a set S { ^S1 , S2 ,..., Sm ` of the m coalitions that are formed. The set S is a partition of N that satisfies three conditions: S j z ‡, j Si ˆ S j

1,..., m ‡, for all i z j , and ‰S j

N.

These conditions state that each player belongs to one and only one of the m non-empty coalitions within the coalition structure, and also specifies that none of

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the players in any coalition m is connected to other players not in the coalition; finally, the mutually exclusive union of all coalitions m forms the grand coalition.

2.10 Characteristic Function and Imputation von Neumann and Morgenstern [23] introduced the term characteristic function for the first time. More formally, we can define that: Definition 3. For each subset S of N, the characteristic function Q of a game gives the biggest amount Q ( S ) that the members of S can be sure of receiving if they act together and form a coalition, without any help from other players not in S. A restriction on this definition is that the value of the game to the empty coalition is zero, that is, Q (‡ ). A further requirement that is generally made is called superadditivity. Superadditivity can be expressed as follows:

Q (S ‰ T ) t Q (S )  Q (T ) for all S , T Ž N such that S ˆ T ‡ . This means that the total payoff for the grand coalition is collectively rational, because the total payoff to the players is always as much as what they would get individually. This suggests the following definition. Definition 4. A game in characteristic function form consists of a set of players, together with a function Q defined for all subsets of N, such that

Q ( S ‰ T ) t Q ( S )  Q (T ) whenever S and T are disjoint coalitions of players. Games in which at least one possible coalition can increase the total payoff of its members are called essential, and those in which there is no coalition that improves the total payoff are called inessential. Mathematically, an essential game is one in which at least one of the superadditive inequalities Q (S ‰ T ) t Q (S )  Q (T ) is strict. The specific actions that players have to take to create this value are not specified: the characteristic function only defines the total value that can be created by utilising all players’ resources. Hence, players are free to form any coalitions that are beneficial to them and no player is endowed with power of any sort. We will further restrict our attention to the transferable utility games in which the outcome of the game is described by real numbers S i , i 1,..., N showing how the total created value (or utility or pie)

S (N )

¦

N i 1

S i was divided among players. Of course, one could offer a very

simple rule prescribing division of the value; for example, a fixed fraction of the total pie can be allocated to each player. However, such rules are often too simplistic to be a good solution concept. A much more frequently used solution concept of the cooperative game theory is the core of the game. This concept can be compared to the NE for non-cooperative games:

Game Theory: an Overview

21

Definition 5. The utility vector S 1 ,..., S N is in the core (and will be called imputation) of the cooperative game if it satisfies S ( N ) Q ( N ), group rationality, and xi t Q (^i`), individual rationality. The core of the game, introduced by Gillies in 1953 [24], can be interpreted through the added-value principle. Define (N\S) as a set of players excluding those in coalition S (coalition can include just one player). Then the contribution of a coalition S can be calculated as Q ( N ) Q ( N \ S ). Clearly, no coalition should be able to capture more than its contribution to the coalition (otherwise the remaining N\S players would be better off without the coalition S). Definition 5 clearly satisfies the added-value principle. Typically, when analysing a game, one has to calculate an added value from each player: if the value is zero, the player is not in the core of the game. If the core is non-empty, the added values of all players in the core comprise the total value that the players create. As is true for NE, the core of the game may not exist (i.e., it may be empty) and the core is often not unique. When the core is non-empty, the cooperative demands of every coalition can be granted, but when the core is empty, at least one coalition will be dissatisfied. Shubik [7] noted that a game with a non-empty core is sociologically neutral, i.e. every cooperative demand by every coalition can be granted, and there is no need to resolve conflicts. On the other hand, in a coreless game, the coalitions are too strong for any mechanism to satisfy every coalitional demand. However, a core set with too many elements is not desirable, and it has little predictive power [25]. Imputations in the core, where they exist, have a certain stability, because no player or subset of players has any incentive to leave the grand coalition. But since many games have empty core, the core fails to provide a general solution for nperson games in characteristic form. von Neumann and Morgenstern [24] proposed a different solution concept more generally applicable than the core. That proposal is called the von Neumann Morgenstern solution or the stable set. The stable set is based on the concept of dominance, which is explained as follows. One imputation is said to dominate another if there is a subset of players who prefer the first to the second and can enforce it by forming a coalition.

2.11 Shapley Value The concept of the core, though intuitively appealing, also possesses some unsatisfying properties. As we mentioned, the core might be empty or quite large or indeterministic. As it is desirable to have a unique NE in non-cooperative games, it is desirable to have a solution concept for cooperative games that results in a unique outcome and hence has a reasonable predictive power. Shapley [26] offered an axiomatic approach to the solution concept that is based on three rather intuitive axioms. First, the value of the player should not change due to permutations of players, i.e., only the role of the player matters and not names or indices assigned to players. Second, if a player’s added value to the coalition is zero then this player should not get any profit from the coalition, or in other words

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only players generating added value should share the benefits. Finally, the third axiom requires additivity of payoffs: for any two characteristic functions Q 1 and Q 2 it must be that S (Q 1  Q 2 , N ) S (Q 1 , N )  S (Q 2 , N ). The surprising result obtained by Shapley is that there is a unique equilibrium payoff (called the Shapley value) that satisfies all three axioms. Theorem 3. There is only one payoff function S that satisfies the three axioms. It is defined by the following expressions for i  N and all Q : |S|!(|N ||S|1)! S i (Q ) ¦ Q  S ‰ ^i` Q  S . |N |! S Ž N \i





The Shapley value assigns to each player his marginal contribution

Q  S ‰ ^i` Q  S when S is a random coalition of agents preceding i and the ordering is drawn randomly. To explain further, (see Myerson [14]), suppose players are picked randomly to enter into a coalition. There are |N|! different orderings for all players, and for each set S that does not contain player i there are |S|!(|N ||S|1)! ways to order players so that all of the players in S are picked ahead of player i. If the orderings are equally likely, there is a probability of |S|!(|N ||S|1)! |N |! that when player i is picked he will find S players in the coalition already. The marginal contribution of adding player i to coalition S is Q  S ‰ ^i` Q (S ) . Hence, the Shapley value is nothing more than a marginal (expected) contribution of adding player i to the coalition. Due to its uniqueness, the concept of the Shapley value has found numerous applications in economics and political sciences.

2.12 The Bargaining Game Model To better understand the negotiational mechanism and theory, which will be shown more specifically in the next chapter, we here consider the former approach to this issue showing how to face the problem addressed by the bargaining in cooperative game theory. In order to do this, consider a group of two or more agents facing with a set of feasible outcomes, any one of which will be the result if it is accepted by unanimous agreement of all participants. In the event that no unanimous agreement is reached, a given disagreement outcome is the result. If the feasible outcomes are such that each participant can do better than the disagreement outcome, then there is an incentive to reach an agreement; however, so long as at least two of the participants differ over which outcome is most preferable, there is a need for bargaining and negotiation over which outcome should be agreed upon. Note that since unanimity is required, each participant has the ability to veto any outcome different from the disagreement outcome. To model this atomic negotiation process, we use the cooperative bargaining process initiated by Nash [27]. It is pertinent to mention that experimental bargaining theory indicates stronger empirical evidence of this bargaining theory than any others. Nash engaged in an axiomatic derivation of the bargaining solution. The solution refers

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to the resulting payoff allocation that each of the participants unanimously agrees upon. The axiomatic approach requires that the resulting solution should possess a list of properties. The axioms do not reflect the rationale of the agents or the process in which an agreement is reached but only attempts to put restrictions on the resulting solution. Further, the axioms do not influence the properties of the feasible set. Before listing the axioms, we will now describe the construction of the feasible set of outcomes. Formally, Nash defined a two-person bargaining problem (which can be extended easily to more than two players) as consisting of a pair F , d where F is a closed convex subset of R 2 , and d  d1 , d2 is a vector in R 2 . F is convex, closed, non-empty, and bounded. Here, F, the feasible set, represents the set of all feasible utility allocations and d represents the disagreement payoff allocation or the disagreement point. The disagreement point may capture the utility of the opportunity profit. Nash looked for a bargaining solution, i.e., an outcome in the feasible set that satisfied a set of axioms. The axioms ensure that the solution is symmetric (identical players receive identical utility allocations), feasible (the sum of the allocations does not exceed the total pie), Pareto optimal (it is impossible for both players to improve their utilities over the bargaining solutions), the solution be preserved under linear transformations and be independent of “irrelevant” alternatives. Due to constraints on space, the reader can refer to Roth [28] for a very good description of the solution approach and a more detailed explanation of the axioms. The remarkable result due to Nash is that there is a bargaining solution that satisfies the above axioms and it is unique. Theorem 4 [27]. There is a unique solution that satisfies all the “axioms”. This solution, for every two-person bargaining game F , d is obtained by

solving: arg

max

x ( x1 , x2 )F , x t d

( x1  d1 )( x2  d2 ) .

The axiomatic approach, though simple, can be used as a building block for much more complex bargaining problems. Even though the axiomatic approach is prescriptive, descriptive non-cooperative models of negotiation such as the Nash demand game [29] and the alternating offer game [30], reach similar conclusions as Nash bargaining. This somehow justifies the Nash bargaining approach to model negotiations. In our discussion, we have only provided a description of the bargaining problem and its solution between two players. However, this result can easily be generalised to any number of players simultaneously negotiating for allocations in a feasible set.

2.13 References [1] [2] [3]

von Neumann J, Morgenstern O (1944) Theory of games and economic behaviour. Princeton University Press Nash JF (1950) Equilibrium points in n-person games. Proc Nat Acad Sci USA 36, 48–49 Kuhn HW (1953) Extensive games and the problem of information. In Contributions to the theory of games, vol II, Kuhn HW, Tucker AW, editors. Princeton University Press. 193–216

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[4]

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

[27] [28] [29] [30]

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Aumann RJ (1959) Acceptable points in general cooperative n-person games. In Contributions to the theory of games, vol IV, Kuhn HW, Tucker AW, editors. Princeton University Press. 287–324 Shubik M (1962) Incentives, decentralized control, the assignment of joint costs and internal pricing. Manag Sci 8:325–343 Vickrey W (1961) Counter speculation, auctions, and competitive sealed tenders. J Finance, Vol.16, 8–37 Shubik M (2002) Game theory and operations research: some musings 50 years later. Oper Res 50:192–196 Friedman JW (1986) Game theory with applications to economics. Oxford University Press Fudenberg D, Tirole J (1991) Game theory. MIT Press. Varian H (1980) A model of sales. Am Econ Rev 70:651–659 Lal R (1990) Price promotions: limiting competitive encroachment. Market Sci 9:247–262 Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4:25–55 Kreps D, Wilson R (1982) Sequential equilibria. Econometria 50:863–894 Myerson RB (1997) Game theory. Harvard University Press. Border KC (1999) Fixed point theorems with applications to economics and game theory. Cambridge University Press. Lederer P, Li L (1997) Pricing, production, scheduling, and delivery-time competition. Oper Res 45:407–420 Debreu D (1952) A social equilibrium existence theorem. Proc Nat Acad Sci USA 38:886–893 Stackelberg H (1934) Markform and gleichgewicht. Vienna: Julius Springer. Selten R (1965) Spieltheoretische behaundlung eines oligopolmodells mit nachfragetragheit. Z gesamte staatswiss 12:301–324 Hall J, Porteus E (2000) Customer service competition in capacitated systems. Manuf Serv Oper Manag 2:144–165 van Mieghem J, Dada M (1999) Price versus production postponement: capacity and competition. Manag Sci 45:1631–1649 Aumann RJ (1989) Game theory. In Eatwell J, Milgate M, Newman P (eds), The new Palgrave, New York, Norton, 8–9 von Neumann J, Morgenstern O (1947) Theory of games and economic behaviour. 2nd ed, Princeton, NJ, Princeton University Press Gillies DB (1953) Some theorems on n-person games. PhD dissertation, Department of Mathematics, Princeton University, Princeton, NJ Kahan JP, Rapoport A (1984) Theories of coalition formation, Lawrence Erlbaum, Hillsdale, NJ Shapley LL (1953) A value for n-person games. In Kuhn HW, Tucker W (eds), Contributions to the theory of games II. Ann Math Studies n.28. Princeton NJ, Princeton University Press Nash JF (1951) Noncooperative games. Ann Math 54:286–295 Roth A (1979) Axiomatic models in bargaining, Springer-Verlag Roth A (1995) Handbook of experimental economics, Princeton University Press Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica 50:97–110

3 Negotiation: an Overview

3.1 Introduction The variety of involved disciplines and perspectives in the conflict resolution field has created different terminologies, definitions, notations and formulations about the concept of negotiation. As a result, interdisciplinary cooperation among the relevant fields of study would suffer from inconsistencies and contradictions [1]. Yet, negotiation requires an interdisciplinary approach because of their psychological, social and cultural character; economic, legal and political considerations involved; quantitative and qualitative aspects; strategic, tactical and managerial perspectives. Clearly, interdisciplinary approaches provide richer and more comprehensive models of negotiators and negotiations. A schematic representation of the different perspectives and influences on negotiation research is presented in Figure 3.1: law and social sciences are the main contributors to the prescriptive and descriptive models, heuristics and qualitative studies of negotiation processes and negotiators’ behaviour [2, 3]; economics and management science concentrated on the construction of formal models and procedures of negotiations, rational strategies and the prediction of outcomes [4, 5]. Computer science and information systems contributions include construction of electronic negotiation tables, decision and negotiation support systems (DSS, NSS), artificial negotiating software agents (NSA) and software platforms for bidding and auctioning [6–9]. The four arrows depicted in Figure 3.1 connect areas of studies with the corresponding common results. The bi-directional arrow indicates that the negotiation tools, agents and platforms are often based on the results of the studies in economic and social sciences [10–13], and also that, increasingly, computational models and systems influence the construction of negotiation techniques, models and procedures [14, 15]. The latter phenomenon marks a fairly recent development in negotiation research: increased computing and networking power and findings from computer science and information systems feed back into negotiation models and procedures giving them more flexibility.

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Computer Science Computational linguistics Data mining and KDD Artificial Intelligence Distributed AI Autometed negotiation Autonomous negotiation agents Negotiation expert systems Distributed negotiations Negotiation software platforms

Negotiation media and systems: tools, agents and platforms

Information systems Decision support Group and negotiation support Workflows models Electronic commerce Decision support systems Negotiation support systems Electronic negotiation tables Negotiation support agents Electronic markets Electronic auctions

Economic sciences and management Econometrics Experimental economics Management science Decision science Bargaining theory Auction theory Game theory Negotiation analysis

Negotiation procedures and models: strategies, tactics and techniques

Law and social sciences Law Psychology Sociology Linguistics Political sciences Mediation and facilitation Models of attitude and perceptions Process models Cultural influences Cognitive models

Figure 3.1. Negotiation research areas, their results and reciprocal key influences1

Most traditional negotiations have been conducted face-to-face; others have been conducted using mail, fax and telephone. Mail-based and email-based negotiations share many similarities in that they are difficult to manage, are time consuming, and prone to misunderstanding [16]. Yet, the impact of information technologies on negotiation is not limited to the use of electronic communication. Information technology changes the way a negotiation problem can be represented 1

M. Bichler, G. Kersten, S. Strecker: Towards a Structured Design of Electronic Negotiations. Group Decision and Negotiation, 2003, Vol. 12, No. 4, 311–335.

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27

and a negotiation process is structured. The use of Internet-based information systems allows us to implement many more activities undertaken during the negotiation process, including efficient matching of potential negotiators; exchange, comparison and categorisation of rich data; and the use of tools for data collection, problem structuring and analysis, and interpretation of offers. These new possibilities have led to the birth of formal negotiation procedures and protocols, which are a prerequisite to extending the use of rich and expressive information technologies from the messages exchange to the other negotiation process phases. Initiated by the commercial exploration of the Internet as a global communication and “negotiation” infrastructure [17], electronic negotiation (enegotiation) has started to contribute in manifold ways-from web-based NSS [18], to on-line auctions [19], to automated agent-based negotiations [20], in both research studies and business applications [21]. Examples of new negotiation protocols include auction protocols with combinatorial bids on product bundles [22–24], automated negotiations among software agents [9, 25, 26] as well as protocols supporting bi- and multi-lateral negotiations among human negotiators [27, 28]. The computerisation of negotiation processes increasingly affects the way organisations (and individuals) interact with each other. E-negotiation promises higher levels of process efficiency and effectiveness, and most importantly, a higher quality and faster agreement achievement. The potential economic impact leads to an increased demand for appropriate electronic negotiations for specific negotiation situations (e.g. electronic tenders for electronic procurement). Yet, both the design of suitable electronic negotiation protocols and the implementation of relevant electronic negotiation media largely lack systematic, traceable and reproducible approaches and thus they remain more an art than a science. Recent developments created an opportunity for mutual fertilisation of research studies and approaches, and for integration of different perspectives on negotiations into an interdisciplinary research effort to develop an engineering approach to electronic negotiations, similar to, for example, system or process engineering which brings together the findings about negotiators and negotiation processes from the different research areas. The phenomenal growth in computer science and information systems has led to the design of models and systems that often use little from the behavioural and economic body of knowledge: people’s biases, misconceptions and misunderstandings that have been shown to occur very often are not considered and countered in negotiation design. There is thus a strong need to consider the many results of economic and social sciences for the design and development of negotiation media and systems. However, both the new methods for data processing, representation and presentation, and many pragmatic approaches to providing user-oriented support that have been proposed by computer science and information systems should not be discarded by social scientists. Hence, there should be, as shown in Figure 3.1, a feedback between results of the two groups of negotiation research areas, that is, between the negotiation media and systems, and the models and procedures. To achieve this a multi-disciplinary approach towards the field of electronic negotiation is necessary, one which is much broader and more encompassing than often implemented.

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3.2 Negotiation and Rational Self-interested Agents One the greatest challenges for computer science is to build computer systems that can work together. The integration of automated systems has always been a challenge, but as computers have become more sophisticated, the demands for coordination and cooperation have become increasingly important and pervasive. Cooperation and coordination are required for those complex problems that cannot be solved by a single system in isolation, but require several systems to work together interactively. Furthermore, there are heterogeneous intelligent systems that were built in isolation that need to cooperate to achieve a new common goal. In other situations, several autonomous systems may work in the same environment, on different goals, and may need to coordinate their activities or to share resources. They may also benefit from cooperation. Problems of coordination and cooperation are not unique to automated systems; they exist at multiple levels of activity in a wide range of populations. There have been several attempts to define an agent [29–31]. For example Etzioni and Weld [32] require an agent to be goal-oriented, collaborative, flexible, and capable of making independent decisions on when to act. In addition, they determined that an agent should be a continuously running process and be able to engage in complex communication with other agents, including people. It should automatically customise itself to the preferences of its user and to changes of the environment. The strategic-negotiation model was developed to address problems in distributed artificial intelligence (DAI), an area concerned with how automated agents can be constructed to interact in order to solve problems effectively. Subrahmanian et al. [30] concentrate on the interaction of an agent with other agents and the environment. They define a software agent as a body of software that: ƒ provides one or more useful services that other agents may use under specified conditions; ƒ includes a description of the service offered by the software, which may be accessed and understood by agents; ƒ includes the ability to act autonomously without requiring explicit direction from a human being; ƒ includes the ability to describe succinctly and declaratively how an agent determines what actions to take even though this description may be kept hidden from other agents, and ƒ includes the ability to interact with other agents either in a cooperative or in a adverse manner, as appropriate. The agent designer faces two critical questions: what is the architecture of each agent, and how do they interconnect, coordinate their activities, and cooperate? There are many approaches to the development of a single agent. These approaches can be divided into three principal categories [31]: deliberative, reactive and hybrid architectures. A deliberative architecture is one that contains an explicitly represented, symbolic model of the world, and one in which decisions (e.g., about what actions to perform) are made via logical reasoning, based on pattern matching and symbol manipulation. The main criticism of this approach is that the computational complexity of symbol manipulation is very high, and some key problems appear to be intractable. A reactive architecture is usually defined as one

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that does not include any kind of central symbolic world model and does not use any complex symbolic reasoning. These types of agents work efficiently when they are faced with many “routine” activities. Many researchers suggest that neither a completely deliberate nor a completely reactive approach is suitable for building agents. They use hybrid systems, which attempt to combine the deliberate an the reactive approaches.

3.3 Negotiation Models Negotiation process implementation has been done using DAI application tools both in distributed problem solving (DPS), where the agents are cooperative, and in multi agent systems (MAS), where the agents are self-interested. Several studies in DPS use negotiation for distributed planning and distributed search for possible solutions for hard problems. For example, Conry et al. [33] suggest multi-stage negotiation to solve distributed constraint satisfaction problems when no central planner exists. Moehlman et al. [34] use negotiation as a tool for distributed planning: each agent has certain important constraints, and it tries to find a feasible solution using a negotiation process. They applied this approach in the Phoenix fireman array. Lander and Lesser [35] use a negotiation search, which is a multistage negotiation as a means of cooperation while searching and solving conflicts among the agents. For the MAS environments, Rosenschein and Zlotkin [9] identified three distinct domains where negotiation is applicable and found a different strategy for each domain: Task-oriented domain: the strategy consists in finding ways in which agents can negotiate to come to an agreement, and allocating their tasks in a way that is beneficial to everyone; State-oriented domain: the strategy consists in finding actions which change the state of the “world” and serve the agents’ goals; and Worth-oriented domain: same strategy as State-Oriented Domain above, but, in this domain, the decision is taken according to the maximum utility the agents gain from the states. Sycara [36, 37] presented a model of negotiation that combines case-based reasoning and optimisation of multi-attribute utilities. In her work agents try to influence the goals and intentions of their opponents. Kraus and Lehmann [28] developed an automated diplomacy player that negotiates and plays well in actual games against human players. Sierra et al. [38] present a model of negotiation for autonomous agents to reach agreements about the provision of service by one agent to another. Their model defines a range of strategies and tactics, borrowed from good behavioural practice in human negotiation, that agents can employ to generate offers and evaluate proposals. Zeng and Sycara [39] consider negotiation in a marketing environment with a learning process in which the buyer and the seller update their beliefs about the opponent’s reservation price using the Bayesian rule. Sandholm and Lesser [40] discuss issues, such as levels of commitment, that arise in automated negotiation among self-interested agents whose rationality is bounded by computational complexity.

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3.4 Underlying Principle for Electronic Negotiation The computerisation of negotiation processes increasingly affects the way customers, suppliers, and other business partners interact. Traditionally, firms conducted negotiations with a counterpart in a bilateral manner: face-to-face, or using media support. Such negotiations are difficult to manage, time consuming, prone to misunderstanding, and require significant cognitive efforts [16]. Traditional negotiations suffer asymmetry of information about negotiated issues (e.g. price transparency), acknowledged solution domain limitation (i. e. a shortage of offers and counter-offers), an ex ante restricted number of potential counterparts (since, for example, the human capacity for handling multiple telephone calls simultaneously is limited) and high transaction costs (i.e. implicit transaction costs such as large bid offer spreads and explicit transaction costs such as labour and equipment) [41]. Those negotiation processes are rarely efficient and often lead to inefficient compromises [42]. Meanwhile e-negotiation offers the possibility of higher levels of process efficiency and effectiveness, including the exchange of quantitatively and qualitatively improved information during the negotiation process. Most importantly, e-negotiations promise higher quality and faster achievement of negotiated agreements. This is because the participants can use negotiation support tools that are able to support and advise negotiation actors who can then formulate offers and counter-offers based on a wider knowledge about negotiation issues. In traditional negotiations, the use of negotiation support tools is difficult and awkward because all information exchanged face-to-face must be entered manually during the process. The design of e-negotiation media, support systems and software agents that matches the diversity of users, and the richness and complexity of negotiation situations, requires categorisation and structuring of the latter, and, also, specification of concepts and constructs. This effort led to the creation of a taxonomy of electronic negotiations comprising types of processes and terms used to describe different types in detail. Raiffa [43] in his seminal work discourages “devising a taxonomy of disputes, in which the listing would be reasonably exhaustive and in which overlaps among categories would be rare. This was possible, I found, only after developing a host of abstract constructs — and even then the taxonomy was not very useful”. Noting this caveat, it is obvious that such efforts need to be made. This is because new information technologies are increasingly being used to construct media for engagement in social and economic processes such as negotiation in parallel and independently of the behavioural and normative models of these processes. Results of social sciences should be taken into account in the design of these media as well as their implications for the processes themselves. In addition, a taxonomy allows for the establishment of a common, unique terminology across disciplines, classification of models and systems, identification of their possible extensions and for the identification of new constructs and negotiation protocols.

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3.5 Electronic Negotiation Protocols The implementation of every model in an information system brings forth certain rules of interaction that those who use this medium must follow. These rules need to be specified so that agents (human or artificial) know the permissible set of actions. An e-negotiation protocol is a negotiation protocol in which at least some activities are supported or performed by information systems and the negotiation steps are conducted by means of an electronic medium. The e-negotiation protocol may be complex and with many rules governing the parties as they move through different stages and phases of the negotiation process. For example, an enegotiation may begin with an auction and, after three winning bidders have been identified, move on to a bilateral bargaining protocol among the three winners. Typically, designers try to achieve certain goals for the outcome of a negotiation and for the negotiation process itself, such as Pareto optimality of the result, maximisation of the bid taker’s revenue/utility, stability, and speed of convergence [44]. These objectives are achieved through: ƒ specification of the structure of the negotiation problem and process; ƒ specification of rules of feasible activities, and their sequencing and timing; and ƒ imposition of limitations on the form and content of information exchange. Every e-negotiation protocol restricts the negotiators’ freedom in order to meet one or more of the above objectives. A closed e-negotiation protocol does not allow us to add new rules throughout the negotiation. A closed negotiation protocol can cover various negotiation situations but the set of rules is fixed and the rules cannot be modified. Implementations of traditional auction formats such as the Dutch or English auction are good examples of a closed e-negotiation protocol. An open enegotiation protocol does not contain all rules required for the negotiation; they may be constructed by the participants or by mechanisms during the negotiation process. In both cases, this involves learning about the participants, problem and process; the results of learning are new rules that were not present prior to the enegotiation. Complex electronic negotiation protocols often involve a combination of two or more different classes of negotiation protocols and thus exhibit the characteristics of multiple negotiation models in either sequential or parallel execution. For example, in financial markets continuous double auction protocols have been combined with bilateral chat markets [45] where a trader can select an offer and engage in a bilateral chat with the respective counterpart.

3.6 Characteristics that Differentiate Negotiations Protocols Evaluation of the results of negotiations is not easy. Since the agents are selfinterested, when a negotiation is said to be successful we must ask “successful for whom?” since each agent is concerned only with its own benefits or looses from the resolution of the negotiation. Nevertheless, there are some parameters that can be used to evaluate different protocols:

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ƒ

Distribution: the decision-making process should be distributed. There should be no central unit or agent required to manage the process. ƒ sSmmetry: the coordination mechanism should not treat agents differently in light of non-relevant attributes. In the considered situations, the agents’ utility functions and their role in the encounter are the relevant attributes. All other attributes, such as an agent’s name, characteristic, or manufacturer, are not relevant. That is, symmetry implies that given a specific situation, the replacement of an agent with another that is identical with respect to the above attributes will not change the outcome of the negotiation. The following parameters can be used to evaluate the results of the negotiation: ƒ Negotiation time: negotiations that end without delay are preferable to negotiations that are time-consuming. This consideration is based on the assumption that a delay in reaching an agreement causes an increase in the cost of communication and computation time spent on the negotiation. ƒ Efficiency: it is preferred that the outcome of the negotiations will be efficient. It increases the number of agents that will be satisfied by the negotiation results and the agents’ satisfaction levels from the negotiation results. Thus it is preferable that the agents reach Pareto optimal agreements. ƒ Simplicity: negotiation processes that are simple and efficient are preferable to complex processes. Being a “simple strategy” means that it is feasible to build it into an automated agent. A “simple strategy” is also one that an agent will be able to compute in a reasonable amount of time. ƒ Stability: a set of negotiation strategies for a given set of agents is stable if, given that all the other agents included in the set are following their strategies, it is beneficial to an agent to follow its strategy too. Negotiation stable strategies are more useful in multi-agent environments than unstable ones: if there are stable strategies, we can recommend to all agent designers to build only relevant strategies into their agents.

3.7 Modelling Approaches and Solution Concepts Many formal approaches and solution concepts have already been utilised in negotiation modelling mainly because of the variety of disciplines involved in the negotiation study. Obviously, further work is needed regarding the comparison and categorisation of numerous results from behavioural research, which concentrated on such issues as the relationships between the social and individual context and the negotiation process and its outcomes, the communication patterns, interpretations and misinterpretations, and the relationships between individual characteristics and the process. In the following the most relevant approaches and solution concepts are briefly reported.

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3.7.1 Decision Theory Negotiations require that the participants make many decisions. The offers, counter-offers and concessions they make in an effort to search for an agreement result from the individual decisions they make. The issues of formulating and solving decision problems, including the specification of feasible alternatives, formulation of decision criteria and preferences, are the subject of decision theory. There is a vast literature on decision theory, rationality and decision making. One of the central tenets of decision theory is rationality and its use in the assessment of decision alternatives and choice. It is well known that people often do not conform to the rationality principles even if they are aware of breaking the axioms [46, 47]. Decision theorists assert that even if full rationality is not attainable, instrumental rationality is, which requires a rigorous approach to gathering and using information about the problem and about the decisionmaker [48]. This is perhaps one of the most important contributions of decision theory which also is directly applicable to negotiations. The three central topics of decision theory are: decision alternatives, multiple conflicting objectives, and uncertainty of the decision outcomes. Uncertainty of decision outcomes is an important issue which needs to be considered in every decision problem, including negotiations. With the exception of the participants having a different opinion about uncertainty, it is an issue that is inherent to the problem and not subject to negotiation. The specification of decision alternatives, either implicitly (i.e., with the use of constraints) or explicitly, is performed during the structuring of the decision problem. The development of the problem’s analytic structure involves the consideration of decision attributes and objectives. The attributes, objectives and alternatives are key aspects of negotiations; decision theory provides a number of well-defined approaches for structuring [49] as well as specific techniques, including decision trees, influence diagrams, and decision tables. Many decisions are difficult because the decision maker has multiple and conflicting objectives. Decision theory suggests the of use a preference elicitation scheme and a function defined on the objectives and preferences [48]. This function, under rationality conditions, determines the subjective value of a decision alternative in the case of certainty or the utility, in the case of uncertainty. This function is a measurement function defined on the set of alternatives and it allows us to compare them. In particular it allows us to compare offers and counter-offers, determine differences among them and select concessions that may increase the utility for the counterpart while limiting the decrease of one’s own utility. The use of utility functions allows for the achievement of efficient compromises providing that the participants or a third party knows these functions and the set of feasible alternatives. Given the set of feasible alternatives, it is the utility function that assigns a unique value to each alternative, making the choice of the optimal alternative automatic. Because of the importance of utility, decision analysts propose that this function be constructed carefully and with the help of experts. They apply a preference elicitation and utility construction approach so that the assumptions underlying the multi-attribute utility theory (MAUT) are met. The strict requirements of MAUT and difficulties in utility construction led to the use of proxy functions neglecting rationality principles.

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3.7.2 Game Theory Game theory is the most rigorous approach towards conflict resolution and allows for formal problem analysis and the specification of well-defined solutions. Gametheoretical bargaining models assume rationality of agents and that each agent’s choice can be uniquely described by utility function. Linhart [50] summarises several restrictions regarding game-theoretical models of bargaining. For example, human agents are often constrained by their bounded rationality, which is one reason why game-theoretical models of bargaining do not fit in explaining real human negotiating behaviour. Yet, game-theoretical models of bargaining are seen as potential candidates for automated negotiations among software agents. 3.7.3 Negotiation Analysis The limitations of game theory are well known and rooted in its normative orientation caused mainly by the strict rationality assumptions that are not met in reality. Therefore, game-theoretica models do not allow for prescriptive advice that is sought by negotiators and their advisers. From the analyst’s perspective it may be reasonable to assume that the negotiator whom the analyst advises is rational however; it is not reasonable to assume the same about the opponent. This observation, together with a weakened rationality assumption, provided the basis for negotiation analysis [51]. With the objective of providing advice to one party, negotiation analysis takes prescriptive/descriptive orientation in that it assumes rationality of one party but not necessarily of the other [52]. Negotiation analysis integrates decision analysis and game theory in order to provide formal and meaningful support. The goal of negotiation analysis is to bridge the gap between descriptive qualitative models and normative game-theoretical models of bargaining. It adopts a number of behavioural concepts and incorporates them in quantitative models. This way significantly extends the expressiveness of the models and their capability of describing various negotiation situations. It also allowed analysts and advisors to conduct formal analysis of negotiations and to provide support. Negotiation analysis tends to adopt the application of gametheoretical solution concepts or efforts to find unique equilibrium outcomes. Instead, negotiation analysts generally focus on changes in perceptions in the zone of possible agreement and the distribution of possible negotiated outcomes, conditional on various actions [51, 53]. The contributions of negotiation analysis include: a subjective perspective on the process and outcomes, concentration on the possible agreements rather than search for one equilibrium point, and acceptance of goal-seeking rather than game-theoretical rationality. This makes an asymmetric perspective possible [44, 52]. Other approaches have a symmetrical orientation: behavioural studies focus on descriptions of the parties and their interactions, game theory and optimisation assume that the parties are rational hence they have symmetrically prescriptive orientation. In contrast, negotiation analysis is used to generate prescriptive advice to the supported party given a descriptive assessment of the opposing parties. In other words, negotiation analysis reconciled several important concepts of behavioural research and game theory.

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3.8 Strategic Negotiation The strategic-negotiation model is based on Rubinstein’s model of alternating offers [54]. In the strategic model there are N agents, Agents ^ A1 ,..., AN ` . The agents need to reach an agreement on a given issue. It is assumed that the agents can take actions in the negotiation only at certain times in the set T ^0,1, 2...` that are determined in advance and are known to the agents. In each period t  T of the negotiation, if the negotiation has not terminated earlier, the eligible agent makes an offer suggesting a possible agreement (with respect to the specific negotiation issue), and each of the other agents may either accept the offer (choose Yes), reject it (choose No), or opt out of the negotiation (choose Opt). If an offer is accepted by all the agents (i.e., all of them choose Yes), then the negotiation ends, and this offer is implemented. If at least one of the agents opts out of the negotiation, then the negotiation ends and a conflictual outcome results. If no agent has chosen “Opt”, but at least one of the agents has rejected the offer, the negotiation proceeds to period t + 1, and the next eligible agent makes a counter-offer, the other agents respond, and so on. We assume that an agent responding to an offer is not informed of the other responses during the current negotiation period. We call this protocol a simultaneous response protocol. In the strategic-negotiation model there are no rules which bind the agents to any specific strategy. We do not make any assumptions about the offers the agents make during the negotiation. In particular, the agents are not bound to any previous offers that have been made. After an offer is rejected, an agent can choose whether to make the same offer again, or to propose a new offer. The protocol only provides a framework for the negotiation process and specifies the termination condition, but there is no limit on the number of periods. A fair and reasonable method for deciding on the order in which agents will make offers is to arrange them randomly in a specific order before the negotiation begins. The agents’ time preferences and the preferences between agreements and opting out are the driving force of the model. They will influence the outcome of the negotiation. In particular, agents will not reach an agreement which is not at least as good as opting out for all of them. Otherwise, the agent who prefers opting out over the agreement, will opt out.

3.9 Negotiation Strategies In the literature special cases of negotiation strategies are discussed [55, 56]. One important family of strategies consists of functions which are only time-dependent, i.e. strategy changes as the time goes on. Other families of strategies which could be employed by the agents are resource-dependent and behaviour-dependent strategies. In a resource-dependent strategy the scarcer the resource the more urgent the need for an agreement to be reached. Taking a behaviour-dependent strategy an agent has to analyze the behaviour of other agents involved in the negotiation and generate his counter-offers according to the dynamics of how the values are changed in the offers he has received. In general, a strategy of an agent

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can be very complex, switching over time between time-dependent, resourcedependent, behaviour-dependent, and other strategies. From a strategic point of view, an agent’s negotiation strategy specifies for the agent what to do next, for each sequence of offers s0 ,..., st . In other words, for the agent whose turn it is to make an offer, it specifies which offer to make next. That is, it indicates to the agent which offer to make at t  1 , if in periods 0 until t the offers s0 ,..., st had been made and were rejected by at least one of the agents, but none of them has opted out. Similarly, in time periods when it is the agent’s turn to respond to an offer, the strategy specifies whether to accept the offer, reject it or opt out of the negotiation. A strategy profile is a collection of strategies, one for each agent [57]. The main question is how a rational agent will choose its negotiation strategy. A useful notion is the Nash equilibrium [58, 59]. This means that if all the agents use the strategies specified for them in the strategy profile of the Nash equilibrium, then no agent has a motivation to deviate and use another strategy. However, the use of the Nash equilibrium in a model of alternating offers leads to absurd Nash equilibria [60]: an agent may use a threat that would not be carried out if the agent were put in the position to do so, since the threat fulfilment would give to the agent lower payoff than it would get by not undertaking the threatened action. This is because Nash equilibrium strategies may be in equilibrium only in the beginning of the negotiation, but may be unstable in intermediate stages. The concept of subgame perfect equilibrium [57], which is a stronger concept, is defined as a strategy profile if the strategy profile induced in every subgame is a Nash equilibrium of that subgame. This means that at any step of the negotiation process, no matter what the history is, no agent has motivation to deviate and use any strategy other than that defined in the strategy profile. In situations of incomplete information there is no proper subgame. The sequential equilibrium [61], which takes the beliefs of the agents into consideration, can be used in the incomplete information situations.

3.10 References [1] [2] [3] [4] [5] [6] [7] [8] [9]

Gulliver PH (1979) Disputes and negotiations: a cross-cultural perspective. Orlando, FL, Academic Bell DE, Raiffa H (1991) Decision making: descriptive, normative, and prescriptive interactions. Cambridge, Cambridge University Pruitt DG (1981) Negotiation behaviour. New York, Academic Nash JF (1950) The bargaining problem. Econometrica 18: 155–162 Roth AE (1979) Axiomatic models of bargaining. Berlin, Springer-Verlag Holsapple CW, Lai H (1998) A formal basis for negotiation support system research. Group Decis Negot 7(3): 199–202 Kersten GE (1997) Support for group decisions and negotiations: an overview. In multicriteria analysis, Climaco J (ed), Springer Verlag, pp 332–346 Maes P, Guttman R (1999) Agents that buy and sell: transforming commerce as we know it. Commun ACM 42(3): 81 Rosenschein JS, Zlotkin G (1994) Rules of encounter: designing conventions for automated negotiation among computers. Cambridge, MA, MIT Press

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[10] Bui T, Yen J (2001) A multi-attribute negotiation support system with market signaling for electronic markets. Group Decis Negot 10(6): 515–537 [11] Hamalainen RP (1995) Special issue on dynamic game modeling in bargaining and environmental negotiations. Group Decis Negot 4(1) [12] Raiffa H (1996) Lectures on negotiation analysis. Cambridge, MA, PON [13] Teich JE, Wallenius H. (1994) Advances in negotiation science. Trans Oper Res 6: 55–94 [14] Kumar M, Feldman S (1998) Internet auctions. Proc 3rd USENIX Workshop on Electronic Commerce. Boston, MA: 49–60 [15] Pennock DM, Horvitz E (2000). Collaborative filtering by personality diagnosis: a hybrid memory and model-based approach. Proc 16th conference on uncertainty in artificial intelligence, San Francisco, USA [16] Thompson L (2001) The mind and heart of the negotiator. Upper Saddle River, NJ, Prentice Hall [17] Raisch W (2000) The emarketplace: strategies for success in B2B ecommerce, McGraw-Hill [18] Shim J Hsiao N (1999) A literature review on web-based negotiation support system. Documentation for web-based negotiation training [19] Vakali A, Angelis L (2001) Internet based auctions: a survey on models and applications. SIGe-com, Exch Newsl ACM Spec Interest Group E-Commerce 2(2): 6– 15 [20] Jennings NR, Faratin P (2001). Automated negotiations: prospects, methods and challenges. Group Decis Negot 10(2): 199–215 [21] Edwards J (2001) Working the wiggle room. Line 56(April): 50–55 [22] Parkes D (1999) iBundle: an efficient ascending price bundle auction. Proc 1st ACM conference on electronic commerce (EC–99): 148–157 [23] Rassenti S, Smith VL (1982) A combinatorial auction mechanism for airport time slot allocations. Bell J Econ 13: 402–417 [24] Rothkopf MH, Pekec A (1998). Computationally manageable combinatorial auctions. Manag Sci 44(8): 1131–1147 [25] Guttman R (1998) Agent-mediated integrative negotiation for retail electronic commerce. Proc workshop on agent-mediated electronic trading (AMET'98) [26] Wurman P, Wellman M (1998). The michigan internet auctionbot: a configurable auction server for human and software agents. Proc 2nd int conf on autonomous agents (Agents–98) [27] Stroebel M (2002). A design and implementation framework for multi-attribute negotiation: intermediation in electronic markets. University of St. Gallen, Switzerland. [28] Kraus S, Lehmann D (1995) Designing and building a negotiating automated agent. Comput Intell 11(1):132–171 [29] Teich JE, Wallenius H (2001) Designing electronic auctions: an internet-based hybrid procedure combining aspects of negotiations and auctions. Electron Commerce Res 1(1): 301–314 [30] Subrahmanian VS, Bonatti P Dix J, Eiter T, Kraus S, Ozean F, Ross R (2000). Heterogenous agent systems: theory and implementation. MIT Press, Cambridge, Massachusetts [31] Wooldrige MJ, Jennings NR (1995). Intelligent agents. Springer-Verlag, Berlin [32] Etzioni O, Weld DS (1995) Intelligent agents on the internet: fact, fiction and forecast. IEEEE Expert 10(4):44–49 [33] Conry SE, Kuwabara K, Lesser VR, Meyer RA (1991) Multistage negotiation for distributed satisfaction. IEEE Trans Syst Man Cybern, Special issue on distributed artificial intelligence, 21(6):1462–1477

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[34] Moehlman T, Lesser V, Buteau B (1992) Decentralized negotiation: An approach to the distributed planning problem. Group Decision and Negotiation, 2:161–191 [35] Lander SE, Lesser VR (1992). Customizing distributed search among agents with heterogeneous knowledge. Proc. 1st int conf information knowledge management Baltimore pp 335–344 [36] Sycara KP (1987) Resolving adversarial conflicts: an approach to integrating casebased and analytic methods. PhD thesis, School of Information and Computer Science, Georgia Institute of Technology [37] Sycara KP (1990). Persuasive argumentation in negotiation. Theory Decis, 28:203– 242 [38] Sierra C, Faratin P, Jennings N (1997) A service-oriented negotiation model between autonomous agents. Proc. 8th european workshop on modeling autonomous agents in a multi-agent world (MAAMAW–97), Ronneby, Sweden pp 17–35 [39] Zeng D., Sycara K (1998) Bayesian learning in negotiation. Int J Hum Comput Studi, 48:125–141 [40] Sandholm TW, Lesser VR (1995) Issues in automated negotiation and electronic commerce: extending the contract net framework. 1st int conf multiagent systems (ICMAS–95), San Fransisco pp 328–335 [41] Weinhardt C, Gomber P (1999) Agent-mediated off-exchange trading. Proc 32nd Hawaii conference on system sciences [42] Kersten G, Mallory G (1999) Rational inefficient compromises in negotiations. Ottawa, Canada, Interneg, Carleton University [43] Raiffa H (1982) The Art and Science of Negotiation. Cambridge, MA, Harvard University Press [44] Jackson MO (2000) Mechanism theory. Pasadena, CA, Humanities and social sciences, California institute of technology [45] Budimir M, Holtmann C (2001) The design of innovative securities markets: the case of asymmetric information. In e-Finance: innovative problemlösungen für Informationssysteme in der Finanzwirtschaft. Buhl HU, Kreyer N, Steck W (eds) Berlin, Heidelberg, New York, Springer: 175–196 [46] Kahneman D, Slovic P (1982) Judgement under uncertainty: heuristics and biases. Cambridge, MA:, Cambridge Univ Press [47] Nozick R (1993) The nature of rationality. Princeton, NJ, Princeton University Press. [48] French S (1998) Decision theory: an introduction to the mathematics of rationality. New York, Ellis Hoerwood [49] Keeney R (1992) Value-focused thinking: a path to creative decision making. Cambridige, Harvard University Press [50] Linhart PB, Radner R (1992). Bargaining with incomplete information. San Diego, CA, Academic Press [51] Sebenius JK (1992) Negotiation analysis: a characterization and review. Manag Sci 38(1): 18–38 [52] Young HP (1991) Negotiation analysis. Ann Arbor, The University of Michigan Press [53] Clyman DR (1995) Measures of joint performance in dyadic mixed-motive negotiations. Organ Behav Hum Decis Proces 64(1): 38–48 [54] Rubinstein A (1982) Perfect equilibrium in a bargaining model. Econometrica, 50(1):97–109 [55] Jennings NR, Parsone S, Sierra C, Faratin P (2000) Automated negotiation. Proc 5th int conf on the practical application of intelligent agents and multi-agent technology (PAAM–2000), pp 23–30 [56] Teuteberg F, Kurbel K (2002) Anticipating agents negotiation strategies in an emarketplace using belief models. Proc 5th int conf business information systems BIS 2002, Poznan, Poland

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[57] Osborne MJ, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge, Massachusetts [58] Nash JF (1953) Two-person cooperative games. Econometrica, 21:128–140 [59] Luce RD, Raiffa H (1957) Games and decisions. Wiley, New York [60] Tirole J (1988) The theory of industrial organization. MIT Press, Cambridge, MA [61] Kreps D, Wilson R (1982) Sequential equilibria. Econometrica, 50:863–894

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4 Multiple-agent Systems: an Overview

4.1 Introduction Most researchers in artificial intelligence (AI) to date have dealt with developing theories, techniques, and systems to study and understand the behaviour and reasoning properties of a single cognitive entity. AI has matured, and it endeavours to face more complex, realistic, and large-scale problems. Such problems are beyond the capabilities of an individual agent. The capacity of an intelligent agent is limited by its knowledge, its computing resources, and its perspective. This bounded rationality is one of the underlying reasons for creating problem-solving organisations. The most powerful tools for handling complexity are modularity and abstraction. Multi-agent systems (MASs) offer modularity. If a problem domain is particularly complex, large, or unpredictable, then the only way it can reasonably be addressed is to develop a number of functionally specific and (nearly) modular components (agents) that are specialised at solving a particular problem aspect. This decomposition allows each agent to use the most appropriate paradigm for solving its particular problem. When interdependent problems arise, the agents in the system must coordinate with each other one to ensure that interdependencies are properly managed. Research in MASs is concerned with the study, behaviour, and construction of a collection of possibly pre-existing autonomous agents that interact with each other and their environments. Study of such systems goes beyond the study of individual intelligence to consider, in addition, problem solving that has social components. A MAS can be defined as a loosely coupled network of problem solvers that interact to solve problems that are beyond the individual capabilities or knowledge of each problem solver [1]. These problem solvers, often called agents, are autonomous and can be heterogeneous or homogenous. The characteristics of MASs are that: ƒ each agent has incomplete information or capabilities for solving the problem and, thus, has a limited viewpoint; ƒ there is no system global control; ƒ data are decentralised; and ƒ computation is asynchronous.

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The motivations for the increasing interest in MAS research include the ability of MASs to do the following. First, MASs are able to solve problems that are too large for a centralized systems because of resource limitations or the sheer risk of having one centralized system that could be a performance bottleneck or could fail at critical times. Second, MASs can allow interconnection and interoperation of multiple existing legacy systems. To keep pace with changing business needs, legacy systems must periodically be updated. Completely rewriting such software tends to be prohibitively expensive and is often simply impossible. Therefore, in the short to medium term, the only way that such legacy systems can remain useful is to incorporate them into a wider cooperating agent community in which they can be exploited by other pieces of software. Third, MASs provide solutions to problems that can naturally be regarded as a society of autonomous interacting components agents. For example, in meeting scheduling, a scheduling agent that manages the calendar of its user can be regarded as autonomous and interacting with other similar agents that manage calendars of different users [2, 3]. Such agents also can be customised to reflect the preferences and constraints of their users. Other examples include air-traffic control [4] and multi-agent bargaining for buying and selling goods on the internet. Fourth, MASs could provide solutions that efficiently use information sources that are spatially distributed. Examples of such domains include sensor networks [5], seismic monitoring [6], and information gathering from the Internet [7]. Fifth, MASs could provide solutions in situations where expertise is distributed. Examples of such problems include concurrent engineering [8], health care, and manufacturing. Sixth, MASs could enhance performance along the dimensions of computational efficiency because concurrency of computation is exploited (as long as communication is kept minimal, for example, by transmitting high-level information and results rather than low-level data); reliability, that is, graceful recovery of component failures, because agents with redundant capabilities or appropriate interagent coordination are found dynamically (for example, taking up responsibilities of agents that fail); extensibility because the number and the capabilities of agents working on a problem can be altered; robustness, the system’s ability to tolerate uncertainty, because suitable information is exchanged among agents; maintainability because a system composed of multiple components agents is easier to maintain because of its modularity; responsiveness because modularity can handle anomalies locally, not propagate them to the whole system; flexibility because agents with different abilities can adaptively organise to solve the current problem; and reuse because functionally specific agents can be reused in different agent teams to solve different problems.

4.2 Applications MASs are now a research reality and are rapidly having a critical presence in many human computer environments. A very challenging application domain for MASs technology is the Internet. Today the Internet has developed into a highly

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distributed open system where heterogeneous software agents come and go, there are no well-established protocols or languages on the agent level (higher than TCP/IP), and the structure of the network itself keeps on changing. In such an environment, MAS technology can be used to develop agents that act on behalf of a user and are able to negotiate with other agents in order to achieve their goals. Auctions on the Internet and electronic commerce are such examples [9, 10]. One can also think of applications where agents can be used for distributed data mining and information retrieval. MASs can also be used for traffic control where agents (software or robotic) are located in different locations, receive sensor data that are geographically distributed, and must coordinate their actions in order to ensure global system optimality [11]. Other applications are in social sciences where MAS technology can be used for simulating interactivity and other social phenomena [12], in robotics where a frequently encountered problem is how a group of robots can localise themselves within their environment [13], and in virtual reality and computer games where the challenge is to build agents that exhibit intelligent behaviour [14].

4.3 Challenging Issues The transition from single-agent systems to MASs offers many potential advantages but also raises challenging issues. Some of these are: ƒ how to decompose a problem, allocate subtasks to agents, and synthesise partial results; ƒ how to handle the distributed perceptual information. How to enable agents to maintain consistent shared models of the world; ƒ how to implement decentralised control and build efficient coordination mechanisms among agents; ƒ how to design efficient multi-agent planning and learning algorithms; ƒ how to represent knowledge. How to enable agents to reason about the actions, plans and knowledge of other agents; ƒ how to enable agents to communicate. What communication languages and protocols to use. What, when, and with whom should an agent communicate; ƒ how to enable agents to negotiate and resolve conflicts; ƒ how to enable agents to form organisational structures like teams or coalitions; ƒ how to assign roles to agents; ƒ how to ensure coherent and stable system behaviour. Clearly the above problems are interdependent and their solutions may affect each other. For example, a distributed planning algorithm may require a particular coordination mechanism, learning can be guided by the organisational structure of the agents, and so on. In the following sections we will try to provide answers to some of the above questions.

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4.4 Individual Agent Reasoning Sophisticated individual agent reasoning can increase MAS coherence because each individual agent can reason about non-local effects of local actions, form expectations of the behaviour of others, or explain and possibly repair conflicts and harmful interactions. Numerous works in AI research try to formalise a logical axiomatisation for rational agents. This axiomatisation is accomplished by formalising a model for agent behaviour in terms of beliefs, desires, goals, and so on. An agent will be considered rational if it always selects an action that optimises an appropriate performance measure, given what the agent knows so far. The performance measure is typically defined by the user (the designer of the agent) and reflects what the user expects from the agent in a specific task. The problem of optimal decision making of an agent was first studied in optimal control [15]. For the purpose of this discussion, it will be considered a discrete set of time steps t 1,..., n , in each of which the agent must choose an action from a finite set of actions A that it has available. Intuitively, in order to act rationally, an agent should take both the past and the future into account when choosing an action. The past refers to what the agent has perceived and what actions it has taken until time t, and the future refers to what the agent expects to perceive and do after time t. If we denote by oW the perception of an agent at time W , then the above implies that in order for an agent to optimally choose an action at time t, it must in general use its complete history of perceptions oW and actions aW for W d t. The function S ( o1 , a1 , o2 , a2 ,..., ot ) at that maps the complete history of perception action pairs up to time t to an optimal action at is called the policy of the agent. As long as we can find a function S that implements the above mapping, the part of optimal decision making that refers to the past is solved. However, defining and implementing such a function is problematic; the complete history can consist of a very large (even infinite) number of perception action pairs, which can vary from one task to another. Merely storing all perceptions would require very large memory, aside from the computational complexity for actually computing S . This fact calls for simpler policies. One possibility is for the agent to ignore all its percept history except for the last perception ot . In this case its policy takes the form S ( ot ) at which is a mapping from the current perception of the agent to an action. An agent that simply maps its current perception ot to a new action at , thus effectively ignoring the past, is called a reflex agent, and its policy is called reactive or memoryless.

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4.5 Observable Worlds From the discussion above it is clear that the terms agent and environment are coupled, so that one cannot be defined without the other. In fact, the distinction between an agent and its environment is not always clear, and it is sometimes difficult to draw a line between these two [16]. To simplify things it will be assumed hereafter the existence of a world in which one or more agents are embedded, and in which they perceive, think, and act. The collective information that is contained in the world at any time step t, and that is relevant for the task at hand, will be called a state of the world and denoted by st . The set of all states of the world will be denoted by S. Depending on the nature of problem, a world can be either discrete or continuous. A discrete world can be characterised by a finite number of states. Examples are the possible board configurations in a chess game. On the other hand, a continuous world can have infinitely many states. A fundamental property that characterises a world from the point of view of an agent is related to the perception of the agent. We will say that the world is (fully) observable to an agent if the current perception ot of the agent completely reveals the current state of the world, that is, st ot . On the other hand, in a partially observable world the current perception ot of the agent provides only partial information about the current world state in the form of a conditional probability distribution P(st |ot ) over states. This means that the current perception ot does not fully reveal the true world state, but to each state st the agent assigns probability P(st |ot ) that st is the true state (with 0 d P(st |ot ) d 1 and

¦

st S

P(st |ot ) 1). Here we treat st as a random variable that can take all

possible values in S. Partial observability can be attributed, for instance, to an inherent property of the environment referred to as perceptual aliasing: different states may produce identical perceptions to the agent at different time steps. In other words, two states may look the same to an agent, although the states are different from each other. Partial observability is much harder to handle than full observability, and algorithms for optimal sequential decision making in a partially observable world can easily become intractable [17].

4.6 Stochastic Transitions and Utilities As mentioned above, at each time step t the agent chooses an action at from a finite set of actions A. When the agent takes an action, the world changes as a result of this action. A transition model (sometimes also called world model) specifies how the world changes when an action is executed. If the current world state is st and the agent takes action at , we can distinguish the following two cases:

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ƒ

in a deterministic world, the transition model maps a state action pair ( st ; at ) to a single new state st  1 . In chess for example, every move changes the configuration on the board in a deterministic manner; ƒ in a stochastic world, the transition model maps a state action pair ( st ; at ) to a probability distribution P(st  1 |st , at ) over states. As in the partial observability case above, st  1 is a random variable that can take all possible values in S, each with corresponding probability P(st  1 |st , at ) . Most real-world applications involve stochastic transition models, for example, robot motion is inaccurate because of wheel slip and other effects. In classical AI, a goal for a particular task is a desired state of the world. Accordingly, planning is defined as a search through the state space for an optimal path to the goal. When the world is deterministic, planning comes down to a graph search problem for which a variety of methods exist, see for example [17]. In a stochastic world, however, planning can not be done by simple graph search because transitions between states are non-deterministic. The agent must now take the uncertainty of the transitions into account when planning. To see how this can be realised, note that in a deterministic world an agent prefers by default a goal state to a non-goal state. More generally, an agent may hold preferences between any world states. A way to formalise the notion of state preferences is by assigning to each state s a real number U(s) that is called the utility of state s for that particular agent. Formally, for two states s and s’ U(s) > U(s’) holds if and only if the agent prefers state s to state s’, and U(s) = U(s’) if and only if the agent is indifferent between s and s’. Intuitively, the utility of a state expresses the desirability of that state for the particular agent; the larger the utility of the state, the better the state is for that agent. Equipped with utilities, the question now is how an agent can efficiently use them for its decision making. Let us assume that the world is stochastic with transition model P(st  1 |st , at ), ,and is currently in state st , while an agent is pondering how to choose its action at . Let U(s) be the utility of state s for the particular agent (we assume there is only one agent in the world). Utility-based decision making is based on the premise that the optimal action a* of t

the agent should maximise expected utility, that is, a * arg max ¦ P(st  1 |st , at ) t

at A

st  1

where it is summed over all possible states st  1  S the world may transition to, given that the current state is st and the agent takes action at . In words, to see how good an action is, the agent has to multiply the utility of each possible resulting state with the probability of actually reaching this state, and sum up the resulting terms. Then the agent must choose the action a * that gives the highest t

sum. If each world state has a utility value, then the agent can do the above calculations and compute an optimal action for each possible state. This provides the agent with a policy that maps states to actions in an optimal sense. In particular,

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given a set of optimal (i.e., highest attainable) utilities U*(s) in a given task, the greedy policy S * (s ) arg max ¦ P(s '|s , a)U * (s ') a

s'

is an optimal policy for the agent.

4.7 Distributed Decision Making As previously discussed, a distinguishing feature of a multi-agent system is the fact that the decision making of the agents can be distributed. This means that there is no central controlling agent that decides what each agent must do at each time step, but each agent is to a certain extent responsible for its own decisions. The main advantages of such a decentralised approach over a centralised one are efficiency, due to the asynchronous computation, and robustness, in the sense that the functionality of the whole system does not rely on a single agent. In order for the agents to be able to take their actions in a distributed fashion, appropriate coordination mechanisms must be additionally developed. Coordination can be regarded as the process by which the individual decisions of the agents result in good joint decisions for the group. A typical situation where coordination is needed is among cooperative agents that form a team, and through this team they make joint plans and pursue common goals. In this case, improving coherence by planning their actions ensures that the agents do not obstruct each other when taking actions, and moreover that these actions serve the common goal. Planning for a single agent is a process of constructing a sequence of actions considering only goals, capabilities, and environmental constraints. However, planning in a MAS environment also considers the constraints that the other agents’ activities place on an agent’s choice of actions, the constraints that an agent’s commitments to others place on its own choice of actions, and the unpredictable evolution of the world caused by other unmodeled agents. Most early work in distributed artificial intelligence (DAI) has dealt with groups of agents pursuing common goals (for example, [18-21]). Agent interactions are guided by cooperation strategies meant to improve their collective performance. Most work on multi-agent planning assumes an individual sophisticated agent architecture that enables them to do rather complex reasoning. Early work on distributed planning took the approach of complete planning before action. To produce a coherent plan, the agents must be able to recognise subgoal interactions and avoid them or resolve them. Another direction of research in cooperative multi-agent planning has been focused on modelling teamwork explicitly. Explicit modelling of teamwork is particularly helpful in dynamic environments where team members might fail or be presented with new opportunities. In such situations, it is necessary that teams monitor their performance and reorganise based on the situation. In many practical applications, however, we have to deal with self-interested agents, for example, agents that act on behalf of some owner that wants to maximise his or her own profit. A typical example is a software agent that participates in an electronic auction. Developing an algorithm or protocol for such a system is a much more challenging task than in the cooperative case. First, an agent has to be motivated to participate in the

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protocol, which is not a priori the case. Second, we have to take into account the fact that an agent may try to manipulate the protocol for its own interest, leading to suboptimal results. The latter includes the possibility that the agent may lie, if needed. The development of protocols that are stable (non-manipulable) and individually rational for the agents is the subject of mechanism design or implementation theory.

4.8 Recognising and Resolving Conflicts Because MASs lack global viewpoints, global knowledge, and global control, there is the potential for disparities and inconsistencies in agents’ goals, plans, knowledge, beliefs, and results. To achieve coherent problem solving, these disparities must be recognised and resolved. Disparities can be resolved by making an agent omniscient so it can see the states of all agents and determine where the disparities lie and how to resolve them. This approach is limiting because it makes this agent a bottleneck and a single point of failure. To detect and correct disparities and conflicts using only local perspective is difficult. To facilitate detection and resolution of conflicts, agents can rely on models of the world and other agents. Disparity resolution can be influenced by the organisational structure of the agent society and an agent’s role within it, the kinds of models an agent has, and the agent’s reasoning algorithms. The main approach for resolving disparities in a MAS is negotiation. Negotiation is seen as a method for coordination and conflict resolution (for example, resolving goal disparities in planning, resolving constraints in resource allocation, resolving task inconsistencies in determining organisational structure). Negotiation has also been used as a metaphor for communication of plan changes, task allocation, or centralised resolution of constraint violations. The main characteristic of negotiation that is necessary for developing applications in the real world is the presence of some sort of conflict that must be resolved in a decentralised manner by self-interested agents under conditions of bounded rationality and incomplete information. Furthermore, the agents communicate and iteratively exchange proposals and counterproposals.

4.9 Communicating Agents Multi-agent interaction is often associated with some form of communication that involves several levels of abstraction. On the lower, network level, one would like to make sure that the messages that are communicated among the agents arrive safely and promptly at their destination. For that, several well-studied formalisms and protocols exist in the distributed systems literature [22]. On an intermediate, language level, one would like to have a basic set of language primitives and a standardised format for exchanging these primitives, so that agents that speak the same language can easily understand each other. Finally, on a high, application level, one would like to effectively use communication for solving standard multiagent problems, like coordination or negotiation. A formal way to describe

Multiple-agent Systems: an Overview

49

communication is by treating each communication primitive as an action that updates the knowledge of an agent about aspects of the state like those described above. The communication primitives that are exchanged among agents are typically referred to as communicative acts or speech acts. Some of the most common types of communicative acts are the following: ƒ informing about some aspects of the current state, ƒ querying aspects of the state that are hidden, ƒ committing to a particular action, ƒ directing an agent to do an action. Each communicative act can affect the knowledge of the agents in a different way. For example, an informing act can reduce the uncertainty of an agent about the current state by eliminating candidate states from his information set. A committing communicative act, on the other hand, can be used for informing about the chosen course of action of an agent. Similar interpretations can be given to the other communicative acts. For instance, a prohibiting act can have the same effect as a role, by enforcing the deactivation of some actions of an agent in a particular situation. Several agent communication languages have been proposed in the agent community, aiming at standardizing the multi-agent communication process [23]. The two most notable ones are Knowledge Query and Manipulation Language (KQML) and Foundation for Intelligent Physical Agents (FIPA), each using a slightly different syntax and set of communicative acts. Unfortunately, many dialects of these languages have already appeared, and the languages seem not to conform with other standards (for example, in the Internet). As is often the case, we might see in the future new language standards emerging directly from applications.

4.10 References [1]

[2]

[3]

[4]

[5]

[6]

[7]

Durfee EH, Lesser V (1989) Negotiating task decomposition and allocation using partial global planning. In Gasser L, Huhns M (eds) Distributed artificial intelligence, Vol 2, ,. San Francisco, CA, Morgan Kaufmann pp 229–244 Dent L, Boticario J, McDermott J, Mitchell T, Zabowski D (1992) A personal learning apprentice. Proc 10th national conference on artificial intelligence. Menlo Park, CA, American Association for Artificial Intelligence, pp 96–103 Garrido L, Sycara K (1996) Multiagent meeting scheduling: preliminary experimental results. Proc 2nd int conf on multiagent systems, Menlo Park, CA, American Association for Artificial Intelligence, pp 95–102. Kinny D, Ljungberg M, Rao A, Sonenberg E, Tidhard G, Werner E (1992) Planned team activity. In Castelfranchi C, Werner E (eds) Artificial social systems, New York, Springer-Verlag Corkill DD, Lesser VR (1983) The use of metalevel control for coordination in a distributed problem solving network. Proc 8th int joint conf on artificial intelligence (IJCAI–83), Menlo Park, CA, pp 767–770 Mason C, Johnson R (1989) DATMS: A framework for distributed assumption-based reasoning. In Huhns M, Gasser L (eds) Distributed artificial intelligence, Vol 2,. San Francisco, CA, Morgan Kaufmann, pp 293–318 Sycara K, Decker K, Pannu A, Williamson M, Zeng D (1996) Distributed intelligent agents. IEEE Expert 11(6): 36–46

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[8]

Lewis CM, Sycara K (1993) Reaching informed agreement in multispecialist cooperation. Group Decis Negot 2(3): 279–300 Sandholm T (1999) Distributed rational decision making. In Weiss G (ed) Multiagent systems: a modern introduction to distributed artificial intelligence, MIT Press pp 201–258 Noriega P, Sierra C (1999) Agent mediated electronic commerce. Lecture notes in Artificial Intelligence 1571, Springer Lesser VR, Erman LD (1980) Distributed interpretation: a model and experiment. IEEE Trans Comput 29(12):1144–1163 Gilbert N, Doran J (1994) Simulating societies: the computer simulation of social phenomena. UCL Press, London Roumeliotis SI, Bekey GA (2002) Distributed multi-robot localization. IEEE Trans Rob Autom 18(5):781–795 Terzopoulos D (1999) Artificial life for computer graphics. Commun ACM, 42(8): 32–42 Bellman R (1961) Adaptive control processes: a guided tour. Princeton University Press. Sutton RS, Barto AG (1998) Reinforcement learning: an introduction. MIT Press, Cambridge, MA Russell SJ, Norvig P (2003) Artificial intelligence: a modern approach. Prentice Hall, 2nd edition Conry SE, Meyer RA, Lesser VR (1988) Multistage negotiation in distributed planning. In Bond AH, Gasser L, (eds) Readings in distributed artificial intelligence. San Francisco, CA, pp 367–384 Durfee EH (1988) Coordination of distributed problem solvers. Boston, Kluwer Academic Lesser VR (1991) A retrospective view of FA/C distributed problem solving. IEEE Trans Syst Man Cybern 21(6): 1347–1363 Lesser VR, Durfee EH, Corkill D (1989) Trends in cooperative distributed problem solving. IEEE Trans Knowl Data Eng 1(1): 63–83 Tanenbaum AS, van Steen M (2001) Distributed systems: principles and paradigms. Prentice Hall Labrou Y, Finin T, Peng Y (1999). Agent communication languages: the current landscape. IEEE Intell Syst 14(2):45–52

[9]

[10] [11] [12] [13] [14] [15] [16] [17] [18]

[19] [20] [21] [22] [23]

5 Distributed Production Planning in Reconfigurable Production Networks

5.1 Introduction Mass customisation consequences, such as shorter product life cycles and low-cost variety, have brought critical pressures to improve production efficiency, responsiveness to market changes, and substantial cost reduction. Scientific papers, as well as government and industry expectations, seem to acknowledge that the challenge keyword is “reconfiguration” [1]. Specifically, two major industrial responses to mass customisation can be acknowledged: reconfigurable manufacturing and reconfigurable enterprise (RE). From a manufacturing perspective, the most agreed response to mass customisation is connected to the concepts of modularity and reconfigurability of production systems. Thanks to their modularity reconfigurable manufacturing systems (RMS) allow us to achieve low customisation cost [2]. On the other hand, from an organizational point of view, reconfigurable production networks or enterprises are nowadays considered the industrial response to the great challenges conveyed by this new era characterised by the global market and the impressive advances of information and communication technology [3]. Market globalisation, indeed, has offered to companies the possibility to split geographically their production capacity; business opportunities lead companies to work together in temporary organisations; in the same firm, business units behave as autonomous profit centres and compete with each other for the production capacity allocation. In other words, REs represent production networks made of different and geographically dispersed plants which can be reconfigured in order to gather a specific production process or product family. However, the RE members need to be properly coordinated to achieve reduction in lead times and costs, alignment of interdependent decisionmaking processes, and improvement in the overall performance of each member, as well as of the RE. In this context, operations management and coordination of RE and RMS are challenging issues involving distributed problem-solving tasks. Specifically, in production planning (PP) and control activities, the concern with internal production planning is replaced by the complexity of the external supply

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chain. Indeed, as soon as a manufacturing unit tries to achieve coordination with its partners, it quickly faces difficulties associated with different operational conventions, locally specific constraints, software legacy and properties, conflicting objectives and misaligned incentives. This task becomes even more difficult when each RE member or plant consists of reconfigurable production systems; indeed, the reconfiguration capability makes planning and scheduling processes even more complex. The distributed production system (DPS), which has been considered here, is linked to the semiconductor component industry; indeed, usually, companies belonging to such an iIndustry sector are often organised into reconfigurable manufacturing units geographically dispersed. In particular, companies have a divisional organisation and the top management is called a corporate; each division is called a group and is in charge of a product family commercialisation. Each group can fulfil the collected orders thanks to the reconfigurable characteristics of the manufacturing plants. Reconfiguration skills offer clear advantages to the enterprise while increasing the complexity of production planning activities. Multi-agent systems (MAS) techniques have been largely used for their suitability in modelling complex systems involving multiple autonomous agents with internal knowledge and reasoning engines which communicate and interact with each other by exchanging messages according to specific protocols. This technology is strictly required for approaching this kind of complex problem, as will be shown in the following sections.

5.2 Production Planning in DPS As a DPS aims at achieving goals deriving from localisation, the production management policies cannot adopt a classical hierarchical approach. System complexity and time-based competition push towards sharing the decision making process. Production planning and control usually adopt different tools depending on the considered time horizon maintaining, anyway, a centralised vision. This chapter presents a decision support system for production planning activities structured in different steps (time horizons) according to a classical viewpoint, providing for each step the active involvement of the interested actors. This last requirement promotes decision autonomy: autonomy becomes a key issue in achieving the dynamic dimension required in the actual market scenario. Also, as stated above, the decision-making process adopted in production planning activities should always guarantee a global satisfaction level by means of coordination and this is accomplished by negotiation and game theory as explained in the following sections. 5.2.1 Context of the Semiconductor Industry Production capacity is the most significant portion of capital investment in semiconductor wafer manufacturing. Effective utilisation and expansion of production capacity have significant cost implications, and arguably drive the profitability of the operation. Capacity management in the industry typically entails long-term strategic planning and short-term operational planning organised in a

Distributed Production Planning in Reconfigurable Production Networks

53

hierarchical manner. Strategic planning decisions include how much of the aggregate microelectronics technology is produced in what facilities, and which capacity element to expand within what timeframe so as to meet the projected demands. Operational planning determines capacity adjustment or reconfigurations when more accurate demand and capacity information becomes available. Operational planning is frequent and dynamic so as to accommodate weekly production wafer “starts” to be released to manufacturing. One important characteristic of semiconductor capacity planning is that both product demands and manufacturing capacity are sources of uncertainty. As is the case in most hi-tech industries, the semiconductor market has a demand structure that is intrinsically volatile. A microelectronic chip that faces high demands today, may be quickly outdated in a few months with the introduction of a next-generation chip requiring an enhanced manufacturing process. New manufacturing processes create high variability in the yields, and consequently uncertainty on the manufacturing throughput, which in turn lead to uncertainty in capacity estimation. Since the production volumes are typically high (for the interests of achieving economies of scale), extreme outcomes on demand and capacity realisations can lead to very undesirable business consequences. Therefore, while capacity configuration and allocation are important decisions for any manufacturing company, a few factors make this problem especially crucial to the semiconductor industry. First is the high cost and long lead time for equipment procurement and cleanroom construction. The semiconductor wafer fabrication process requires state-ofthe-art manufacturing equipment, costing many millions and which must be ordered up to twelve months in advance. Because of the long lead time involved, capacity expansion decisions must be made far in advance. A wrong decision, either over- or under- estimation, could have major impacts on profitability. On the other hand, if the capacity for a certain technology is not expanded promptly to meet market demands, a significant loss of market share may result. A second factor that exacerbates the impact of capacity planning in the semiconductor industry is the rapid advancement of fabrication technologies and the pace of transition from old technologies to new. Semiconductor technologies can be defined in several ways, one of which is the space between features on a semiconductor die, known in the industry as line width. The most expensive and crucial pieces of equipment in the wafer fabrication line are used in the photolithography process, where the chip features are defined on a silicon wafer. With each advancement in photolithography technology, new and more expensive equipment must be purchased so that features with smaller line widths can be produced. Although the equipment is more expensive, it allows the manufacturer to either make smaller chips or fit more features on the same-size chip, effectively reducing manufacturing costs for a given chip functionality. Another factor in the advancement of semiconductor manufacturing technology is the size of the wafers. Equipment manufacturers are continuously trying to increase the wafer size, which increases the number of chips to be made at once and produces higher yields, which in turn reduces the unit manufacturing cost. As semiconductor technologies advance, the company must be prepared to switch manufacturing capability to the newer technologies. These transitions take time and they must be anticipated correctly. A premature transition will lead to costly underutilisation, or forcing

54

Production Planning in Production Networks

manufacturing to use newer, more expensive equipment to manufacture older technologies that do not generate expected revenue. An overdue transition leads to missed market opportunities, which also lead to lower ROI for the capital investment. A third factor unique to the semiconductor industry is that manufacturing capacity often suffers high variability. The aggregate notion of manufacturing capacity used during strategic planning is in reality an approximation at best. Given a particular capacity configuration for each clean room, the manufacturing manager still has much flexibility in how that capacity is utilised, and his/her decision will determine what the “effective capacity” ultimately is. For instance, newer equipment can typically be used to manufacture older technologies, albeit at a lower cost efficiency. Further, the “effective capacity” to manufacture the same technology is different in each location, depending upon the technology mixture (capacity configuration), the wafer size made in that facility, skill level of the labour, and myriad other factors. This is further complicated by contractual requirements with the customer, which may dictate that certain products must be made in certain locations. Finally, if there is not enough capacity in the companyowned facilities, outsourced foundry capacity can be bought for some technologies, at a higher cost. If wafers are made at a foundry, there may be contractual requirements that specify a minimum purchase. All of these factors come into play to make manufacturing capacity a significant source of uncertainty at the point of strategic planning. 5.2.2 PP in the Considered Industrial Case Semiconductor manufacturing operations consist of two main stages: the “frontend” operation of wafer fabrication, and the “back-end” operation of assembly and testing. The front-end operation is typically the bottleneck as the process involves a 6–12-week manufacturing lead time, while the back-end requires 2–4 days. Moreover, the wafer fabrication facilities are extremely capital intensive while requiring significant lead-time to build. Demand in high-tech industry is known to be volatile and particularly sensitive to economic cycles. Managing wafer fabrication capacity is one of the most crucial activities for semiconductor firms. In the considered case, every year, the corporate level assigns to the company groups (responsible for product families such as electronic memories) a certain level of the total production capacity (called capacity ownership) based on long-term demand forecasting and product strategic positioning. Every three months the groups, after having collected backlog and forecast orders coming from the regional divisions, according to the ownership they hold and to the demand they have to supply, make their capacity allocation plan. If the group capacity ownership is not enough to supply the demand orders, then the group can negotiate a portion of capacity with the other groups whose assigned ownership exceeds their actual demand [4]. In practice, such negotiation and consequent possible exchange of capacity turn out into a re-assignment of some production plants to a different group. Plants assigned at the beginning of the year to the production of components belonging to a specific product family, could be reconfigured throughout the year for producing different types of components, i.e. different product families. Also, within the

Distributed Production Planning in Reconfigurable Production Networks

55

annual quarter in which the assignment of plants to a group remains fixed, orders of products belonging to a product family (group) must be allocated to the different plants temporary assigned to the group [5]. Plants represent reconfigurable production systems able to be reconfigured in the short period (within the three months) in order to manufacture different types of product of the same part family. Such a brief description demonstrates how the PP process in a distributed organisation can become complex, multi-period, multi-decision and multi-issue when a somewhat reconfigurable capability is considered. Figure 5.1 reports an IDEF0 view of the PP process in reconfigurable enterprises made up of DPS. 5.2.3 IDEF0 Architecture Each PP activity reported in Figure 5.1 is related to a different time horizon and concerns different PP levels. The top PP level (activity A1) starting from information on groups’ one-year demand forecast and priorities for each group deriving from strategy considerations, assigns the global ownership to groups. The output of activity A1 represents an input for A2 together with groups’ priorities and demand forecast for the considered time horizon of three months. Reducing the forecast horizon, future estimates are more reliable and at the high PP level a tuning in ownership allocation is allowed. The medium PP level focuses on the same time horizon of the previous level but it aims at assigning each plant to a group basing on the output of activity A2. Low level and shop-floor level both concern real-time planning issues: the first one allocates, for each group, orders to one of the plants assigned to that group, while the second one is responsible, for each plant, for the allocation of jobs to resources. 5.2.4 Agent Architecture Actors involved in the distributed production planning (DPP) are: corporate, groups, plants, orders, jobs, resources. They have characteristics proper of agents: goal-oriented, collaborative, flexible and capable of making independent decisions on when to act [6]. Actually, involved agents pursue their own goals, interact with other agents to bargain a common action plan, get different roles along the process and know when and how to act in accordance with the concerted protocol. In the following sections we will characterise the collaboration/cooperation among those actors by identifying for each level of PP: ƒ the objective of the cooperation, that is, the goal to be reached; ƒ the bargain issue; ƒ the actors involved; ƒ the roles actors play in the bargaining.

NODE:

Groups priorities

1 year demand forecast

TOP PP Level

A0

TITLE:

3-months Groups demand forecasts

Tools to allocate ownership

Global Ownership

1

Production Planning activities

Group Orders

3

Parts production caacity

MEDIUM PP Level

Re-modulated ownership

4

NO.:

5

SHOP FLOOR PP Level

Order/Plant allocation

Job/Resources allocation

Reseource technology characteristics

Tools to allocate Resources to Jobs

Tools to allocate orders to plants

LOW PP Level

Plant/Group allocation matrix

Figure 5.1. A layered architecture for the PP problem

Parts characteristics

2

HIGH PP Level

Tools to remodulate ownership

Groups Ownership

56 Production Planning in Production Networks

Distributed Production Planning in Reconfigurable Production Networks

57

5.3 Top PP Level At this level, the corporate behaves as a seller while groups act as buyers. In fact, the corporate behaves like the owner that splits ownership to groups, based on the priorities assessed at the strategic level and communicated to all the groups. Each group aims at maximising the negotiated ownership and in order to limit this approach and to achieve company goals, the corporate concedes ownership obliging groups to assure to the company a certain profit level according to their priorities. The conditions of the contracts signed in this phase would be used as control for groups’ performance at the end of the year. These considerations represent guidelines to keep in mind to set, in related future research, strategies and tactics. Protocol variables to be set (the others are common for all the levels) concern the communication channel (cc): there exist a cc for offer and counteroffer and an informational cc to broadcast groups’ priorities. Summarising, at this level the objective is the ownership assignment; this goal is obtained through a bargaining process among the corporate and the groups, which are of course the actors involved in the bargaining process acting respectively as seller and buyers. Such a situation is depicted in Table 5.1. Table 5.1. DPP variables for top PP level Objective

Bargaining issue

Actors

Roles

Ownership assignment

Ownership and requested credits

Corporate & groups

Buyer (Groups) Seller (Corporate)

5.4 High PP Level Once ownership is assigned and the time horizon reduced, it might happen that some groups have been assigned an excess of capacity when compared with the actual demand they actually need to face, while other groups might be, for opposite reasons, in a capacity defecient position. At this level then, groups can assume an opportunistic behaviour changing over ownership in the current quarter to contract options for receiving it back in the future. To do that, it is necessary to introduce a lateral payment using an expedient: credits. Credits correspond to a virtual production capacity and are equally distributed to the groups at the beginning of each year; they are used to buy capacity, then groups with a great number of credits have a great contractual power to obtain capacity. At this level, based on the quarter forecast and on the ownership assigned at the previous level, groups can take on a buyer or a seller behaviour. If the workload related to the forecasted demand is higher than the ownership, the group wants to buy production capacity; in the opposite condition it is interested in selling the extra capacity and receiving credits. Sellers and buyers adopt a time-dependent tactic and use different generative functions for the order and counter-order formulation. Summarising, at

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this level the objective is to balance capacity among groups; this goal is obtained through a bargaining process among the groups, which acting as sellers and buyers, depending on the available capacity, negotiate by exchanging capacity ownership and credits. DPP characterisation at this level is reported in Table 5.2. Table 5.2. DPP variables for high PP level Objective

Bargaining issue

Actors

Roles

Balance capacity among groups

Capacity ownership and credits

Groups

Buyer (Groups) Seller (Groups)

5.5 Medium PP Level Once groups have balanced capacity with their workload, they need to acquire actual production capacity by buying it from plants. This level has the same time horizon as the previous one, but involves plants as active participants which have to be assigned to groups, based on the ownership obtained at the end of the HIGH PP level. Assuming that each plant is a cost and profit centre, it is interested to be assigned to the most promising group in terms of future profit; the plant, to avoid imperfect commitment, asks a price to assure its availability. Moreover it can happen that group catches plants at different rounds offering an increasing price, but giving it a lower priority in term of guaranteed workload. So plants assess a risk attitude, which is therefore used in their reactive function formulation. Summarising, at this level the objective is to acquire actual production capacity through plant assignment to groups; this goal is obtained through a bargaining process among the groups, which act as buyers, and the plants, which act as sellers of capacity. They bargain by exchanging plant production capacity and price the buyers pay to gain it. DPP characterisation at this level is reported in Table 5.3.

Table 5.3. DPP variables for medium PP level Objective

Bargaining issue

Actors

Roles

Plant assignment

Plant production capacity and price

Groups and plants

Buyer (Groups) Seller (Plants)

5.6 Low PP Level This level considers a real time horizon; each group collects orders from its divisions, and each order is assigned to one of the plants gained at the previous level. Therefore, at this level the problem is to assign each order to one of the

Distributed Production Planning in Reconfigurable Production Networks

59

plants of the group in order to satisfy orders’ and groups’ objectives. The assignment will obviously consider, as in the previous level, logistic (the distance between the plant and the final customer), economic (the demand elasticity of each product) and technology issues (plant/product efficiency matrix and reconfiguration costs) but, at this level, time is most of all a scarce resource. Moreover, each plant does not know the price agreed with other plants, nor their priorities; therefore, two situations can occur: orders or plants can represent the scarce resource. It can happen that workload is greater than available capacity and vice versa; however, this information is not known because it results as a private one. Because a third part (a mediator) is not present, it can be argued that plants behave as sellers and order as buyers because of the promises deriving from previous level (price and priority). Summarising, at this level the objective is to allocate actual orders to plants assigned to groups (order assignment); this goal is obtained through a bargaining process among plants, which act as capacity sellers, and the orders, which act as buyers. They bargain by exchanging production capacity and price. DPP characterisation at this level is reported in Table 5.4. Table 5.4. DPP variables for low PP level Objective

Bargaining issue

Actors

Roles

Assign orders to plant

Plant capacity/order assignment and price

Orders and plants

Buyer (Orders) Seller (Plants)

5.7 Shop-floor PP Level This last level concerns resources assignment within each plant. Indeed, once each order has been assigned to a plant it is decomposed into jobs characterised by volumes and due-date. Therefore this is a common scheduling problem, which is faced in a distributed fashion. Indeed, scheduling is obtained through a bargaining process among the plant resources (sellers), which sell their production availability to jobs (buyers) obtaining a price in exchange. At the shop-floor level the global perspective is the plant perspective and the local interests arise from the opposite goal pursued by jobs and resources. Table 5.5 summarises the DPP at this level. Table 5.5. DPP variables for shop-floor level Objective

Bargaining issues

Actors

Roles

Scheduling problem: jobs to resources Assignment

Job/resources assignment and price

Job and resources

Buyer (jobs) Seller (resources)

Table 5.6 summarises the proposed hierarchical distributed production planning approach. As the reader can see it associates the planning horizon, the planning

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objective, the makers involved and information/issues considered in the bargaining mechanism. Table 5.6. Production planning levels Planning horizon

Planning objective

Makers

Information/issues in the bargaining mechanism

1 year

Assign capacity ownership to groups

Corporate agent and groups’ agents

Long-term demand forecasting and product strategic positioning

3 months

Remodulate ownership among groups

Groups’ agents

Unbalancing level among medium-term forecasting plus backorder and assigned ownership

Medium

3 months

Assign plants to group

Groups’ agents and plants’ agent

Plants’ skills in producing a given part family, geographical position of the plants

Low

Realtime

Allocate orders to plant

Order agents and plants’ agent

Plant’s skills in processing the order and geographical position of the plants

Shopfloor

Realtime

Allocate jobs to production resources

Job agents and resources’ agent

Resources’ skills in processing the job, reconfiguration time required to process the job

Planning level

Top

High

The specific focus of this book concerns the design of a negotiational and game theoretical mechanism for solving bargaining at the high and medium-level production planning. This results will be benchmarked with a centralised model with full knowledge. Low and shop-floor levels will be neglected because several decentralised approaches have been proposed in the literature which can find application in our proposal.

5.8 References [1] [2] [3] [4]

US National Research Council (1998) Visionary manufacturing challenges for 2020. National Academy, Washington, DC: 13–36 Koren Y, Heisel U, Jovane F, Moriwaki T, Pritschow G, Ulsoy G, Van Brussel H (1999) Reconfigurable manufacturing systems. Ann CIRP, 48/2: 527–541 Wiendahl HP, Lutz S (2002) Production networks. Ann CIRP 51/2: 573–587 Argoneto P, Bruccoleri M, Lo Nigro G, Perrone G, Noto La Diega S, Renna P, Sudhoff W (2006) High level planning of reconfigurable enterprises: a game theoretic approach. Ann CIRP Vol. 55/1

Distributed Production Planning in Reconfigurable Production Networks

[5]

[6]

61

Argoneto P, Perrone G, Renna P, 2006. Medium level planning of reconfigurable enterprises: a game theoretical approach. MITIP, 11–12 September, Budapest, Hungary Etzioni O, Weld D (1995) Intelligent agents on the internet: fact, fiction, and forecast. IEEE Expert, August 1995, 44–49

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6 Distributed Models for Planning Capacity of Reconfigurable Production Networks at Medium Term

6.1 Introduction Production planning at medium term consists in what we have referred to as highlevel production planning in the previous chapter. As mentioned, at this level the planning objective is to balance capacity among groups; this goal is obtained through a bargaining process among groups, which, by acting as sellers and buyers and accordin to the available capacity, negotiate for exchanging capacity ownership and credits. In the following sections a set of distributed production planning models for such a purpose will be discussed.

6.2 Initial State Let us assume that the total planning horizon T is divided into N sub-periods t (t = 1,2,…,N); at the beginning of each sub-period, each group receives a new workload from the respective divisions and a group priority level from the corporate level; on the other hand, it has been assumed that the group capacity ownership, assigned to the top level phase, remains the same along the whole planning horizon T. Each group (g = 1,…,G) receives, from its divisions, data about the workload WL it needs to sustain and, from the corporate agent, data regarding the priority level, P , that the specific group has been assigned to according to the customer importance. Let us define with Pg [0, 1] the priority index that the corporate agent has assigned to the g-th group. Groups compute their respective available capacity C g according to the ownership they have got. Specifically: C g (Og ˜ Ctot )/100

(6.1)

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Production Planning in Production Networks

where Og is the ownership of the group g and Ctot is the total capacity the company holds. At this point each group computes an index S g measuring the difference between workloads and capacity: Sg WLg  C g Subsequently groups are OG ^1,..., i ,..., N` , i.e.

^1,..., j ,..., M` , with

UG

identified and groups with

(6.2) classified as overloaded Sg ! 0, or underloaded

Sg  0. Afterwards, OG and UG groups compute

respectively the required capacity RC g and the offered capacity OC g : RC g

WLg  C g

(6.3)

OC g

C g  WLg

(6.4)

6.3 The Centralised Model First of all, it is very important to distinguish between the centrality of the problem and the centrality of the solution. In the following centralised model, the assumption is that all the relevant planning data can be centrally located and can be analysed by a centralised algorithm. If the approach is distributed, like in the following sections, the statement is that it is physically impossible (or unwanted) to locate all the actual data at one physical host (e.g. logistics planning). Actually, a given problem can be centralised only to some extent (e.g. only meta-information such as from where to where is the semi-product transported is centrally available while the exact location of the truck is unknown). Obviously, centralised problems are more likely to be solved by centralised algorithms, while distributed problems would require more negotiation-based approaches. In this model, after having classified groups in OG and UG, the corporate-centralised decision maker’s goal is to minimise the capacity backlog by minimising the OG groups requested capacities. That is achieved through the following steps: 1. a matrix L(OG u UG ), whose elements cij RC i  OC j are the differences between the requested capacity  RC i by the i-th group and



residual capacity OC j

L 2.

§ ... ... ... · ¨ ¸ ¨ ... cij ... ¸ ; ¨ ... ... ... ¸ © ¹

which is offered by the j-th group is computed: (6.5)

in order to achieve the best capacity allocation solution, the following MILP model is solved: § N M · min ¨ ¦¦ c ij ˜ xij ¸ , (6.6) ¨i 1 j 1 ¸ © ¹

Distributed Models for Planning Capacity at Medium Term

65

subject to: N

¦x

ij

1;

i 1

ij

1, if N = M

(6.7)

ij

d 1, if N < M

(6.8)

1, if N >M

(6.9)

j 1

N

¦x

M

¦x

ij

1;

i 1

M

¦x j 1

N

¦ xij d 1; i 1

M

¦x

ij

j 1

where xij is a binary variable such that ­1, if group j capacity is assigned to i ® ¯0, otherwise Once each group i  OG is associated to a group j  UG , i.e. xij xij

(6.10)

1, their requested

and offered capacities are respectively updated as RC i ' RC i  min( RC i , OC j )

(6.11)

OC j ' OC j  min( RCi , OC j )

(6.12)

6.4 The Negotiation Model The negotiational mechanism has the same goal as the centralised one: to minimise the unallocated capacity. The method, this time, is a decentralised one and is referred to as a one-to-many negotiation approach. During this process, negotiators are allowed to communicate their respective desires and they will compromise to reach mutually beneficial agreements. This strategic negotiation is a process that will include several iterations of offers and counter-offers. In particular, the algorithm works as follows: 1. for each step s the group i *  OG , which is in charge of starting the negotiation, is the one with the maximum value of workload excess, i.e for s = 1: RC s* / i * i

2.

^

i

i

(6.13)

each group j  UG computes the capacity to be offered and its own credits threshold according to M rj min( RC r* , OC r ), (6.14) i

t rj

j

r 1· § M rj ˜ ¨ 1  PL j  ¸, r ¹ ©

where: PL j 3.

`

max RC s ;

C

j

(6.15)



 WL j /C j ;

the i*-th group formulates its bid to gain the requested capacity; specifically, it offers a certain number of credits to each j  UG , according to

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Production Planning in Production Networks

^

OC ir* j

`

min CC ir* , M rj ˜ (1  r ˜ PLi * ) ,

(6.16)

r i*

where CC are the i*-th group current credits. Indeed, each group i  OG is requested to provide a certain number of credits to j in order to bargain some amount of capacity; 4. the negotiation continues until r rmax , unless group i* and one group j belonging to UG come to an agreement; the agreement condition assures a maximum allocation capacity from i*-th’s point of view, i.e.: max OC ir* j t t rj , (6.17) j

5.

if at r

^

`

rmax no agreement is reached, a sub-optimal accord is sought:

group i and j come together if OC ir t t rj for a generic j. If there exist

6.

several j groups able to satisfy the previous condition, the one minimising the residual i’s requested capacity is selected; if no agreement is reached, accordingly step s is increased and a new i *  OG , corresponding to the group having the maximum value of workload excess for the new step s, is selected, according to Condition 6.18, and the negotiation starts again from point 2; 1 s1 i * /max ^RCi  ( RCmax ,..., RCmax )` (6.18) i

If, also in this case, no agreement is reached, the process begins again from point 6 until the end of i  OG ; 7. the residual capacity and credits are uploaded for each group and the mechanism starts from point 6.

6.5 The Game-theoretical Model In this case, the problem can be formalised like an assignment game. The outline is a quadruple AG r (OG r , UG r , T r , O r ) , where:

^1,..., i ,..., N `

ƒ

OG r

r

ƒ

overloaded group at round r; UG r 1,..., j ,..., M r is a vector whose element j represents an

ƒ

underloaded group at round r; T r is a vector whose element, t j r , represents the credits (threshold of)

ƒ

that the j-th group, belonging to UG r , assigns to its own residual capacity. This value is calculated through Equation 6.15; O r is a matrix whose generic element OC ij r is the value the

^

is a vector whose element i represents an

`

group i  OG r assigns to the capacity offered by each player j  UGr using Equation 6.16.

Distributed Models for Planning Capacity at Medium Term

67

A transfer utility game [1] is associated to the quadruple AG r ; in that, the whole set of players are given by * r OG r ‰ UG r and the characteristic function Ȟ is defined, given that groups i and j make a coalition, as follows: ­°OCij r  t j r if OC ij r  t j r ! 0 v (i , j ) hij r ® (6.19) if OC ij r  t j r d 0 °¯0 The set of hij r defines the following assignment problem: § N M · max ¨ ¦¦ hij r ˜ zij ¸ ¨i 1 j 1 ¸ © ¹ where,

^

(6.20)

1, if i and j makes a coalition

zij

0, otherwise

,

(6.21)

subject to,  i  OG r ,  j  UG r , the following constraints: N

¦z

ij

1;

i 1 N

¦z

ij

¦ i 1

ij

1 , if N = M

(6.22)

d 1 , if N < M

(6.23)

1 , if N >M

(6.24)

j 1

1;

i 1 N

M

¦z

zij d 1;

M

¦z

ij

j 1 M

¦z

ij

j 1

After having created the assignment among groups, it is necessary to consider two different situations depending on the value of the characteristic function. 6.5.1 Case 1: Characteristic Function hij > 0 In this case, the coalitional players are required to split the game surplus. In order to do this, the optimising charge belonging to the core game [2] is computed. According to Owen’s theorem [3], a core allocation, in a linear assignment game, can be obtained starting from an optimal solution of the dual programming model: M § N · min ¨ ¦ yi  ¦ y j ¸ , (6.25) ¨i 1 ¸ j 1 © ¹ subject to: yi  y j t hij r . (6.26) Once the solution is calculated, the credits and the capacities are uploaded for all the coalitional players. 6.5.2 Case 2: Characteristic Function hij < 0 Here, in comparison to that explained above, different considerations have to be made. This is the situation in which the group i offers a non-sufficient (scarce) number of credits to the group j. Obviously, there is no surplus to be shared among

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players but, on the contrary, it is necessary, if possible, to reach an agreement, to share the penalty given by the spread existing by the offered and requested credits: OCij r  t j r . This is the typical situation that can be formalised like a bargaining situation between two players. The adopted solution concept is referred to as the Nash formulation [4], which satisfies a set of axioms (invariance to equivalent utility representations, symmetry, independence of irrelevant alternatives and Pareto efficiency) for a “fair” bargain. In order to reach this goal, it is necessary to build up two different kinds of utility functions for the involved players. 6.5.2.1 Utility Function for the OG Group From the perspective of this group, two different parameters have to be taken into consideration: one taking into account their unallocated capacity (whit respect to the one needed), and the other one considering the numbers of credits they are willing to pay in order to obtain it. ƒ The first parameter is indicated by: ci  callir 1 [i , (6.27) ci

callir 1 being the capacity already allocated to the group until the round r  1. The higher is [ i , the more the group is willing to increase the numbers of credits to pay to obtain the capacity it needs. Having defined with ' the percentage of the difference OC ij r  t j r the group i-th has still to fill up to

ƒ

reach the agreement, it is obvious that the more ' increases, the more its own associated utility decreases with a slope depending on [ i ; The second parameter to be considered is: Cri Ki , (6.28) Crtot

Cri being the residual credits owned by the i-th group at step r and Crtot the credits it owned at the beginning of the game. Also in this case, increasing ' means decreasing utility, with slope depending on Ki . Consequently, the utility function of the groups belonging to the OG set will be a parametric function that will be modelled by using a Bézier’s curve. In particular, the Bezier curve control points (Pi), those constituting the characteristic polygon, are: P1 {  0 , 1 , P2 { P3 { P4 { Xi ,Xi , P5 {  1 , 0 , , where:

Xi

Ki ˜ [ i .

(6.29) This choice allows us to obtain a utility function equal to the minimum value, 0, when the penalty to pay is ' 100% , and equal to 1 when ' 0% Alsothe choice to have three overlapping points is useful to obtain a curve that best approximates the characteristic polygon. Therefore, the utility function for the OG group will be written as:

Distributed Models for Planning Capacity at Medium Term

UiOG

5

¦Pb

B  u

i t ,5

u ,

0 d u d 1,

69

(6.30)

t 0

where the polynomials § 5· 5 t bt ,5  u ¨ ¸ ut  1  u ,t 0 ,..., 5 , t © ¹ are known as Bernstein basis polynomials of degree t.

(6.31)

U iOG 1

Qi 0,8 0,6 0,4 0,2

' 0 0

0,2

0,4

0,6

0,8

1

1,2

Figure 6.1. Utility function of the groups belonging to the OG set

6.5.2.2 Utility Function of the Groups Belonging to the UG Set From the perspective of the groups belonging to the UG set, the reasoning to build up their utility function is rather more intuitive. Given the above assumptions, they are required to assign their own capacity M rj to the OG groups. Therefore, they just wish to gain more credits as possible. This means that the more the offered



payment OC ij r





is close to the requested one t j r , the more the utility value

tends, with linear approximation, to the maximum value. Using the same notation as above, we simply have: U jUG ' . (6.32)

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Production Planning in Production Networks

U jUG

1

0

'

0

1

Figure 6.2. Utility function of the groups belonging to the UG set

6.5.3 The Bargaining Solution Figure 6.3 shows the bargaining set. Graphically it is the area under the curve and is easy to see that, as requested by the Nash solution, it is convex and bounded.

U iOG

1

Bargaining set

0 0

U jUG Figure 6.3. The bargaining set

1

Distributed Models for Planning Capacity at Medium Term

71

The Nash bargaining solution is given by the following equation:

NBS

max U







max ª U iOG  U iOG ˜ U jUG  U jUG º , ¬ ¼

(6.33)

U iOG , U jUG being the fall-back outcome, i.e. the outcome that can result if the bargaining breaks down. In this specific case, it is easy to demonstrate that U iOG U jUG 0 . The result will be the additional credits (OC ij r  t j r ) ˜ ' the

group i has to pay to the group j and, complementarily, the number of credits the jth group has to give up, (OC ij r  t j r ) ˜ (1  ' ), in order to reach the agreement. Finally, the actual exchanged credits will be OC ij r ' :

OC ij r '

OC ij r ˜ (1  ' )  (t rj ˜ ' ),

(6.34) If the j-th group do not have a sufficient amount of credits to allow the exchange, the bargaining falls and the i-th group does not assign the requested capacity. Finally, the residual capacity and credits are uploaded for each group involved in the game and the algorithm starts again (r = r + 1), formulating a new assignment game, until one of the following two conditions (no more workload excess or no more extra capacity) comes true: N

¦ RC i 1

r i

0;

M

¦ OC j 1

r j

0;

(6.35)

6.6 Simulation Case Study A discrete event simulator, based on a multi-agent system (MAS) approach, has been developed and used for testing the proposed models. Optimisation has been performed through the LINGO® package conveniently linked to the simulation environment. The simulation environment has been entirely developed using JAVA. The choice of using JAVA has led to obtaining high flexibility in the development of the simulator, especially thanks to the advantages related to the object-oriented philosophy. To reduce development time and cost related to the simulator development, the open source Java Development Kit (JDK) has been utilised. The adopted formalism is related to all possible interactions existing between the considered agents. The simulation carries on considering an events timetable of the agents’ actions: this schedule is utilised to define the sequence of the events. In the following, UML class and sequence diagrams will be utilised to describe the simulation environment. 6.6.1 The Simulation Environment

For all different approaches above explained, the simulation environment consists of the following objects, as also depicted in the class diagram of Figure 6.4: ƒ the Schedule: this object is in charge of managing the sequence of events during the simulation time;

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Production Planning in Production Networks

ƒ

the Model: this object handles the signals related to the orders, coordinates the activities among groups and selects the strategies; ƒ the Group: this object possesses all groups’ information and functions necessary to the implementation; ƒ the Statistical Analysis: this last object is dedicated to the analysis of the groups’ performance. The sequence diagram shown in Figure 6.5 refers to a generic order o, o = 1,..,O:

Figure 6.4. Class diagram of the high production planning level

Distributed Models for Planning Capacity at Medium Term

73

Figure 6.5. Sequence diagram, high production planning level

The Schedule object activates all other objects as explained below. ƒ o-th order: order number o is communicated to Model object; Model object updates information related to o-th order (workload, ownership, priority); ƒ Initialize o-th order: information related to o-th order is communicated to groups; Groups calculate the capacity they need or their capacity excess: this evaluation leads the groups to choose the OG or the UG set; ƒ Select set (OG or UG): the selected set, for each group, is communicated to the Model; Depending on the selected set, Model needs information by the groups (e.g. offered or required capacity, utility values); ƒ Values request: needed values are requested to groups; Groups calculate the requested values; ƒ Values communication: requested values are communicated by Group to Model;

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Production Planning in Production Networks

Model applies one of the three different approaches, depending on the considered model: centralised, negotiational, game theoretical; ƒ Performances communication: groups’ performances are communicated to Statistical Analysis object; ƒ Assignment communication: new capacity adjustment among groups is communicated to Group object; Statistical Analysis object uploads performances and carries on with the simulation until the defined confidence interval is reached; ƒ Statistical performances: performances are communicated to Schedule object; Schedule verifies the performances’ statistical significance; Significance communication: Schedule communicates to Statistical Analysis object the result of the previous check; Final report is written. 6.6.2 Simulation Case Study

The test environment consists of 27 different combinations of input parameters: workloads (WL), priorities (P) and ownerships C indexes. In particular, the o-th order is given by a vector (WL,P,C): each index is considered randomly drawn from three normal distributions N(µ;ı) whose parameters are reported in Table 6.1 as Small (S), Bounded (B) or Wide (W) with relation to the standard deviations size: Table 6.1. Distributions of parameters Small (S)

Bounded (B)

Wide (W)

Workload (WL)

N(17; 0.85)

N(17; 5.1)

N(17; 10.2)

Ownership (C)

N(17; 3.4)

N(17; 6.8)

N(17; 10.2)

Priority (P)

N(0.5; 0.025)

N(0.5; 0.3)

N(0.5; 0.5)

Simulation experiments have been conducted with an infinite number of credits and, for each replication, the average of unallocated capacity (Avg UC) with a confidence interval equal to 95% has been evaluated.

6.7 Results The results of 27 different combinations of input parameters for the centralised (C.A.), negotiational (N.A.) and game-theoretical (G.A.) approaches are reported in Appendix A. They are represented in tables indicating: the average value of the unallocated capacity (Avg), the (half-width) confidence interval (Half), the number of replications needed by the simulator to calculate it (Rep), the extreme values (Min, Max) of the estimated parameter and the variance (Var). In order to analyse the results, two different techniques have been used: the two-way Analysis of

Distributed Models for Planning Capacity at Medium Term

75

Variance (ANOVA) and the Design of Experiment (DoE). The first one leads us to understand the dependence of the results from the variance of input parameters, from the three different proposed models and, if there is, from their interaction. Instead, DoE will lead us to comprehend the direction of this dependence. 6.7.1

Two-way Analysis of Variance

The two-way Analysis of Variance is an extension of the one-way ANOVA: here, there are two independent variables. The classical assumptions are the following: ƒ the population from which the data are drawn must be normally or approximately normally distributed; ƒ data must be independent; ƒ the variances of the populations must be equal; ƒ the groups must have the same sample size. In the analysed case, the independent variables are: the different models, C.A., N.A. and G.A., and the different input parameters, WL, P and C. The obtained results, rearranged in the following tables, can be considered like a set of data drawn from three normal distributions with means equal to the three different grand means (depending on the utilised model) and the same (unknown) variance. This last assumption assures us that all the above constraints are satisfied. Each factor (i.e. each independent variable) has three different levels. Respectively, the factor “model” has levels corresponding to C.A., N.A., G.A.; the factor “input parameters” has levels equal to W, B, S. The goal is to understand in which way the input parameters affect the estimated performance. In order to do this, it is necessary to consider only one parameter at time. The first one will be the workload. 6.7.1.1 Workload Table 6.2 reports the result of the unallocated capacity (given in Appendix A) rearranged to highlight the workload parameter. Indeed, in the rows of Table 6.2 we have the different levels of workload for each set of model; in the columns, on the other hand, we have the levels for combinations of the other two input parameters, P and C. Such combinations have been obtained as indicated in Table 6.3. Table 6.4 shows all the mean values for each model and for each level of WL. Data from this table suggest that the two decentralised approaches perform slightly worse than the centralised approach, there being a reduction in performance of about a  20% on average. This is quite a good performance considering the distributed nature of the two approaches. Also no great difference is evident between the two distributed approaches, namely the negotiation and the game theoretical one.

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Production Planning in Production Networks

Table 6.2. Replications values for WL

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

9

W

51.04

39.16

34.10

51.54

34.31

34.65

49.98

38.84

34.48

B

41.88

27.78

21.69

42.72

28.20

20.66

41.86

26.32

21.48

S

36.43

18.49

8.74

35.88

18.67

8.45

35.22

18.59

8.76

W

67.73

43.64

35.61

64.86

35.10

34.89

67.27

43.12

36.15

B

59.13

31.56

21.57

58.88

31.65

21.73

57.86

30.51

21.74

S

52.23

20.90

8.53

55.79

21.77

8.35

53.60

20.53

9.05

W

65.83

43.88

36.76

66.14

43.32

34.80

67.61

44.58

36.66

B

58.57

31.37

22.00

57.72

30.19

21.93

57.64

30.40

21.06

S

51.28

20.53

8.56

53.39

21.07

8.60

53.23

20.61

8.69

Table 6.3. Combination of P and C 1

2

3

4

5

6

7

8

9

Priority (P) variance

W

W

W

B

B

B

S

S

S

Ownership (C) variance

W

B

S

W

B

S

W

B

S

Table 6.4. Mean values for WL W

B

S

Mean

C.A.

40.90

30.29

21.03

30.74

G.A.

47.60

37.18

27.86

37.55

N.A.

48.84

36.76

27.33

37.65

Mean

45.78

34.74

25.41

35.31

Let us investigate deeply the obtained results. In particular, there are three sets of hypotheses to consider; indeed, the null and alternative hypotheses for each set are the following: 1.

2.

for the population means of the first factor (the model) PC PN PG ­° H 0 : ; (6.36) ®H : not all Pi^C , N ,G` are equal °¯ 1 for the population means of the second factor (the workload): P W P B PS °­ H 0 : ; (6.37) ®H : not all P j^W , B ,S` are equal °¯ 1

Distributed Models for Planning Capacity at Medium Term

77

3.

considering the interdependencies: there are no interactions ­H0 : . (6.38) ® there are interactions ¯ H1 : Before explaining the components of the two-way ANOVA table, it is necessary to the notation generalise a little bit. Let us assume that: ƒ the main effect A has a levels (and a  1 degrees of freedom), ƒ the main effect B has b levels (and b  1 degrees of freedom), ƒ c is the sample size of each treatment, and ƒ N = abc is the total sample size, then we have: Table 6.5. Two-way ANOVA table Source of Variation

Sum square (SS)

Degree of freedom (df)

Main Effect A

SSA

A1

s 2A

SSA /( a  1)

s 2A / sr2

Main Effect B

SSB

b1

sB2

SSB /(b  1)

sB2 / sr2

Interaction effect

SSi

(a  1)(b  1)

SSi /( a  1)(b  1)

si2 / sr2

Within

SSr

ab(c  1)

SSr / ab(c  1)

–

Total

SST

abc  1

ƒ

ƒ ƒ

ƒ

Mean square (MS)

si2 sr2

F-test

–

The main effect involves the independent variables one at a time. The interaction is ignored for this part. Just the rows or just the columns are used, not a mixture. This is the part which is similar to the one-way ANOVA. The interaction effect is the effect that one factor has on the other factor. The degrees of freedom here are the product of the two degrees of freedom for each factor. The within variation is the sum of squares within each treatment group. The total number of groups’ treatment is given by the product of the number of the levels multiplied for each factor. The within variance is the within variation divided by its degrees of freedom. The within group is also called the error. At the last column, there is an F-test for each of the hypotheses: it is the mean square for each main effect and the interaction effect divided by the within variance. The numerator degrees of freedom come from each effect, and the denominator degrees of freedom are the degrees of freedom for the within variance in each case.

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Production Planning in Production Networks

In the specific case we have: Table 6.6. Two-way ANOVA table for WL SS

df

MS

F-test

F-crit D

5%

First factor

846.45

2

423.23

1.87

F2 , 72

3,13

Second factor

5616.96

2

2808.48

12.43*

F2 , 72

3,13

Interaction effect

10.90

4

2.72

0.01

F4 , 72

2 , 51

Within

16272.65

72

226.01

–

Total

22746.96

80

–

–

From the above results, we can see that only the main effect of WL is significant, but the interaction between the input factors is not. That is, only the second null hypothesis can be rejected: the workload means are not all equal. Often, the best way of interpreting and understanding an interaction is by a graph. A two-factor ANOVA with no significant interaction can be represented by two approximately parallel lines, whereas a significant interaction results in a graph with non-parallel lines. Because two lines will rarely be exactly parallel, the significance test on the interaction is also a test of whether the two lines diverge significantly from being parallel. If there is no interaction between the considered factors, each mean reported in Table 6.7 is influenced only by the two factors in an additive way. For example, if we consider A (factor in the column) and B (factor in the row) without any interaction, we should obtain: X ai b j

X ai  X b j  X

(6.39)

i.e.: if no interaction exists between A and B, each single considered value of the table of means results from the sum of the means in the row and in the column minus the grand mean. In other words, the values shown in Table 6.4 are the observed values, while the estimated values calculated using Equation 6.40 are reported in Table 6.7. Table 6.7. Estimated values for WL W

B

S

Mean

C.A.

41.21

30.17

20.83

30.74

G.A.

48.02

36.98

27.64

37.55

N.A.

48.12

37.08

27.74

37.65

Mean

45.78

34.74

25.41

35.31

Therefore, it is possible to plot these values and verify that the obtained lines are all parallel in both cases; this will confirm that there is no interaction between the factors.

Distributed Models for Planning Capacity at Medium Term

79

50 45 40 Centralised

35

GameTheory Negotiation

30 25 20 0

1 W

2 B

3 S

4

Figure 6.6. Observed values

50 45 40

Centralised 35

Game Theory Negotiation

30 25 20 0

1 W

2B

3S

4

Figure 6.7. Estimated values

The ANOVA test has been used to find out if there is a significant difference between the considered groups of means. However, the ANOVA analysis simply indicates if there is a difference between two or more groups of means, but it does not tell us what mean is significantly different from the others. To understand it, in the following, the Tukey test has been used. This is a test properly designed to perform a pairwise comparison of the means to see where the significant difference is; according to it, it is sufficient to simply calculate the T-critical value and the

80

Production Planning in Production Networks

difference between all possible pairs of means. Each difference is then compared to the T-critical value: if this difference is larger than the Tukey value, the comparison is significant (and it will be indicated by “*”), otherwise it is not. T  critical

sr2 , k

QD , p , ab( c 1) ˜

(6.40)

being: ƒ ƒ ƒ ƒ ƒ ƒ

D is the selected probability (5%); p is the simultaneously number of compared means; ab(c  1) is the df of the within variance; sr2 is the within variance; Q is the studentised range statistic; k is the number of the data necessary to calculate the means we are going to compare. Referring to the analysed case, we have: Table 6.8. T-values for WL Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

3.39 ˜

226.66 27

9.80

Column factor

Q0.05 ,3,72

3.39

T

3.39 ˜

226.66 27

9.80

Interaction

Q0.05 ,9 ,72

4.53

T

4.53 ˜

226.66 9

22.70

Using data reported in Table 6.4, we obtain: Table 6.9. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

6.81

0

0

N.A.

6.91

0.10

0

Table 6.10. Differences of means for column factors Column factor

W

B

S

B

11.04*

0

0

S

20.37*

9.34

0

Distributed Models for Planning Capacity at Medium Term

81

It is easy to see that, also with this test, the only influence on means we observe is due to the column factor, i.e. to the main effect of WL, like already observed by refusing the only null hypothesis 6.38. Table 6.11 is referred to the groups’ interaction analysis: Table 6.11. Groups’ means differences for column factors W-C.A.

B-C.A.

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

B-C.A.

10.61

–

–

–

–

–

–

–

S-C.A.

19.87

9.26

–

–

–

–

–

–

W-G.A.

6.70

17.31

26.57*

–

–

–

–

–

B-G.A.

3.72

6.89

16.15

10.42

–

–

–

–

S-G.A.

13.04

2.43

6.83

19.73

9.32

–

–

–

W-N.A.

7.94

18.55

27.82*

1.25

11.66

20.98

–

–

B-N.A.

4.14

6.47

15.74

10.83

0.42

8.90

12.08

–

S-N.A.

13.57

2.96

6.30

20.27

9.85

0.53

21.52

9.44

Finally, given these results with a confidence interval equal to 5%, we can draw the following conclusions: ƒ there is no statistical difference between the performance estimated using the centralised model and the two decentralised ones; ƒ the variability of the input data (WL) significantly influences the response of the systems; ƒ the input data variability and the different models does not interact significantly. This considerations lead us to a very important result: given a fixed variability of input data (WL) there is, statistically speaking, no difference between the three proposed models. This is a really good result considering the distributed nature of the negotiational and game-theoretical approaches. 6.7.1.2 Priority Similarly to the workload analysis, we start filling in Table 6.12. In the rows we have the levels of both considered factors (priority and type of model) and, in the columns, the values of nine different replications obtained combining the other two input parameters (WL, C) in a combinatorial way, as explained in Table 6.13.

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Production Planning in Production Networks

Table 6.12. Replication values for P

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

9

W

51.04

39.16

34.10

41.88

27.78

21.69

36.43

18.49

8.74

B

51.54

34.31

34.65

42.72

28.20

20.66

35.88

18.67

8.45

S

49.98

38.84

34.48

41.86

26.32

21.48

35.22

18.59

8.76

W

67.73

43.64

35.61

59.13

31.56

21.57

52.23

20.90

8.53

B

64.86

35.10

34.89

58.88

31.65

21.73

55.79

21.77

8.35

S

67.27

43.12

36.15

57.86

30.51

21.74

53.60

20.53

9.05

W

65.83

43.88

36.76

58.57

31.37

22.00

51.28

20.53

8.56

B

66.14

43.32

34.80

57.72

30.19

21.93

53.39

21.07

8.60

S

67.61

44.58

36.66

57.64

30.40

21.06

53.23

20.61

8.69

Table 6.13. Combination of WL and C 1

2

3

4

5

6

7

8

9

Workload (WL) variance

W

W

W

B

B

B

S

S

S

Ownership (C) variance

W

B

S

W

B

S

W

B

S

The next step is again to refer to a table with all the mean values for each model and for each level of P. Again, observing Table 6.12 and 6.14, it is possible to conclude that the two distributed approaches perform slightly worse than the centralised one by about 20%. Also in this case the performances of the two decentralised approaches are very close each other. Table 6.14. Mean values for P W

B

S

Mean

C.A.

31.03

30.57

30.62

30.74

G.A.

37.88

37.00

37.76

37.55

N.A.

37.64

37.46

37.83

37.65

Mean

35.52

35.01

35.40

35.31

Also in this case, there are three sets of hypotheses to consider, as shown above for WL. The null and alternative hypothesis for each sets are given, also in this case, by Equations 6.37 –6.39. Using the same notation asTable 6.5, we have:

Distributed Models for Planning Capacity at Medium Term

83

Table 6.15. Two-way ANOVA table for P SS

df

MS

F-Test

F-crit D

5%

First factor

846.45

2

423.23

1.39

F2.72

3.13

Second factor

3.81

2

1.91

0.006

F2.72

3.13

Interaction effect

2.03

4

0.51

0.002

F4.72

2.51

Within

21894.67

72

304.09

–

Total

22746.96

80

–

–

From the above results, we can see that no factor is significant, including the interaction between them. Therefore, in this case, it is pointless to use a Tukey test, because it is impossible to reject all null hypotheses reported in Equations 6.37– 6.39. To further support this consideration, the observed and the estimated values2 are plotted in Figures 6.8 and 6.9. Table 6.16. Estimated values for P W

B

S

Mean

C.A.

30.95

30.44

30.83

30.74

G.A.

37.75

37.25

37.64

37.55

N.A.

37.85

37.35

37.74

37.65

Mean

35.52

35.01

35.40

35.31

It is possible verify that the obtained lines are all parallel in both cases: this confirms that there is no interaction between factors. These conclusions lead us to understand, considering a confidence interval equal to 5%, that: ƒ there is no statistical difference between the performance estimated using the centralised model and the two decentralised ones; ƒ the variability of the input data (P) does not significantly influence the response of the systems; ƒ the input data variability and the different used models does not interact significantly. These considerations lead us to another very important result: regardless the variability of input data (P) there is, statistically speaking, difference between the three proposed models.

2

This values are calculated using (6.40).

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Production Planning in Production Networks

39,00 38,00 37,00 36,00

Centralised

35,00

Game theory

34,00

Negotiation

33,00 32,00 31,00 30,00 W

B

S

Figure 6.8. Observed values

39 38 37 36

Centralised

35

Game theory

34

Negotiation

33 32 31 30 W

B

S

Figure 6.9. Estimated values

6.7.1.3 Ownership The last parameter to investigate is the ownership. Table 6.17 reports the data as usual. In the rows we have the levels of both factors and, in the columns, the values of nine different replications obtained combining the other two input parameters (WL, P) in a combinatorial way, like explained in Table 6.18.

Distributed Models for Planning Capacity at Medium Term

85

Table 6.17. Replication values for C

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

9

W

51.04

51.54

49.98

41.88

42.72

41.86

36.43

35.88

35.22

B

39.16

34.31

38.84

27.78

28.20

26.32

18.49

18.67

18.59

S

34.10

34.65

34.48

21.69

20.66

21.48

8.74

8.45

8.76

W

67.73

64.86

67.27

59.13

58.88

57.86

52.23

55.79

53.60

B

43.64

35.10

43.12

31.56

31.65

30.51

20.90

21.77

20.53

S

35.61

34.89

36.15

21.57

21.73

21.74

8.53

8.35

9.05

W

65.83

66.14

67.61

58.57

57.72

57.64

51.28

53.39

53.23

B

43.88

43.32

44.58

31.37

30.19

30.40

20.53

21.07

20.61

S

36.76

34.80

36.66

22.00

21.93

21.06

8.56

8.60

8.69

Table 6.18. Combination of WL and P 1

2

3

4

5

6

7

8

9

Workload (WL) variance

W

W

W

B

B

B

S

S

S

Priority (P) variance

W

B

S

W

B

S

W

B

S

Table 6.19 shows the means for the data reported in Table 6.17. Also in this case decentralised approaches perform slightly worse than the centralised and no great difference can be measured between the two approaches. Table 6.19. Mean values for C Ownership

W

B

S

Mean

C.A.

42.95

27.82

21.45

30.74

G.A.

59.70

30.97

21.96

37.55

N.A.

59.05

31.77

22.12

37.65

Mean

53.90

30.19

21.84

35.31

Again, the null hypothesis sets to consider is given by Equations 6.37–6.39. Referring to Table 6.5, we have:

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Production Planning in Production Networks

Table 6.20. Two-way ANOVA table for C SS

df

MS

F-test

F-crit D

5%

First factor

846.45

2

423.23

4.99*

F2.72

3.13

Second factor

14937.46

2

7468.73

88.04*

F2.72

3.13

Interaction effect

854.84

4

213.71

2.52*

F4.72

2.51

Within

6108.21

72

84.84

–

Total

22746.96

80

–

–

From the above results, we can see that all three null hypotheses can be rejected. This means that not only the main effects of considered factors are significant, but also their interaction. In this case, if we consider the observed values plot, we will not obtain parallel lines, but divergent ones. Obviously, this consideration is not true for the estimated values. Table 6.21 Estimated values for C Ownership

W

B

S

Mean

C.A.

49.33

25.62

17.27

30.74

G.A.

56.14

32.42

24.08

37.55

N.A.

56.24

32.52

24.18

37.65

Mean

53.90

30.19

21.84

35.31

70 60 50

Centralised

40

Game theory Negotiation

30 20 10 0 W

B

S

Figure 6.10. Observed values

Distributed Models for Planning Capacity at Medium Term

87

60 50 Centralised

40

Game theory 30

Negotiation

20 10 0 W

B

S

Figure 6.11. Estimated values

To better understand where this significant difference is located, we will use the Tukey test. Referring to Formula 6.40, we have: Table 6.22. T-values for C Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

3.39 ˜

84.84 27

6.01

Column factor

Q0.05 ,3,72

3.39

T

3.39 ˜

84.84 27

6.01

Interaction

Q0.05 ,9 ,72

4.53

T

4.53 ˜

84.84 9

13.91

Using data reported in Table 6.19, we obtain: Table 6.23. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

6,81*

0

0

N.A.

6,91*

0,10*

0

Table 6.24. Differences of means for column factors Column factor

W

B

S

B

23.71*

0

0

S

32.06*

8.35*

0

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Production Planning in Production Networks

It is easy to see that, also using this test, the influence on means we observe is due by either the column and row factor, i.e. the main effect of O and the main effect of the models, like already view refusing all hypotheses 6.37-6.39. Table 6.25 refers to the interaction: Table 6.25. Groups means’ differences for column factors W-C.A.

B-C.A.

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

B-C.A.

15.13 *

–

–

–

–

–

–

–

S-C.A.

21.50*

6.37

–

–

–

–

–

–

W-G.A.

16.75*

31.89*

38.26*

–

–

–

–

–

B-G.A.

11.98

3.16

9.53

28.73*

–

–

–

–

S-G.A.

20.99*

5.86

0.51

37.75*

9.02

–

–

–

W-N.A.

16.09*

31.23*

37.60*

0.66

28.07*

37.09*

-

-

B-N.A.

11.18

3.95

10.33

27.93*

0.80

9.81

27.27*

-

S-N.A.

20.83*

5.70

0.67

37.59*

8.86

0.16

36.93*

9.65

At the end, considering a confidence interval equal to 5%, it is possible to conclude that: ƒ there is statistical difference between the performance estimated using the centralised model and the two decentralised ones and also between them; ƒ the variability of the input data (C) significantly influences the response of the systems; ƒ the input data variability and the different models used interact significantly. These results lead us to consider that, in this case, it makes sense to select the most favourable combination of input parameters to obtain the minimisation of the UC. In particular, with a small variability of the ownership, the centralised model returns the minimum value of unallocated capacity. 6.7.2

Design of Experiment (DoE)

In this section, DoE is used to investigate in which way the input variables influence the response, i.e. the unallocated capacity. In particular, a full factorial experiment has been conducted: responses have been measure3 at all combinations of the experimental factor levels. Factorial design allows us to understand the simultaneous effects that several factors may have on the results. MinitabTM has been used to analyse the obtained values and to build the following graphs.

3

Each response is the estimated average of the unallocated capacity as explained in the Section 6.6. Relative tables are reported in Appendix A.

Distributed Models for Planning Capacity at Medium Term

89

6.7.2.1 Centralised Model In the centralised model, our benchmark, a full factorial analysis has been conducted to investigate the influence of the input parameters. In Figures 6.12 and 6.13 it is possible to observe the main effects and the interaction between WL, P and C. Priority main effect is not significant: the output parameter is scarce conditioned by it. Also, its interaction with others (WL, C) is irrelevant. Different consideration can be made for workload and ownership. Both their main effect and their interaction strongly influence the unallocated capacity (UC). In particular, if we observe the output parameter minimum values, reported in the Table 6.26, we can see that reached results are indifferent to P but require WL and C at their low level: Table 6.26. UC minimum values Centralised

WL

P

C

UC

S

W

S

8.744

S

B

S

8.451

S

S

S

8.762

WL

P

Mean of Unallocated Capacity

40 35 30 25 20 W

B C

S

W

B

S

W

40 35 30 25 20

Figure 6.12. Main effects plot

B

S

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Production Planning in Production Networks

W

B

S

W

B

S 50

30

WL

WL W B S

10 50

30

P

P W B S

10

C

Figure 6.13. Interaction plot

Finally, Figure 6.14 shows the surface response of the models. The first pair of plots shows how when the workload increases in variance, passing from S to W, the performance deteriorates since the unallocated capacity grows. This trend is confirmed observing the second pair of plots: in this case the two factors act jointly. In the last situation only the Ownership, when its variance increases, leads to bad performance; Priority does not have any influence on the surface response. 6.7.2.2 Negotiation Adopting the same methodology, it is possible to analyse the negotiational model performance as summarized in Figures 6.15 and 6.16.

Distributed Models for Planning Capacity at Medium Term

C

W

91

S

Figure 6.14. Surface plot for UC (C.A.)

Priority, also in this case, plays a very marginal role. Its main effect and its interaction is unimportant; therefore, the considerations made in the previous case are still valid. In the following table the minimum values of UC are indicated and the surfaces plots are reported in Figure 6.17. Also in this case, as the uncertainty of the workload and the ownership increases, the negotiation model performance deteriorates. It is to be noticed that performance deteriorates in a worse way than in the centralised model. Therefore, the negotiation model is more sensitive to the uncertainty.

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Production Planning in Production Networks

WL

P

60

Mean of Unallocated Capacity

50 40 30 20 W

B C

S

W

B

S

W

B

S

60 50 40 30 20

Figure 6.15. Main effects plot

W

B

S

W

B

S

50 WL

WL W B S

25

0

50 P 25

0

C

Figure 6.16. Interaction plot

P W B S

Distributed Models for Planning Capacity at Medium Term

93

Table 6.27. UC minimum values Negotiation

WL

P

C

UC

S

W

S

8.557

S

B

S

8.603

S

S

S

8.688

C

W

S

Figure 6.17. Surface plot for UC (N.A.)

6.7.2.3 Game Theory In this last case, we obtain the effects and interactions reported in the following figures and tables. Also in this case the game-theoretical model is very sensitive to variance of input data.

Production Planning in Production Networks

WL

P

60 50

Mean of Unallocated Capacity

94

40 30 20 W

B C

S

W

B

S

W

B

S

60 50 40 30 20

Figure 6.18. Main effects plot

W

B

S

W

B

S

50 WL

WL W B S

25

0 50 P

25

0

C

Figure 6.19. Interaction plot Table 6.28. UC minimum values Game theory

WL

P

C

UC

S

W

S

8.528

S

B

S

8.353

S

S

S

9.052

P W B S

Distributed Models for Planning Capacity at Medium Term

C

W

95

S

Figure 6.20. Surface plot for UC (G.A.)

6.8 References [1] [2] [3] [4]

Fernandez FR, Hinojosa MA, Puerto J (2004) Set-valued TU-games. Eur J Oper Res 159 181–195 Shapley LS, Shubik M (1972) The assignment game: the core. Int J Game Theory 1:111–130 Owen G (1975) On the core of linear production games. Math Progr 9 358–37 Nash J (1950) The bargaining problem. Econometrica, 18(2):155–162

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7 Distributed Models for Plant Capacity Allocation

7.1 Introduction Plant capacity allocation consists in what we have referred as medium-level production planning in the general overview discussed in Chapter 5. As mentioned, at this level the objective is to acquire actual production capacity through plant assignment to groups; this goal is obtained through a bargaining process among the groups, which act as buyers, and the plants, which act as a sellers of capacity. They bargain by exchanging plant production capacity and price, which the buyers pay to gain capacity.

7.2 Initial State Let us suppose that the company produces g different (g = 1,…,G) product families, each associated to a group. Such products are produced in m plants (m = 1,…,M) each geographically dispersed and reconfigurable in order to produce all of the different product families. The reconfiguration is allowed once every three months and, at that time, the m plants are assigned to the g groups for accomplishing the annual quarter production of the associated product family. The group–plant association is addressed, as in the previous chapter, using a centralised model and two decentralised ones: a negotiational and a game-theoretical approach. The group–plant association can be interpreted as an internal transaction. The group will be requested to pay a certain number of fictitious credits to the plant in order to receive the needed capacity. The number of credits, pgm ˜ Vgm , is the product of the negotiated transaction price, pgm, that the g-th group pays to the plant m at the end of the quarter, times the volume of product family g that will be produced by the plants m, Vgm. Indeed, credits might be thought of as a virtual production capacity and they are equally distributed to the groups at the beginning of each year.

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Production Planning in Production Network

7.3 The Centralised Model In the following centralised model, we will keep the assumption that all the relevant planning data can be located centrally and can be analysed by a centralised algorithm. Given the annual quarter capacity ownership  Cg , assigned to each group as an outcome of the high planning level, the agents of groups (hereafter GAs) look for an agreement with the agents of plants (hereafter PAs) in order to gain a sufficient number of plants for supplying the required C g . In particular, we consider that the goal is to minimise the incurring costs of groups using the following objective function: § · u ª¬ xgm ˜ Qm ˜ ( Dgm  V gm )  ( Rc ˜ wgm ) º¼  ¦ Wg ˜ ¨¨ Rg  ¦ xgm ˜ Qm ¸¸ (7.1) ¦¦ g m g m © ¹ where: ƒ Qm is the production capacity of the m-th plant; ƒ

u is a matrix whose elements are the unit transport cost needed to Dgm

ƒ

sustain products transfer from the m-th plant to the g-th regional division; V gm is a measure of the efficiency of plant m in producing products for

ƒ

group g; it represents the plant m variable costs for producing this specific product; Rc is the reconfiguration cost of plant m when it is allocated to a different group with respect to that of the previous quarter; Wg is the unit cost the g-th group needs to sustain when the acquired

ƒ

capacity is different with respect to the one it needs. It can be interpreted like a production failure cost when the obtained capacity is less than the required one and like a unutilised capacity cost on the opposite situation; R is the capacity requested by group g.

ƒ

g

The constraints to take into account are: x gm

^

1, if plant m is assigned to g 0, otherwise

(7.2)

G

¦x

1

gm

(7.3)

g 1

wgm t x gm  k gm

(7.4)

where: k gm

^

1, if in the antecedent annual quarter m was allocated to g 0, otherwise

(7.5)

Constraint 7.2 defines the decision variables; Constraint 7.3 expresses that each plant can be assigned only to one group, while Constraints 7.4 and 7.5 are binary variables that lead us to take into account the reconfiguration costs.

Distributed Models for Plant Capacity Allocation

99

7.4 The Negotiation Model The negotiational mechanism has the same goal as the centralised one: minimise the incurring costs of groups. The negotiation is bilateral and simultaneous: all of the GAs, every three months, concurrently submit their own offer to the PAs. During the negotiation, the GAs make offers, based on their generative functions, to the PAs and these, based on their reactive functions, evaluate such proposals and decide whether they want to sign the contract with a given group or to ask for a new offer. There is a fixed number of rounds, rmax, for the negotiation being concluded. If at the end of rmax some of the plants are not allocated to any group, they are automatically allocated to the groups that in the last round proposed the best offer. A given plant can be assigned only to one group. The negotiation issue is the price p gm that the g-th group offers to the m-th plants at each round r. The GA’ generative functions for the price offer computation and the PA’ reactive functions for the price offer evaluation are described in what follows. 7.4.1 Generative Function

The generative function allows us to compute the price as defined below: u § r 1 · Rg  callgr 1 Dgmax  Dgm 1 pr p min  ˜ ( p gmax  p r 1 ) ˜ ¨   max ¸, (7.6) min ¨ rmax  1 Rg 3 Dg  Dg ¸¹ © where: ƒ r is the negotiation round; ƒ pgmax is the fixed maximum price (reservation price) that each group can gm

g

gm

ƒ

offer; p min is the minimum offered price. It is a constant guaranteeing fair

ƒ

(based on common value) offer by the groups to the plants; Rg , Du have been already defined;

g

gm

r 1 g

ƒ

call r  1;

is the capacity already allocated by the g-th group until round

ƒ

Dgmax

u max Dgm and Dgmin m

^ `

^ `

u min Dgm . m

As can be observed in Equation 7.6, the price the group g offers to the plant m increases according to the round of the negotiation (time-dependent tactic), while it decreases according to the capacity already allocated by the group itself (resourcedependent tactic). 7.4.2 Reactive Function

The reactive function is represented as: Wr pm* ˜ U mr ˜ V gm  Rc, gm

where:

(7.7)

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Production Planning in Production Network

ƒ

r W gm is the reactive function threshold. The plant m accepts the offer of r r group g if and only if pgm t W gm ;

ƒ ƒ ƒ

pm* is a constant defined by plant m; Rc is the reconfiguration cost, already defined; V gm is the plant m variable costs, already defined;

ƒ

the last term U mr can be expressed as:

U mr

ª

U m ˜ «1  Imr ˜ ¬

Gr

Imr

r Igm

¦G

r

r º » , where rmax ¼

, and

(7.8) (7.9)

g 1

r r 1 p gm  pgm

r Igm

(7.10) . It represents the risk attitude of plant m at round r; it increases according to the round itself (time-dependent tactic) and decreases with the average gradient Imr (imitative tactic) between the prices that the groups offer at rounds r and r  1. r 1 pgm

G r is the number of groups that are still negotiating at round r; U m is a constant depending on the plant and it characterizes the decisional parameter related to its own risk attitude. The risk is related to a first in, first out (FIFO) discipline in allocating orders to the plants; that is, throughout the annual quarter the groups allocate the actual production orders first to those plants which have signed earliest the contract with the group itself. Thus, plants that are allocated last have a higher risk of remaining underloaded. Equation 7.7 shows that the utility threshold of the plant agents increases according with their risk attitude (agent state), their efficiency in producing product g, and the reconfiguration cost.

ƒ ƒ

7.5 The Game-theoretical Model The game-theoretical model is quite similar to the one already discussed for production planning at the high level, which has been presented in Chapter 6. Here we will put into evidence the main differences with that discussed before. This time, the outline of the game is a quadruple AG r (G r , P r , T r , O r ) , where: ƒ

Gr

ƒ

round r; P r ^1,..., m ,..., M` is a vector whose element m represents a plant at

^1,..., i ,..., G` is a vector whose element g represents a group at

round r;

Distributed Models for Plant Capacity Allocation

101

ƒ

r T r is a vector whose elements W gm , represent the credits requested by the

ƒ

plant m to sell its own capacity to the g-th group. This value is calculated by the reactive function 7.7; Or is a matrix whose generic element prgm is the value the group g assigns

to the capacity offered by each player m  Pr using the generative function of Equation 7.6. A transfer utility game can be associated to the quadruple AG r ; here the whole players are given by G r ‰ P r and the characteristic function Ȟ is defined, given that group g and plant m make a coalition, as in Equation 7.11: ­° p gm r  W gmr if p gm r  W gmr ! 0. r v ( g , m) h gm ® (7.11) if p gm r  W gm r d 0. °¯0 The consequent assignment problem is solved using, also this time, the core game solution, and the related Owen’s theorem, if hgm > 0, or the Nash bargaining approach otherwise. Obviously, the utility functions which has been used is reported in Equation 7.12 for the GAs and Equation 7.13 for the PAs:

Ug

5

¦P ˜b t

t ,5

u ,

0 d u d 1,

(7.12)

t 0

Um

'

(7.13) Ug being the Bernstein basis polynomials of degree t with Pt points and ' a parameter that has been already defined in Equation 6.33.

7.6 The Simulation As in the high-level production planning case of Chapter 6, a discrete event simulator, based on a multi-agent system (MAS), has been developed using JAVA. Optimisation has been performed through the LINGO® package conveniently linked to the simulation environment. The simulation carries on considering an events timetable of the agents’ actions: this schedule is utilised to define the sequence of the events. In the following, class and sequence diagrams will be utilised to show the simulation environment. 7.6.1 The Simulation Environment

The simulation environment considers, for all different approaches explained above, the following objects: ƒ Schedule: this object is in charge of managing the sequence of events during the simulation time; ƒ Model: this object handles the signals related to the orders, coordinates the activities among groups and plants and selects the strategies; ƒ Group: this object represents the GAs set; it has all the information, algorithms and functions needed for their implementation;

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Production Planning in Production Network

ƒ

Plant: this object represents the PAs set; it has all the information, algorithms and functions needed for their implementation; ƒ Statistical Analysis: this last object is dedicated to the analysis of groups’ and plants’ performance analysis. The class diagram of Figure 7.1 shows the most relevant methods utilised in the Java code to obtain the described simulation environment.

Figure 7.1. Class diagram of the medium production planning level

On the other hand, Figure 7.2 shows the sequence diagram; it is referred to a generic order o, o = 1,..,O:

Distributed Models for Plant Capacity Allocation

103

Figure 7.2. Sequence diagram of the medium production planning level

Schedule object activates all other objects shown here. ƒ o-th order: order number o is communicated to the Model object; Model object updates information related to o-th order; ƒ Initialsze o-th order: information related to o-th order is communicated to groups; Groups calculate the capacity they need; ƒ Needed capacity: groups’ requested capacity is communicated to the Model; ƒ Requested capacity: requested capacity is communicated to the plants; Plants calculate their utilities values related to the offered credits; ƒ Utilities values: utilities values are communicated by Plant to Model; Model applies one of the three different approaches, depending on the considered model: centralized, negotiational, game-theoretical; ƒ Group/Plant Assignments: Group/Plant assignment is communicated to Group and Plants objects; ƒ Performance communication: groups’ and plants’ performance measures are communicated to Statistical Analysis object; Statistical Analysis object uploads performance measures and carries on with the simulation until the defined confidence interval is reached; ƒ Statistical performances: performance measures are communicated to Schedule object;

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Production Planning in Production Network

Schedule verifies the performance measures statistical significance; ƒ Significance communication: Schedule communicates to Statistical Analysis object the result of the previous check; Final report is written. 7.6.2 The Simulation Case Study

The test environment consists of 27 different combination of input parameters: capacity ownership of groups (C), risk attitude ( R { U m ) and capacities (Q) of





plants. In particular, the o-th order is given by a vector (C,R,Q): each index is considered randomly drawn from three normal distributions N(µ;ı) whose parameters are reported in Table 7.1 as, Small (S), Bounded (B), Wide (W) with relation to the standard deviations size. Table 7.1. Distributions of input parameters Small (S)

Bounded (B)

Wide (W)

Ownership (C)

N(25;5)

N(25;10)

N(25;15)

Risk attitude (R)

N(0.5;0.025)

N(0.5;0.3)

N(0.5;0.5)

Capacity (Q)

N(10;0.5)

N(10;3)

N(10;8)

An infinite number of credits have been supposed for each simulation experiment; for each replication the following performance variables have been computed with a confidence interval equal to 95%: G

V

¦¦ Q

m

˜ V gm ,

(7.14)

m

u ˜ Dgm ,

(7.15)

g 1 mg G

G

¦¦ Q g 1 mg G

-

M

¦¦ w

gm

,

(7.16)

g 1m 1

§ · ˜ ¨ Rg  ¦ Qm ¸ . (7.17) ¨ ¸ g 1 mg © ¹ Equation 7.14 expresses the efficiency of the models in assigning plants to groups. The lower its value, the higher the ability of the models to properly assign plants to the respective group. On the other hand, Equation 7.15 takes into account the transportation costs related to the distances existing between plants and regional divisions, while Equation 7.16 provides a measure of the mean number of reconfigurations obtained during the simulation. The last performance, i.e. the one reported in Equation 7.17, is the cost of the difference between the requested and the actual allocated capacity. G

Z

¦W

g

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105

7.7 Results The simulation parameters as well as the results of the 27 different combinations of input parameters for the centralised (C.A.), negotiational (N.A.) and gametheoretical (G.A.) approaches are reported in Appendix B. Also in this case, a proper analysis of the results has been carried out through the two-way Analysis of Variance (ANOVA) and the Design of Experiment (DoE) for each estimated performance. The obtained results, rearranged in the following tables, can be considered like a set of data drawn from three normal distributions with mean equal to the three different grand mean (depending on the utilized model) and the same variance. Each factor (i.e. independent variable) has three different levels; respectively, the factor model has three levels corresponding to C.A., N.A., G.A. and factor input parameter has also three levels corresponding to the variance wideness of the input data, i.e. wide (W), bounded (B) and small (S) variance. The goal is to understand in which way the input parameters affect the estimated performance measures. 7.7.1 Efficiency Performance Analysis: Two-way ANOVA

7.7.1.1 Ownership Table 7.2 reports in the rows the levels of the input factors and in the columns the values of the nine different combinations of the other two input parameters (R, Q) as explained in Table 7.3. Table 7.2. Replication values of V

C.A.

G.A

N.A

W

1 14.45

2 14.43

3 12.18

4 14.36

5 14.44

6 12.16

7 14.43

8 14.50

9 14.47

B

14.67

14.95

12.07

14.75

15.05

12.03

14.77

14.95

12.06

S

15.16

15.59

12.68

15.07

15.66

12.92

15.12

15.10

12.65

W

17.34

17.56

17.72

17.35

17.58

17.67

17.34

17.59

17.31

B

17.64

17.85

18.04

17.59

17.96

17.89

17.48

17.96

17.99

S

17.94

18.77

18.77

17.90

18.65

19.16

17.91

17.92

18.94

W

15.60

15.74

15.95

15.64

15.70

15.71

15.68

15.60

16.50

B

15.90

16.02

15.86

15.54

15.62

15.62

16.93

17.32

17.09

S

15.91

15.92

15.91

15.56

15.62

15.26

17.17

17.03

17.50

Table 7.3. Combination of R and Q 1

2

3

4

5

6

7

8

9

Risk attitude (R) variance

W

W

W

B

B

B

S

S

S

Production capacity (Q) variance

W

B

S

W

B

S

W

B

S

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Production Planning in Production Network

Table 7.4 reports all the mean values for each model and for each level of the ownership variance. This results will be deeply investigated in what follows. Table 7.4. Average values of V W

B

Mean

S

C.A.

13.94

13.92

14.44

14.10

G.A.

17.49

17.82

18.44

17.92

N.A. Mean

15.79

16.21

16.21

16.07

15.74

15.98

16.36

16.03

Specifically, there are three sets of hypotheses to consider. The null and alternative hypotheses for each set are the following: 1.

2.

for the population means of the first factor (the model): PC PN PG ­H0 : ° ; ®H : not all P are equal i^C ,N ,G` °¯ 1

(7.18)

for the population means of the second factor (the variance): PW P B PS ­H0 : ° ; ®H : not all P are equal j^ W ,B,S` °¯ 1

(7.19)

and, finally, for the interdependencies: there is no interaction °­H 0 : ; ® there is interaction °¯H1 : Table 7.5 reports the results of the ANOVA test for V. 3.

(7.20)

Table 7.5. Two-way ANOVA table for V

First factor

SS

df

MS

F-test

F-crit D

197.021

2

98.510

143.081*

F2.72

3.13 3.13 2.51

Second factor

5.302

2

2.651

3.850*

F2.72

Interaction Effect

1.458

4

0.364

0.529

F4.72

Within

49.572

72

0.688

–

Total

253.352

80

–

5%

–

From the analysis of Table 7.4, that only the main effects are significant, while the interaction between them is not. That is, only the first two null hypotheses 7.18 and 7.19 can be rejected. Figures 7.3 and 7.44 report respectively the observed and

Distributed Models for Plant Capacity Allocation

107

estimated values, whose numerical values are reported in Table 7.6, from which it is possible to observe the absence of interaction. Table 7.6. Estimated values for V

C.A. G.A. N.A.

Mean

W

B

S

Mean

13.81

14.05

14.43

14.10

17.63

17.87

18.25

17.92

15.78

16.03

16.40

16.07

15.74

15.98

16.36

16.03

19,00 18,00 17,00 Centralised

16,00

Game Theory

15,00

Negotiation

14,00 13,00 12,00 0

W 1

B 2

S3

4

Figure 7.3. Observed values

19,00 18,00 17,00

Centralised

16,00

Game Theory Negotiation

15,00 14,00 13,00 0

1 W

B2

3S

4

Figure 7.4. Estimated values

As also done in Chapter 6, Table 7.7 shows the T-critical test values.

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Production Planning in Production Network

Table 7.7. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

0.541

Column factor

Q0.05 ,3,72

3.39

T

0.541

Interaction

Q0.05 ,9 ,72

4.53

T

1.253

Using data reported in Table 7.4, we obtain: Table 7.8. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

3.819*

0

0

N.A.

1.972*

1.848*

0

Table 7.9. Differences of means for column factors Column factor

W

B

S

B

0.244

0

0

S

0.622*

0.377

0

Last Table is referred to the group’s interaction analysis. Table 7.10. Groups means differences for column factors W–C.A.

B–C.A.

S–C.A.

W–G.A.

B–G.A.

S–G.A.

W–N.A.

B–N.A.

–

–

–

–

–

–

–

B–C.A.

0.014

S–C.A.

0.504

0.518

–

–

–

–

–

–

W–G.A.

3.559*

3.573*

3.056*

–

–

–

–

–

B–G.A.

3.886*

3.900*

3.382*

0.327

–

–

–

–

S–G.A.

4.503*

4.517*

4.000*

0.944

0.617

–

–

–

W–N.A.

1.855*

1.869*

1.352*

1.704*

2.031*

2.648*

–

–

B–N.A.

2.276*

2.290*

1.773*

1.283*

1.609*

2.227*

0.421

–

S–N.A.

2.273*

2.288*

1.770*

1.286*

1.612*

2.229*

0.418

0.003

By comparing data from Table 7.7 and Tables 7.8, 7.9 and 7.10 it appears quite evident that, given a confidence interval equal to 5%, ƒ there is statistical difference between the value of efficiency estimated using the centralised model and the two decentralised ones and between them also;

Distributed Models for Plant Capacity Allocation

109

ƒ

the variability of the input data (C) significantly influences the response of the system passing from W to S; ƒ the input data variability and the different models used do not interact significantly. This considerations lead us to conclude that the centralised model really behaves better with respect to the decentralised ones in the optimisation of the efficiency. At the same time, a high variability of C seems to have a positive influence on the estimated performance. 7.7.1.2 Risk Attitude Similarly to the ownership analysis, Table 7.11 reports the values of the performance V depending on the two factors levels on the rows, i.e. model typology and risk attitude variance wideness, and on the combinations of the other two input parameters (C,Q) in the columns. The association of the combination of C and Q to the number reported in the columns of Table 7.11 is reported in Table 7.12. Table 7.11. Replication values for V

C.A.

G.A.

N.A.

W

1 14.45

2 14.43

3 12.18

4 14.67

5 14.95

6 12.07

7 15.16

8 15.59

9 12.68

B

14.36

14.44

12.16

14.75

15.05

12.03

15.07

15.66

12.92

S

14.43

14.50

14.47

14.77

14.95

12.06

15.12

15.10

12.65

W

17.34

17.56

17.72

17.64

17.85

18.04

17.94

18.77

18.77

B

17.35

17.58

17.67

17.59

17.96

17.89

17.90

18.65

19.16

S

17.34

17.59

17.31

17.48

17.96

17.99

17.91

17.92

18.94

W

15.60

15.74

15.95

15.64

15.70

15.71

15.68

15.60

16.50

B

15.90

16.02

15.86

15.54

15.62

15.62

16.93

17.32

17.09

S

15.68

15.60

16.50

16.93

17.32

17.09

17.17

17.03

17.50

Table 7.12. Combination of C and Q 1

2

3

4

5

6

7

8

9

Ownership (C) variance

W

W

W

B

B

B

S

S

S

Production capacity (Q) variance

W

B

S

W

B

S

W

B

S

Table 7.13 reports the mean values for the performance V. As the reader can notice from Table 7.13 also in this case efficiency seems to be smaller with the centralised models. This conclusion will be deeply investigated by performing a two-way ANOVA.

110

Production Planning in Production Network Table 7.13. Average values of V

C.A. G.A. N.A.

Mean

W

B

S

Mean

14.02

14.05

14.23

14.10

17.96

17.97

17.83

17.92

15.79

16.21

16.76

16.25

15.92

16.08

16.27

16.09

Specifically, there are three sets of hypothesis to consider. The null and alternative hypothesis for each sets are given, also in this case, by 7.18–7.20. Table 7.14 reports the results of the ANOVA analysis in this case. Table 7.14. Two-way ANOVA table for R

SS

Df

MS

F-Test

F-crit D

198.034

2

99.017

133.140*

F2.72

3.13

Second factor

1.646

2

0.823

1.107

F2.72

3.13

Interaction Effect

2.939

4

0.735

0.988

F4.72

2.51

–

First factor

Within

53.547

72

0.744

Total

256.166

80

–

5%

–

From the above results it is quite evident that only one factor is significant for the estimated parameter. In order to strongly support this conclusion, also in this case, the Tukey test has been carried out. The T-values of the test are reported in Table 7.15, while Table 7.16 reports the differences of the means. The Tukey test confirms the result of the two-way ANOVA. Finally, Table 7.17 reports the interaction analysis. Table 7.15. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

0.563

Column factor

Q0.05 ,3,72

3.39

T

0.563

Interaction

Q0.05 ,9 ,72

4.53

T

1.302

Using data reported in Table 7.13, we obtain: Table 7.16. Differences of the means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

3.820*

0

0

N.A.

2.155*

1.665*

0

Distributed Models for Plant Capacity Allocation

111

The last table refers to the group’s interaction analysis. Table 7.17. Group’mean differences for column factors W–C.A.

B–C.A.

S–C.A.

W–G.A.

B–G.A.

S–G.A.

W–N.A.

B–N.A.

B–C.A.

0.029

–

–

–

–

–

–

–

S–C.A.

0.208

0.179

–

–

–

–

–

–

W–G.A.

3.936*

3.907*

3.729*

–

–

–

–

–

B–G.A.

3.953*

3.924*

3.745*

0.017

–

–

–

–

S–G.A.

3.806*

3.777*

3.598*

0.130

0.147

–

–

–

W–N.A.

1.771*

1.742*

1.563*

2.165*

2.182*

2.035*

–

–

B–N.A.

2.192*

2.163*

1.984*

1.744*

1.761*

1.614*

0.421

–

S–N.A.

2.739*

2.710*

2.531*

1.197

1.214

1.067

0.968

0.547

The obtained results, with a confidence interval equal to 5%, lead us to the following conclusions: ƒ there is a statistical difference between the performance estimated using the centralised model and the two decentralised ones and between them also; ƒ the variability of the input data (R) does not significantly influence the response of the system; ƒ the input data variability and the different models used do not interact significantly. These considerations lead us to conclude that, also in this case, high variability of input data and the centralised model minimize the efficiency. 7.7.1.3 Plants Capacity The last parameter to be investigated is the plant capacity Q. The performance measure V values are reported in Table 7.18, while average values are reported in Table 7.20, which is the basis of the two-way ANOVA. As the reader can notice, also for the this input variable, the centralised approach seems to perform better that the decentralised ones. The two null hypotheses are 7.18 and 7.20. ANOVA analysis results are reported in Table 7.21. The next step is to calculate a table with all the mean values for each model and for each level of Q. Again, the null hypothesis set is given by 7.18–7.20. In this case, from Table 7.21, we can see that all three null hypotheses can be rejected. This means that not only the main effects of considered factor are significant, but also their interaction. In this case, if we consider the observed values plot, we will not obtain parallel lines, but divergent ones. Obviously, this consideration is not true for the estimated values reported in Table 7.22. This graphical analysis is reported in Figures 7.5 and 7.6.

112

Production Planning in Production Network Table 7.18. Replication values of V

C.A.

G.A.

N.A.

W

1 14.45

2 14.36

3 14.43

4 14.67

5 14.75

6 14.77

7 15.16

8 15.07

9 15.12

B

14.43

14.44

14.50

14.95

15.05

14.95

15.59

15.66

15.10

S

12.18

12.16

14.47

12.07

12.03

12.06

12.68

12.92

12.65

W

17.34

17.35

17.34

17.64

17.59

17.48

17.94

17.90

17.91

B

17.56

17.58

17.59

17.85

17.96

17.96

18.77

18.65

17.92

S

17.72

17.67

17.31

18.04

17.89

17.99

18.77

19.16

18.94

W

15.60

15.64

15.68

15.90

15.54

16.93

15.91

15.56

17.17

B

15.74

15.70

15.60

16.02

15.62

17.32

15.92

15.62

17.03

S

15.95

15.71

16.50

15.86

15.62

17.09

15.91

15.26

17.50

Table 7.19. Combination of C and R 1

2

3

4

5

6

7

8

9

Ownership (C) variance

W

W

W

B

B

B

S

S

S

Risk attitude (R) variance

W

B

S

W

B

S

W

B

S

Table 7.20. Average values of V

C.A. G.A. N.A.

Mean

W

B

S

Mean

14.75

14.96

12.58

14,10

17.61

17.98

18.17

17,92

15.99

16.06

16.16

16,07

16,12

16,33

15,63

16,03

Table 7.21. Two-way ANOVA table for Q

First factor Second factor Interaction effect Within Total

SS 197.021

df 2

MS 98.510

F-Test 302.927*

F-crit D

F2.72

3.13

6.961

2

3.480

10.703*

F2.72

3.13

25.956

4

6.489

19.954*

F4.72

2.51

23.414

72

0.325

-

253.352

80

-

-

5%

Distributed Models for Plant Capacity Allocation

113

Table 7.22. Estimated values of V

C.A. G.A. N.A.

Mean

W

B

S

Mean

14.19

14.40

13.70

14.10

18.01

18.22

17.52

17.92

16.16

16.38

15.68

16.07

16.12

16.33

15.63

16.03

Also in this case, the Tukey test, whose T-values are reported in Table 7.23, is carried out to better understand where this significance is located. Mean differences are reported in Tables 7.24 and 7.25 and the test brings us to conclude that the influence on means is due to either the column or the row factor; this allows us to reject the main effects null hypothesis. On the other hand, Table 7.26 reports the interaction data and it allows us to reject also the interaction null hypotheses. 19 18 17

Centralised Game theory

16

Negotiation

15 14 13 12 W

B

S

Figure 7.5. Observed values

19 18 17

Centralised Game theory

16

Negotiation

15 14 13 12 W

B

S

Figure 7.6. Estimated values

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Production Planning in Production Network

Table 7.23. T-values for Q Factor

Q-value

T-value

Row factor

Q0 , 05 ,3,72

3, 39

T

0 , 372

Column factor

Q0 , 05 ,3,72

3, 39

T

0 , 372

Interaction

Q0 , 05 ,9 ,72

4, 53

T

0 , 861

Using data reported in Table 7.17, we obtain: Table 7.24. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

3,820*

0

0

N.A.

1,972*

1,848*

0

Table 7.25. Differences of means for column factors Column factor

W

B

S

B

0,216

0

0

S

0,485*

0,701*

0

It is easy to see that, also using this test, the influence on means we observe is due to either the column or the row factor, as already viewed refusing all hypotheses 7.18–7.20. The last table refers to the interaction. Table 7.26. Group’ mean differences for column factors W – C.A.

B –C.A.

S –C.A.

W –G.A.

B –G.A.

S –G.A.

W –N.A.

B –N.A.

B –C.A.

0.209

–

–

–

–

–

–

–

S –C.A.

2.175*

2.384*

–

–

–

–

–

–

W –G.A.

2.857*

2.648*

5.032*

–

–

–

–

–

B –G.A.

3.225*

3.016*

5.400*

0.368

–

–

–

–

S –G.A.

3.411*

3.202*

5.586*

0.554

0.186

–

–

–

W –N.A.

1.238*

1.029*

3.413*

1.619*

1.987*

2.173*

–

–

B –N.A.

1.309*

1.100*

3.483*

1.548*

1.916*

2.102*

0.071

–

S –N.A.

1.403*

1.194*

3.578*

1.454*

1.822*

2.008*

0.165

0.094

With a confidence interval equal to 5%. it has been highlighted that: ƒ there is a statistical difference between the efficiency estimated using the centralised model and the two decentralised ones and it is relevant also incomparing N.A. and G.A.;

Distributed Models for Plant Capacity Allocation

ƒ ƒ

115

the variability of the input data (Q) significantly influences the response of the systems, specifically when the variance diminishes passing from W to S and from B to S; the input data variability and the different models interact significantly combining their effects on the estimated performance.

7.7.2 Efficiency Performance Analysis: DoE

In the following analysis, DoE is used to investigate in which way the input variables influence the efficiency V . 7.7.2.1 Centralised Model In the centralised model, our benchmark, a full factorial analysis has been conducted in order to investigate the influence of the input parameters. Figure 7.7 reports the main effect’ impact on the efficiency, Figure 7.8 the interaction effect. C

R

11,5 11,0

Mean of Distances

10,5 10,0 9,5 W

B Q

S

W

B

S

W

11,5 11,0 10,5 10,0 9,5

Figure 7.7. Main effects plot (C.A.)

B

S

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W

C

B

S

W

B

C W B S

S 15,0

13,5

12,0

R

R W B S

15,0

13,5

12,0

Q

Figure 7.8. Interaction plot (C.A.)

As the reader can notice, efficiency, V, rapidly decreases when the production capacity of plants (Q) has a small variance. The other two parameters, and their interaction, have less influence. These results are reported in the surface response curve of Figure 7.9.

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117

Figure 7.9. Surface plot for efficiency (C.A.)

7.7.2.2 Negotiation Model Adopting the same approach, it is possible to analyse the negotiational model. The main effects are reported in Figure 7.10, and the interaction effect in Figure 7.11. As the reader can notice, a wide variance of the three input variable can cause a significant reduction of the efficiency. On the other hand, if variance wideness decreases the performance approach rises. Figure 7.12 shows the surface curves in this case.

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Production Planning in Production Network

C

R

16,8 16,5 16,2

Mean Efficiency

15,9 15,6 W

B

S

W

B

S

Q

16,8 16,5 16,2 15,9 15,6 W

B

S

Figure 7.10. Main effects plot (N.A.)

W

C

B

S

W

C W B S

B

S

16,8

16,2

15,6

R

R W B S

16,8

16,2

15,6

Figure 7.11. Interaction plot (N.A.)

Distributed Models for Plant Capacity Allocation

R

W

119

S

Figure 7.12. Surface plot for efficiency (N.A.)

7.7.2.3 Game Theory In this last case we obtain the effects shown in Figure 7.13, and interactions in Figure 7.14. Groups’ ownership and plants’ production capacity play, within this model, a very significant role: by diminishing the variance of this data input, the estimated parameters increase in a quasi linear way. Also their interaction leads to a very strong influence on the output. Finally, the surface plots are reported in Figure 7.15.

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Production Planning in Production Network

C

18,50

R

18,25

Mean of Efficiency

18,00 17,75 17,50 W

B

S

W

B

S

Q

18,50 18,25 18,00 17,75 17,50 W

B

S

Figure 7.13. Main effects plot (G.A.)

W

C

B

S

W

B

C W B S

S

18,6

18,0

17,4

R

R W B S

18,6

18,0

17,4

Q

Figure 7.14. Interaction plot (G.A.)

Distributed Models for Plant Capacity Allocation

121

W

R

B

W

S

Figure 7.15. Surface plot for efficiency (G.A.)

7.7.3 Distance Performance Analysis: Two-way ANOVA

7.7.3.1 Ownership Table 7.27 reports, in the rows, the levels of the input factors while in the columns the values of the nine different combinations of the other two input parameters (R, Q) as explained in Table 7.3. Table 7.28 reports all the mean values for each model and for each level of the ownership variance. This results will be deeply investigated in what follows. The null and alternative hypotheses have been already defined in 7.18–7.20. The ANOVA table we obtain is reported in Table 7.29.

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Table 7.27. Replication values of G

C.A.

G.A.

N.A.

W

1 11.09

2 11.13

3 9.48

4 11.02

5 11.09

6 9.47

7 11.04

8 11.14

9 11.08

B

11.30

11.50

9.40

11.32

11.61

9.35

11.26

11.54

9.30

S

11.67

11.99

9.59

11.58

12.07

9.89

11.61

11.62

9.64

W

13.23

13.36

13.47

13.24

13.52

13.61

13.23

13.40

13.26

B

13.52

13.58

13.59

13.41

13.65

13.70

13.33

13.62

13.81

S

13.50

13.57

13.67

13.47

13.60

13.69

13.74

13.48

13.60

W

12.15

12.22

12.30

12.19

12.25

12.18

12.19

12.07

13.08

B

12.37

12.48

12.47

12.18

12.24

12.35

13.39

13.59

13.54

S

12.46

12.63

12.60

12.29

12.38

12.24

13.57

13.54

13.80

Table 7.28. Average values of G

C.A. G.A. N.A.

Mean

W

B

S

Mean

10.73

10.73

11.07

10.84

13.37

13.58

13.59

13.51

12.29

12.74

12.83

12.62

12,13

12,35

12,50

12,33

Table 7.29. Two-way ANOVA table for C

First factor

SS 99.713

df 2

MS 49.856

F-test 126.228*

F-crit D

5%

F2.72

3.13

Second factor

1.873

2

0.936

2.371

F2.72

3.13

Interaction effect

0.617

4

0.154

0.391

F4.72

2.51

Within

28.438

72

0.395

–

Total

130.640

80

–

–

From the analysis of Table 7.28, only the main effect related to the first factor is significant: the first null hypothesis 7.18 can be rejected. Considering the values reported in Table 7.28 we can graphically observe the absence of interaction (lines are approximately parallel) in the Figure 7.16.

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123

14,00 13,00 Centralised Game Theory

12,00

Negotiation 11,00 10,00 0

1 W

2 B

3S

4

Figure 7.16. Observed values

Referring to the analysed case we can calculate the T-critical values and the following table. Table 7.30a. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

0.410

Interaction

Q0.05 ,9 ,72

4.53

T

0.949

Using data reported in Table 7.28, we obtain: Table 7.30b. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

2.69*

0

0

N.A.

1.777*

0.892*

0

Table 7.31 refers the group’s interaction analysis. Finally, given these results with a confidence interval equal to 5%, we can draw the following conclusions: there is a statistical difference between the performance estimated using the centralised model and the two decentralised ones and between them also; ƒ the variability of the input data (C) does significantly influence the response of the systems; ƒ the input data variability and the different models do not interact significantly. Statistically speaking, there are differences between the three proposed models. Ample discrepancy is present considering the distance existing between the

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benchmark (C.A.) and the two decentralised approaches, but no great difference can be measured among G.A. and N.A. Table 7.31. Group’ means differences for column factors W-C.A. B -C.A.

0.004

S -C.A.

0.346

W -G.A.

B-C.A. -

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

-

-

-

-

-

-

0.342

-

-

-

-

-

-

2.641*

2.637*

2.295*

-

-

-

-

-

B -G.A.

2.851*

2.847*

2.505*

0.210

-

-

-

-

S -G.A.

2.866*

2.862*

2.519*

0.225

0.014

-

-

-

W -N.A.

1.566*

1.562*

1.220*

1.075*

1.285*

1.300*

-

-

B -N.A.

2.010*

2.006*

1.664*

0.631

0.841

0.856

0.444

-

S -N.A.

2.106*

2.102*

1.760*

0.535

0.745

0.759

0.540

0.096

7.7.3.2 Risk Attitude Similarly to the ownership analysis, Table 7.32 reports the values of the performance G depending on the two factor levels in the rows, i.e. model typology and risk attitude variance wideness, and on the combinations of the other two input parameters ( C , Q ) in the columns. The association of the combination of C and Q to the number reported in the columns of Table 7.32 is reported in Table 7.12. Table 7.32. Replication values for G

C.A.

G.A.

N.A.

1

2

W

11.09

11.13

B

11.02

S

11.04

W B S W B S

3

4

5

9.48

11.30

11.50

11.09

9.47

11.32

11.14

11.08

11.26

13.23

13.36

13.47

13.24

13.52

13.23

6

7

8

9

9.40

11.67

11.99

9.59

11.61

9.35

11.58

12.07

9.89

11.54

9.30

11.61

11.62

9.64

13.52

13.58

13.59

13.50

13.57

13.67

13.61

13.41

13.65

13.70

13.47

13.60

13.69

13.40

13.26

13.33

13.62

13.81

13.74

13.48

13.60

12.15

12.22

12.30

12.37

12.48

12.47

12.46

12.63

12.60

12.19

12.25

12.18

12.18

12.24

12.35

12.29

12.38

12.24

12.19

12.07

13.08

13.39

13.59

13.54

13.57

13.54

13.80

Table 7.33 reports the mean values for the performanceG. As the reader can notice from Table 7.33 also in this case distance seems to be smaller with the centralised models. This conclusion will be deeply investigated by performing a two-way ANOVA.

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125

Table 7.33. Average values of G

C.A. G.A. N.A.

Mean

W

B

S

Mean

10.79

10.82

10.92

10.84

13.50

13.54

13.50

13.51

12.41

12.26

13.20

12.62

12.23

12.21

12.54

12.33

The hypothesis set is the same as given by Equations 7.18 –7.20. Table 7.33. Two-way ANOVA table for R SS

df

MS

F-test

F-crit D

5%

First factor

99.713

2

49.856

136.771*

F2.72

3.13

Second factor

1.800

2

0.900

2.469

F2.72

3.13

Interaction effect

2.882

4

0.720

1.977

F4.72

2.51

Within

26.246

72

0.365

–

Total

130.640

80

–

–

From the above results it is quite evident that only one factor is significant for the estimated parameter. In order to strongly support this conclusion, also in this case, the Tukey test has been carried out. The T-values of the test are reported in Table 7.35, while Table 7.35 reports the differences of the means. The Tukey test confirms the result of the two-way ANOVA. Finally, Table 7.37 reports the interaction analysis. Table 7.35. T-values Factor

Q-value

T-value

Row factor

Q0.05,3,72

3.39

T

0.394

Interaction

Q0.05,9 ,72

4.53

T

0.912

Using data reported in Table 7.33, we obtain: Table 7.36. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

2.669*

0

0

N.A.

1.777*

0.892*

0

The last table refers to the groups’ interaction analysis.

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Production Planning in Production Network

Table 7.37. Group’ means differences for column factors W-C.A.

B-C.A.

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

B –C.A.

0.031

–

–

–

–

–

–

–

S –C.A.

0.123*

0.092*

–

–

–

–

–

–

W –G.A.

2.707*

2.676*

2.584*

–

–

–

–

–

B –G.A.

2.752*

2.721*

2.628*

0.044

–

–

–

–

S –G.A.

2.703*

2.672*

2.580*

0.004

0.049

–

–

–

W –N.A.

1.618*

1.586*

1.494*

1.090*

1.134*

1.086*

–

–

B –N.A.

1.463*

1.432*

1.340*

1.244*

1.288*

1.240*

0.154

–

S –N.A.

2.405*

2.374*

2.282*

0.302

0.346

0.298

0.788

0.942

By plotting data shown in Table 7.33 we obtain:

14,00

13,00 Centralised 12,00

Game Theory Negotiation

11,00

10,00 0

1 W

B2

S3

4

Figure 7.17. Observed values

It is easy to see that there is some interaction between factors but, with the utilised confidence interval, it cannot be considered relevant for our analysis. Consequently, with a confidence interval equal to 5%, we can draw the same conclusions as reported in the previous section. 7.7.3.3 Plant Capacity The last parameter to be investigated is the plant capacity Q. As usual the performance measure G values are reported in Table 7.38, while average values are reported in Table 7.39, which is the basis of the two-way ANOVA. Also for this input variable, the centralised approach seems to perform better than the decentralised ones. The two null hypotheses are 7.18 and 7.20. ANOVA analysis results are reported in Table 7.40.

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127

Table 7.38. Replication values of G

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

9

W

11.09

11.02

11.04

11.30

9.40

9.35

11.67

9.59

9.89

B

11.13

11.02

11.04

11.50

11.32

11.26

11.99

11.58

11.61

S

9.48

9.47

11.14

9.40

9.35

11.54

9.59

9.89

11.62

W

13.23

13.24

13.23

13.52

13.41

13.33

13.50

13.47

13.74

B

13.36

13.52

13.40

13.58

13.65

13.62

13.57

13.60

13.48

S

13.47

13.61

13.26

13.59

13.70

13.81

13.67

13.69

13.60

W

12.15

12.19

12.19

12.37

12.18

13.39

12.46

12.29

13.57

B

12.22

12.25

12.07

12.48

12.24

13.59

12.63

12.38

13.54

S

12.30

12.18

13.08

12.47

12.35

13.54

12.60

12.24

13.80

The next step is to create a table with all the mean values. Table 7.39. Average values of G

C.A. G.A. N.A.

Mean

W

B

S

Mean

10.48

11.38

10.16

10.68

13.41

13.53

13.60

13.51

12.53

12.60

12.73

12.62

12.14

12.50

12.16

12.27

In this case, from Table 7.40, we can see that all three null hypotheses can be rejected. This means that not only the main effects of the considered factor are significant, but also their interaction. In this case, if we consider the observed values plot, we will not obtain parallel lines, but divergent ones. Obviously, this consideration is not true for the estimated values reported in Table 7.41. This graphical analysis is reported in Figures 7.18 and 7.19. Table 7.40. Two-way ANOVA table for Q

First factor Second factor Interaction Effect Within Total

SS 113.571

df 2

MS 56.786

F-test 174.283*

F-crit D

F2.72

3.13

2.237

2

1.118

3.432*

F2.72

3.13

5.306

4

1.326

4.071*

F4.72

2.51

23.459

72

0.326

–

144.573

80

–

Table 7.41. Estimated values of G

–

5%

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Production Planning in Production Network

C.A. G.A. N.A.

Mean

W

B

S

Mean

10.55

10.91

10.57

10.68

13.38

13.75

13.41

13.51

12.49

12.86

12.52

12.62

12.14

12.50

12.16

12.27

Also in this case, the Tukey test, whose T-values are reported in Table 7.42, is carried out to better understand where this significance is located. Mean differences are reported in Tables 7.43 and 7.44 and the reader can notice how the test brings us to conclude that the influence on means is due to either the column or the row factor; this allows us to reject the main effects null hypothesis. On the other hand, Table 7.45 reports the interaction data and it allows us to reject also the interaction null hypotheses. Table 7.42. T-values for Q Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

0.372

Column factor

Q0.05 ,3,72

3.39

T

0.372

Interaction

Q0.05 ,9 ,72

4.53

T

0.861

14,00

13,00 Centralised 12,00

Game Theory Negotiation

11,00

10,00 0

W 1

2B

3S

Figure 7.18. Observed values

4

Distributed Models for Plant Capacity Allocation

129

14,00

13,00 Centralised 12,00

Game Theory Negotiation

11,00

10,00 0

1 W

2B

3 S

4

Figure 7.19. Estimated values

Using data reported in Table 7.39, we obtain: Table 7.43. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

2.836*

0

0

N.A.

1.944*

0.892*

0

Table 7.44. Differences of means for column factors Column factor

W

B

S

B

0.374*

0

0

S

0.024

0.340

0

Table 7.45 refers to the interaction. In this case, the conclusions we can draw are: ƒ there is a statistical difference between the performance estimated using the centralised model and the two decentralised ones. There is a difference also considering the G.A. and the N.A.; ƒ the variability of the input data (Q) significantly influences the response of the systems, especially passing from W to B; ƒ the input data variability and the different models interact significantly and, therefore, influence the value of the estimated performance. Specifically, the centralised approach with a small variance of input data leads us to obtain the minimum value of the performance. The negotiational approach works better with respect to the game-theoretical one: the variability of Q does not significantly influence the results.

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Production Planning in Production Network

Table 7.45. Group’ mean differences for column factors W-C.A.

B-C.A.

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

–

–

–

–

–

–

–

B-C.A.

0.90*

S-C.A.

0.32

1.22*

–

–

–

–

–

–

W-G.A.

2.93*

2.02*

3.24*

–

–

–

–

–

B-G.A.

3.05*

2.15*

3.37*

0.12

–

–

–

–

S-G.A.

3.12*

2.22*

3.44*

0.19

0.07

–

–

–

W-N.A.

2.05*

1.15*

2.37*

0.87*

0.99*

1.07*

–

–

B-N.A.

2.12*

1.22*

2.44*

0.81

0.93*

1.00*

0.07

–

S-N.A.

2.25*

1.35*

2.57*

0.68

0.80

0.87*

0.19

0.13

7.7.4 Distance Performance Analysis: DoE

In the following analysis, DoE is used to investigate in which way the input variables influence the distance. 7.7.4.1 Centralised Model Figure 7.20 reports the main effects impact on the efficiency, and Figure 7.21 the interaction effect.

C

R

11,5 11,0

Mean of Distances

10,5 10,0 9,5 W

B Q

S

W

B

S

W

11,5 11,0 10,5 10,0 9,5

Figure 7.20. Main effects plot (C.A.)

B

S

Distributed Models for Plant Capacity Allocation

W

B

S

W

B

131

S 12

C W B S

11

10 12

R

R W B S

11

10

Q

Figure 7.21. Interaction plot (C.A.)

As the reader can notice, distance rapidly diminishes when the production capacity of plants (Q) have a small variance. The other two parameters, and their interaction, have less influence on the estimated performance. In Figure 7.22 the surfaces response of the model are depicted. 7.7.4.2 Negotiation Model Adopting the same approach, it is possible to analyse the negotiational model. The main effects are reported in Figure 7.23, interaction effects in Figure 7.24. A wide variance of the three input variables can cause a significant reduction of the distance performance. Figure 7.25 shows the surface curves in this case.

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Production Planning in Production Network

S R

B

Figure 7.22. Surface plot for distance (C.A.)

W

S

Distributed Models for Plant Capacity Allocation

C

R

13,2 12,9

Mean of Distances

12,6 12,3 W

B Q

S

W

B

S

W

B

S

13,2 12,9 12,6 12,3

Figure 7.23. Main effects plot (N.A.) W

C

B

S

W

B

C W B S

S 13,5

13,0

12,5

R

13,5

R W B S

13,0

12,5

Q

Figure 7.24. Interaction plot (N.A.)

133

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Production Planning in Production Network

R

B

W

S

Figure 7.25. Surface plot for efficiency (N.A.)

7.7.4.3 Game Theory In this last case we obtain the following effects, Figure 7.26, and interactions, Figure 7.27. Groups’ ownership and plants’ production capacity play, within this model, a very significant role: by diminishing the variance of this data input, the estimated parameters increase. Also their interaction leads to a very strong influence on the output. Finally, the surface plots are reported in Figure 7.28.

Distributed Models for Plant Capacity Allocation

C

135

R

13,60 13,55 13,50

Mean of Distances

13,45 13,40 W

B Q

S

W

B

S

W

B

S

13,60 13,55 13,50 13,45 13,40

Figure 7.26. Main effects plot (G.A.) W

C

B

S

W

C W B S

B

S

13,6

13,4

13,2

R

R W B S

13,6

13,4

13,2

Figure 7.27. Interaction plot (G.A.)

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Production Planning in Production Network

R

W

S

Figure 7.28. Surface plot for distance (G.A.)

7.7.5 Number of Reconfigurations Performance Analysis: Two-way ANOVA

7.7.5.1 Ownership Table 7.46 reports, in the rows, the levels of the input factors and in the columns the values of the nine different combinations of the other two input parameters (R,Q) as explained in Table 7.3. Next step is to calculate a table with all the mean values for each model and for each level of C : in this way it will be easier to formulate and verify the hypotheses 7.18 –7.20 related to the two-way ANOVA. From the analysis of Table 7.48 results that not only the main effects are significant, but the interaction between them also. That is, all the three null hypothesis 7.18 –7.20 can be rejected. Figures 7.29 and 7.30 report respectively the observed and estimated values, whose numerical values are reported in Table 7.49, from which it is possible to observe the absence of interaction. The estimated values are reported in Table 7.49.

Distributed Models for Plant Capacity Allocation Table 7.46. Replication values of 1

2

3

4

5

6

7

8

9

W

8.63

8.83

6.68

8.58

8.77

6.76

8.61

8.74

8.51

B

8.44

8.57

5.70

8.48

8.62

5.88

8.51

8.65

5.67

S

8.36

8.72

4.53

8.48

8.69

4.54

8.35

8.37

4.59

W

7.31

7.30

7.06

7.30

7.17

7.05

7.31

7.11

7.35

B

6.09

5.47

5.20

6.02

5.51

5.41

6.08

5.39

5.21

S

4.37

3.03

2.44

4.37

3.01

2.72

4.38

4.41

2.49

W

8.39

8.52

8.62

7.29

7.56

7.37

7.51

7.29

6.04

B

7.22

7.35

7.16

6.00

5.97

6.09

4.60

4.35

4.16

S

6.25

6.12

6.18

4.83

4.56

4.71

3.19

3.15

2.30

C.A.

G.A.

N.A.

Table 7.47. Mean values of -

C.A. G.A. N.A.

Mean

W

B

S

Mean

8.23

7.61

7.18

7.68

7.22

5.60

3.47

5.43

7.62

5.88

4.59

6.03

7.69

6.36

5.08

6.38

Table 7.48. Two-way ANOVA table for -

SS

df

MS

F-test

F-crit D

5%

First factor

73.123

2

36.561

27.633*

F2.72

3.13

Second factor

92.136

2

46.068

34.817*

F2.72

3.13

Interaction effect

18.285

4

4.571

3.455*

F4.72

2.51

Within

95.265

72

1.323

-

Total

278.809

80

-

-

Table 7.49. Estimated values for -

C.A. G.A. N.A.

Mean

W

B

S

Mean

8.99

7.66

6.38

7.68

6.74

5.41

4.13

5.43

7.34

6.01

4.73

6.03

7.69

6.36

5.08

6.38

137

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Production Planning in Production Network

Referring to the analysed case we can calculate the T-critical values reported in Table 7.50. Table 7.50. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

0.750

Column factor

Q0.05 ,3,72

3.39

T

0.750

Interaction

Q0.05 ,9 ,72

4.53

T

1.374

Using data reported in Table 7.47, we obtain what is reported in Table 7.51 and 7.52. Table 7.51. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

2.247*

0

0

N.A.

1.648*

0.599

0

9 8 7 Centralised Game Theory

6

Negotiation

5 4 3 0

1 W

2B

3S

Figure 7.29. Observed values

4

Distributed Models for Plant Capacity Allocation

139

10 9 8 Centralised

7

Game Theory 6

Negotiation

5 4 3 0

1 W

2 B

3S

4

Figure 7.30. Estimated values Table 7.51. Differences of means for column factors Column factor

W

B

S

B

1.329*

0

0

S

2.612*

1.284*

0

Last table is referred to the groups’ interaction analysis. Table 7.53. Groups’ means differences for column factors W-C.A.

B-C.A.

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

B-C.A.

0.621

-

-

-

-

-

-

-

S-C.A.

1.053

0.432

-

-

-

-

-

-

W-C.A.

1.016

0.395

0.038

-

-

-

-

-

B-G.A.

2.637*

2.016*

1.583

1.621

-

-

-

-

S-G.A.

4.764*

4.143*

3.710*

3.748*

2.127*

-

-

-

W-N.A.

0.613

0.008

0.441

0.403

2.024*

4.151*

-

-

B-N.A.

2.357*

1.736

1.304

1.342

0.279

2.407*

1.744*

-

S-N.A.

3.648*

3.027*

2.595*

2.633*

1.011

1.116

3.035*

1.291

Given these results with a confidence interval equal to 5%, we can draw the following conclusions: ƒ there is a statistical difference between the performance estimated using C.A. and the two decentralized ones but not between G.A. and N.A.;

140

Production Planning in Production Network

ƒ

the variability of the input data (C) significantly influences the response of the system; ƒ the input data variability and the different models used interact significantly in the performance evaluation. These considerations lead us to a very important result: there is, statistically speaking, no difference between the two proposed decentralised models. The better one proves to be the G.A., and in a particular case, i.e. for small variability of input data, this approach leads us to obtain the minimum value of the number of reconfigurations. This is a really good result considering the distributed nature of this approach: it can be utilised to reduce to the minimum value the global number of reconfigurations. 7.7.5.2 Risk Attitude Table 7.54 reports the values of the performance depending on the two factor levels in the rows, i.e. model typology and risk attitude variance wideness, and on the combinations of the other two input parameters (C,Q) on the columns. The association of the combination of C and Q to the number reported in the columns of Table 7.54 is reported in Table 7.12. Table 7.55 reports the mean values for the performance - . As the reader can notice, in this case, - seems to be higher with the centralised model. This conclusion will be deeply investigated by performing a two-way ANOVA. Table 7.54. Replication values for -

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

9

W

8.63

8.83

6.68

8.44

8.57

5.70

8.36

8.72

4.53

B

8.58

8.77

6.76

8.48

8.62

5.88

8.48

8.69

4.54

S

8.61

8.74

8.51

8.51

8.65

5.67

8.35

8.37

4.59

W

7.31

7.30

7.06

6.09

5.47

5.20

4.37

3.03

2.44

B

7.30

7.17

7.05

6.02

5.51

5.41

4.37

3.01

2.72

S

7.31

7.11

7.35

6.08

5.39

5.21

4.38

4.41

2.49

W

8.39

8.52

8.62

7.22

7.35

7.16

6.25

6.12

6.18

B

7.29

7.56

7.37

6.00

5.97

6.09

4.83

4.56

4.71

S

7.51

7.29

6.04

4.60

4.35

4.16

3.19

3.15

2.30

Table 7.55. Average values of -

C.A. G.A. N.A.

Mean

W

B

S

Mean

7.61

7.65

7.78

7.68

5.36

5.40

5.53

5.43

7.31

6.04

4.73

6.03

6.76

6.36

6.01

6.38

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141

The null and alternative hypotheses to consider for each set are given, also in this case, by 7.18 –7.20. Table 7.56. Two-way ANOVA table for R SS

df

MS

F-test

F-crit D

5%

First factor

73.123

2

36.561

15.009*

F2.72

3.13

Second factor

7.626

2

3.813

1.565

F2.72

3.13

Interaction effect

22.668

4

5.667

2.586*

F4.72

2.51

Within

175.391

72

2.436

–

Total

278.809

80

–

–

From the above results it is evident that only one factor is significant for the estimated parameter, but also the interaction plays a very important role. In order to strongly support this conclusion, also in this case, the Tukey test has been carried out. The T-values of the test are reported in Table 7.58, while Table 7.59 reports the differences of the means. The Tukey test confirms the result of the twoway ANOVA. Finally, Table 7.60 reports the interaction analysis. Table 7.57. Estimated values for -

C.A. G.A. N.A.

Mean

W

B

S

Mean

8.06

7.66

7.31

7.68

5.81

5.41

5.06

5.43

6.41

6.01

5.66

6.03

6.76

6.36

6.01

6.38

Figure 7.32, whose numerical values are reported in Table 7.57, shows the absence of interaction given by the plotted parallel lines. Table 7.58. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

1.018

Column factor

Q0.05 ,3,72

3.39

T

1.018

Interaction

Q0.05 ,9 ,72

4.53

T

2.357

Using data reported in Table 7.55, Table 7.59 can be created.

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Production Planning in Production Network

Table 7.59. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

2.247*

0

0

N.A.

1.648*

0.599

0

8

7 Centralised 6

Game Theory Negotiation

5

4 0

1 W

2 B

3S

4

Figure 7.31. Observed values

9 8 Centralised

7

Game Theory 6

Negotiation

5 4 0

1 W

2B

3S

Figure 7.32. Estimated values

Table 7.60 refers to the groups’ interaction analysis.

4

Distributed Models for Plant Capacity Allocation

143

Table 7.60. Groups’ mean differences for column factors W-C.A.

B-C.A.

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

B-C.A.

0.038

-

-

-

-

-

-

-

S-C.A.

0.169

0.130

-

-

-

-

-

-

W-G.A.

2.243

2.281

2.412*

-

-

-

-

-

B-G.A.

2.210

2.248

2.379*

0.033

-

-

-

-

S-G.A.

2.082

2.120

2.250

0.161

0.128

-

-

-

W-N.A.

0.293

0.332

0.462

1.949

1.916

1.788

-

-

B-N.A.

1.567

1.605

1.735

0.676

0.643

0.515

1.273

-

S-N.A.

2.876*

2.915*

3.045*

0.634

0.666

0.795

2.583*

1.310

Finally, given these results with a confidence interval equal to 5%, we can draw the following conclusions: ƒ there is a statistical difference between the performance estimated using the centralised model and the two decentralised ones but not between G.A. and N.A.; ƒ the variability of the input data (R) does not significantly influence the response of the systems; ƒ the input data variability and the different models interact significantly. These considerations lead us to the same conclusions as shown in the previous section: G.A. is the best approach in optimising the global number of reconfiguration. 7.7.5.3 Plant Capacity The last parameter to be investigated is the plant capacity Q . The performance measure - values are reported in Table 7.61, while average values are reported in Table 7.61, which is the basis of the two-way ANOVA. Also for this input variable, the centralised approach seems to perform worst that the decentralised ones. However, also in this case the two null hypotheses are 7.18–7.20. ANOVA analysis results are reported in Table 7.63.

144

Production Planning in Production Network Table 7.61. Replication values of -

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

9

W

8.63

8.58

8.61

8.44

8.48

8.51

8.36

8.48

8.35

B

8.83

8.77

8.74

8.57

8.62

8.65

8.72

8.69

8.37

S

6.68

6.76

8.51

5.70

5.88

5.67

4.53

4.54

4.59

W

7.31

7.30

7.31

6.09

6.02

6.08

4.37

4.37

4.38

B

7.30

7.17

7.11

5.47

5.51

5.39

3.03

3.01

4.41

S

7.06

7.05

7.35

5.20

5.41

5.21

2.44

2.72

2.49

W

8.39

7.29

7.51

7.22

6.00

4.60

6.25

4.83

3.19

B

8.52

7.56

7.29

7.35

5.97

4.35

6.12

4.56

3.15

S

8.62

7.37

6.04

7.16

6.09

4.16

6.18

4.71

2.30

The next step is to calculate a table with all the mean values for each model and for each level of Q . Table 7.62. Average values of -

C.A. G.A. N.A.

Mean

W

B

S

Mean

8.49

8.66

5.87

7.68

5.92

5.38

4.99

5.43

6.14

6.10

5.85

6.03

6.85

6.71

5.57

6.38

Again, the nulls and alternative hypotheses for each set are given by 7.18 –7.20. From the results reported in figure 7.63, we can see that only the main effects are significant, but the interaction between them is not (even if its value is very close to the one of the threshold. This nearness has its effect on the graphs of mean values). That is, only the first two null hypotheses 7.18 and 7.19 are rejected. The estimated values are reported in Table 7.64, while the graphical representations of Tables 7.62 and 7.643 are reported, respectively, in Figures 7.33 and 7.34. Table 7.63. Two-way ANOVA table for SS 73.123

df 2

MS 36.561

F-test 16.731*

F-crit D

First factor

F2.72

3.13

Second factor

26.631

2

13.316

6.093*

F2.72

3.13

Interaction effect

21.714

4

5.429

2.484

F4.72

2.51

Within

157.340

72

2.185

–

Total

278.809

80

–

–

5%

Distributed Models for Plant Capacity Allocation

145

Table 7.64. Estimated values for -

C.A. G.A. N.A.

Mean

W

B

S

Mean

8.15

8.01

6.87

7.68

5.90

5.76

4.62

5.43

6.50

6.36

5.22

6.03

6.85

6.71

5.57

6.38

Also in this case, the Tukey test, whose T-values are reported in Table 7.65, is carried out to better understand where this significance is located. Table 7.65. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

0.964

Column factor

Q0.05 ,3,72

3.39

T

0.964

Interaction

Q0.05 ,9 ,72

4.53

T

2.232

Mean differences are reported in Tables 7.66 and 7.67 and the reader can notice how the test allows us to conclude that the influence on means is due to either the column or the row factors; this brings us to reject main effects null hypotheses. Table 7.66. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

2.247*

0

0

N.A.

1.648*

0.599

0

Table 7.67. Differences of means for column factors Column factor

W

B

S

B

0.138

0

0

S

1.279*

1.141*

0

146

Production Planning in Production Network

9 8 7

Centralised Game Theory

6

Negotiation

5 4 0

1 W

2 B

3S

4

Figure 7.33. Observed values

9 8 7

Centralised Game Theory

6

Negotiation

5 4 0

1 W

2 B

3 S

Figure 7.34. Estimated values

4

Distributed Models for Plant Capacity Allocation

147

Table 7.68 refers to group’s interaction analysis. Table 7.68. Groups’ mean’ differences for column factors W-C.A.

B-C.A.

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

B-C.A.

0.168

-

-

-

-

-

-

-

S-C.A.

2.621*

2.789*

-

-

-

-

-

-

W-G.A.

2.579*

2.747*

0.042

-

-

-

-

-

B-G.A.

3.115*

3.283*

0.494

0.536

-

-

-

-

S-G.A.

3.502*

3.670*

0.881

0.923

0.387

-

-

-

W-N.A.

2.352*

2.520*

0.269

0.227

0.763

1.150

-

-

B-N.A.

2.398*

2.566*

0.223

0.180

0.716

1.103

0.046

-

S-N.A.

2.647*

2.815*

0.026

0.068

0.468

0.855

0.295

0.248

Also in this last case, the conclusions we can highlight are the same as the previous two sections. The only difference consists in the influence that the variability of input data plays in the estimation of the performance. Passing from W to S and from B to S there is a relevant reduction of the number of reconfigurations. 7.7.6 Number of Reconfigurations Performance Analysis: DoE

In the following analysis, DoE is used to investigate in which way the input variables influence the number of reconfigurations. 7.7.6.1 Centralised Model In the centralised model, regarding the main effects on the estimated performance, we have the main effect impact reported in Figure 7.35, while in Figure 7.36 the interaction effects are shown.

148

Production Planning in Production Network

C

9

R

Mean of Reconfigurations

8 7 6 W

B

S

W

B

S

Q

9 8 7 6 W

B

S

Figure 7.35. Main effects plot (C.A.) W

C

B

S

W

B

C W B S

S 8

6

4

R

R W B S

8

6

4

Q

Figure 7.36. Interaction plot (C.A.)

The number of reconfigurations rapidly diminishes when the production capacity of plants (Q) and the ownership of groups have a small variance. The risk attitude of plants influences less the estimated parameter. Given this result, it is interesting

Distributed Models for Plant Capacity Allocation

149

to show some graphs representing the surface response of the model reported in Figure 7.37.

B

R

B

W

S

Figure 7.37. Surface plot for reconfiguration (C.A.)

7.7.6.2 Negotiation Model Adopting the same approach, it is possible to analyse the negotiational model. Main effects are reported in Figure 7.38, and interaction effects in Figure 7.39. A wide variance of the C and R input variables can cause a significant increase in the performance. Figure 7.40 shows the surface curves in this case.

150

Production Planning in Production Network

C

8

R

Mean of Reconfiguration

7 6 5 W

B

S

W

B

S

Q

8 7 6 5 W

B

S

Figure 7.38. Main effects plot (N.A.)

W

C

B

S

W

B

C W B S

S 8 6 4

R

R W B S

8 6 4

Q

Figure 7.39. Interaction plot (N.A.)

Distributed Models for Plant Capacity Allocation

151

W S R

B

W

S

Figure 7.40. Surface plot for reconfiguration (N.A.)

7.7.6.3 Game Theory In this last case we obtain the effects shown in Figure 7.41 and interactions shown in Figure 7.42. Groups’ ownership and plants’ production capacity play, within this model, a very significant role: by increasing the variance of this data input, the estimated parameters increase in a quasi linear way. Also their interaction leads to a very strong influence on the output. Finally, the surface plots are reported in Figure 7.43.

152

Production Planning in Production Network

C

R

7 6

Mean of Reconfiguration

5 4 3 W

B Q

S

W

B

S

W

B

S

7 6 5 4 3

Figure 7.41. Main effects plot (G.A.)

W

C

B

S

W

C W B S

B

S

6

4

2

R

R W B S

6

4

2

Figure 7.42. Interaction plot (G.A.)

Distributed Models for Plant Capacity Allocation

R

B

W

153

S

Figure 7.43. Surface plot for reconfiguration (G.A.)

7.7.7 Absolute Residual Performance Analysis: Two-way ANOVA

7.7.7.1 Ownership To start with the analysis, Table 7.69 has been taken into consideration. In the rows we have the levels of both considered factors and, in the columns, the values of the nine different replications obtained combining the other two input parameters (R,Q) in a combinatorial way. The next step is to calculate a table with all the mean values for each model and for each level of C : in this way it will be easier to formulate and verify the hypothesis related to the two-way ANOVA.

154

Production Planning in Production Network Table 7.69. Replication values of Z

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

9

W

52.84

46.06

54.88

53.34

45.95

56.81

53.36

46.86

54.36

B

43.52

33.20

44.26

42.49

32.36

43.91

43.39

32.65

44.52

S

31.64

19.68

35.95

31.06

19.80

36.80

32.12

32.09

35.35

W

86.18

93.29

86.18

116.24

92.72

86.32

114.49

93.14

115.08

B

97.42

71.64

64.13

96.17

71.42

64.39

96.28

71.70

65.64

S

86.26

57.34

51.57

84.64

57.83

52.35

84.89

85.35

51.77

W

99.45

78.21

72.40

99.62

79.56

73.29

100.04

79.07

100.97

B

90.71

66.76

60.80

90.96

67.99

60.64

91.44

69.22

62.96

S

80.35

54.68

48.36

80.85

54.54

48.01

80.98

79.68

50.35

Table 7.70. Average values of Z W

B

S

Mean

C.A.

51,61

40,03

30,50

40,71

G.A.

98,18

77,64

68,00

81,28

N.A.

86,96

73,50

64,20

74,89

78,92

63,72

54,23

65,62

Mean

The centralised model seems to perform better in comparison to the two decentralised ones. The nulls and alternative hypotheses are given by 7.18–7.20. The ANOVA table we obtain is reported in Table 7.71. Table 7.71. Two-way ANOVA table for Z df 2

MS 12842.917

F-test 86.727*

F-crit D

First factor

SS 25685.835

F2.72

3.13

Second factor

8370.568

2

4185.284

28.263*

F2.72

3.13

274.463

4

68.616

0.463

F4.72

2.51

Within

10662.069

72

148.084

–

Total

44992.935

80

Interaction Effect

5%

–

–

From the results reported in Table 7.71, we can see that only both main effects are significant, their interaction is not. That is, only first two null hypothesis 7.18 and 7.19 are rejected. The estimated values are reported in the following Table 7.72.

Distributed Models for Plant Capacity Allocation

155

Table 7.72. Estimated values for Z C.A. G.A. N.A.

Mean

W

B

S

Mean

54.00

38.81

29.32

40.71

94.57

79.38

69.88

81.28

88.18

72.99

63.49

74.89

78.92

63.72

54.23

65.62

By plotting these values, see Figures 7.44 and 7.45, it is easy to observe the absence of interaction as highlighted above. Referring to the analyzed case we can calculate the T-critical values reported in Table 7.73. Table 7.73. T-Values Factor

Q-value

T-value

Row factor

Q0.05 ,3,72

3.39

T

7.939

Column factor

Q0.05 ,3,72

3.39

T

7.939

Interaction

Q0.05 ,9 , 72

4.53

T

18.375

105 95 85 75

Centralised

65

Game Theory

55

Negotiation

45 35 25 0

1 W

2B

3S

4

Figure 7.44. Observed values

156

Production Planning in Production Network

105 95 85 75

Centralised

65

Game Theory

55

Negotiation

45 35 25 0

W 1

2B

3S

4

Figure 7.45. Estimated values

Using data reported in Table 7.70, we obtain: Table 7.74. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

40.56*

0

0

N.A.

34.17*

6.39

0

Table 7.75. Differences of means for column factors Column factor

W

B

S

B

15.19*

0

0

S

24.68*

9.49*

0

Table 7.76 reports group’s interaction analysis. Finally, given these results with a confidence interval equal to 5%, we can draw the following conclusions. ƒ there is a statistical difference between the performance estimated using the centralised model and the two decentralised ones. Between the last two, the existing difference is not relevant; ƒ the variability of the input data (C) significantly influence the response of the systems; ƒ the input data variability and the different used models do not interact significantly. The centralised model proves to be the best one in the optimization of this performance. N.A. acts, statistically speaking, in the same way as G.A.. For all

Distributed Models for Plant Capacity Allocation

157

three approaches, the reduction of variability of C leads to the best values of absolute residual. Table 7.76. Groups’ mean differences for column factors W-C.A.

B-C.A. –

S-C.A.

W-G.A.

B-G.A.

–

–

–

–

–

S-G.A.

W-N.A.

B-N.A.

–

–

–

–

–

–

–

B-C.A.

11 .574

S-C.A.

21.106*

9.532

W-G.A.

46.577*

58.151*

67.683*

–

–

–

–

–

B-G.A.

26.037*

37.611*

47.144*

20.540*

–

–

–

–

S-G.A.

16.395

27.969*

37.501*

30.182*

9.643

–

–

–

W-N.A.

35.352*

46.926*

56.458*

11.225

9.314

18.957*

–

–

B-N.A.

21.892*

33.466*

42.998*

24.685*

4.145

5.497

13.460

–

S-N.A.

12.594

24.168*

33.700*

33.983*

13.444

3.801

22.758*

9.298

7.7.7.2 Risk Attitude We start filling in Table 7.77. The values on the columns are the nine different replications obtained combining the other two input parameters (C, Q) in a combinatorial way. Table 7.77. Replication values for Z

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

9

W

52.84

46.06

54.88

43.52

33.20

44.26

31.64

19.68

35.95

B

53.34

45.95

56.81

42.49

32.36

43.91

31.06

19.80

36.80

S

53.36

46.86

54.36

43.39

32.65

44.52

32.12

32.09

35.35

W

86.18

93.29

86.18

97.42

71.64

64.13

86.26

57.34

51.57

B

116.24

92.72

86.32

96.17

71.42

64.39

84.64

57.83

52.35

S

114.49

93.14

115.08

96.28

71.70

65.64

84.89

85.35

51.77

W

72.40

90.71

66.76

60.80

80.35

54.68

48.36

99.45

78.21

B

99.62

79.56

73.29

90.96

67.99

60.64

80.85

54.54

48.01

S

100.04

79.07

100.97

91.44

69.22

62.96

80.98

79.68

50.35

The next step is to create Table 7.78 with all the mean values for each model and for each level of R : in this way it will be easier to formulate and verify the hypothesis related to the two-way ANOVA whose results are reported in Table 7.79.

158

Production Planning in Production Network Table 7.78. Average values of Z

C.A. G.A. N.A.

Mean

W

B

S

Mean

40.23

40.28

41.63

40.71

77.11

80.23

86.48

81.28

72.41

72.83

79.41

74.89

63.25

64.45

69.18

65.62

There are three sets of hypothesis to consider, like above showed for Ownership. The null and alternative hypothesis for each set are given, also in this case, by Equations 7.18–7.20. Table 7.79. Two-way ANOVA table for R

First factor Second factor Interaction effect

5%

SS

df

MS

F-test

F-crit D

25685.803

2

12842.902

49.692*

F2.72

3.13

530.221

2

265.116

1.027

F2.72

3.13

F4.72

2.51

168.583

4

42.145

0.163

Within

18608,295

72

258,448

–

Total

44992,934

80

–

–

From the above results, we can see that only the first factor is significant with its main effect. Given this result we can plot the (means of) observed values and their estimated ones (see Figures 7.46 and 7.47). Table 7.80. Estimated values for Z W

B

S

Mean

C.A.

38.34

39.54

44.26

40.71

G.A.

78.90

80.10

84.83

81.28

N.A.

72.51

73.71

78.44

74.89

Mean

63.25

64.45

69.18

65.62

Distributed Models for Plant Capacity Allocation

159

90 80 70

Centralised

60

Game Theory Negotiaton

50 40 30 0

W 1

B 2

S3

4

Figure 7.46. Observed values

90 80 70

Centralised

60

Game Theory Negotiation

50 40 30 0

W 1

2B

3S

4

Figure 7.47. Estimated values

Referring to the analysed case we can calculate the T-critical values reported in Table 7.81. Table 7.81. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.72

3.39

T

10.488

Interaction

Q0.05 ,9.72

4.53

T

24.275

Using data reported in Table 7.78, we obtain data of Table 7.88.

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Production Planning in Production Network

Table 7.82. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

40.56*

0

0

N.A.

34.17*

6.39

0

Table 7.83 reports groups’ interaction analysis. Table 7.83. Groups’ mean differences for column factors W-C.A.

B-C.A.

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

B-C.A.

0.055

S-C.A.

1.408

1.353

W-G.A.

36.887*

36.833*

35.479*

B-G.A.

40.006*

39.951*

38.598*

3.119

S-G.A.

46.259*

46.204*

44.851*

9.372

6.253

-

-

-

W-N.A.

32.188*

32.134*

30.781*

4.699

7.818

14.070

-

-

B-N.A.

32.605*

32.551*

31.198*

4.282

7.401

13.653

0.417

-

S-N.A.

39.187*

39.132*

37.779*

2.300

0.819

7.072

6.999

6.581

Given these results with a confidence interval equal to 5%, we can conclude that the only relevant difference is between the performance estimated using the centralised model and the two decentralised ones. G.A. and N.A. are statistically equivalent. 7.7.7.3 Plant Capacity In the rows of Table 7.84 we have the levels of both considered factors and, in the columns, the values of the nine different replications obtained combining the other two input parameters (C , R ) in a combinatorial way. The next step is to create Table 7.85 with all the mean values for each model and for each level of Q : in this way it will be easier to formulate and verify the hypotheses related to the two-way ANOVA whose results are reported in Table 7.86.

Distributed Models for Plant Capacity Allocation

161

Table 7.84. Replication values of Z

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

9

W

52.84

53.34

53.36

43.52

42.49

43.39

31.64

31.06

32.12

B

46.06

45.95

46.86

33.20

32.36

32.65

19.68

19.80

32.09

S

54.88

56.81

54.36

44.26

43.91

44.52

35.95

36.80

35.35

W

86.18

116.24

114.49

97.42

96.17

96.28

86.26

84.64

84.89

B

93.29

92.72

93.14

71.64

71.42

71.70

57.34

57.83

85.35

S

86.18

86.32

115.08

64.13

64.39

65.64

51.57

52.35

51.77

W

99.45

99.62

100.04

90.71

90.96

91.44

80.35

80.85

80.98

B

78.21

79.56

79.07

66.76

67.99

69.22

54.68

54.54

79.68

S

72.40

73.29

100.97

60.80

60.64

62.96

48.36

48.01

50.35

Table 7.85. Average values of Z

C.A. G.A. N.A.

Mean

W

B

S

Mean

42.64

34.29

45.20

40.71

95.84

77.16

70.83

81.28

90.49

69.97

64.20

74.89

76.32

60.47

60.08

65.62

There are three sets of hypotheses to consider. The null and alternative hypotheses for each set are given by Equations 7.18–7.20. The ANOVA table we obtain is: Table 7.86. Two-way ANOVA table for Z

5%

df 2

MS 12842.92

F-test

F-crit D

First factor

SS 25685.83

75.55*

F2.72

3.13

Second factor

4638.21

2

2319.11

13.64*

F2.72

3.13

Interaction effect

2429.22

4

607.30

3.57*

F4.72

2.51

Within

12239.67

72

170.00

-

Total

44992.93

80

-

-

From the above results, we can see that interactions between factors and their main effects are relevant. That means that all three null hypothesis 7.18–7.20 are rejected. In Table 7.87 the estimated values are reported.

162

Production Planning in Production Network Table 7.87. Estimated values for Z

C.A. G.A. N.A.

Mean

W

B

S

Mean

51.41

35.56

35.16

40.71

91.97

76.13

75.73

81.28

85.58

69.73

69.34

74.89

76.32

60.47

60.08

65.62

Referring to the analysed case, the Tukey test results are reported in Table 7.87. Table 7.88. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.72

3.39

T

7.939

Column factor

Q0.05 ,3.72

3.39

T

7.939

Interaction

Q0.05 ,9.72

4.53

T

18.375

Using data reported in Table 7.85, Table 7.89 and 7.90 can be built. Table 7.89. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

40.56*

0

0

N.A.

34.17*

6.39

0

100 90 80 Centralised Game Theory

70 60

Negotiation

50 40 30 0

W 1

B 2

3S

Figure 7.48. Observed values

4

Distributed Models for Plant Capacity Allocation

163

100 90 80 70

Centralised

60

Game Theory Negotiation

50 40 30 W 1

0

2B

3S

4

Figure 7.49. Estimated values Table 7.90. Differences of means for column factors Column factor

B S

W

B

S

15.85*

0

0

16.25*

0.40

0

Table 7.91 reports groups’ interaction analysis. Table 7.90. Groups’ mean differences for column factors W-C.A.

B-C.A.

S-C.A.

W-G.A.

B-G.A.

S-G.A.

W-N.A.

B-N.A.

B-C.A.

8.345

-

-

-

-

-

-

-

S-C.A.

2.564

10.910

-

-

-

-

-

-

W-G.A.

53.202*

61.547*

50.638*

-

-

-

-

-

B-G.A.

34.520*

42.865*

31.956*

18.682

-

-

-

-

S-G.A.

28.186*

36.532*

25.622*

25.016*

6.333

-

-

-

W-N.A.

47.850*

56.195*

45.286*

5.352

13.330

19.663

-

-

B-N.A.

27.328*

35.673*

24.764*

25.874*

7.192

0.858

20.522*

-

S-N.A.

21.559*

29.904*

18.995

31.643*

12.961

6.628

26.291*

5.769

164

Production Planning in Production Network

Finally, given these results with a confidence interval equal to 5%, we can draw the following conclusions: ƒ the only relevant difference is between the performance estimated using the centralised model and the two decentralised ones. G.A. and N.A. are statistically equivalent; ƒ the variability of the input data (Q) significantly influences the response of the systems, especially passing from W to B and from W to S; ƒ the input data variability and the different used models interact significantly in the estimation of the performance. 7.7.8 Absolute Residual Performance Analysis: DoE

In this analysis, DoE is used to investigate in which way the input variables influence the absolute residual. 7.7.8.1 Centralised Model In the centralised model, regarding the main effects on the estimated performance, simulation results are reported in Figures 7.50 and 7.51.

C

R

50

Mean of Absolute Residual

45 40 35 30 W

B Q

S

W

B

S

W

50 45 40 35 30

Figure 7.50. Main effects plot (C.A.)

B

S

Distributed Models for Plant Capacity Allocation

W

C

B

S

W

B

C W B S

165

S

50

35

20

R

R W B S

50

35

20

Q

Figure 7.51. Interaction plot (C.A.)

The absolute residual rapidly diminishes when the capacity ownership of groups has a small variance. The dependence by the risk attitude of plants is bounded while, regarding the capacity of plants, the minimum value of the estimated parameter is reached when the variance is at level B. This trend is also observable in the interaction plot. The surface responses of the model are reported in Figure 7.52.

166

Production Planning in Production Network

W S R

B

W

S

Figure 7.52. Surface plot for absolute residual (C.A.)

7.7.8.2 Negotiation Model Adopting the same methodology, it is possible to analyse the negotiational model (Figures 7.53 and 7.54).

Distributed Models for Plant Capacity Allocation

C

167

R

90

Mean of Absolute Residual

80

70

W

B Q

S

W

B

S

W

B

S

90

80

70

Figure 7.53. Main effects plot (N.A.) W

C

B

S

W

B

S 100

C W B S

75

R

50 100

R W B S

75

50

Q

Figure 7.54. Interaction plot (N.A.)

Wide variance of ownership and capacity of plants leads us to obtain the higher values of the absolute residual. The diminishing wideness of this data input means

168

Production Planning in Production Network

that in an approximately linear way, the value of the estimated performance diminishes. Surface plots are reported in Figure 7.55.

W

R

Figure 7.55. Surface plot for absolute residual (N.A.)

W

S

Distributed Models for Plant Capacity Allocation

169

7.7.8.3 Game Theory In this last case we obtain the effects and interactions reported in Figures 7.56 and 7.57. C

100

R

Mean of Absolute Residual

90 80 70 W

B

S

W

B

S

Q

100 90 80 70 W

B

S

Figure 7.56. Main effects plot (G.A.) W

C

B

S

W

B

C W B S

S 100

75

50

R

R W B S

100

75

50

Q

Figure 7.57. Interaction plot (G.A.)

170

Production Planning in Production Network

Wide variance of ownership and capacity of plants leads us to obtain higher values of the absolute residual. Diminishing wideness of this data input means that, in an approximately linear way, the value of the estimated performance diminishes. Surface plots are reported in Figure 7.58.

R

W

Figure 7.58. Surface plot for absolute aesidual (G.A.)

S

8 Distributed Production Planning Models: an Integrated Approach

8.1 Introduction In Chapter 5 a hierarchical model for planning production through a distributed approach in a complex reconfigurable production network has been presented. That approach is articulated in five hierarchical steps of planning, each one referring to a specific time horizon and a specific planning objective. It is also quite evident how each lower level receives input from the upper level and therefore how a functional integration is needed between each couple of models. The aim of this chapter is to show how this integration is actually possible between, at least, two levels of the hierarchical structure presented in Chapter 5. This will be obtained by carrying out an integrated simulation of two models. Therefore, the final aim of this chapter is to show how by integrating distributed production planning approaches is possible to obtain a complete tool for planning production at different levels of a corporation. The IDEF0 diagram of Figure 8.1 shows the functional integration of the two production planning models at high and medium levels. As the reader can notice the interdependence between the two models is represented by the “balanced ownership”, that is, the output of activity A1. Indeed, as the reader will remember, the aim of A1 is to balance the ownership, initially located at the Top level, to the groups. As described in Chapter 6 this activity is obtained by bargaining ownership between overloaded and underloaded groups. Once the balanced ownership is defined, this becomes the input of the planning at medium level, whose objective is to allocate plants to groups as described in Chapter 7.

172

Production Planning in Production Networks

Global Ownership

Re-modulated Ownership

Groups Ownership 3 Months groups demand forecast

High PP Level A1

Tools to re-modulate ownership

Medium PP Level A2

Plants characteristics

Plants/Groups allocation matrix

Tools to allocate plants to groups

Figure 8.1. The integration of the high PP and medium PP levels

As already mentioned, both the high level and medium level focus on the same time horizon. In this chapter a simulation for the two distributed models, the gametheoretical and the negotiational one, and for the benchmark, the centralised one, is carried out. The simulation has been carried out through the agent-based simulation environment described in the previous chapters. The case study consists of six groups and 16 plants whose characteristics have been reported in Appendix C.

8.2 The Simulation Case Study The test environment consists of 16 different combination of four input parameters, i.e.: groups’ workloads ( WLg ) and ownerships ( C g ), plants’ production capacities ( Qm ) and the risk attitude ( Rm ). Specifically, each input parameter is considered randomly drawn from a normal distribution N(µ;ı) whose parameters are reported in Table 7.1. Again, like in the other chapters, small, S, and wide, W, refer to the standard deviation size of the normal distribution. Table 8.1. Distributions of input parameters Small (S)

Wide (W)

N(27; 5.4)

N(27; 21.6 )

Ownership C g

N(27;1.35)

N(27; 16.2)

Capacity  Qm

N(10; 0.5)

N(10; 6)

Risk attitude  Rm

N(0.5; 0.025)

N(0.5; 0.5 )



Workloads WL g





Distributed Production Planning Models: an Integrated Approach

173

Simulation experiments have been conducted with an infinite number of credits. The aim of each replication is to evaluate some performance measures such as: the efficiency (V), the distance (G) and the absolute residual ( Z ), respectively defined in the previous chapter in Section 7.6.2. Each performance measure has been evaluated with a confidence interval equal to 90%.

8.3 Results Appendix C reports the fixed coefficients used for the simulation and the results grouped for the 16 different combinations of the input parameters. To better analyse this results, also in this case, the two-way Analysis of Variance (ANOVA) and the Design of Experiment (DoE) have been used. 8.3.1 Efficiency Performance Analysis: Two-way ANOVA

Two factors have been used, i.e. the “kind of model”, whose levels are C.A. (centralised model), N.A. (negotiation model), G.A. (game-theoretical model) and the “input parameter” whose levels refer the wideness of the variance range, i.e. wide (W) and small (S). The “input parameters” are those reported in Table 8.1. 8.3.1.1 Workload WLg Table 8.2 reports the value of the parameter efficiency, V, obtained when the levels of the input parameters varies, in the rows, and when the other input parameters ( Cg , Qm , Rm ) are combined as in the columns. The reader should refer to Table 8.3 for the association of the combinations to the numbers in the rows of Table 8.2. Table 8.2. Replication values of V 1

2

3

4

5

6

7

8

W

7.928

8.297

7.903

8.332

7.980

8.534

7.454

7.577

S

7.984

8.369

8.026

8.470

7.901

8.546

7.474

7.317

W

8.880

8.940

8.974

8.943

9.434

9.418

9.200

9.200

S

8.717

8.865

8.967

8.859

9.428

9.298

9.200

9.200

W

7.956

8.460

7.928

8.296

7.928

8.437

7.476

7.546

S

7.998

8.248

7.869

8.427

7.950

8.536

7.430

7.318

C.A.

G.A.

N.A.

174

Production Planning in Production Networks Table 8.3. Combination of C g , Qm , Rm 1

2

3

4

5

6

7

8

Cg

W

W

W

W

S

S

S

S

Qm

W

W

S

S

W

W

S

S

Rm

W

S

W

S

W

S

W

S

Table 8.4 reports the mean values of the numbers reported in Table 8.2, while the last column and row of Table 8.2 report the average of the means. Table 8.4. Average values of V W

S

Mean

C.A.

8.00

8.01

8.01

G.A.

9.12

9.07

9.10

N.A.

8.00

7.97

7.99

Mean

8.38

8.35

8.36

Table 8.4 shows that efficiency in the negotiation model is quite similar to the centralised one, while the game-theoretical one seems to perform worse than the other two models. This conclusion will be deeply investigated with the two-way ANOVA. Specifically, there are three sets of hypotheses to be considered. As in the previous chapters, we consider the null  H 0 and the alternative hypothesis

 H1 for each set: for the population means of the factor “kind of model” PC PN PG °­ H 0 : ; ®H : not all Pi^C , N ,G` are equal °¯ 1 for the population means of the factor “workload”: P W P B PS ­° H 0 : ; ®H : not all P j^W , B ,S` are equal ¯° 1 for the interdependencies: there is no interaction ­H0 : ; ® there is interaction ¯ H1 :

(8.1)

(8.2)

(8.3)

Table 8.5 reports the ANOVA results. Only the main effect relative to the first factor is significant: therefore, only the first null hypothesis 8.1 is rejected. This actually means that the model influences the performance parameter efficiency, while the workload does not. This also means that the difference between the

Distributed Production Planning Models: an Integrated Approach

175

performance of the C.A. and N.A. models and the G.A. one is quite significant, i.e. the game theoretical model performs worse than the other two, while the negotiation model performs similarly to the centralised one. Also in this case the Ttest has been carried out; Table 8.6 reports the T-values, and Table 8.7 the differences. Table 8.5. Two-way ANOVA table for V

5%

SS

df

MS

F-test

F-crit D

First factor

12.873

2

6.437

49.352*

F2.42

3.252

Second factor

0.008

1

0.008

0.062

F1.42

4.102

Interaction effect

0.009

2

0.005

0.035

F2.42

3.252

Within

5.478

42

0.130

–

Total

18.368

47

–

–

Table 8.6. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

0.254

Column factor

Q0.05 , 2.42

2.86

T

0.258

Interaction

Q0.05 , 6.42

4.23

T

0.540

Table 8.7. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

1.089*

0

0

N.A.

0.018

1.107*

0

Table 8.8. Groups’ mean differences for column factors W-C.A.

S-C.A.

W-G.A.

S-G.A.

W-N.A.

S–C.A.

0.010

–

–

–

–

W–G.A.

1.123*

1.113*

–

–

–

S–G.A.

1.066*

1.056*

0.057

–

–

W–N.A.

0.003

0.008

1.120*

1.064*

–

S–N.A.

0.029

0.039

1.152*

1.095*

0.031

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Production Planning in Production Networks

From the analysis of the T-test it is possible to conclude that the estimated differences among the models are relevant only between C.A. and G.A. and between G.A. and N.A., but not among the centralised approach and the negotiational one. Finally, Table 8.8 reports the interaction analysis. It confirms that the models, combined with the WL input parameter, influence less the estimated performance. 8.3.1.2 Capacity Ownership  Cg

Similarly to the workload analysis, Table 8.9 reports the efficiency performance, V, results obtained when the input parameter levels vary as shown in the table rows and when the other parameters ( WLg , Qm , R ) vary their combination according to m

Table 8.3 where WLg takes the place of Cg. Table 8.10 reports the mean values of the numbers in Table 8.9. Table 8.9. Replication values of V 1

2

3

4

5

6

7

8

W

7.928

8.297

7.903

8.332

7.984

8.369

8.026

8.470

S

7.980

8.534

7.454

7.577

7.901

8.546

7.474

7.317

W

8.880

8.940

8.974

8.943

8.717

8.865

8.967

8.859

S

9.434

9.418

9.200

9.200

9.428

9.298

9.200

9.200

W

7.956

8.460

7.928

8.296

7.998

8.248

7.869

8.427

S

7.928

8.437

7.476

7.546

7.950

8.536

7.430

7.318

C.A.

G.A.

N.A.

Table 8.10. Average values of V W

S

Mean

C.A.

8.16

7.85

8.01

G.A.

8.89

9.30

9.10

N.A.

8.15

7.83

7.99

Mean

8.40

8.32

8.36

As the reader can notice also in this case the centralised and the negotiational model perform similarly and better than the game-theoretical one. This conclusion will be deeply investigated with the ANOVA, whose results are reported in Table 8.11. Also in this case the first null hypothesis can be rejected, leading to considerations very similar to those of the previous section, i.e. the model typologies influence the efficiency. In this case, however, also the third null hypothesis is rejected meaning that some interactions between the two factors are present. Such interactions can be evaluated observing data from Table 8.12 and Figures 8.2 and 8.3.

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177

Table 8.11. Two-way ANOVA table for V 5%

SS

df

MS

F-test

F-crit D

First factor

12.873

2

6.437

67.017*

F2.42

3.252

Second factor

0.071

1

0.071

0.744

F1.42

4.102

Interaction effect

1.390

2

0.695

7.235*

F2.42

3.252

Within

4.034

42

0.096

–

Total

18.368

47

–

–

Table 8.12. Estimated values for V W

S

Mean

C.A.

8.04

7.97

8.01

G.A.

9.13

9.06

9.10

N.A.

8.03

7.95

7.99

Mean

8.40

8.32

8.36

10,00 9,50 9,00 Centralised Game Theory

8,50

Negotiation 8,00 7,50 7,00 W 1

2S

Figure 8.2. Observed values

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Production Planning in Production Networks

10,00

9,50

9,00 Centralised Game Theory

8,50

Negotiation 8,00

7,50

7,00 W 1

2S

Figure 8.3. Estimated values

Also in this case a T-test analysis has been carried out and results have been reported in Tables 8.13–8.14. The T-test analysis confirms the ANOVA results. Finally, Table 8.15 reports the groups’ interaction analysis whose results support the previous conclusions. Table 8.13. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

0.218

Column factor

Q0.05 , 2.42

2.86

T

0.222

Interaction

Q0.05 , 6.42

4.23

T

0.463

Table 8.14. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

1.089*

0

0

N.A.

0.018

1.108*

0

Distributed Production Planning Models: an Integrated Approach

179

Table 8.15. Groups’ mean differences for column factors

W–C.A.

S–C.A.

W–G.A.

S–G.A.

W–N.A.

S-C.A.

0.316

–

–

–

–

W-G.A.

0.729*

1.045*

–

–

–

S-G.A.

1.134*

1.449*

0.404

–

–

W-N.A.

0.016

0.300

0.746*

1.150*

–

S-N.A.

0.336

0.020

1.065*

1.470*

0.320

8.3.1.3 Plant Production Capacity (Qm) In this section the influence of the input factor Qm is investigated. Of course, in this case the other input combinations reported in Table 8.16 refer to the input parameters WLg , C g , Rm ; combinations are obtained as indicated in Table 8.3. Table 8.16. Replication values of V 1

2

3

4

5

6

7

8

W

7.928

8.297

7.980

8.534

7.984

8.369

7.901

8.546

S

7.903

8.332

7.454

7.577

8.026

8.470

7.474

7.317

W

8.880

8.940

9.434

9.418

8.717

8.865

9.428

9.298

S

8.974

8.943

9.200

9.200

8.967

8.859

9.200

9.200

W

7.956

8.460

7.928

8.437

7.998

8.248

7.950

8.536

S

7.928

8.296

7.476

7.546

7.869

8.427

7.430

7.318

C.A.

G.A.

N.A.

Also in this case the two-way ANOVA is carried out, where the null hypotheses are the same of the previous section. Data for the ANOVA analysis are reported in Tables 8.17–8.18. The two first null hypotheses are rejected; this means that both the input factors, model typology and Qm, impact the efficiency means. Again it is confirmed that the negotiation and centralised approaches perform quite similarly and somewhat better than the game-theoretical. Table 8.17. Average values of V W

S

Mean

C.A.

8.19

7.82

8.01

G.A.

9.12

9.07

9.10

N.A.

8.19

7.79

7.99

Mean

8.50

8.22

8.36

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Production Planning in Production Networks

Table 8.18. Two-way ANOVA table for V

5%

SS

df

MS

F-test

F-crit D

First factor

12.873

2

6.437

63.204*

F2.42

3.252

Second factor

0.920

1

0.920

9.029*

F1.42

4.102

Interaction effect

0.298

2

0.149

1.464

F2.42

3.252

Within

4.277

42

0.102

–

Total

18.368

47

–

–

Table 8.19. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

0.224

Column factor

Q0.05 , 2.42

2.86

T

0.228

Interaction

Q0.05 , 6.42

4.23

T

0.477

Table 8.20. Differences of means for column factors Column factor

W

S

S

0.277*

0

Table 8.21. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

1.089*

0

0

N.A.

0.018

1.108*

0

Table 8.22. Groups’ mean differences for column factors W-C.A.

S-C.A.

W-G.A.

S-G.A.

W-N.A.

S–C.A.

0.373

–

–

–

–

W–G.A.

0.930*

1.303*

–

–

–

S–G.A.

0.876*

1.249*

0.055

–

–

W–N.A.

0.003

0.370

0.933*

0.879*

–

S–N.A.

0.406

0.033

1.336*

1.282*

0.403

Distributed Production Planning Models: an Integrated Approach

181

T-test analysis, whose results are reported in Tables 8.19–8.21, confirms these conclusions. Finally, interactions analysis reported in Table 8.22 confirms that there is not any significant interaction between the two factors. 8.3.1.4 Risk Attitude  Rm

Finally, the influence of the Rm input parameter has been analysed. Of course, in this case, the other input parameters, whose combinations are reported in the columns of Table 8.23, are WLg , Cg , Qm . Also in this case a two-way ANOVA analysis has been carried out whose data and results are reported in Tables 8.23– 8.25. In this case the first and third null hypotheses can be rejected. This means again that model typologies influence efficiency; in particular C.A. and N.A. models perform quite similarly and better than the G.A. one. Furthermore, an interaction between the two input factors is present. T-test analysis (see Tables 8.26–8.27) and interaction analysis, Table 8.28, confirm this conclusion. Referring to efficiency it is possible to conclude that model typology impacts efficiency performance in the integrated uses of high and medium levels models. The negotiation distributed approach (N.A.) performs quite similarly to the centralised one (C.A.), while the game-theoretical shows performances a bit worse than the other two approaches. Table 8.23. Replication values of V 1

2

3

4

5

6

7

8

W

7.928

7.903

7.980

7.454

7.984

8.026

7.901

7.474

S

8.297

8.332

8.534

7.577

8.369

8.470

8.546

7.317

W

8.880

8.974

9.434

9.200

8.717

8.967

9.428

9.200

S

8.940

8.943

9.418

9.200

8.865

8.859

9.298

9.200

W

7.956

7.928

7.928

7.476

7.998

7.869

7.950

7.430

S

8.460

8.296

8.437

7.546

8.248

8.427

8.536

7.318

C.A.

G.A.

N.A.

Table 8.24. Average values of V W

S

Mean

C.A.

7.83

8.18

8.01

G.A.

9.10

9.09

9.10

N.A.

7.82

8.16

7.99

Mean

8.25

8.48

8.36

182

Production Planning in Production Networks

Table 8.25. Two-way ANOVA table for V 5%

SS

df

MS

F-test

F-crit D

First factor

12.873

2

6.437

59.540*

F2.42

3.252

Second factor

0.619

1

0.619

5.721*

F1.42

4.102

Interaction effect

0.336

2

0.168

1.554

F2.42

3.252

Within

4.541

42

0.108

–

Total

18.368

47

–

–

Table 8.26. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

0.235

Column factor

Q0.05 , 2.42

2.86

T

0.235

Interaction

Q0.05 , 6.42

4.23

T

0.492

Table 8.27. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

1.089*

0

0

N.A.

0.018

1.108*

0

Table 8.28. Groups’ mean differences for column factors W-C.A.

S-C.A.

W-G.A.

S-G.A.

W-N.A.

S–C.A.

0.349

–

–

–

–

W–G.A.

1.269*

0.920*

–

–

–

S–G.A.

1.259*

0.910*

0.010

–

–

W–N.A.

0.014

0.364

1.283*

1.274*

–

S–N.A.

0.327

0.022

0.942*

0.932*

0.342

8.3.2 Efficiency Performance Analysis: DoE

DoE is used to investigate in which way the input variables influence the estimated efficiency.

Distributed Production Planning Models: an Integrated Approach

183

8.3.2.1 Centralised Model Figure 8.4 reports the main effects, i.e. how input parameter levels influence the efficiency performance, V. As the reader can notice, the workload variance has no influence on the performance, while capacity ownership and plant production capacity variances negatively influence the efficiency; on the other hand, risk attitude variance positively influences the efficiency. Figure 8.5 shows the input parameter interactions. Only C g vs. Q shows a discordant interaction.

WL

Cg

8,2 8,1

Mean of Efficiency

8,0 7,9 7,8 S

W

S

W

Q

R

8,2 8,1 8,0 7,9 7,8 S

W

S

W

Figure 8.4. Main effects plot (C.A.) S

W

S

W

S

W 8,5

8,0

WL

WL S W

7,5 8,5

8,0

Cg

Cg S W

7,5 8,5

8,0

Q

7,5

R

Figure 8.5. Interaction plot (C.A.)

Q S W

184

Production Planning in Production Networks

To better understand which factor is actually relevant, a normal probability plot and a Pareto chart have been considered. The first one, see Figure 8.6, is a graphical method, based on a subjective visual examination of the data, for acknowledging whether estimated data match the hypothesised (normal) distribution. In the normal probability plot of the effects, points not falling near the line usually signal important effects. To understand not only the importance but also the magnitude of such effects, a Pareto analysis has also been considered (Figure 8.7). The related chart displays the absolute value of the effects by setting a reference line: any effect extending its influence beyond this reference line is potentially important. Both the analyses confirm that significant and important effects are Q, Cg, R and the joint effect of Cg and Q. 99 C

95 90

B

Factor A B C D

80 Percent

Effect Type Not Significant Significant

70 60 50 40 30 20 D

10 5

BC

1 -0,5

-0,4

-0,3

-0,2

-0,1 0,0 Effect

0,1

0,2

0,3

0,4

Figure 8.6. Normal probability plot of the effects (C.A.)

Name WL Cg Q R

Distributed Production Planning Models: an Integrated Approach

185

Term

0,1785 Factor A B C D

BC C D B BCD CD AB BD ACD ABCD ABC ABD AD A AC 0,0

0,1

0,2 Effect

0,3

Name WL Cg Q R

0,4

Figure 8.7. Pareto chart of the effects (C.A.)

The above results are also confirmed by the surface curves of Figure 8.8. Concluding, for the centralised model, the efficiency, V, is negatively influenced by the capacity ownership, Cg, and by the plant capacity, Q, positively by the risk attitude, R, and by the interaction effects of the Cg and Q variances. 8.3.2.2 Negotiation Model By adopting the same methodology, the impact of the input factor variance on the efficiency performance has been evaluated within the negotiation model. By observing Figure 8.9, which reports the main effects, Figure 8.10, which reports the interactions, the normal probability plot (Figure 8.11), the Pareto chart (Figure 8.12) and the interaction response curves of Figure 8.13 it is possible to conclude that the efficiency performance in the negotiation model is negatively influenced by the capacity ownership, Cg, and by the plant capacity variances (the most important parameter according to Figure 8.12), Q, while it is positively influenced by the risk attitude variance, R. Also, significant and important interactions are those among Q, Cg and R and between Q and Cg.

186

Production Planning in Production Networks

R

Figure 8.8. Surface plot for efficiency (C.A.)

W

S

Distributed Production Planning Models: an Integrated Approach

WL

8,2

Cg

8,1

Mean of Efficiency

8,0 7,9 7,8 L

H

L

H

Q

R

8,2 8,1 8,0 7,9 7,8 L

H

L

H

Figure 8.9. Main effects plot (N.A.)

L

H

L

H

L

H 8,5

8,0

WL

WL L H

7,5 8,5

8,0

Cg

Cg L H

7,5 8,5

8,0

Q

7,5

R

Figure 8.10. Interaction plot (N.A.)

Q L H

187

Production Planning in Production Networks

99

90

B

Factor A B C D

BCD

80 Percent

Effect Type Not Significant Significant

C

95

70 60 50 40 30

Name WL Cg Q R

20 10

D

5

BC

1 -0,4

-0,3

-0,2

-0,1

0,0 0,1 Effect

0,2

0,3

0,4

0,5

Figure 8.11. Normal probability plot of the effects (N.A.)

0,1203

Term

188

Factor A B C D

C BC D B BCD CD ABCD ABC BD A ACD AD AC AB ABD 0,0

0,1

0,2 Effect

0,3

Figure 8.12. Pareto chart of the effects (N.A.)

0,4

Name WL Cg Q R

Distributed Production Planning Models: an Integrated Approach

R

W

189

S

Figure 8.13. Surface plot for efficiency (N.A.)

8.3.2.3 Game Theory As the reader can notice by observing Figures 8.14–8.18 a different situation emerges for the game-theoretical model; indeed, only the capacity ownership variance has a significant and also an important (according to the Pareto chart) effect on efficiency performance, V. Furthermore, also the interaction effect of Cg and Q has some significant and important effect on the efficiency.

Production Planning in Production Networks

WL

Cg

9,3 9,2 9,1

Mean of Efficiency

190

9,0 8,9 S

W

S

W

Q

R

9,3 9,2 9,1 9,0 8,9 S

W

S

W

Figure 8.14. Main effects plot (G.A.)

S

W

S

W

S

W 9,4 9,2

WL

WL S W

9,0

9,4 9,2 Cg

Cg S W

9,0

9,4 9,2 Q 9,0

R

Figure 8.15. Interaction plot (G.A.)

Q S W

Distributed Production Planning Models: an Integrated Approach

99

Effect Type Not Significant Significant

95 90

Factor A B C D

80 Percent

191

70 60 50 40 30

Name WL Cg Q R

20 BC

10 5 1

B

-0,4

-0,3

-0,2

-0,1

0,0

0,1

Effect

Figure 8.16. Normal probability plot of the effects (G.A.)

0,0980 Factor A B C D

B BC BCD A C

Name WL Cg Q R

Term

ABCD AC BD CD AB ABD AD D ACD ABC

0,0

0,1

0,2 Effect

0,3

0,4

Figure 8.17. Pareto chart of the effects (G.A.)

This allows us to draw an interesting conclusion about the use of the proposed models in an integrated way at high and medium level planning. Although the game-theoretical approach seems to perform a bit worse of the centralised and the negotiation one, it is the less sensible to the input factors uncertainty.

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Production Planning in Production Networks

Therefore, when the input factors are known in a quite certain way, the negotiation approach should be used to plan in a distributed way production at high and medium level; on the other hand, when input factors uncertainty is high, the game-theoretical approach should be considered.

R

W

S

Figure 8.18. Surface plot for efficiency (G.A.)

8.3.3 Distance Performance Analysis: Two-way ANOVA

8.3.3.1 Workload ( WLg )

Table 8.29 reports the values of the distance performance when input factors “kind of model” and “workload variance” vary according to their levels (table rows) and when the other input parameters, namely Cg , Qm , Rm , are combined according to Table 8.3.

Distributed Production Planning Models: an Integrated Approach

193

Table 8.29. Replication values of G 1

2

3

4

5

6

7

8

W

5.957

6.294

5.937

6.328

6.150

6.540

5.855

6.282

S

6.004

6.387

6.075

6.482

6.107

6.498

5.790

5.936

W

6.707

6.677

6.792

6.729

6.897

6.690

6.600

6.600

S

6.522

6.711

6.791

6.747

6.818

6.590

6.600

6.600

W

6.033

6.357

5.984

6.353

6.090

6.482

5.790

6.257

S

5.957

6.242

5.954

6.492

6.121

6.632

5.831

5.936

C.A.

G.A.

N.A.

Table 8.30 reports the average value of numbers in Table 8.29, while Table 8.31 reports the results of the ANOVA. Tables 8.32 the T-test and Table 8.33 the interaction analysis. From the analysis of the results it is possible to conclude that only the null hypothesis 1 can be rejected: i.e. the factor “kind of model” has a significant impact on the distance performance. Specifically. again the model C.A. and N.A. seem to perform quite similarly. while the model G.A. performs a bit worse than the others. Table 8.30. Average values of G W

S

Mean

C.A.

6.17

6.16

6.16

G.A.

6.71

6.67

6.69

N.A.

6.17

6.15

6.16

Mean

6.35

6.33

6.34

Table 8.31. Two-way ANOVA table for G 5%

SS

df

MS

F-test

F-crit D

First factor

3.013

2

1.507

31.818*

F2.42

3.252

Second factor

0.006

1

0.006

0.137

F1.42

4.102

Interaction Effect

0.002

2

0.001

0.021

F2.42

3.252

Within

1.989

42

0.047

–

Total

5.010

47

–

–

194

Production Planning in Production Networks

8.3.3.2 Capacity Ownership  Cg

Tables 8.34–8.41 show the usual data for the capacity ownership factor. Again only the null hypothesis 8.1 can be rejected, that is, the factor “kind of model” influences the distance performance. Also in this case models C.A. and N.A. perform quite similarly, while the model G.A. performs a bit worse than the others. Table 8.32. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

0.153

Column factor

Q0.05 , 2.42

2.86

T

0.1556

Interaction

Q0.05 , 6.42

4.23

T

0.325

Table 8.33. Groups’ mean differences for column factors W-C.A.

S-C.A.

W-G.A.

S-G.A.

W-N.A.

S-C.A.

0.008

-

-

-

-

W-G.A.

0.544*

0.552*

-

-

-

S-G.A.

0.504*

0.512*

0.039

-

-

W-N.A.

0.000

0.008

0.543*

0.504*

-

S-N.A.

0.022

0.,014

0.566*

0.527*

0.023

Table 8.34. Replications values of G

C.A.

G.A.

N.A.

1

2

3

4

5

6

7

8

W

5.957

6.294

5.937

6.328

6.004

6.387

6.075

6.482

S

6.150

6.540

5.855

6.282

6.107

6.498

5.790

5.936

W

6.707

6.677

6.792

6.729

6.522

6.711

6.791

6.747

S

6.897

6.690

6.600

6.600

6.590

6.600

6.600

6.577

W

6.033

6.357

5.984

6.353

5.957

6.242

5.954

6.492

S

6.090

6.482

5.790

6.257

6.121

6.632

5.831

5.936

Distributed Production Planning Models: an Integrated Approach

Table 8.35. Average values of G W

S

Mean

C.A.

6.18

6.14

6.16

G.A.

6.71

6.64

6.68

N.A.

6.17

6.14

6.16

Mean

6.35

6.31

6.33

Table 8.36. Two-way ANOVA table for G 5%

SS

df

MS

F-test

F-crit D

First factor

2.845

2

1.423

30.408*

F2.42

3.252

Second factor

0.023

1

0.023

0.498

F1.42

4.102

Interaction Effect

0.003

2

0.001

0.030

F2.42

3.252

Within

1.965

42

0.047

–

Total

4.836

47

–

–

Table 8.37. Estimated values for G W

S

Mean

C.A.

6.18

6.14

6.16

G.A.

6.71

6.64

6.68

N.A.

6.17

6.14

6.16

Mean

6.35

6.31

6.33

Table 8.38. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

0.152

Column factor

Q0.05 , 2.42

2.86

T

0.155

Interaction

Q0.05 , 6.42

4.23

T

0.323

195

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Production Planning in Production Networks

Table 8.39. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

0.521*

0

0

N.A.

0.007

0.522*

0

Table 8.40. Differences of means for column factors Column factor

W

S

S

0.004

0

Table 8.41. Groups’ mean differences for column factors W-C.A.

S-C.A.

W-G.A.

S-G.A.

W-N.A.

S–C.A.

0.038

–

–

–

–

W–G.A.

0.527*

0.565*

–

–

–

S–G.A.

5.462*

5.499*

4.935*

–

–

W–N.A.

0.011

0.027

0.538*

5.473*

–

S-N.A.

0.040

0.002

0.567*

5.502*

0.029

8.3.3.3 Plant Production Capacity ( Qm ) Tables 8.42–8.47 show the usual data for the plant production capacity factor. Again only the null hypothesis 8.1 can be rejected, that is, the factor “kind of model” influences the distance performance. Also in this case models C.A. and N.A. perform quite similarly, while the model G.A. performs a bit worse than the others. Table 8.42. Replication values of G 1

2

3

4

5

6

7

8

W

5,957

6,294

6,150

6,540

6,004

6,387

6,107

6,498

S

5,937

6,328

5,855

6,282

6,075

6,482

5,790

5,936

W

6,707

6,677

6,897

6,690

6,522

6,711

6,818

6,590

S

6,792

6,729

6,600

6,600

6,791

6,747

6,600

6,600

W

6,033

6,357

6,090

6,482

5,957

6,242

6,121

6,632

S

5,984

6,353

5,790

6,257

5,954

6,492

5,831

5,936

C.A.

G.A.

N.A.

Distributed Production Planning Models: an Integrated Approach

197

Table 8.43. Average values of G W

S

Mean

C.A.

6,24

6,09

6,16

G.A.

6,70

6,68

6,69

N.A.

6,24

6,07

6,16

Mean

6,39

6,28

6,34

Table 8.44. Two-way ANOVA table for G 5%

SS

df

MS

F-test

F-crit D

First factor

3.013

2

1.507

35.359*

F2.42

3.252

Second factor

0.154

1

0.154

3.619

F1.42

4.102

Interaction effect

0.053

2

0.027

0.627

F2.42

3.252

Within

1.789

42

0.043

–

Total

5.010

47

–

–

Table 8.45. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

0.224

Column factor

Q0.05 , 2.42

2.86

T

0.228

Interaction

Q0.05 , 6.42

4.23

T

0.477

Table 8.46. Differences of means for row factors Row factor

C.A.

G.A.

N.A.

G.A.

0.528*

0

0

N.A.

0.007

0.535*

0

8.3.3.4 Risk Attitude ( Rm ) Finally, tables 8.48–8.55 show data for the risk attitude factor. This time all the three null hypotheses 8.1 can be rejected. This means: a) the factor “kind of model” influences the distance performance; b) the factor “risk attitude variance” influences the distance performance; c) there is an interaction effect of the input factors on performance values.

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Production Planning in Production Networks

Table 8.47. Groups’ mean differences for column factors W-C.A.

S-C.A.

W-G.A.

S-G.A.

W-N.A.

S-C.A.

0.157

–

–

–

–

W-G.A.

0.459*

0.616*

–

–

–

S-G.A.

0.440*

0.597*

0.019

–

–

W-N.A.

0.003

0.154

0.462*

0.443*

–

S-N.A.

0.167

0.011

0.627*

0.608*

0.164

Table 8.48. Replication values of G 1

2

3

4

5

6

7

8

W

7.928

7.903

7.980

7.454

7.984

8.026

7.901

7.474

S

8.297

8.332

8.534

7.577

8.369

8.470

8.546

7.317

W

8.880

8.974

9.434

9.200

8.717

8.967

9.428

9.200

S

8.940

8.943

9.418

9.200

8.865

8.859

9.298

9.200

W

7.956

7.928

7.928

7.476

7.998

7.869

7.950

7.430

S

8.460

8.296

8.437

7.546

8.248

8.427

8.536

7.318

C.A.

G.A.

N.A.

Table 8.49. Average values of G W

S

Mean

C.A.

5.98

6.34

6.16

G.A.

6.72

6.67

6.69

N.A.

5.97

6.34

6.16

Mean

6.22

6.45

6.34

Table 8.50. Two-way ANOVA table for G 5%

SS

df

MS

F-test

F-crit D

First factor

3.013

2

1.507

69.240*

F2.42

3.252

Second factor

0.625

1

0.625

28.732*

F1.42

4.102

Interaction effect

0.458

2

0.229

10.527*

F2.42

3.252

Within

0.914

42

0.022

–

Total

5.010

47

–

–

Distributed Production Planning Models: an Integrated Approach

Table 8.51. Estimated values for G W

S

Mean

C.A.

6.05

6.28

6.16

G.A.

6.58

6.81

6.69

N.A.

6.04

6.27

6.16

Mean

6.22

6.45

6.34

Table 8.52. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

0.104

Column factor

Q0.05 , 2.42

2.86

T

0.105

Interaction

Q0.05 , 6.42

4.23

T

0.221

Table 8.53. Differences of means for row factors

Row factor

C.A.

G.A.

N.A.

G.A.

0.528*

0

0

N.A.

0.007

0.535*

0

Table 8.54. Differences of means for column factors

Column factor

W

S

S

0.228249*

0

6,80 6,70 6,60 6,50 6,40

Centralised

6,30

Game Theory

6,20

Negotiation

6,10 6,00 5,90 5,80

W 1

2S

Figure 8.19. Observed values

199

200

Production Planning in Production Networks

6,80 6,70 6,60 6,50 6,40

Centralised

6,30

Game Theory

6,20

Negotiation

6,10 6,00 5,90 5,80 1W

S 2

Figure 8.20. Estimated values Table 8.55. Groups’ mean differences for column factors W-C.A.

S-C.A.

W-G.A.

S-G.A.

W-N.A.

S-C.A.

0.359*

–

–

–

–

W-G.A.

0.731*

0.373*

–

–

–

S-G.A.

0.683*

0.325*

0.048

–

–

W-N.A.

0.014

0.373*

0.746*

0.698*

–

S-N.A.

0.359*

0.001

0.372*

0.324*

0.374*

8.3.4 Distance Performance Analysis: DoE

Again, DoE is used to investigate in which way the input variables influence the estimated efficiency. 8.3.4.1 Centralised Model Figure 8.21 reports the main effects, i.e. how input parameter levels influence the distance performance. The workload and capacity ownership variances have no influence on the performance; on the other hand, plant production capacity variance negatively influences the efficiency and the risk attitude variance positively influences the distance parameter. From the interaction side, Figure 8.22, it seems that the interaction of Q and Cg might have influence on the distance performance. This is confirmed by the normal probability plot, Figure 8.23, and by the Pareto chart, Figure 8.24, which shows how the risk attitude variance is the most important factor in this case. Finally, Figure 8.25 reports the response curves analysis.

Distributed Production Planning Models: an Integrated Approach

WL

Cg

6,3

Mean of Distance

6,2 6,1 6,0 S

W

S

W

Q

R

6,3 6,2 6,1 6,0 S

W

S

W

Figure 8.21. Main effects plot (C.A.)

S

W

S

W

S

W 6,4

6,2

WL

WL S W

6,0 6,4

6,2

Cg

Cg S W

6,0 6,4

6,2

Q

6,0

R

Figure 8.22. Interaction plot (C.A.)

Q S W

201

202

Production Planning in Production Networks 99

Effect Type Not Significant Significant

C

95 90

Factor A B C D

80

Percent

70 60 50 40

Name WL Cg Q R

30 20

BC

10 5

D

1 -0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

Effect

Figure 8.23. Normal probability plot of the effects (C.A.)

Term

0,1298 Factor A B C D

D BC C AB ABC ABD ACD B BCD ABCD AD AC BD CD A 0,0

0,1

0,2 Effect

0,3

Figure 8.24. Pareto chart of the effects (C.A.)

0,4

Name WL Cg Q R

Distributed Production Planning Models: an Integrated Approach

R

W

203

S

Figure 8.25. Surface plot for distance (C.A.)

8.3.4.2 Negotiation Figure 8.26 reports the main effects, i.e. how input parameter levels influence the distance performance when using the N.A. model. Again, the workload and capacity ownership variances have no influence on the performance; on the other hand, plant production capacity variance negatively influences the efficiency and the risk attitude variance positively influences the distance parameter. From the interaction side, Figure 8.27, it seems that the interaction of Q and Cg might have influence on the distance performance. This is confirmed by the normal probability plot, Figure 8.28, and by the Pareto chart, Figure 8.29, that puts on evidence how the risk attitude variance is the most important factor in this case. Finally, Figure 8.30 reports the response curves analysis.

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Production Planning in Production Networks

WL

6,4

Cg

6,3 6,2

Mean of Distance

6,1 6,0 S

W

S

W

Q

6,4

R

6,3 6,2 6,1 6,0 S

W

S

W

Figure 8.26. Main effects plot (N.A.)

S

W

S

W

S

W 6,4 6,2

WL

WL S W

6,0

6,4 6,2

Cg

Cg S W

6,0 6,4

6,2

Q

6,0

R

Figure 8.27. Interaction plot (N.A.)

Q S W

Distributed Production Planning Models: an Integrated Approach

99

Effect Type Not Significant Significant

C

95 90

Factor A B C D

Percent

80 70 60 50 40 30

Name WL Cg Q R

20 BC

10 5

D

1 -0,4

-0,3

-0,2

-0,1 Effect

0,0

0,1

0,2

Figure 8.28. Normal probability plot of the effects (N.A.)

Term

0,0996 Factor A B C D

D BC C ABC ABCD BCD ABD ACD B A AC AD BD CD AB 0,0

0,1

0,2 Effect

0,3

Figure 8.29. Pareto chart of the effects (N.A.)

0,4

Name WL Cg Q R

205

206

Production Planning in Production Networks

R

W

S

Figure 8.30. Surface plot for distance (N.A.)

8.3.4.3 Game Theory Figure 8.31 reports the main effects when using the G.A. model. This time, all the input parameter variances have a significant influence on the performance. Also from the interaction side, Figure 8.32, it seems that the interaction of WL and Q, Q and Cg, and of Cg and R might have influence on the distance performance. However, the normal probability plot, Figure 8.33, and the Pareto chart show us that only the interaction among Q and Cg is significant and important.

Distributed Production Planning Models: an Integrated Approach

WL

207

Cg

6,71 6,70

Mean of Distance

6,69 6,68 6,67 S

W

S

W

Q

R

6,71 6,70 6,69 6,68 6,67 S

W

S

W

Figure 8.31. Main effects plot (G.A.)

S

W

S

W

S

W

6,72 WL

6,66

WL S W

6,60

6,72 Cg

6,66

Cg S W

6,60

6,72 Q

6,66 6,60

R

Figure 8.32. Interaction plot (G.A.)

Q S W

Production Planning in Production Networks

99

Effect Type Not Significant Significant

95 90

Factor A B C D

Percent

80 70 60 50 40 30

Name WL Cg Q R

20 10 5

BC

1 -0,15

-0,10

-0,05

0,00 Effect

0,05

0,10

Figure 8.33. Normal probability plot of the effects (G.A.)

0,1157 Factor A B C D

BC BCD BD D AC A

Term

208

B ABD ABCD AD ACD CD C AB ABC 0,00

0,02

0,04

0,06

0,08

0,10

0,12

Effect

Figure 8.34. Pareto chart of the effects (G.A.)

0,14

Name WL Cg Q R

Distributed Production Planning Models: an Integrated Approach

209

W S R

W

S

R

W

S

Figure 8.35. Surface plot for distance (G.A.)

The analysis allows us to draw a conclusion that is quite similar to that drawn for the efficiency performance. Indeed, also for the distance performance although the game-theoretical approach, G.A., seems to perform a bit worse of the centralised, C.A., and the negotiation, N.A., one, it is the last sensitive to the input factor uncertainty. Therefore, when the input factors are known in a quite certain way, the negotiation approach should be used to plan in a distributed way production at high and medium level; on the other hand, when input factors uncertainty is high, the game-theoretical approach should be used. 8.3.5 Absolute Residual Performance Analysis: Two-ways ANOVA

8.3.5.1 Workload WLg

Table 8.57 reports the values of the absolute residual performance when input factors “kind of model” and “workload variance” vary according to their levels (Table rows) and when the other input parameters, namely Cg , Qm , Rm , are

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Production Planning in Production Networks

combined according to Table 8.3. Table 8.57 reports the average value of numbers in Table 8.56, while Table 8.58 reports the results of the ANOVA. As the reader can notice, none of the null hypotheses can be rejected, therefore the T-test analysis is pointless. None of the input factors, nor their interaction influences the performance values. Table 8.56. Replication values of Z 1

2

3

4

5

6

7

8

W

51.761

57.385

42.226

43.216

36.641

36.289

22.450

23.298

S

53.674

52.864

46.744

49.030

36.917

37.535

22.347

23.184

W

58.791

59.694

53.668

47.153

38.927

40.474

24.000

25.500

S

58.485

58.392

49.213

50.741

40.141

39.182

23.667

24.750

W

53.296

55.043

46.694

47.156

36.947

37.937

22.336

24.037

S

50.415

54.864

44.223

45.007

36.096

39.126

23.353

23.168

C.A.

G.A.

N.A.

Table 8.57. Average values of Z W

S

Mean

C.A.

39.16

40.29

39.72

G.A.

43.53

43.07

43.30

N.A.

40.43

39.53

39.98

Mean

41.04

40.96

41.00

Table 8.58. Two–way ANOVA table for Z 5%

SS

df

MS

F–test

F–crit D

First factor

127.251

2

63.626

0.391

F2.42

3.252

Second factor

0.068

1

0.068

0.000

F1.42

4.102

Interaction effect

9.090

2

4.545

0.028

F2.42

3.252

Within

6827.921

42

162.570

–

Total

6964.330

47

–

–

8.3.5.2 Capacity Ownership ( Cg )

Tables 8.59–8.64 show the usual data for the capacity ownership factor. This time, only the null hypothesis 8.2 can be rejected, that is, the factor “capacity ownership variance” influences the absolute residual performance. Also in this case, however, it is possible to conclude that the factor “kind of model” does not influence the

Distributed Production Planning Models: an Integrated Approach

211

performance under investigation. Of course, T-test analysis refers in this case to the second input factor. Table 8.59. Replication values of Z 1

2

3

4

5

6

7

8

W

51.761

57.385

42.226

43.216

53.674

52.864

46.744

49.030

S

36.641

36.289

22.450

23.298

36.917

37.535

22.347

23.184

W

58.791

59.694

53.668

47.153

58.485

58.392

49.213

50.741

S

38.927

40.474

24.000

25.500

40.141

39.182

23.667

24.750

W

53.296

55.043

46.694

47.156

50.415

54.864

44.223

45.007

S

36.947

37.937

22.336

24.037

36.096

39.126

23.353

23.168

C.A.

G.A.

N.A.

Table 8.60. Average values of Z W

S

Mean

C.A.

49.61

29.83

39.72

G.A.

54.52

32.08

43.30

N.A.

49.59

30.38

39.98

Mean

51.24

30.76

41.00

Table 8.61. Two-way ANOVA table for Z 5%

SS

df

MS

F-test

F-crit D

First factor

127.251

2

63.626

1.500

F2.42

3.252

Second factor

5031.374

1

5031.374

118.585*

F1.42

4.102

Interaction effect

23.709

2

11.855

0.279

F2.42

3.252

Within

1781.995

42

42.428

–

Total

6964.330

47

–

–

Table 8.62. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

4.657

Column factor

Q0.05 , 2.42

2.86

T

4.574

Interaction

Q0.05 , 6.42

4.23

T

9.741

212

Production Planning in Production Networks

Table 8.63. Differences of means for column factors Column factor

W

S

S

20.476*

0

Table 8.64. Groups’ mean differences for column factors W-C.A.

S-C.A.

W-G.A.

S-G.A.

W-N.A.

S-C.A.

19.780*

–

–

–

–

W-G.A.

4.904

24.684*

–

–

–

S-G.A.

17.532*

2.247

22.437*

–

–

W-N.A.

0.025

19.755*

4.930

17.507*

–

S-N.A.

19.237*

0.542

24.142*

1.705

19.212*

8.3.5.3 Plants Production Capacity ( Qm ) Tables 8.65–8.69 show the usual data for the plants production capacity factor. Also in this case, only the null hypothesis 8.2 can be rejected. However, it is possible to conclude that the factor “kind of model” does not influence the performance under investigation. Of course, T-test analysis, also in this case, refers to the second input factor. Table 8.65. Replications values of Z 1

2

3

4

5

6

7

8

W

51.761

57.385

36.641

36.289

53.674

52.864

36.917

37.535

S

42.226

43.216

22.450

23.298

46.744

49.030

22.347

23.184

W

58.791

59.694

38.927

40.474

58.485

58.392

40.141

39.182

S

53.668

47.153

24.000

25.500

49.213

50.741

23.667

24.750

W

53.296

55.043

36.947

37.937

50.415

54.864

36.096

39.126

S

46.694

47.156

22.336

24.037

44.223

45.007

23.353

23.168

C.A.

G.A.

N.A.

Table 8.66. Average values of Z W

S

Mean

C.A.

45.38

34.06

39.72

G.A.

49.26

37.34

43.30

N.A.

45.47

34.50

39.98

Mean

46.70

35.30

41.00

Distributed Production Planning Models: an Integrated Approach

213

Table 8.67. Two-way ANOVA table for Z 5%

SS

df

MS

F-test

F-crit D

First factor

127.251

2

63.626

0.507

F2.42

3.252

Second factor

1560.857

1

1560.857

12.429*

F1.42

4.102

Interaction effect

1.868

2

0.934

0.007

F2.42

3.252

Within

5274.354

42

125.580

–

Total

6964.330

47

–

–

Table 8.68. T-values Factor

Q-value

T-value

Row factor

Q0.05 ,3.42

3.44

T

7.868

Column factor

Q0.05 , 2.42

2.86

T

8.012

Interaction

Q0.05 , 6.42

4.23

T 16.759

Table 8.69. Differences of means for column factors

Column factor

W

S

S

11.405*

0

8.3.5.4 Risk Attitude ( Rm ) Finally, tables 8.70–8.72 show the usual data for the risk attitude factor. Also in this case, like for the workload, none of the null hypotheses can be rejected. This means, in particular that the performance Z is not influenced by the typology of the model used. Of course, in this case T-test analysis is pointless. Table 8.70. Replication values of Z 1

2

3

4

5

6

7

8

W

51.761

42.226

36.641

22.450

53.674

46.744

36.917

22.347

S

57.385

43.216

36.289

23.298

52.864

49.030

37.535

23.184

W

58.791

53.668

38.927

24.000

58.485

49.213

40.141

23.667

S

59.694

47.153

40.474

25.500

58.392

50.741

39.182

24.750

W

53.296

46.694

36.947

22.336

50.415

44.223

36.096

23.353

S

55.043

47.156

37.937

24.037

54.864

45.007

39.126

23.168

C.A.

G.A.

N.A.

214

Production Planning in Production Networks Table 8.71. Average values of Z W

S

Mean

C.A.

39.10

40.35

39.72

G.A.

43.36

43.24

43.30

N.A.

39.17

40.79

39.98

Mean

40.54

41.46

41.00

Table 8.72. Two-way ANOVA table for Z

8.3.6

5%

SS

df

MS

F-test

F-crit D

First factor

127.251

2

63.626

0.392

F2.42

3.252

Second factor

10.094

1

10.094

0.062

F1.42

4.102

Interaction effect

6.795

2

3.397

0.021

F2.42

3.252

Within

6820.189

42

162.385

–

Total

6964.330

47

–

–

Absolute Residual Performance Analysis: DoE

8.3.6.1 Centralised Model Figure 8.36 reports the main effects, i.e. how input parameter levels influence the Z performance. As the reader can notice, the workload and the risk attitude variances have no influence on the performance; on the other hand, capacity ownership and plant production capacity variances positively influence the performance. From the interaction side, Figure 8.37, it seems that no interaction has influence on the performance. This is confirmed by the normal probability plot, Figure 8.38, and by the Pareto chart, Figure 8.39. Finally, Figure 8.40 reports the response curves analysis.

Distributed Production Planning Models: an Integrated Approach

WL

Cg

50 45

Mean of Absolute Residual

40 35 30 S

W

S

W

Q

R

50 45 40 35 30 S

W

S

W

Figure 8.36. Main effects plot (C.A.)

S

W

S

W

S

W 50 40

WL

WL S W

30

50 40

Cg

Cg S W

30

50 40

Q

30

R

Figure 8.37. Interaction plot (C.A.)

Q S W

215

216

Production Planning in Production Networks

99

Effect Type Not Significant Significant

B

95 90

C

Factor A B C D

Percent

80 70 60 50 40 30

Name WL Cg Q R

20 10 5 1 0

5

10

15

20

Effect

Figure 8.38. Normal probability plot of the effects (C.A.)

Term

3,25 Factor A B C

B C BC ABC AC D A ABCD ACD AB BD ABD AD BCD CD 0

5

10 Effect

15

Figure 8.39. Pareto chart of the effects (C.A.)

20

Name WL Cg Q

Distributed Production Planning Models: an Integrated Approach

R

W

217

S

Figure 8.40. Surface plot for absolute residual (C.A.)

8.3.6.2 Negotiation Model Figure 8.41 reports the main effects, i.e. how input parameter levels influence the Z performance. Also in this case, the workload and the risk attitude variances have no influence on the performance; on the other hand, capacity ownership and plant production capacity variances negatively influence the performance. From the interaction side, Figure 8.42, it seems that none interaction has influence on the performance. This is confirmed by the normal probability plot, Figure 8.43, and by the Pareto chart, Figure 8.44. Finally, Figure 8.45 reports the response curves analysis.

218

Production Planning in Production Networks

WL

Cg

50

Mean of Absolute Residual

45 40 35 30 S

W

S

W

Q

R

50 45 40 35 30 S

W

S

W

Figure 8.41. Main effects plot (N.A.)

L

H

L

H

L

H 50 40

WL

WL L H

30

50 40

Cg

Cg L H

30

50 40

Q

30

R

Figure 8.42. Interaction plot (N.A.)

Q L H

Distributed Production Planning Models: an Integrated Approach

99

Effect Type Not Significant Significant

B

95 90

C

Factor A B C D

80

Percent

70 60 50

Name WL Cg Q R

40 30 20

D

10 5

BC

1

0

5

10

15

20

Effect

Figure 8.43. Normal probability plot of the effects (N.A.)

Term

1,46 Factor A B C D

B C BC D AB CD A ACD AD ABD BCD BD AC ABCD ABC 0

5

10 Effect

15

Figure 8.44. Pareto chart of the effects (N.A.)

20

Name WL Cg Q R

219

220

Production Planning in Production Networks

R

W

S

Figure 8.45. Surface plot for absolute residual (N.A.)

8.3.6.3 Game Theory Finally, Figure 8.46 reports the main effects influence on the Z performance. Also in this case, the workload and the risk attitude variances have no influence on the performance; on the other hand, capacity ownership and plant production capacity variances positively influence the performance. From the interaction side, Figure 8.47, it seems that no interaction has influence on the performance. This is not confirmed by the normal probability plot, Figure 8.48, which indicates that there is a significant interaction between Q and Cg, even if its importance is just beyond the threshold, which as reported in the Pareto chart, Figure 8.49. Finally, Figure 8.50 reports the response curves analysis.

Distributed Production Planning Models: an Integrated Approach

WL

221

Cg

55 50

Mean of Absolute Residual

45 40 35 S

W

S

W

Q

55

R

50 45 40 35 S

W

S

W

Figure 8.46. Main effects plot (G.A.)

S

W

S

W

S

W

50 WL

35

WL S W

20 50 Cg

35

Cg S W

20 50 Q

35

20

R

Figure 8.47. Interaction plot (G.A.)

Q S W

Production Planning in Production Networks

99

Effect Type Not Significant Significant

B

95 90

C

Factor A B C D

80

Percent

70 60 50

Name WL Cg Q R

40 30 20 10 5

BC

1

0

5

10 Effect

15

20

25

Figure 8.48. Normal probability plot of the effects (G.A.)

1,91

Term

222

Factor A B C D

B C BC ACD ABD BCD BD ABCD AD CD A ABC AB D AC 0

5

10

15

20

Effect

Figure 8.49. Pareto chart of the effects (G.A.)

25

Name WL Cg Q R

Distributed Production Planning Models: an Integrated Approach

R

W

223

S

Figure 8.50. Surface plot for absolute residual (G.A.)

As far as absolute residual performance, Z, is concerned we need to stress that there is no statistical evidence that the model typology influences the performance. The influence of the input parameter variance on the models is, on the other hand, quite similar among the three models.

8.4 Conclusions This chapter shows the integrated use of the planning models we have discussed in the previous chapters for planning production at high and medium levels. The aim of the chapter was to understand what kind of indications could be drawn from an integrated use of the models. The main conclusion is that the negotiation model, N.A., performs quite similarly to the centralised one, C.A., as far as efficiency and distance are concerned, while the game-theoretical model, G.A., performs a bit worse than the others.

224

Production Planning in Production Networks

However, the G.A. model is less sensitive to input parameters uncertainty. Therefore, when the uncertainty is low, the N.A. model should be used; conversaly, if the uncertainty is high the G.A. model should be used. The absolute residual performance measure is not influenced by the model typology.

9 Conclusions

9.1 Summary This book presents a study that has been conducted about the opportunity to utilise a set of decentralised models to face the difficulty emerging in the capacity allocation problem which concerns the semiconductor industry or, more generally, high-tech, high-volume Industries with a relevant numbers of groups (product families) and plants geographically dispersed. We modelled the decision problem using the viewpoints of the managers of the organisation. We first showed that decentralisation requires the additional restriction of maintaining private information, which creates unavoidable degradation of overall performance. Under the private information assumption, we showed that coordination is achievable between groups’ and plants’ agents using an information exchange scheme. We proposed two coordination mechanisms using this framework and proved that these approaches converge to the optimal global solution defined by the centralised (benchmark) model. From a mathematical standpoint, we modelled the information exchange as an auction between the competing decision-makers. This approach reflects the actual opposite agents’ local utility functions and it tries to regulate the local decisions to match the other side’s needs as closely as possible. The negotiational and game-theoretical theories provided a flexible analytical framework to develop this coordination device. Finally, we tested the proposed mechanisms in an agent-based ad hoc developed discrete-event simulation environment by using generated numerical data. As far as we know, this research is one of the very few studies that utilises the application of the game-theory and negotiational framework to study this kind of issue. It appears that future research will needed to apply this approach to both optimisation and coordination problems that can be solved by mathematical-simulative composition. In what follows, we summarise the main scientific contributions of this book and some directions for future work.

226

Production Planning in Production Networks

9.2 Major Scientific Contributions of This Book The main contributions of this book are: ƒ In Chapter 5 an innovative distributed and hierarchical approach for planning capacity at different levels of time horizon has been introduced. For each level bargaining objective, actors, information and roles have been located. In synthesis the model presented in Chapter 5 represents a general framework for distributed production planning of complex and multi-plant manufacturing companies in high-tech, high-volume industries; ƒ A structured model of agents with bounded rationality that interact to solve, in a competitive way, a capacity allocation problem. We presented models framework for the high-level production planning (Chapter 6) and for the medium-level production planning (Chapter 7). We defined the utility functions introducing performance analysis and concepts that are feasible to be adopted in practice and that can lead to better drive the agents’ coordination and the control decisions compared to other approaches from the scientific literature. ƒ A formal game-theoretical and negotiational model for this goal has been developed. We proposed incorporating the deliberation actions of agents in a mathematical setting. In the first part of Chapters 6 and 7 we presented definitions of strategies for agents with resources limitations, and introduced the concept of equilibrium using a fictitious credits. ƒ We also discussed new strategic behaviours which may arise among agents. In particular, we introduced the Bargaining solution in an allocation cooperative game that drastically increases the performance of this approach. ƒ Analysis of negotiation mechanisms has also been conducted. In Chapters 6 and 7 we also analysed negotiation protocols, using the Rubinstein equilibrium as the solution concept. ƒ A simulation environment has been utilised to simulate the behaviour of the defined agents and to estimate the established performance measures. In particular, the last part of Chapter 6 showed that, regarding the unallocated capacity of the high level, no relevant differences can be highlighted among the three different proposed approaches. Considering a confidence interval equal to 5% the existing discrepancies are not significantly relevant: this is a really good result considering the decentralised nature of the negotiational and game-theoretical proposed framework. ƒ In Chapter 7 we highlighted the importance of each model with respect to different performance indicators. In this case the centralised one proves to be the better one in optimising the utility function and no difference, statistically speaking, can be considered between the two decentralised approaches. Instead, by considering one performance measure at time, we noticed that game theory models approaches better the global plants’ number of reconfigurations.

Conclusions

ƒ

227

Chapter 8 showed how the system acts when the proposed models work in an integrated way. This simulation leads us to highlight a statistically relevant difference between the two decentralised models and between the centralised and the game-theoretical one, but not between the centralised and negotiational one. This result leads us to a very important conclusion: in optimising the capacity allocation and the allocation to the groups of the production plants the negotiation model acts very well, quite reaching the same performance obtained in the centralised case. The noticed differences are only due to the randomness of data input. However, when the data uncertainty is high, the game-theoretical approach is more robust in its performance than the negotiational one.

9.3 Directions for Future Work This book opens up new interesting directions for future research in this area. ƒ In Chapters 6 and 7 we proposed a set of properties which we believed that the designed mechanisms should exhibit. In particular, we believed that mechanisms should be provided with an infinite number of credits, that agents should act in a cooperative way and that agents should not have incentives to deceive others about their computed results. Unfortunately, it is realistic to consider that these properties are, together, not attainable. Therefore, at least one property must be relaxed. ƒ Regarding the number of credits, it might be possible to consider some incentives schemes that can be introduced in such a model. For example, appropriate incentives will be needed to get the agent to solve the problems, and to share the information with all agents to which it pertains. Naturally, some problems that need to be addressed include: o What sort and how much information should the agents reveal through the mechanism? o Is there a minimal amount of information that needs to be revealed to the mechanism so that strategic-advantage disappears? o What sort of incentives are required before agents will truthfully reveal this information? o What type of feedback should the mechanism provide to the agents? ƒ Regarding the integrated model it might be interesting to evaluate the influence of the mathematical models on the results. This can be obtained by modifying the proposed mechanism considering not only the input/output relationship between high and medium levels working sequentially, but the mathematical integration too.

This page intentionally blank

A Simulation Results Related to Chapter 6

Table A.1. Centralised model results Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WL

P

C

51.040

2.552

0

343

3541

58.973

5.00%

W

W

W

39.160

1.958

0

227

2369

37.009

5.00%

W

W

B

34.098

1.704

0

136

1785

27.957

5.00%

W

W

S

51.539

2.577

0

354

3645

60.417

5.00%

W

B

W

34.315

1.715

0

143

1670

27.218

5.00%

W

B

B

34.649

1.732

0

149

1662

27.421

5.00%

W

B

S

49.983

2.498

0

340

3725

59.218

5.00%

W

S

W

38.840

1.942

0

228

2389

36.857

5.00%

W

S

B

34.475

1.723

0

175

1751

28.006

5.00%

W

S

S

41.878

2.094

0

409

4621

55.275

5.00%

B

W

W

27.777

1.389

0

177

3100

30.026

5.00%

B

W

B

21.694

1.085

0

96

1791

17.826

5.00%

B

W

S

42.722

2.136

0

344

4529

55.816

5.00%

B

B

W

28.200

1.410

0

169

3165

30.796

5.00%

B

B

B

20.659

1.033

0

118

1984

17.863

5.00%

B

B

S

41.863

2.093

0

355

4703

55.744

5.00%

B

S

W

26.324

1.316

0

183

3359

29.624

5.00%

B

S

B

21.485

1.074

0

111

1875

18.062

5.00%

B

S

S

36.434

1.822

0

361

6000

54.796

5.00%

230

Appendix A Table A.1 (continued) S

W

W

18.486

0.924

0

184

5363

26.280

5.00%

S

W

B

8.744

0.437

0

52

2616

8.683

5.00%

S

W

S

35.884

1.794

0

388

5962

53.790

5.00%

S

B

W

18.671

0.933

0

214

5418

26.683

5.00%

S

B

B

8.451

0.422

0

50

2475

8.163

5.00%

S

B

S

35.216

1.761

0

339

6094

53.379

5.00%

S

S

W

18.595

0.930

0

173

5269

26.207

5.00%

S

S

B

8.762

0.438

0

45

2396

8.327

5.00%

S

S

S

51.040

2.552

0

343

3541

58.973

5.00%

Table A.2. Negotiational model results Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WL

P

C

65.826

3.291

0

368

1937

56.246

5.00%

W

W

W

43.880

2.193

0

216

1826

36.400

5.00%

W

W

B

36.765

1.838

0

137

1413

26.826

5.00%

W

W

S

66.142

3.306

0

357

1875

55.596

5.00%

W

B

W

43.321

2.165

0

268

1817

35.823

5.00%

W

B

B

34.803

1.739

0

187

1551

26.604

5.00%

W

B

S

67.615

3.380

0

352

1913

57.413

5.00%

W

S

W

44.582

2.228

0

211

1628

34.912

5.00%

W

S

B

36.657

1.832

0

150

1520

27.732

5.00%

W

S

S

58.570

2.928

0

340

2330

54.883

5.00%

B

W

W

31.366

1.568

0

185

2554

30.778

5.00%

B

W

B

21.999

1.099

0

97

1642

17.301

5.00%

B

W

S

57.717

2.886

0

412

2361

54.453

5.00%

B

B

W

30.194

1.509

0

167

2487

29.229

5.00%

B

B

B

21.926

1.096

0

149

1779

17.952

5.00%

B

B

S

57.637

2.881

0

357

2382

54.603

5.00%

B

S

W

30.400

1.519

0

208

2582

29.983

5.00%

B

S

B

21.064

1.053

0

99

1812

17.403

5.00%

Appendix A

231

Table A.2. (continued) B

S

S

51.281

2.563

0

322

2619

50.947

5.00%

S

W

W

20.532

1.026

0

146

3941

25.021

5.00%

S

W

B

8.557

0.428

0

51

2549

8.388

5.00%

S

W

S

53.387

2.668

0

360

2497

51.784

5.00%

S

B

W

21.071

1.054

0

181

3940

25.681

5.00%

S

B

B

8.603

0.430

0

51

2573

8.472

5.00%

S

B

S

53.231

2.660

0

354

2600

52.679

5.00%

S

S

W

20.605

1.030

0

204

4022

25.365

5.00%

S

S

B

8.688

0.434

0

67

2632

8.651

5.00%

S

S

S

65.826

3.291

0

368

1937

56.246

5.00%

Table A.3. Game-theoretical model results Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WL

P

C

67.729

3.386

0

383

1838

56.378

5.00%

W

W

W

43.636

2.181

0

261

1840

36.328

5.00%

W

W

B

35.612

1.780

0

173

1530

27.038

5.00%

W

W

S

64.861

3.241

0

338

1951

55.600

5.00%

W

B

W

35.100

1.754

0

149

1535

26.692

5.00%

W

B

B

34.894

1.744

0

146

1452

25.811

5.00%

W

B

S

67.269

3.362

0

387

1898

56.885

5.00%

W

S

W

43.118

2.155

0

195

1757

35.081

5.00%

W

S

B

36.148

1.806

0

150

1486

27.035

5.00%

W

S

S

59.126

2.956

0

433

2125

52.916

5.00%

B

W

W

31.559

1.578

0

191

2436

30.463

5.00%

B

W

B

21.575

1.078

0

91

1771

17.621

5.00%

B

W

S

58.884

2.944

0

357

2346

55.138

5.00%

B

B

W

31.647

1.582

0

185

2483

30.616

5.00%

B

B

B

21.730

1.086

0

98

1840

18.092

5.00%

B

B

S

57.858

2.892

0

367

2251

53.292

5.00%

B

S

W

30.507

1.525

0

210

2497

29.600

5.00%

B

S

B

21.739

1.087

0

90

1760

17.708

5.00%

232

Appendix A Table A.3. (continued) B

S

S

52.230

2.611

0

300

2486

50.555

5.00%

S

W

W

20.899

1.045

0

155

3866

25.223

5.00%

S

W

B

8.528

0.426

0

59

2671

8.556

5.00%

S

W

S

55.789

2.789

0

416

2561

54.811

5.00%

S

B

W

21.772

1.089

0

184

3904

26.413

5.00%

S

B

B

8.353

0.418

0

49

2523

8.145

5.00%

S

B

S

53.596

2.679

0

427

2552

52.552

5.00%

S

S

W

20.534

1.027

0

190

4091

25.500

5.00%

S

S

B

9.052

0.452

0

57

2367

8.549

5.00%

S

S

S

67.729

3.386

0

383

1838

56.378

5.00%

B Simulation Input Parameters and Results Related to Chapter 7

Table B.1. Values of V gm parameter Group 1

Group 2

Group 3

Plant 1

0.9

0.2

0.5

Plant 2

0.8

0.2

0.9

Plant 3

0.4

0.5

0.7

Plant 4

0.6

0.7

0.9

Plant 5

1.0

0.2

0.8

Plant 6

0.8

0.5

0.6

Plant 7

0.3

0.4

0.6

Plant 8

0.8

0.9

0.3

Table B.2. Values of Dgm parameter Group 1

Group 2

Group 3

Plant 1

0.6

0.3

0.8

Plant 2

0.8

1.1

0.3

Plant 3

0.9

1.2

0.7

Plant 4

1.1

0.4

0.9

Plant 5

0.5

0.7

1.2

Plant 6

0.6

0.5

0.7

Plant 7

0.6

1.2

1.0

Plant 8

1.1

0.9

0.9

234

Appendix B

Table B.3. Utility functions data

p min

p max g

pm*

Rc

0.5

2

0.5

0.6

g

Table B.4. Centralised model results for efficiency Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

14.447

0.127

5.7

23.6

3068

2.725

< 5%

W

W

W

14.435

0.143

6.4

22.4

1698

2.283

< 5%

W

W

B

12.180

0.238

6.2

18.6

565

2.194

< 5%

W

W

S

14.363

0.125

4.1

23.0

3008

2.665

< 5%

W

B

W

14.435

0.147

7.0

21.8

1552

2.242

< 5%

W

B

B

12.156

0.224

6.3

19.0

571

2.076

< 5%

W

B

S

14.430

0.129

5.8

23.1

2903

2.706

< 5%

W

S

W

14.503

0.144

7.0

21.8

1588

2.235

< 5%

W

S

B

14.473

0.130

4.7

22.8

2938

2.745

< 5%

W

S

S

14.670

0.112

4.6

23.9

3908

2.729

< 5%

B

W

W

14.945

0.128

8.4

22.8

1813

2.121

< 5%

B

W

B

12.070

0.265

7.3

17.8

370

1.978

< 5%

B

W

S

14.754

0.111

6.0

24.0

3964

2.713

< 5%

B

B

W

15.051

0.128

8.6

22.5

1918

2.171

< 5%

B

B

B

12.030

0.270

7.0

18.1

342

1.937

< 5%

B

B

S

14.771

0.112

4.1

23.7

3996

2.745

< 5%

B

S

W

14.946

0.132

6.9

21.9

1670

2.102

< 5%

B

S

B

12.056

0.289

7.3

19.1

350

2.103

< 5%

B

S

S

15.161

0.106

4.8

24.0

4743

2.841

< 5%

S

W

W

15.592

0.121

8.8

22.8

1830

2.015

< 5%

S

W

B

12.679

0.432

7.8

18.2

149

2.049

< 5%

S

W

S

15.070

0.102

6.2

24.0

4828

2.760

< 5%

S

B

W

15.664

0.124

8.9

22.4

1826

2.051

< 5%

Appendix B

Table B.4. (continued) S

B

B

12.917

0.386

7.6

18.0

172

1.968

< 5%

S

B

S

15.121

0.109

5.5

24.4

4642

2.879

< 5%

S

S

W

15.096

0.104

6.0

24.4

4783

2.806

< 5%

S

S

B

12.653

0.402

8.8

17.3

147

1.891

< 5%

S

S

S

14.447

0.127

5.7

23.6

3068

2.725

< 5%

Table B.5. Centralised model results for distance Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

11.091

0.101

4.1

18.4

3068

2.167

< 5%

W

W

W

11.130

0.110

5

18.4

1698

1.764

< 5%

W

W

B

9.475

0.184

4.4

15.5

565

1.696

< 5%

W

W

S

11.016

0.100

2.9

18

3008

2.129

< 5%

W

B

W

11.093

0.111

5

18.2

1552

1.697

< 5%

W

B

B

9.473

0.178

4.3

15.6

571

1.653

< 5%

W

B

S

11.043

0.102

3.3

17.7

2903

2.135

< 5%

W

S

W

11.140

0.113

5.3

17.1

1588

1.745

< 5%

W

S

B

11.078

0.103

2.4

18.1

2938

2.173

< 5%

W

S

S

11.296

0.088

2.8

18.6

3908

2.148

< 5%

B

W

W

11.496

0.100

5.3

17.6

1813

1.654

< 5%

B

W

B

9.397

0.213

4.9

14.5

370

1.592

< 5%

B

W

S

11.320

0.089

4.6

18.9

3964

2.180

< 5%

B

B

W

11.611

0.101

5.9

17.5

1918

1.718

< 5%

B

B

B

9.354

0.222

4.7

14.4

342

1.591

< 5%

B

B

S

11.264

0.089

2.3

18.8

3996

2.190

< 5%

B

S

W

11.545

0.105

5.6

17.8

1670

1.659

< 5%

B

S

B

9.295

0.226

4.9

13.9

350

1.641

< 5%

B

S

S

11.669

0.086

3.4

19.2

4743

2.289

< 5%

S

W

W

11.988

0.096

4.9

18.4

1830

1.603

< 5%

S

W

B

9.586

0.346

5.4

14.4

149

1.642

< 5%

S

W

S

11.584

0.081

2.4

18.8

4828

2.192

< 5%

235

236

Appendix B

Table B.5. (continued) S

B

W

12.070

0.099

6.1

17.7

1826

1.637

< 5%

S

B

B

9.887

0.305

5.7

14.3

172

1.552

< 5%

S

B

S

11.607

0.086

2.8

19.6

4642

2.273

< 5%

S

S

W

11.625

0.084

3.5

19.1

4783

2.254

< 5%

S

S

B

9.641

0.301

6.1

12.7

147

1.417

< 5%

S

S

S

11.091

0.101

4.1

18.4

3068

2.167

< 5%

Table B.6. Centralised model results for reconfiguration Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

8.632

0.148

0

22

3068

3.180

< 5%

W

W

W

8.830

0.189

1

19

1698

3.019

< 5%

W

W

B

6.681

0.290

1

19

565

2.681

< 5%

W

W

S

8.575

0.148

0

19

3008

3.150

< 5%

W

B

W

8.771

0.194

1

19

1552

2.970

< 5%

W

B

B

6.764

0.289

1

17

571

2.680

< 5%

W

B

S

8.607

0.149

0

25

2903

3.125

< 5%

W

S

W

8.738

0.183

1

19

1588

2.837

< 5%

W

S

B

8.511

0.148

0

19

2938

3.114

< 5%

W

S

S

8.444

0.139

0

20

3908

3.380

< 5%

B

W

W

8.574

0.183

1

20

1813

3.023

< 5%

B

W

B

5.697

0.269

0

16

370

2.757

< 5%

B

W

S

8.483

0.137

0

22

3964

3.357

< 5%

B

B

W

8.619

0.178

0

20

1918

3.030

< 5%

B

B

B

5.883

0.259

0

18

342

2.576

< 5%

B

B

S

8.511

0.139

0

20

3996

3.401

< 5%

B

S

W

8.646

0.188

0

20

1670

2.985

< 5%

B

S

B

5.666

0.175

0

15

350

2.724

< 5%

B

S

S

8.357

0.151

0

23

4743

4.029

< 5%

S

W

W

8.721

0.213

0

20

1830

3.531

< 5%

S

W

B

4.530

0.052

0

15

149

2.481

< 5%

Appendix B

237

Table B.6. (continued) S

W

S

8.484

0.150

0

23

4828

4.040

< 5%

S

B

W

8.692

0.205

0

20

1826

3.404

< 5%

S

B

B

4.541

0.182

0

14

172

2.962

< 5%

S

B

S

8.354

0.154

0

25

4642

4.086

< 5%

S

S

W

8.367

0.152

0

23

4783

4.073

< 5%

S

S

B

4.585

0.189

0

13

147

2.774

< 5%

S

S

S

8.632

0.148

0

22

3068

3.180

< 5%

Table B.7. Centralised model results for absolute residual Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

52.838

1.788

0

291

3068

38.459

< 5%

W

W

W

46.064

1.967

0

215

1698

31.479

< 5%

W

W

B

54.876

2.437

8

170

565

22.500

< 5%

W

W

S

53.339

1.810

1

276

3008

38.541

< 5%

W

B

W

45.952

1.962

0

173

1552

30.020

< 5%

W

B

B

56.807

2.581

12

185

571

23.949

< 5%

W

B

S

53.358

1.802

0

279

2903

37.706

< 5%

W

S

W

46.858

2.000

0

216

1588

30.946

< 5%

W

S

B

54.362

1.869

1

301

2938

39.337

< 5%

W

S

S

43.519

1.372

1

332

3908

33.309

< 5%

B

W

W

33.201

1.420

0

161

1813

23.475

< 5%

B

W

B

44.257

2.015

13

110

370

15.054

< 5%

B

W

S

42.485

1.319

1

270

3964

32.239

< 5%

B

B

W

32.359

1.374

0

255

1918

23.362

< 5%

B

B

B

43.912

1.995

15

89

342

14.326

< 5%

B

B

S

43.392

1.370

0

249

3996

33.642

< 5%

B

S

W

32.646

1.393

1

134

1670

22.109

< 5%

B

S

B

44.517

2.011

16

94

350

14.613

< 5%

B

S

S

31.641

1.180

0

250

4743

31.562

< 5%

S

W

W

19.685

0.961

0

108

1830

15.963

< 5%

238

Appendix B

Table B.7. (continued) S

W

B

35.946

1.792

15

62

149

8.495

< 5%

S

W

S

31.063

1.159

0

241

4828

31.275

< 5%

S

B

W

19.797

0.970

0

129

1826

16.090

< 5%

S

B

B

36.802

1.831

13

59

172

9.326

< 5%

S

B

S

32.122

1.201

0

213

4642

31.768

< 5%

S

S

W

32.088

1.200

0

207

4783

32.230

< 5%

S

S

B

35.354

1.758

14

68

147

8.276

< 5%

S

S

S

52.838

1.788

0

291

3068

38.459

< 5%

Table B.8. Negotiation model results for efficiency Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

15.600

0.124

9.1

21.5

2001

2.149

< 5%

W

W

W

15.738

0.145

7.9

21.1

1330

2.054

< 5%

W

W

B

15.954

0.153

9.8

21.5

1102

1.971

< 5%

W

W

S

15.638

0.130

7.6

22.3

2001

2.253

< 5%

W

B

W

15.697

0.136

8.9

21.1

1358

1.942

< 5%

W

B

B

15.710

0.151

8.7

21.1

1107

1.954

< 5%

W

B

S

15.677

0.131

6.7

22.1

2001

2.283

< 5%

W

S

W

15.599

0.147

8.6

22.0

1315

2.073

< 5%

W

S

B

16.504

0.136

9.1

23.4

2001

2.367

< 5%

W

S

S

15.901

0.120

8.1

22.1

2001

2.087

< 5%

B

W

W

16.020

0.121

10.9

21.6

1456

1.788

< 5%

B

W

B

15.864

0.140

9.9

20.9

1048

1.757

< 5%

B

W

S

15.542

0.115

8.5

21.4

2001

1.996

< 5%

B

B

W

15.621

0.121

10.0

21.5

1368

1.744

< 5%

B

B

B

15.617

0.128

10.2

20.9

1147

1.684

< 5%

B

B

S

16.931

0.125

7.8

22.6

2001

2.179

< 5%

B

S

W

17.319

0.135

9.9

22.8

1335

1.910

< 5%

B

S

B

17.093

0.150

11.4

22.7

1000

1.848

< 5%

B

S

S

15.910

0.113

8.8

22.3

2001

1.960

< 5%

Appendix B

Table B.8. (continued) S

W

W

15.919

0.099

10.8

21.0

1714

1.586

< 5%

S

W

B

15.913

0.137

11.6

20.6

800

1.506

< 5%

S

W

S

15.556

0.113

7.2

21.5

2001

1.962

< 5%

S

B

W

15.618

0.099

11.4

21.2

1660

1.562

< 5%

S

B

B

15.260

0.132

12.3

21.1

743

1.394

< 5%

S

B

S

17.173

0.124

9.1

22.6

2001

2.162

< 5%

S

S

W

17.035

0.120

10.4

23.2

2001

2.088

< 5%

S

S

B

17.499

0.174

12.8

22.6

728

1.823

< 5%

S

S

S

15.600

0.124

9.1

21.5

2001

2.149

< 5%

Table B.9. Negotiation model results for distance Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

12.150

0.095

5.8

17.1

2001

1.652

< 5%

W

W

W

12.222

0.106

6.2

16.8

1330

1.507

< 5%

W

W

B

12.304

0.117

7.1

17

1102

1.505

< 5%

W

W

S

12.187

0.096

5.8

17.1

2001

1.667

< 5%

W

B

W

12.251

0.104

6.9

16.5

1358

1.483

< 5%

W

B

B

12.182

0.114

7.6

17.3

1107

1.476

< 5%

W

B

S

12.191

0.097

5.2

17.5

2001

1.681

< 5%

W

S

W

12.072

0.110

5.6

17.4

1315

1.553

< 5%

W

S

B

13.077

0.101

6.9

18.7

2001

1.746

< 5%

W

S

S

12.374

0.090

5.4

17.7

2001

1.568

< 5%

B

W

W

12.482

0.088

7.8

16.7

1456

1.303

< 5%

B

W

B

12.472

0.099

7.5

16.1

1048

1.240

< 5%

B

W

S

12.183

0.088

6.4

16.8

2001

1.524

< 5%

B

B

W

12.240

0.088

6.9

16.6

1368

1.262

< 5%

B

B

B

12.349

0.099

7.4

16.5

1147

1.300

< 5%

B

B

S

13.393

0.091

7

18.7

2001

1.589

< 5%

B

S

W

13.594

0.098

8.1

18

1335

1.391

< 5%

B

S

B

13.544

0.112

8.3

18.3

1000

1.372

< 5%

239

240

Appendix B

Table B.9. (continued) B

S

S

12.460

0.082

5.9

17.3

2001

1.426

< 5%

S

W

W

12.627

0.071

8.4

16.6

1714

1.136

< 5%

S

W

B

12.597

0.091

9.3

15.8

800

0.997

< 5%

S

W

S

12.293

0.083

5.9

17

2001

1.438

< 5%

S

B

W

12.375

0.070

8.8

16.2

1660

1.111

< 5%

S

B

B

12.239

0.094

9.8

16.1

743

0.996

< 5%

S

B

S

13.570

0.084

6.7

18.2

2001

1.467

< 5%

S

S

W

13.536

0.086

8.1

19.5

2001

1.501

< 5%

S

S

B

13.799

0.112

10.2

17.1

728

1.170

< 5%

S

S

S

12.150

0.095

5.8

17.1

2001

1.652

< 5%

Table B.10. Negotiation model results for reconfiguration Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

8.392

0.166

0

22

2001

2.889

< 5%

W

W

W

8.521

0.195

2

19

1330

2.755

< 5%

W

W

B

8.624

0.218

1

19

1102

2.812

< 5%

W

W

S

7.290

0.162

0

23

2001

2.813

< 5%

W

B

W

7.557

0.191

1

19

1358

2.740

< 5%

W

B

B

7.371

0.204

0

16

1107

2.640

< 5%

W

B

S

7.508

0.163

0

19

2001

2.825

< 5%

W

S

W

7.290

0.187

0

20

1315

2.628

< 5%

W

S

B

6.041

0.155

0

17

2001

2.689

< 5%

W

S

S

7.224

0.149

1

16

2001

2.584

< 5%

B

W

W

7.350

0.171

0

18

1456

2.537

< 5%

B

W

B

7.163

0.190

0

14

1048

2.391

< 5%

B

W

S

6.000

0.139

0

16

2001

2.414

< 5%

B

B

W

5.966

0.154

1

16

1368

2.210

< 5%

B

B

B

6.085

0.168

0

14

1147

2.214

< 5%

B

B

S

4.600

0.125

0

12

2001

2.163

< 5%

B

S

W

4.352

0.142

0

13

1335

2.016

< 5%

Appendix B

241

Table B.10. (continued) B

S

B

4.155

0.160

0

12

1000

1.960

< 5%

B

S

S

6.246

0.144

0

15

2001

2.499

< 5%

S

W

W

6.122

0.148

0

15

1714

2.375

< 5%

S

W

B

6.183

0.223

0

15

800

2.451

< 5%

S

W

S

4.831

0.127

0

14

2001

2.206

< 5%

S

B

W

4.556

0.128

0

13

1660

2.026

< 5%

S

B

B

4.708

0.188

0

13

743

1.990

< 5%

S

B

S

3.187

0.103

0

11

2001

1.786

< 5%

S

S

W

3.148

0.103

0

14

2001

1.783

< 5%

S

S

B

2.297

0.149

0

10

728

1.560

< 5%

S

S

S

8.392

0.166

0

22

2001

2.889

< 5%

Table B.11. Negotiation model results for absolute residual Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

99.455

1.930

17

285

2001

33.534

< 5%

W

W

W

78.208

1.915

18

224

1330

27.127

< 5%

W

W

B

72.400

1.876

23

154

1102

24.189

< 5%

W

W

S

99.618

1.911

16

266

2001

33.194

< 5%

W

B

W

79.564

1.960

15

214

1358

28.053

< 5%

W

B

B

73.294

1.940

20

170

1107

25.064

< 5%

W

B

S

100.044

1.970

25

282

2001

34.220

< 5%

W

S

W

79.071

1.917

27

200

1315

27.003

< 5%

W

S

B

100.966

1.946

28

301

2001

33.806

< 5%

W

S

S

90.706

1.662

31

262

2001

28.873

< 5%

B

W

W

66.760

1.299

25

171

1456

19.246

< 5%

B

W

B

60.804

1.328

21

130

1048

16.700

< 5%

B

W

S

90.959

1.642

27

224

2001

28.520

< 5%

B

B

W

67.991

1.366

17

160

1368

19.617

< 5%

B

B

B

60.643

1.260

23

121

1147

16.569

< 5%

B

B

S

91.443

1.630

27

222

2001

28.320

< 5%

B

S

W

69.216

1.384

24

177

1335

19.645

< 5%

242

Appendix B

Table B.11. (continued) B

S

B

62.961

1.405

23

132

1000

17.253

< 5%

B

S

S

80.350

1.562

19

197

2001

27.142

< 5%

S

W

W

54.680

0.901

18

120

1714

14.481

< 5%

S

W

B

48.356

1.032

17

93

800

11.341

< 5%

S

W

S

80.850

1.603

23

205

2001

27.839

< 5%

S

B

W

54.541

0.908

20

135

1660

14.364

< 5%

S

B

B

48.015

0.998

13

83

743

10.563

< 5%

S

B

S

80.982

1.629

22

229

2001

28.296

< 5%

S

S

W

79.678

1.619

17

213

2001

28.130

< 5%

S

S

B

50.346

1.105

18

85

728

11.579

< 5%

S

S

S

99.455

1.930

17

285

2001

33.534

< 5%

Table B.12. Game-theoretical model results for efficiency Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

17.336

0.135

8.6

23.9

1973

2.321

< 5%

W

W

W

17.556

0.148

10.4

23.7

1307

2.079

< 5%

W

W

B

17.721

0.166

11.4

23.2

1018

2.061

< 5%

W

W

S

17.351

0.137

9.4

23.5

1933

2.341

< 5%

W

B

W

17.578

0.149

11.8

23.8

1242

2.037

< 5%

W

B

B

17.675

0.157

11.1

23.5

1071

1.997

< 5%

W

B

S

17.342

0.129

8

24.1

2104

2.305

< 5%

W

S

W

17.585

0.150

11.3

23.7

1284

2.081

< 5%

W

S

B

17.310

0.142

7.3

24.4

1921

2.413

< 5%

W

S

S

17.639

0.121

7.9

24.2

2341

2.277

< 5%

B

W

W

17.846

0.134

10.7

23

1334

1.907

< 5%

B

W

B

18.038

0.160

11.3

23.6

962

1.925

< 5%

B

W

S

17.594

0.122

8.4

23.6

2426

2.326

< 5%

B

B

W

17.955

0.132

10.6

24

1405

1.915

< 5%

B

B

B

17.891

0.153

12.1

23.6

948

1.834

< 5%

B

B

S

17.484

0.122

7.2

23.6

2409

2.331

< 5%

B

S

W

17.959

0.138

11.8

23.7

1294

1.931

< 5%

Appendix B

Table B.12. (continued) B

S

B

17.986

0.148

11.5

24.1

974

1.790

< 5%

B

S

S

17.936

0.106

7.6

24.1

2889

2.213

< 5%

S

W

W

18.767

0.116

13.7

24.1

1649

1.826

< 5%

S

W

B

18.766

0.145

14.6

23.3

692

1.486

< 5%

S

W

S

17.905

0.109

7.6

23.8

3053

2.333

< 5%

S

B

W

18.649

0.118

12.5

23.5

1583

1.826

< 5%

S

B

B

19.160

0.173

14.5

23.3

632

1.688

< 5%

S

B

S

17.912

0.098

8.9

24.1

3060

2.115

< 5%

S

S

W

17.915

0.101

8.4

23.8

3285

2.250

< 5%

S

S

B

18.940

0.182

14.2

23.2

645

1.794

< 5%

S

S

S

17.336

0.135

8.6

23.9

1973

2.321

< 5%

Table B.13. Game-theoretical model results for distance Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

13.229

0.108

6.7

19

1973

1.862

< 5%

W

W

W

13.357

0.112

8.3

18.9

1307

1.571

< 5%

W

W

B

13.471

0.124

8.1

18.6

1018

1.537

< 5%

W

W

S

13.237

0.104

5.8

19

1933

1.783

< 5%

W

B

W

13.517

0.116

8.3

18.6

1242

1.588

< 5%

W

B

B

13.613

0.120

8.6

19

1071

1.525

< 5%

W

B

S

13.229

0.104

5.3

18.8

2104

1.848

< 5%

W

S

W

13.401

0.116

6.4

17.9

1284

1.614

< 5%

W

S

B

13.257

0.111

5.1

19

1921

1.891

< 5%

W

S

S

13.518

0.093

5.3

18.7

2341

1.745

< 5%

B

W

W

13.578

0.098

8

17.8

1334

1.386

< 5%

B

W

B

13.592

0.111

10

17.6

962

1.341

< 5%

B

W

S

13.412

0.094

6.2

18.8

2426

1.795

< 5%

B

B

W

13.654

0.101

8.9

17.8

1405

1.472

< 5%

B

B

B

13.695

0.111

9.2

17.9

948

1.323

< 5%

B

B

S

13.327

0.092

5.3

18.3

2409

1.761

< 5%

B

S

W

13.620

0.099

9.4

17.4

1294

1.384

< 5%

243

244

Appendix B

Table B.13. (continued) B

S

B

13.807

0.111

10

17.9

974

1.350

< 5%

B

S

S

13.505

0.077

6.1

18.8

2889

1.612

< 5%

S

W

W

13.571

0.072

9.9

16.6

1649

1.129

< 5%

S

W

B

13.674

0.099

10.1

16.6

692

1.013

< 5%

S

W

S

13.473

0.079

4.9

17.9

3053

1.701

< 5%

S

B

W

13.597

0.073

8.9

16.9

1583

1.126

< 5%

S

B

B

13.695

0.099

9.9

16.3

632

0.966

< 5%

S

B

S

13.735

0.079

7.1

18.9

3060

1.689

< 5%

S

S

W

13.485

0.073

6.3

18.5

3285

1.631

< 5%

S

S

B

13.598

0.094

10.5

16.4

645

0.929

< 5%

S

S

S

13.229

0.108

6.7

19

1973

1.862

< 5%

Table B.14. Game-theoretical model results for reconfiguration Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

7.313

0.170

0

18

1973

2.933

< 5%

W

W

W

7.301

0.213

0

17

1307

2.997

< 5%

W

W

B

7.059

0.225

0

16

1018

2.792

< 5%

W

W

S

7.305

0.172

0

16

1933

2.935

< 5%

W

B

W

7.173

0.206

0

19

1242

2.814

< 5%

W

B

B

7.049

0.229

0

16

1071

2.905

< 5%

W

B

S

7.306

0.165

0

17

2104

2.937

< 5%

W

S

W

7.114

0.206

0

21

1284

2.862

< 5%

W

S

B

7.349

0.172

0

18

1921

2.928

< 5%

W

S

S

6.090

0.154

0

19

2341

2.897

< 5%

B

W

W

5.473

0.187

0

15

1334

2.648

< 5%

B

W

B

5.201

0.216

0

14

962

2.604

< 5%

B

W

S

6.019

0.152

0

18

2426

2.907

< 5%

B

B

W

5.506

0.185

0

16

1405

2.692

< 5%

B

B

B

5.412

0.225

0

16

948

2.695

< 5%

B

B

S

6.078

0.155

0

18

2409

2.947

< 5%

B

S

W

5.389

0.190

0

15

1294

2.653

< 5%

Appendix B

245

Table B.14. (continued) B

S

B

5.213

0.217

0

15

974

2.634

< 5%

B

S

S

4.372

0.137

0

19

2889

2.868

< 5%

S

W

W

3.035

0.131

0

13

1649

2.070

< 5%

S

W

B

2.436

0.116

0

10

692

1.701

< 5%

S

W

S

4.373

0.131

0

17

3053

2.817

< 5%

S

B

W

3.015

0.137

0

13

1583

2.120

< 5%

S

B

B

2.725

0.136

0

11

632

1.811

< 5%

S

B

S

4.384

0.130

0

17

3060

2.802

< 5%

S

S

W

4.409

0.127

0

16

3285

2.821

< 5%

S

S

B

2.488

0.106

0

10

645

1.733

< 5%

S

S

S

7.313

0.170

0

18

1973

2.933

< 5%

Table B.15. Game-theoretical model results for absolute residual Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

C

R

Q

86.182

1.974

29

185

1018

33.084

< 5%

W

W

W

93.291

2.012

26

219

1307

28.247

< 5%

W

W

B

86.182

1.974

29

185

1018

24.458

< 5%

W

W

S

116.245

2.039

42

302

1933

34.808

< 5%

W

B

W

92.717

2.009

26

203

1242

27.502

< 5%

W

B

B

86.317

2.001

31

204

1071

25.425

< 5%

W

B

S

114.494

1.937

38

260

2104

34.506

< 5%

W

S

W

93.139

2.024

30

211

1284

28.170

< 5%

W

S

B

115.081

2.005

30

295

1921

34.134

< 5%

W

S

S

97.415

1.633

28

252

2341

30.692

< 5%

B

W

W

71.642

1.434

22

158

1334

20.343

< 5%

B

W

B

64.127

1.397

25

140

962

16.822

< 5%

B

W

S

96.167

1.538

30

251

2426

29.415

< 5%

B

B

W

71.421

1.341

30

179

1405

19.514

< 5%

B

B

B

64.395

1.396

20

123

948

16.690

< 5%

B

B

S

96.281

1.552

28

240

2409

29.581

< 5%

B

S

W

71.702

1.382

22

156

1294

19.307

< 5%

246

Appendix B

Table B.15. (continued) B

S

B

65.642

1.439

27

146

974

17.443

< 5%

B

S

S

86.261

1.424

22

207

2889

29.718

< 5%

S

W

W

57.344

1.030

18

158

1649

16.243

< 5%

S

W

B

51.568

1.062

21

91

692

10.851

< 5%

S

W

S

84.639

1.342

20

217

3053

28.795

< 5%

S

B

W

57.830

0.980

22

116

1583

15.141

< 5%

S

B

B

52.350

1.157

19

89

632

11.295

< 5%

S

B

S

84.892

1.343

24

221

3060

28.850

< 5%

S

S

W

85.350

1.309

19

222

3285

29.131

< 5%

S

S

B

51.774

1.122

23

85

645

11.069

< 5%

S

S

S

86.182

1.974

29

185

1018

33.084

< 5%

C Simulation Input Parameters and Results Related to Chapter 8

Table C.1. Values of V gm parameter Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Plant 1

0.9

0.2

0.5

–

–

–

Plant 2

0.8

0.2

0.9

–

–

–

Plant 3

0.4

0.5

0.7

–

–

–

Plant 4

0.6

0.7

0.9

–

–

–

Plant 5

1.0

0.2

0.8

–

–

–

Plant 6

0.8

0.5

0.6

–

–

–

Plant 7

0.3

0.4

0.6

–

–

–

Plant 8

0.8

0.9

0.3

–

–

–

Plant 9

–

–

–

0.9

0.2

0.5

Plant 10

–

–

–

0.8

0.2

0.9

Plant 11

–

–

–

0.4

0.5

0.7

Plant 12

–

–

–

0.6

0.7

0.9

Plant 13

–

–

–

1.0

0.2

0.8

Plant 14

–

–

–

0.8

0.5

0.6

Plant 15

–

–

–

0.3

0.4

0.6

Plant 16

–

–

–

0.8

0.9

0.3

248

Appendix C

Table C.2. Values of V gm parameter Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Plant 1

0.6

0.3

0.8

–

–

–

Plant 2

0.8

1.1

0.3

–

–

–

Plant 3

0.9

1.2

0.7

–

–

–

Plant 4

1.1

0.4

0.9

–

–

–

Plant 5

0.5

0.7

1.2

–

–

–

Plant 6

0.6

0.5

0.7

–

–

–

Plant 7

0.6

1.2

1.0

–

–

–

Plant 8

1.1

0.9

0.9

–

–

–

Plant 9

–

–

–

0.6

0.3

0.8

Plant 10

–

–

–

0.8

1.1

0.3

Plant 11

–

–

–

0.9

1.2

0.7

Plant 12

–

–

–

1.1

0.4

0.9

Plant 13

–

–

–

0.5

0.7

1.2

Plant 14

–

–

–

0.6

0.5

0.7

Plant 15

–

–

–

0.6

1.2

1.0

Plant 16

–

–

–

1.1

0.9

0.9

Table C.3. Utility functions data

p min

p max g

pm*

Rc

0.5

2

0.5

0.6

g

Table C.4. Centralised model results for efficiency Input parameters’levels

Avg

Half

Min

Max

Rep

Var

Check

WLg

Cg

Qm

Rm

W

W

W

W

7.928

0.205

2.2

11.7

301

1.38

< 10%

W

W

W

S

8.297

0.236

0.6

11.3

301

1.59

< 10%

W

W

S

W

7.903

0.223

0.8

11

301

1.51

< 10%

W

W

S

S

8.332

0.248

1.4

11.3

301

1.67

< 10%

W

S

W

W

7.980

0.125

5.7

10.6

301

0.84

< 10%

Appendix C

249

Table C.4. (continued) W

S

W

S

8.534

0.112

6.7

10.4

301

0.76

< 10%

W

S

S

W

7.454

0.182

6.2

9.6

120

0.77

< 10%

W

S

S

S

7.577

0.045

6.2

7.6

84

0.16

< 10%

S

W

W

W

7.984

0.231

1.2

11.2

301

1.56

< 10%

S

W

W

S

8.369

0.203

2

11.3

301

1.37

< 10%

S

W

S

W

8.026

0.209

2.8

10.9

301

1.41

< 10%

S

W

S

S

8.470

0.231

0

11.6

301

1.55

< 10%

S

S

W

W

7.901

0.128

5.5

10.3

301

0.86

< 10%

S

S

W

S

8.546

0.129

6.1

10.7

301

0.87

< 10%

S

S

S

W

7.474

0.175

5.9

9.4

121

0.75

< 10%

S

S

S

S

7.317

0.053

6.7

7.4

87

0.19

< 10%

Table C.5. Centralised model results for distance Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WLg

Cg

Qm

Rm

W

W

W

W

5.957

0.164

1.3

8.1

301

1.10

< 10%

W

W

W

S

6.294

0.175

0.5

8.2

301

1.18

< 10%

W

W

S

W

5.937

0.173

0.7

9

301

1.17

< 10%

W

W

S

S

6.328

0.185

1.1

8.6

301

1.25

< 10%

W

S

W

W

6.150

0.081

4.7

8

301

0.55

< 10%

W

S

W

S

6.540

0.076

4.7

8

301

0.51

< 10%

W

S

S

W

5.855

0.105

5.2

7.3

120

0.45

< 10%

W

S

S

S

6.282

0.036

5.2

6.3

84

0.13

< 10%

S

W

W

W

6.004

0.172

0.8

9.2

301

1.16

< 10%

S

W

W

S

6.387

0.157

1.9

8.1

301

1.06

< 10%

S

W

S

W

6.075

0.162

1.3

8.4

301

1.09

< 10%

S

W

S

S

6.482

0.169

0

8.2

301

1.14

< 10%

S

S

W

W

6.107

0.077

4.2

7.8

301

0.52

< 10%

S

S

W

S

6.498

0.077

4.4

7.9

301

0.52

< 10%

S

S

S

W

5.790

0.101

5.2

7

121

0.43

< 10%

S

S

S

S

5.936

0.041

5.6

6

87

0.15

< 10%

250

Appendix C

Table C.6. Centralised model results for absolute residual Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WLg

Cg

Qm

Rm

W

W

W

W

51.761

3.812

10

172

301

25.68

< 10%

W

W

W

S

57.385

4.936

1

188

301

33.26

< 10%

W

W

S

W

42.226

3.988

4

158

301

26.87

< 10%

W

W

S

S

43.216

4.199

1

158

301

28.29

< 10%

W

S

W

W

36.641

1.748

12

73

301

11.78

< 10%

W

S

W

S

36.289

1.687

11

66

301

11.37

< 10%

W

S

S

W

22.450

0.914

12

39

120

3.89

< 10%

W

S

S

S

23.298

0.844

17

32

84

3.01

< 10%

S

W

W

W

53.674

4.659

8

165

301

31.39

< 10%

S

W

W

S

52.864

4.257

6

165

301

28.69

< 10%

S

W

S

W

46.744

4.550

8

161

301

30.66

< 10%

S

W

S

S

49.030

4.293

7

202

301

28.92

< 10%

S

S

W

W

36.917

1.849

8

86

301

12.46

< 10%

S

S

W

S

37.535

1.898

12

83

301

12.79

< 10%

S

S

S

W

22.347

1.005

14

42

121

4.29

< 10%

S

S

S

S

23.184

1.116

16

34

87

4.04

< 10%

Table C.7. Game-theoretical model results for efficiency Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WLg

Cg

Qm

Rm

W

W

W

W

8.880

0.153

3.3

11.6

301

1.35

< 10%

W

W

W

S

8.940

0.152

3.9

11.1

301

1.35

< 10%

W

W

S

W

8.974

0.148

3.8

12

301

1.31

< 10%

W

W

S

S

8.943

0.165

2

11.8

301

1.46

< 10%

W

S

W

W

9.434

0.110

6.7

11.8

179

0.75

< 10%

W

S

W

S

9.418

0.132

7.4

10.9

152

0.83

< 10%

W

S

S

W

9.200

0.000

9.2

9.2

22

–

< 10%

W

S

S

S

9.200

0.000

9.2

9.2

14

–

< 10%

Appendix C

251

Table C.7. (continued) S

W

W

W

8.717

0.170

2.2

11.8

301

1.51

< 10%

S

W

W

S

8.865

0.134

4

11.5

301

1.18

< 10%

S

W

S

W

8.967

0.147

3.5

11.3

301

1.31

< 10%

S

W

S

S

8.859

0.145

4.1

11.3

301

1.28

< 10%

S

S

W

W

9.428

0.124

7.5

10.8

142

0.75

< 10%

S

S

W

S

9.298

0.141

6.7

11.3

121

0.79

< 10%

S

S

S

W

9.200

0.000

9.2

9.2

12

–

< 10%

S

S

S

S

9.200

0.000

9.2

9.2

16

–

< 10%

Table C.8. Game-theoretical model results for distance Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WLg

Cg

Qm

Rm

W

W

W

W

6.707

0.126

2.1

8.8

301

1.12

< 10%

W

W

W

S

6.677

0.143

1.9

8.9

301

1.27

< 10%

W

W

S

W

6.792

0.130

1.8

8.6

301

1.15

< 10%

W

W

S

S

6.729

0.132

0.8

9.2

301

1.16

< 10%

W

S

W

W

6.897

0.076

5.3

8.2

179

0.52

< 10%

W

S

W

S

6.690

0.092

4.8

8

152

0.58

< 10%

W

S

S

W

6.600

0.000

6.6

6.6

22

–

< 10%

W

S

S

S

6.600

0.000

6.6

6.6

14

–

< 10%

S

W

W

W

6.522

0.147

1.4

8.7

301

1.30

< 10%

S

W

W

S

6.711

0.117

2.2

9.4

301

1.03

< 10%

S

W

S

W

6.791

0.126

2.4

9.2

301

1.12

< 10%

S

W

S

S

6.747

0.137

2.4

9

301

1.21

< 10%

S

S

W

W

6.818

0.097

4.6

8.3

142

0.59

< 10%

S

S

W

S

6.590

0.079

5.2

7.7

121

0.44

< 10%

S

S

S

W

6.600

0.000

6.6

6.6

12

–

< 10%

S

S

S

S

6.600

0.000

6.6

6.6

16

–

< 10%

252

Appendix C

Table C.9. Game-theoretical model results for absolute residual Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WLg

Cg

Qm

Rm

W

W

W

W

58.791

3.304

16

214

301

29.24

< 10%

W

W

W

S

59.694

3.349

11

182

301

29.65

< 10%

W

W

S

W

53.668

3.563

7

165

301

31.54

< 10%

W

W

S

S

47.153

3.160

7

156

301

27.97

< 10%

W

S

W

W

38.927

1.922

11

73

179

13.12

< 10%

W

S

W

S

40.474

1.991

10

75

152

12.52

< 10%

W

S

S

W

24.000

2.131

10

35

22

–

< 10%

W

S

S

S

25.500

2.472

19

36

14

–

< 10%

S

W

W

W

58.485

3.267

12

184

301

28.92

< 10%

S

W

W

S

58.392

3.478

11

253

301

30.79

< 10%

S

W

S

W

49.213

2.964

7

143

301

26.24

< 10%

S

W

S

S

50.741

3.238

7

150

301

28.66

< 10%

S

S

W

W

40.141

2.066

13

74

142

12.56

< 10%

S

S

W

S

39.182

1.988

13

64

121

11.16

< 10%

S

S

S

W

23.667

2.267

19

34

12

–

< 10%

S

S

S

S

24.750

2.329

15

33

16

–

< 10%

Table C.10. Negotiation model results for efficiency Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WLg

Cg

Qm

Rm

W

W

W

W

7.956

0.220

0.6

11.2

301

1.48

< 10%

W

W

W

S

8.460

0.223

1.6

12.2

301

1.51

< 10%

W

W

S

W

7.928

0.215

1.9

11.2

301

1.45

< 10%

W

W

S

S

8.296

0.225

0.8

12

301

1.52

< 10%

W

S

W

W

7.928

0.124

5.5

10.3

301

0.84

< 10%

W

S

W

S

8.437

0.126

6.2

10.6

301

0.85

< 10%

W

S

S

W

7.476

0.186

5.9

9.5

113

0.77

< 10%

W

S

S

S

7.546

0.077

6.2

7.6

54

0.22

< 10%

Appendix C

253

Table C.10. (continued) S

W

W

W

7.998

0.222

1.8

11.2

301

1.49

< 10%

S

W

W

S

8.248

0.233

0

12

301

1.57

< 10%

S

W

S

W

7.869

0.207

2.5

10.9

301

1.39

< 10%

S

W

S

S

8.427

0.222

3.1

11.9

301

1.49

< 10%

S

S

W

W

7.950

0.123

5.9

10.2

301

0.83

< 10%

S

S

W

S

8.536

0.124

5.8

10.7

301

0.84

< 10%

S

S

S

W

7.430

0.145

5.9

9.4

153

0.70

< 10%

S

S

S

S

7.318

0.053

6.2

7.4

125

0.23

< 10%

Table C.11. Negotiation model results for distance Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WLg

Cg

Qm

Rm

W

W

W

W

6.033

0.167

0.6

8

301

1.13

< 10%

W

W

W

S

6.357

0.173

1.1

8.5

301

1.17

< 10%

W

W

S

W

5.984

0.173

1.7

8.6

301

1.16

< 10%

W

W

S

S

6.353

0.164

0.6

8

301

1.10

< 10%

W

S

W

W

6.090

0.085

4.2

7.7

301

0.57

< 10%

W

S

W

S

6.482

0.079

5.1

7.9

301

0.53

< 10%

W

S

S

W

5.790

0.099

5.2

6.9

113

0.41

< 10%

W

S

S

S

6.257

0.061

5.2

6.3

54

0.17

< 10%

S

W

W

W

5.957

0.174

0.8

8.6

301

1.17

< 10%

S

W

W

S

6.242

0.181

0

9

301

1.22

< 10%

S

W

S

W

5.954

0.159

1.4

7.9

301

1.07

< 10%

S

W

S

S

6.492

0.173

1.9

9.1

301

1.17

< 10%

S

S

W

W

6.121

0.078

4

7.5

301

0.53

< 10%

S

S

W

S

6.632

0.075

4.8

8.1

301

0.51

< 10%

S

S

S

W

5.831

0.096

5.2

7.6

153

0.46

< 10%

S

S

S

S

5.936

0.041

5.2

6

125

0.18

< 10%

254

Appendix C

Table C.12. Game-theoretical model results for absolute residual Input parameter levels

Avg

Half

Min

Max

Rep

Var

Check

WL g

Cg

Qm

Rm

W

W

W

W

53.296

4.546

3

204

301

30.63

< 10%

W

W

W

S

55.043

5.093

6

276

301

34.31

< 10%

W

W

S

W

46.694

5.109

5

210

301

34.42

< 10%

W

W

S

S

47.156

4.654

1

155

301

31.35

< 10%

W

S

W

W

36.947

1.824

11

82

301

12.29

< 10%

W

S

W

S

37.937

1.752

12

75

301

11.81

< 10%

W

S

S

W

22.336

1.079

12

38

113

4.45

< 10%

W

S

S

S

24.037

1.034

17

32

54

2.95

< 10%

S

W

W

W

50.415

4.249

11

217

301

28.63

< 10%

S

W

W

S

54.864

4.590

11

197

301

30.92

< 10%

S

W

S

W

44.223

4.266

1

150

301

28.74

< 10%

S

W

S

S

45.007

4.335

7

166

301

29.21

< 10%

S

S

W

W

36.096

1.861

7

77

301

12.54

< 10%

S

S

W

S

39.126

1.708

8

76

301

11.51

< 10%

S

S

S

W

23.353

0.941

14

42

153

4.52

< 10%

S

S

S

S

23.168

0.986

13

33

125

4.28

< 10%

Index

added-value principle, 21 advanced planning and scheduling (APS), v agent mechanism design, 49 ANOVA, 77, 107 artificial intelligence, 42 axiomatic approach, 21, 23 behaviour-dependent strategy, 37 Bézier curve, 70 brownian motion model, 5 capacity allocation, 99 capacity balancing, 60 capacity configuration, 55 capacity expansion, 2, 54 capacity management, 2 capacity ownership, 55, 60 capacity planning, 54 centralised planning, 66, 100 characteristic function, 20, 69, 103 class diagram, 74, 104 coalition, 19 empty coalition, 19 grand coalition, 19 coalition structure, 19 communication channel (cc), 59 cooperation, 29 coordination, v, 3, 9, 29, 48, 52, 227 decentralised production network, 1 decision making, 34 corporate level, 53 group level, 53

decision support system, 5 decentralised decision-making, 6 demand uncertainty, 3 design of experiment (DoE), 77, 90, 107 distributed artificial intelligence, 29, 48 distributed decision making, 48 distributed production planning, v, 9, 52, 56, 173 distributed production system, 53 electronic negotiation, 31 electronic procurement, 28 fall-back outcome, 73 Foundation for Intelligent Physical Agents (FIPA), 50 game theory, 13, 68, 102 cooperative game, 18 core game, 69 dynamic game, 17 game payoff, 14, 15 non-cooperative games, 15 transfer utility game, 68, 103 trigger strategy, 18 generative function, 60, 101 graph search problem, 47 heuristic models, 8 high-tech industry, 2, 55, 227 high-volume industry, 2, 9, 227 IDEF0, viii, 56, 173 imitative tactic, 102 implicit collusion, 18

256

Index

Knowledge Query and Manipulation Language (KQML), 50 linear programming, 7 LINGO® package, 73, 103 mathematical programming, 6 multi-agent systems, 30, 42, 103 agent architecture, 56 agent policy, 45 agent reasoning, 45 multi-attribute utility theory (MAUT), 34 Nash equilibrium, 15, 37 negotiation, 26, 67, 101 credits, 59, 99 negotiation protocol, 32 round, 101 news vendor model, 3 on-line auctions, 28 operational planning, 54 optimal control, 45 order assignement, 61 ownership assigment, 59 Pareto chart, 186, 187 Pareto efficiency, 70 Pareto frontier, 16 Pareto optimal, 23, 32 perceptual aliasing, 46 plant assigment, 60 production capacity, 54 production network, v production planning, 53

queuing network model, 8 reactive function, 60, 101 real options theory, 4 reconfigurable enterprise, 9, 52 reconfigurable manufacturing systems, 1, 52 reconfigurable production network, 52, 173 reconfiguration cost, 100 reservation price, 101 resource assignment, 62 resource-dependent strategy, 36 response function, 15 risk attitude, 4, 60, 102, 106 risk sharing, 3 Rubinstein model, 25, 36, 39, 40 semiconductor industry, 2, 54 sequence diagram, 75, 105 Shapley value, 19, 21 simulation, 73, 103 simultaneous response protocol, 36 software agent, 29 stable set, 21 stochastic programming, 6 stochastic transition model, 47 superadditivity, 20 time-dependent tactic, 102 Tukey test, 82 utility function, 47, 70, 71 wafer fabrication, 12, 54, 55