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Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen
Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany
6729
Ying Tan Yuhui Shi Yi Chai Guoyin Wang (Eds.)
Advances in Swarm Intelligence Second International Conference, ICSI 2011 Chongqing, China, June 12-15, 2011 Proceedings, Part II
13
Volume Editors Ying Tan Peking University Key Laboratory of Machine Perception (MOE) Department of Machine Intelligence Beijing, 100871, China E-mail: [email protected] Yuhui Shi Xi’an Jiaotong-Liverpool University Department of Electrical and Electronic Engineering Suzhou, 215123,China E-mail: [email protected] Yi Chai Chongqing University Automation College Chongqing 400030, China E-mail: [email protected] Guoyin Wang Chongqing University of Posts and Telecommunications College of Computer Science and Technology Chongqing, 400065, China E-mail: [email protected] ISSN 0302-9743 e-ISSN 1611-3349 ISBN 978-3-642-21523-0 e-ISBN 978-3-642-21524-7 DOI 10.1007/978-3-642-21524-7 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011928465 CR Subject Classification (1998): F.1, H.3, I.2, H.4, H.2.8, I.4-5 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues
This book and its companion volume, LNCS vols. 6728 and 6729, constitute the proceedings of the Second International Conference on Swarm Intelligence (ICSI 2011) held during June 12–15, 2011 in Chongqing, well known as the Mountain City, the southwestern commercial capital of China. ICSI 2011 was the second gathering in the world for researchers working on all aspects of swarm intelligence, following the successful and fruitful Beijing ICSI event in 2010, which provided a high-level international academic forum for the participants to disseminate their new research findings and discuss emerging areas of research. It also created a stimulating environment for the participants to interact and exchange information on future challenges and opportunities in the field of swarm intelligence research. ICSI 2011 received 298 submissions from about 602 authors in 38 countries and regions (Algeria, American Samoa, Argentina, Australia, Austria, Belize, Bhutan, Brazil, Canada, Chile, China, Germany, Hong Kong, Hungary, India, Islamic Republic of Iran, Japan, Republic of Korea, Kuwait, Macau, Madagascar, Malaysia, Mexico, New Zealand, Pakistan, Romania, Saudi Arabia, Singapore, South Africa, Spain, Sweden, Chinese Taiwan, Thailand, Tunisia, Ukraine, UK, USA, Vietnam) across six continents (Asia, Europe, North America, South America, Africa, and Oceania). Each submission was reviewed by at least 2 reviewers, and on average 2.8 reviewers. Based on rigorous reviews by the Program Committee members and reviewers, 143 high-quality papers were selected for publication in the proceedings with an acceptance rate of 47.9%. The papers are organized in 23 cohesive sections covering all major topics of swarm intelligence research and development. In addition to the contributed papers, the ICSI 2011 technical program included four plenary speeches by Russell C. Eberhart (Indiana University Purdue University Indianapolis (IUPUI), USA), K. C. Tan (National University of Singapore, Singapore, the Editor-in-Chief of IEEE Computational Intelligence Magazine (CIM)), Juan Luis Fernandez Martnez (University of Oviedo, Spain), Fernando Buarque (University of Pernambuco, Brazil). Besides the regular oral sessions, ICSI 2011 had two special sessions on ‘Data Fusion and Swarm Intelligence’ and ‘Fish School Search Foundations and Application’ as well as several poster sessions focusing on wide areas. As organizers of ICSI 2011, we would like to express sincere thanks to Chongqing University, Peking University, Chongqing University of Posts and Telecommunications, and Xi’an Jiaotong-Liverpool University for their sponsorship, to the IEEE Computational Intelligence Society, World Federation on Soft Computing, International Neural Network Society, and Chinese Association for Artificial Intelligence for their technical co-sponsorship. We appreciate the Natural Science Foundation of China for its financial and logistic supports.
VI
Preface
We would also like to thank the members of the Advisory Committee for their guidance, the members of the International Program Committee and additional reviewers for reviewing the papers, and members of the Publications Committee for checking the accepted papers in a short period of time. Particularly, we are grateful to the proceedings publisher Springer for publishing the proceedings in the prestigious series of Lecture Notes in Computer Science. Moreover, we wish to express our heartfelt appreciation to the plenary speakers, session chairs, and student helpers. There are still many more colleagues, associates, friends, and supporters who helped us in immeasurable ways; we express our sincere gratitude to them all. Last but not the least, we would like to thank all the speakers and authors and participants for their great contributions that made ICSI 2011 successful and all the hard work worthwhile. June 2011
Ying Tan Yuhui Shi Yi Chai Guoyin Wang
Organization
General Chairs Russell C. Eberhart Dan Yang Ying Tan
Indiana University - Purdue University, USA Chongqing University, China Peking University, China
Advisory Committee Chairs Xingui He Qidi Wu Gary G. Yen
Peking University, China Tongji University, China Oklahoma State University, USA
Program Committee Chairs Yuhui Shi Guoyin Wang
Xi’an Jiaotong-Liverpool University, China Chongqing University of Posts and Telecommunications, China
Technical Committee Chairs Yi Chai Andries Engelbrecht Nikola Kasabov Kay Chen Tan Peng-yeng Yin Martin Middendorf
Chongqing University, China University of Pretoria, South Africa Auckland University of Technology, New Zealand National University of Singapore, Singapore National Chi Nan University, Taiwan, China University of Leipzig, Germany
Plenary Sessions Chairs Xiaohui Cui James Tin-Yau Kwok
Oak Ridge National Laboratory, USA The Hong Kong University of Science and Technology, China
Special Sessions Chairs Majid Ahmadi Hongwei Mo Yi Zhang
University of Windsor, Canada Harbin Engineering University, China Sichuan University, China
VIII
Organization
Publications Chairs Rajkumar Roy Radu-Emil Precup Yue Sun
Cranfield University, UK Politehnica University of Timisoara, Romania Chongqing University, China
Publicity Chairs Xiaodong Li Haibo He Lei Wang Weiren Shi Jin Wang
RMIT Unversity, Australia University of Rhode Island Kingston, USA Tongji University, China Chongqing University, China Chongqing University of Posts and Telecommunications, China
Finance Chairs Chao Deng Andreas Janecek
Peking University, China University of Vienna, Austria
Local Arrangements Chairs Dihua Sun Qun Liu
Chongqing University, China Chongqing University of Posts and Telecommunications, China
Program Committee Members Payman Arabshahi Carmelo Bastos Christian Blum Leandro Leandro dos Santos Coelho
University of Washington, USA University of Pernambuco, Brazil Universitat Politecnica de Catalunya, Spain
Pontif´ıcia Universidade Cat´ olica do Parana, Brazil Carlos Coello Coello CINVESTAV-IPN, Mexico Oscar Cordon European Centre for Soft Computing, Spain Jose Alfredo Ferreira Costa UFRN Universidade Federal do Rio Grande do Norte, Brazil Iain Couzin Princeton University, USA Xiaohui Cui Oak Ridge National Laboratory, USA Swagatam Das Jadavpur University, India Prithviraj Dasgupta University of Nebraska, USA Kusum Deep Indian Institute of Technology Roorkee, India Mingcong Deng Okayama University, Japan Haibin Duan Beijing University of Aeronautics and Astronautics, China
Organization
Mark Embrechts Andries Engelbrecht Wai-Keung Fung Beatriz Aurora Garro Licon Dunwei Gong Ping Guo Walter Gutjahr Qing-Long Han Haibo He Lu Hongtao Mo Hongwei Zeng-Guang Hou Huosheng Hu Guang-Bin Huang Yuancheng Huang Hisao Ishibuchi Andreas Janecek Zhen Ji Changan Jiang Licheng Jiao Colin Johnson Farrukh Aslam Khan Arun Khosla Franziska Kl¨ ugl James Kwok Xiaodong Li Yangmin Li Fernando Buarque De Lima Neto Guoping Liu Ju Liu Qun Liu Wenlian Lu Juan Luis Fernandez Martinez Wenjian Luo Jinwen Ma Bernd Meyer
IX
RPI, USA University of Pretoria, South Africa University of Manitoba, Canada CIC-IPN, Mexico China University of Mining and Technology, China Beijing Normal University, China University of Vienna, Austria Central Queensland University, Australia University of Rhode Island, USA Shanghai Jiao Tong University, China Harbin Engineering University, China Institute of Automation, Chinese Academy of Sciences, China University of Essex, UK Nanyang Technological University, Singapore Wuhan University, China Osaka Prefecture University, Japan University of Vienna, Austria Shenzhen University, China Kagawa University, Japan Xidian University, China University of Kent, UK FAST-National University of Computer and Emerging Sciences, Pakistan National Institute of Tech. Jalandhar, India ¨ Orebro University, Sweden Hong Kong University of Science and Technology, China RMIT University, Australia University of Macau, China Polytechnic School of Pernambuco, Brazil University of Glamorgan, UK Shandong University, China Chongqing University of Posts and Communications, China Fudan University, China University of Oviedo, Spain University of Science and Technology of China, China Peking University, China Monash University, Australia
X
Organization
Martin Middendorf Mahamed G. H. Omran Jeng-Shyang Pan Pan Shaoning Pang Bijaya Ketan Panigrahi Thomas Potok Radu-Emil Precup Guenter Rudolph Gerald Schaefer Yuhui Shi Michael Small Jim Smith Ponnuthurai Suganthan Norikazu Takahashi Kay-Chen Tan Ying Tan Ke Tang Peter Tino Christos Tjortjis Frans Van Den Bergh Ba-Ngu Vo Bing Wang Guoyin Wang Hongbo Wang Jiahai Wang Jin Wang Lei Wang Ling Wang Lipo Wang Benlian Xu Pingkun Yan Yingjie Yang Hoengpeng Yin Peng-Yeng Yin Dingli Yu Jie Zhang Jun Zhang Lifeng Zhang Qieshi Zhang Qingfu Zhang
University of Leipzig, Germany Gulf University for Science and Technology, Kuwait National Kaohsiung University of Applied Sciences, Taiwan, China Auckland University of Technology, New Zealand IIT Delhi, India ORNL, USA Politehnica University of Timisoara, Romania TU Dortmund University, Germany Loughborough University, UK Xi’an Jiaotong-Liverpool University, China Hong Kong Polytechnic University, China University of the West of England, UK Nanyang Technological University, Singapore Kyushu University, Japan National University of Singapore, Singapore Peking University, China University of Science and Technology of China, China University of Birmingham, UK The University of Manchester, UK CSIR, South Africa The University of Western Australia, Australia University of Hull, UK Chongqing University of Posts and Telecommunications, China Yanshan University, China Sun Yat-sen University, China Chongqing University of Posts and Telecommunications, China Tongji University, China Tsinghua University, China Nanyang Technological University, Singapore Changshu Institute of Technology, China Philips Research North America, USA De Montfort University, UK Chongqing University, China National Chi Nan University, Taiwan, China Liverpool John Moores University, UK Newcastle University, UK Waseda University, Japan Renmin University of China, China Waseda University, Japan University of Essex, UK
Organization
Dongbin Zhao Zhi-Hua Zhou
Institute of Automation, Chinese Academy of Science, China Nanjing University, China
Additional Reviewers Bi, Chongke Cheng, Chi Tai Damas, Sergio Ding, Ke Dong, Yongsheng Duong, Tung Fang, Chonglun Guo, Jun Henmi, Tomohiro Hu, Zhaohui Huang, Sheng-Jun Kalra, Gaurav Lam, Franklin Lau, Meng Cheng Leung, Carson K. Lu, Qiang Nakamura, Yukinori Osunleke, Ajiboye
Qing, Li Quirin, Arnaud Saleem, Muhammad Samad, Rosdiyana Sambo, Francesco Singh, Satvir Sun, Fuming Sun, Yang Tang, Yong Tong, Can V´ azquez, Roberto A. Wang, Hongyan Wang, Lin Yanou, Akira Zhang, Dawei Zhang, X.M. Zhang, Yong Zhu, Yanqiao
Research of Pareto-Based Multi-Objective Optimization for Multi-vehicle Assignment Problem Based on MOPSO . . . . . . . . . . . . . . . . . Ai Di-Ming, Zhang Zhe, Zhang Rui, and Pan Feng
Improved Multiobjective Particle Swarm Optimization for Environmental/Economic Dispatch Problem in Power System . . . . . . . . . Yali Wu, Liqing Xu, and Jingqian Xue
49
A New Multi-Objective Particle Swarm Optimization Algorithm for Strategic Planning of Equipment Maintenance . . . . . . . . . . . . . . . . . . . . . . . Haifeng Ling, Yujun Zheng, Ziqiu Zhang, and Xianzhong Zhou
Multi-robot, Swarm-robot, and Multi-agent Systems A Self-organized Approach to Collaborative Handling of Multi-robot Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tian-yun Huang, Xue-bo Chen, Wang-bao Xu, and Wei Wang
82
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Table of Contents – Part II
An Enhanced Formation of Multi-robot Based on A* Algorithm for Data Relay Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhiguang Xu, Kyung-Sik Choi, Yoon-Gu Kim, Jinung An, and Suk-Gyu Lee WPAN Communication Distance Expansion Method Based on Multi-robot Cooperation Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yoon-Gu Kim, Jinung An, Kyoung-Dong Kim, Zhi-Guang Xu, and Suk-Gyu Lee
91
99
Relative State Modeling Based Distributed Receding Horizon Formation Control of Multiple Robot Systems . . . . . . . . . . . . . . . . . . . . . . . Wang Zheng, He Yuqing, and Han Jianda
Comparison of Ensemble Classifiers in Extracting Synonymous Chinese Transliteration Pairs from Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chien-Hsing Chen and Chung-Chian Hsu
236
Combining Classifiers by Particle Swarms with Local Search . . . . . . . . . . . Liying Yang
244
An Expert System Based on Analytical Hierarchy Process for Diabetes Risk Assessment (DIABRA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mohammad Reza Amin-Naseri and Najmeh Neshat
252
Practice of Crowd Evacuating Process Model with Cellular Automata Based on Safety Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shi Xi Tang and Ke Ming Tang
260
Feature Selection Algorithms Feature Selectionfor Unlabeled Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chien-Hsing Chen Feature Selection Algorithm Based on Least Squares Support Vector Machine and Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . Song Chuyi, Jiang Jingqing, Wu Chunguo, and Liang Yanchun Unsupervised Local and Global Weighting for Feature Selection . . . . . . . . Nadia Mesghouni, Khaled Ghedira, and Moncef Temani
Calculation of Quantities of Spare Parts and the Estimation of Availability in the Repaired as Old Models . . . . . . . . . . . . . . . . . . . . . . . . . . Zhe Yin, Feng Lin, Yun-fei Guo, and Mao-sheng Lai
419
The Design of the Algorithm of Creating Sudoku Puzzle . . . . . . . . . . . . . . Jixian Meng and Xinzhong Lu Research and Validation of the Smart Power Two-Way Interactive System Based on Unified Communication Technology . . . . . . . . . . . . . . . . . Jianming Liu, Jiye Wang, Ning Li, and Zhenmin Chen
427
434
A Micro Wireless Video Transmission System . . . . . . . . . . . . . . . . . . . . . . . . Yong-ming Yang, Xue-jun Chen, Wei He, and Yu-xing Mao
An Integration Process Model of Enterprise Information System Families Based on System of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yingbo Wu, Xu Wang, and Yun Lin
479
XVIII
Table of Contents – Part II
Special Session on Data Fusion and Swarm Intelligence A Linear Multisensor PHD Filter Using the Measurement Dimension Extension Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weifeng Liu and Chenglin Wen
Theoretical Analysis of Swarm Intelligence Algorithms Particle Swarm Optimization: A Powerful Family of Stochastic Optimizers. Analysis, Design and Application to Inverse Modelling . . . . . Juan Luis Fern´ andez-Mart´ınez, Esperanza Garc´ıa-Gonzalo, Saras Saraswathi, Robert Jernigan, and Andrzej Kloczkowski
Using PSO Algorithm for Simple LSB Substitution Based Steganography Scheme in DCT Transformation Domain . . . . . . . . . . . . . . Feno Heriniaina Rabevohitra and Jun Sang
Multi-Objective Optimization for Dynamic Single-Machine Scheduling Li Nie, Liang Gao*, Peigen Li, and Xiaojuan Wang The State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People’s Republic of China [email protected]
Abstract. In this paper, a multi-objective evolutionary algorithm based on gene expression programming (MOGEP) is proposed to construct scheduling rules (SRs) for dynamic single-machine scheduling problem (DSMSP) with job release dates. In MOGEP a fitness assignment scheme, diversity maintaining strategy and elitist strategy are incorporated on the basis of original GEP. Results of simulation experiments show that the MOGEP can construct effective SRs which contribute to optimizing multiple scheduling measures simultaneously. Keywords: multi-objective optimization; gene expression programming; dynamic scheduling; single-machine.
1 Introduction Production scheduling problem is one of the most important tasks carried out in manufacturing systems and has received considerable attention in operations research literature. In this area, it is usually assumed that all the jobs to be processed are available at the beginning of the whole planning horizon. However, in many real situations, jobs may arrive over time due to transportation etc. There are many approaches have been proposed to solve production scheduling problem, such as branch and bound [1], genetic algorithms [2], tabu search [3] etc. However, these methods usually offer good quality solutions with the cost of a huge amount of computational time. Furthermore, these techniques are not applicable in dynamic or uncertain conditions because it is needed to frequently modify the original schedules to respond to the changes of system status. Scheduling with scheduling rules (SRs) that defines only the next state of the system is highly effective in such dynamic environment [4]. Due to inherent complexity and variability of scheduling problem, a considerable effort is needed to develop suitable SRs for the problem at hand. Many researchers have investigated the use of genetic programming (GP) to create problem specific SRs [4][5][6]. In our previous work, we have applied gene expression programming (GEP), a new evolutionary algorithm, on dynamic single-machine scheduling problem (DSMSP) with job release dates and demonstrated that GEP is more promising than GP to create efficient SRs [7]. All the *
work mentioned above has only concentrated on single-objective optimization. Several objectives usually must be considered simultaneously in the real-world production situation and these objectives often conflict with each other. It is not possible to have a single solution which simultaneously optimizes all objectives. To tradeoff between these objectives is necessary, which makes the multi-objective optimization problems (MOPs) more difficult than the single-objective optimization problems. There have been many multi-objective optimization evolutionary algorithm (MOEA) proposed [8][9][10][11][12][13]. However, they can not be employed to construct SRs for DSMSP. In this paper, we propose a multi-objective evolutionary algorithm based on gene expression programming (MOGEP) and apply it on optimizing of several objectives simultaneously for DSMSP. In MOGEP, (1) a fitness assignment scheme which combines Pareto-dominating relation and density information is proposed to guide the search to approximate the Pareto optimal solutions; (2) a diversity maintain strategy is used to adjust non-dominated set of each generation in order to keep the diversity of the non-dominated set; (3) an elitist strategy is used to guarantee the convergence of search. The remainder of the paper is organized as follows. In Section 2, DSMSP with job release date is described. In section 3, the fundamental concepts of multi-objective optimization are stated briefly. In Section 4, MOGEP and its application on DSMSP are elaborated. In Section 5, the experiments are presented. The final conclusions are given in Section 6.
2 Problem Description The DSMSP with job release dates is described as following: The shop floor consists of one machine and n jobs, which are released over time and are processed once on the machine without preemption. The attributes of a job, such as processing time, release date, due date, are unknown in advance till the job is currently available at the machine or arrive in the immediate future. It is assumed that the machine is available all the time and cannot process more than one job simultaneously. The task of scheduling is to determine a sequence of jobs on the machine in order to minimize several optimization criteria simultaneously, in our case, makespan, mean flow time, maximum lateness and mean tardiness. The four performance criteria are defined below.
F1 = Cmax = max(ci , i = 1,..., n) .
F2 = F =
1 n ∑ (ci − ri ) . n i =1
F3 = Lmax = max(ci − d i , i = 1,..., n) .
F4 = T =
1 n ∑ max(ci − di , 0) . n i =1
(1) (2) (3) (4)
Where, ci, ri and di denote the completion time, release date and due date of job i, respectively. n denotes the number of jobs. Cmax, F , Lmax and T denote makespan, mean flow time, maximum lateness and mean tardiness, respectively.
Multi-Objective Optimization for Dynamic Single-Machine Scheduling
3
3 Basic Concepts of Multi-Objective Optimization In the section, we shortly describe several basic concepts of multi-objective optimization and Pareto-optimality those are intensively used in the literature [14]. The multi-objective optimization problem is generally formulated as follow:
minimize X ∈Ω
F ( X ) = ( F1 ( X ), F2 ( X ),..., FL ( X )) .
(5)
Where X is a possible solution, Ω is the feasible solution space, F (i ) is the objective function and Fr (i ) is the rth objective function (for 1 ≤ r ≤ L ). A solution a dominates a solution b (or b is dominated by a), if the following conditions are satisfied:
Fi (a ) ≤ Fi (b), ∀i ∈{1, 2,..., L} .
(6)
Fi (a ) < Fi (b), ∃i ∈{1, 2,..., L} .
(7)
A solution a is indifferent to solution b, if a can not dominate b and b can not dominate a. A solution is called non-dominated solution, if it is not dominated by any other one. Pareto-optimal set is constituted of non-dominated solutions. Pareto-optimal frontier is allocations in the objective space corresponding to the Pareto-optimal set. The goal of multi-objective optimization is to find or approximate the Pareto-optimal set. It is usually not possible to have a single solution which simultaneously optimizes all objectives, therefore, an algorithm that gives a large number of alternative solutions lying on or near the Pareto-optimal front is of great practical value.
4 MOGEP for DSMSP GEP is a new technique of creating computer programs based on principle of evolution, firstly proposed by Ferreira [15]. GEP has been applied in different field, e.g. functions discovery [16], classification rules discovery [17], time series prediction [18], digital elevation model generation [19], and it shows powerful ability to solve complex problems. However, original GEP can only optimize one objective. If several objectives are demanded to be optimized concurrently, some extra steps should be designed specially. Based on original GEP, MOGEP is equipped with a fitness assignment scheme, diversity maintaining strategy and elitist strategy. 4.1 Flow of MOGEP MOGEP is executed in the following steps: Step 1: Iteration counter iter = 0. An initial population Pt which consists of N individuals are randomly generated and an empty external archive NDSet, whose size is M (M
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size M, external archive NDSet is filled with dominated solutions in population Pt; otherwise if the number of non-dominated solutions exceeds archive size M, some member in the archive NDSet is removed according with the diversity maintaining scheme (section 4.3). Step 4: If iter exceeds the maximal number of iteration, the algorithm is ended and NDSet is put out; otherwise, go to the Step 5. Step 5: According with the elitist strategy (section 4.4) the individuals in the external archive NDSet are copied directly to the next population Pt+1. Step 6: Genetic operators (section 4.6) are employed on population Pt and the offspring individuals are saved into the population Pt+1. Pt+1 size is maintained to be N . Then increment iteration counter iter = iter + 1, and go to step 2. 4.2 Fitness Assignment Scheme As for MOPs, assignment scheme of fitness is very important and effective fitness assignment scheme makes sure that the search is directed towards the Pareto-optimal solutions. In this paper, a fitness assignment scheme which combines Pareto dominance relation and the density information is proposed. Each individual at each generation is evaluated according the following steps: (1) Rank for an individual is determined. (2) Density of an individual is estimated. (3) Fitness of an individual is determined through incorporating its density information into its rank. The non-dominated sorting algorithm [9] is used to define a rank for each individual. According Pareto dominance relation the population is splits into different nondominated fronts PF1, PF2,…, PFG, where G is the number of non-dominated fronts. The individual in PFj+1 is dominated by at lease an individual in PFj (j=1,…, G-1). And the individuals in each non-dominated fronts PFj (j=1,…, G) are indifferent to each other. The rank of each individual i in PFj is assigned as below:
R (i ) = j − 1(i ∈ PFj ) .
(8)
Since the individuals in each non-dominated front do not dominate each other and have identical rank, additional density information is necessary to be incorporated to discriminate between them. The density estimation technique [8] is used to define an order among the individuals in PFj (j=1,…,G). Specifically, for each individual i in PFj, the distances to all individuals in PFj are calculated and stored in increasing order. The k k-th element is denoted as di . k is set to be the square root of the front size. The density D(i) corresponding to individual i is defined by:
D (i ) = exp(−2 * dik / d max ) .
(9)
d max = max{d ik , i ∈ PF j } .
(10)
Where
Finally, the fitness f(i) of an individual i is formulated as
f (i ) = G − R (i ) − D (i) .
(11)
Multi-Objective Optimization for Dynamic Single-Machine Scheduling
5
According with the fitness assignment scheme, the fitness of the individuals in PF1 is in the interval of [G-1,G), the fitness of those in PF2 is in the interval of [G-2,G-1), and the fitness of those in PFG is in the interval of [0,1). It is notable that fitness is to be maximized here, in other words, a better individual is assigned a higher fitness so that it may transfer fine genes to offspring with a higher probability. 4.3 Diversity Maintaining Scheme Apart from the population, an external archive, whose size is fixed, is used to save the non-dominated individuals of the population. If the number of non-dominated individuals exceeds the predefined archive size, some individuals is needed to be deleted from the archive. In order to maintain the diversity of the population, the individuals with higher density should be deleted, i.e., the individuals with lower fitness should be deleted from the archive. 4.4 Elitist Strategy Although a number of Pareto-based multi-objective optimizing algorithms, such as MOGA [10], NPGA [11], and NSGA [12] are demonstrated the capability to approximate the set of optimal trade-offs in a single optimization run, they are all Non-elitism approach. Zitzler et al. [13] had shown in his study that elitism helps in achieving better convergence in multi-objective evolutionary algorithm. In the paper, each member of the archive is regarded as an elitist and replicated directly to the next population. Being directly copied to the next population is the only way how an individual can survive several generations in addition to pure reproduction which may occur by chance. This technique is incorporated in order not to lose certain portions of the current nondominated front due to random effects. 4.5 Chromosome Representation Scheme The function set (FS) and terminal set (TS) used to construct SRs are defined as follows. FS = {+, -, *, /}. TS = { p, r, d, sl, st, wt}, where p denotes job’s processing time; r denotes job’s release date; d denotes job’s due date; sl denotes job’s positive slack, sl = max {d − p − max{t, r}, 0}, where t denotes the idle time of the machine; st denotes job’s stay time, st = max {t – r, 0}, where t is defined as above; wt denotes job’s wait time, wt = max {r – t, 0}, where t is defined as above. Chromosomes are encoded according with the stipulations: (1) The head may contain symbols from both FS and TS, whereas the tail consists only of symbols come from TS. (2) The length of head and tail must satisfy the equation tl = hl * (arg − 1) + 1, where hl and tl is the length of head and tail, respectively, and arg is the maximum number of arguments for all operations in FS. An example chromosome expressed with the elements of FS and TS defined in above is illustrated in Fig. 1(a), where underlines are used to indicate the tails. Decoding is the process transferring the chromosomes to SRs. For the example chromosome shown in Fig. 1(a), it is mapped into expression tree (ET) following a depth-first fashion (Fig. 1(b)). The ET is interpreted into a SR with a mathematical form as shown in Fig. 1(c).
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wt
-.*.+.r.d./.sl.p.wt.st.wt.p.sl.
(r+d)*sl/p-wt +
/
(a) chromosome
(c) SR
r d sl p (b) ET
Fig. 1. Encoding and decoding scheme of GEP
4.6 Genetic Operators The genetic operators are carried out on the population are listed below [15]: Selection operator creates a mating pool comprised of individuals selected from current population according to fitness by roulette wheel sampling. The roulette is spun N-M times in order to maintain the population size unchanged (Note that M individuals are copied directly from NDSet). Mutation operator randomly changes symbols in a chromosome. In order to maintain the structural organization of chromosomes, in the head, any symbol can change into any other function or terminals, while symbols in the tail can only change into terminals. Transposition operator (1) IS transposition, i.e., randomly choose a fragment begins with a function or terminal (called IS elements) and transpose it to the head of genes, except for the root of genes; (2) RIS transposition, i.e., randomly choose a fragment begins with a function (called RIS elements) and transpose it to the root of genes. In order not to affect the tail of the gene, symbols are removed from the end of the head to make room for the inserted string. Recombination operator (1) one-point recombination, i.e., split the two randomly chosen parent chromosomes into halves and swap the corresponding sections; (2) two-point recombination, i.e., split the chromosomes into three portions and swap the middle one.
5 Experiments and Results In this section, Simulation experiments are conducted on training and validating sets which comprise of several problem instances to create SRs and evaluate their performance. The problem instances are randomly generated according with the following method. The number of jobs is set to be 100. The processing times of jobs are drawn out of [1,100]. Release dates of jobs are chosen randomly from [0, c*TP], where TP denotes the sum of the processing time of all jobs, c is assigned values of 0.1. Due dates of jobs are drawn out of [r+(TP-r)*(1-T-R/2), r+(TP-r)*(1-T+R/2)], where r denotes release date of jobs, TP denotes the sum of the processing time of all jobs, T and R are assigned values of 0.5, respectively. A train set which consists of 5 problem instances is generated and used to train MOGEP to create SRs. Another 5 instances are generated with the same parameter settings and compose a validating set.
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In the experiments, MOGEP parameter settings are shown below. Population size is 200. The length of head is 10 and thereby the total length of a chromosome is 21. The mutation probability is 0.3. IS and RIS probability are 0.3 and 0.1, respectively. One-point and two-point probability are 0.2 and 0.5, respectively. GEP stops run if it finishes 100 iterations. The SRs created by MOGEP on the training sets are listed below: R1: 2p+wt(r+st)
(12)
R2: p+d+sl*wt
(13)
R3: p+(d+wt)(r+st)-sl
(14)
R4: p+(sl+wt)(r+st)+sl
(15)
To illustrate the efficiency of the SRs created by MOGEP, we compare their results on the validating set with those of the SRs created by GEP [7], which aim to optimize single objective. The results are listed in Table 1. R-Fi denotes the SRs obtained by GEP for single objective of Fi (i = 1,…,4), respectively. Ri denotes the SRs obtained by MOGEP for the objective of Fi (i = 1,…,4) simultaneously. Take the result on instance 1-1 for example. The solutions generated on instance 1-1 with the MOGEP-created SRs R1, R2, R3 and R4 are X1={5167, 1557, 3218, 385}, X2={5167, 2397, 1338, 368}, X3={5167, 2428, 1330, 385}, and X4={5167, 2292, 2601, 234}, respectively. The solution generated on instance 1-1 with the GEPcreated SR R-F1 is X-F1={5167, 2447, 3422, 675}. It is optimal with respect to the objective of makespan in the comparison with the solutions generated by other GEPcreared SRs on instance 1-1, whereas it is dominated by Xi (i=1, 2, 3, 4). The solution generated with R-F2 is X-F2={5168, 1569, 3219, 380}. It is optimal with respect to the objective of flowtime in the comparison with the other solutions generated by GEPcreated SRs. However, X1 is better than X-F2 with respect to the objective of flowtime. Although X2, X3 and X4 is worse than X-F2 with respect to the objectives of flowtime, they are distinct better than X-F2 with regard to the other objectives, respecitvely. The solution generated with R-F3 is X-F3={5167, 2434, 1330, 385}. It is optimal with respect to the objective of lateness. However, it is dominated by X3. Although X1, X2 and X4 is worse than X-F3 with respect to the objectives of lateness, they outperform X-F3 with regard to the other objectives, respectively. The solution generated with RF4 is X-F4={5215, 2270, 2649, 235}. It is optimal with respect to the objective of tardiness in the comparison with the other solutions generated by GEP-created SRs. However, X4 is better than X-F4 with respect to the objective of tardiness. Although X1, X2 and X3 is worse than X-F4 with respect to the objectives of tardiness, they outperform X-F4 with regard to the other at least two objectives, respectively. The results in table 1 demonstrate that the SRs created by GEP can generate optimal solutions on a majority of instances with respect to single objective, whereas other objectives are deteriorated meantime. The SRs created by MOGEP can generate solutions which tradeoff well between multi objectives.
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Ins.
1-1
1-2
1-3
1-4
1-5
Obj. F1 F2 F3 F4 F1 F2 F3 F4 F1 F2 F3 F4 F1 F2 F3 F4 F1 F2 F3 F4
6 Conclusions Considering the fact that jobs usually arrive over time and several optimization objectives must be considered simultaneously in many real scheduling situations, we proposed MOGEP and applied it on the construction of SRs for DSMSP. MOGEP was equipped with a fitness assignment scheme, diversity maintain strategy and elitist strategy on the basis of original GEP. Simulation experiment results demonstrate that MOGEP creates effective SRs which can generate good Pareto optimal solutions for DSMSP. These findings encourage the further improvement of MOGEP and application it on more complex scheduling problems. Acknowledgments. This research is supported by the Natural Science Foundation of China (NSFC) under Grant No. 60973086, 51005088, the Program for New Century Excellent Talents in University under Grant No. NCET-08-0232.
References 1. Balas, E.: Machine scheduling via disjunctive graphs: an implicit enumeration algorithm. Oper. Res. 17, 941–957 (1969) 2. Goldberg, D.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading (1989)
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3. Laguna, M., Barnes, J., Glover, F.: Tabu search methods for a single machine scheduling problem. J. Intell. Mauf. 2, 63–74 (1991) 4. Jakobović, D., Budin, L.: Dynamic Scheduling with Genetic Programming. In: Collet, P., Tomassini, M., Ebner, M., Gustafson, S., Ekárt, A. (eds.) EuroGP 2006. LNCS, vol. 3905, pp. 73–84. Springer, Heidelberg (2006) 5. Atlan, L., Bonnet, J., Naillon, M.: Learning Distributed Reactive Strategies by Genetic Programming for the General Job Shop Problem. In: 7th Annual Florida Artificial Intelligence Research Symposium. IEEE Press, Florida (1994) 6. Miyashita, K.: Job-shop Scheduling with Genetic Programming. In: Genetic and Evolutionary Computation Conference, pp. 505–512. Morgan Kaufmann, San Fransisco (2000) 7. Nie, L., Shao, X.Y., Gao, L., Li, W.D.: Evolving Scheduling Rules with Gene Expression Programming for Dynamic Single-machine Scheduling Problems. Int. J. Adv. Manuf. Tech. 50, 729–747 (2010) 8. Zitzler, E., Thiele, L.: Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach. IEEE T. Evolut. Comput. 3(4), 257–271 (1999) 9. Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A Fast Elitist Nondominated Sorting Genetic Algorithm for Mmulti-objective Optimization: NSGA-II. In: Schoenauer, M., Deb, K., Rudolph, G., Yao, X., Lutton, E., Merelo, J.J., Schwefel, H.-P. (eds.) Parallel Problem Solving from Nature – PPSN VI, pp. 849–858. Springer, Berlin (2000) 10. Fonseca, C.M., Fleming, P.J.: Genetic Algorithms for Multiobjective Optimization: Formulation, Discussion and Generalization. In: 5th International Conference on Genetic Algorithms, pp. 416–423. Morgan Kaufmann, California (1993) 11. Horn, J., Nafpliotis, N., Goldberg, D.E.: A Niched Pareto Genetic Algorithm for Multiobjective Optimization. In: 1st IEEE Conference on Evolutionary Computation, IEEE World Congress on Computational Computation, pp. 82–87. IEEE Press, New Jersey (1994) 12. Srinivas, N., Deb, K.: Multiobjective Optimization Using Nondominated Sorting in Genetic Algorithms. Evol. Comput. 2(3), 221–248 (1994) 13. Zitzler, E., Deb, K., Thiele, L.: Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evol. Comput. 8(2), 173–195 (2000) 14. Kacem, I., Hammadi, S., Borne, P.: Pareto-optimality Approach for Flexible Job-shop Scheduling Problems: Hybridization of Evolutionary Algorithms and Fuzzy Logic. Math. Comput. Simulat. 60, 245–276 (2002) 15. Ferreira, C.: Gene Expression Programming: A New Adaptive Algorithm for Solving Problems. Complex System 13(2), 87–129 (2001) 16. Ferreira, C.: Discovery of the Boolean Functions to the Best Density-Classification Rules Using Gene Expression Programming. In: Foster, J.A., Lutton, E., Miller, J., Ryan, C., Tettamanzi, A.G.B. (eds.) EuroGP 2002. LNCS, vol. 2278, pp. 50–60. Springer, Heidelberg (2002) 17. Zou, C., Nelson, P.C., Xiao, W., Tirpak, T.M.: Discovery of Classification Rules by Using Gene Expression Programming. In: International Conference on Artificial Intelligence, Las Vegas, pp. 1355–1361 (2002) 18. Zuo, J., Tang, C., Li, C., Yuan, C., Chen, A.: Time Series Prediction Based on Gene Expression Programming. In: Li, Q., Wang, G., Feng, L. (eds.) WAIM 2004. LNCS, vol. 3129, pp. 55–64. Springer, Heidelberg (2004) 19. Chen, Y., Tang, C., Zhu, J.: Clustering without Prior Knowledge Based on Gene Expression Programming. In: 3rd International Conference on Natural Computation, pp. 451–455 (2007)
Research of Pareto-Based Multi-Objective Optimization for Multi-Vehicle Assignment Problem Based on MOPSO Ai Di-Ming, Zhang Zhe, Zhang Rui, and Pan Feng Beijing Special Vehicles Research Institute School of Automation, Beijing Institute of Technology (BIT), 5 South Zhongguancun Street, Haidian District BeiJing, 100081 P.R.China [email protected]
Abstract. The purpose of a multi-vehicle assignment problem is to allocate vehicles to complete various missions at different destinations, meanwhile it is required to satisfy all constrains and optimize overall criteria. Combined with MOPSO algorithm, a Pareto-based multi-objective model is proposed, which includes not only the time-cost tradeoff, but also a “Constraint-First-ObjectiveNext" strategy which handles constraints as an additional objective. Numerical experimental results illustrate that it can efficiently achieve the Pareto front and demonstrate the effectiveness. Keywords: Multi-objective vehicle assignment problem, Pareto, Multi-objective Particle swarm optimizer (MOPSO).
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2 Multi-Objective Model of Vehicle Assignment Problem According to various requirements, assignment schemes may be different from each other. In this paper, the first objective is to assign vehicles to minimize the total mission time cost, and the second one is to maximize the total profit for performing tasks on all targets. The third objective is to minimize the constraint violations. 2.1 Minimize Total Time Cost A vehicle assignment problem consists of M vehicles, N destinations and K tasks, subject to all constraints. The time cost function can be defined as follows:
（
J1 = min J CN ,M , ΠK , N , ONK )
(1)
The objective defined in Equation (1) is to minimize the maximum cumulative completion time, where J(|CN,M(·)|) is the maximum value of the cumulative time matrix CN,M. The variables are explained in Table 1. Table 1. Nomenclature Items Explanation Tij = {tij}(N+M)×N, time cost matrix. tij is the flight time from node i to node j C NM = {cn,m}N×M, cumulative time matrix. cn,m stands for the cumulative time of both the n-th target and the m-th vehicle. O NK = {oi-k}1×N·K, target sequence array. oik means the execution of the k-th task at the i-th target Π KN = {лk,n}K×N UAV assignment matrix, лk,m is the vehicle number to perform k-th task at the n-th target
a. Cumulative Time matrix CNM The matrix CNM (in Equation(2)) contains a cumulative time accounting of the mission time. Each row in the CNM matrix corresponds to one destination, and each column corresponds to one vehicle. When a job of the n-th destination is completed by the mth vehicle, the cumulative completion time cn is calculated. Thus the largest value appearing in any cell of matrix CN.M is the maximum elapsed time for the mission, which is the maximum time taken by any of the vehicles. ⎡ c1,1 ... ⎢ C N , M = ⎢ ... cn , m ⎢⎣ cN ,1 ...
c1, M ⎤ ⎥ ... ⎥ cN ,M ⎥⎦ N ×M
(2)
b. Assignment Matrix ПKN The assignment matrix ПKN is a K by N matrix (in Equation(3)). Each row stands for one task and each column stands for one destination. πKN is the element in the k-th row and the n-th column of ПKN and stands for the vehicle number which performs the k-th task of the n-th target.
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.... ⎡ π 1，1 Π K , N = ⎢⎢ ... π k，n ... ⎣⎢π K，1
π 1，N ⎤
... ⎥⎥ π K，N ⎦⎥ K × N
(3)
2.2 Maximize Total Profit Since the importance differences of the tasks at various destinations and implementation capacities of the vehicles, the total profit (see Equation(4)) reflects the effectiveness of each task completion, in terms of the assignment scheme. V1,V2,…Vm are defined as the preferred value for each target. Pij represents that the abilities of the vehicles to perform the task. i is the vehicle sequence array and j is task sequence array. So the total profit for all tasks can be calculated as followed: max J 2 =
∑
i∈M , j∈N
Vi ⋅ Pij
(4)
2.3 Handle Constrains The constraints are described as follows: a. A vehicle leaves a destination at most once. b. Only one vehicle can be assigned to perform ‘Task-2’ at one destination and cannot subsequently be assigned to perform any other tasks at any destinations. c. A vehicle, coming from the outside, can visit the destination at most once. d. A vehicle can perform a task only if the preceding task at the node has been completed. Each constraints function is defined as an objective (see Equation(5)) The “Constraint-First-Objective-Next" technique is introduced. Obviously, the optimal of the vehicle assignment problem not only has a minimum time cost to complete all tasks, but also can satisfy all constraints which means Гi=0, i=1,2…R, that is J3=0. J 3 = min ∑ Γi (Π K , N , ONK )
(5)
3 MOPSO Application Based on the classic Particle Swarm Optimizer (PSO), whose parameters are set in Table 2, with the consideration of the multi-objective optimization problem, some modifications are introduced and the improved PSO is called MOPSO. Table 2. PSO parameter settings Parameters Population size Neighborhood size Pareto pool of pbest
Settings 20 2 10
Parameters Inertia weight c1, c2 Pareto pool of gbest
Settings 0.79 2.05 20
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3.1 Design of Fitness Function Aforementioned multi-objective assignment problem can be defined as follows:
J * = {min J1 (⋅), max J 2 (⋅), min J 3 (⋅)}
(6)
The purpose of the MOP is to find the Pareto optimal sets and Pareto front, so as to provide a series of potential solution to decision makers who can reach the final plan according to synthesize other information and requirements. The MOPSO includes external file maintenance, pbest and gbest information update, and so on. 3.2 Encoding The position vector X of a particle in MOPSO is a K· (N+M) vector with real values(see Equation(7)), which consists of two parts that the first K·N elements correspond to the target sequence array and the last K·M variables stand for the assignment O matrix. X KN is sorted first and then the modules of these sorted serial numbers are calculated to acquire the target sequence array, explained as Table 3. The operation of O X is similar to X KN . Π KM
O Π X =[x1,..., xn,..., xK⋅N , xK⋅N+1,..., xv ,..., xK⋅N+K⋅M ] =[XKN , XKM ]
(7)
Table 3. Target Sequence Array O X KN
O X KN (1)
O X KN (2)
O X KN (3)
X KNO (4)
Real value ONK
12.315 3
10.343 4
55.819 1
19.556 2
3.3 The Maintenance of pBest and gBest The maintenance strategy of Pareto pool for both pBest and gbest is as follows: a. If the inferior solution particles dominate some of the solutions in the Pareto pool, delete the dominated particles, and join the current solution to the Pareto pool. b. The particle which be dominated by the Pareto pool particles is directly ignored. c. If the current particle and the Pareto pool particles have no relation of dominate, and the population of the Pareto pool did not reach the scale of the pool, then the particle will join the Pareto pool directly. Otherwise, calculate the particle’s distance among other particles and remove the particle with the minimum density distance. The pBest for next iteration will be selected by the roulette-wheel selection from the Pareto pool. And all existing gBest in the set will be ranked by the density. The gbest in the set with less density will have a high probability to be selected as the gbest for next generation.
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3.4 The Details of MOPSO Algorithm The Details of MOPSO algorithm as follows: a. Initialize P[i] randomly(P is the population of particles), the speed of each particle V[i], the maximum number of iterations T and the parameters as Table 2, Create external archive A that stores nondominated solutions null; b. If T=0, Store the nondominated vectors found in P into A. c. The values of objective functions based on the position of P[i] are calculated in P and A. d. Updating of pBest. e. Insert all new nondominated solution in P into A, if they are not dominated by any of the stored solutions. All dominated solutions in A by the new solution are removed from A. If A is full, the solution to be replaced is determined by the the crowding distance values of each nondominated solution in A. f. Updating of gBest. g. Calculate the new position and the new velocity. h. The procedure from c to g is repeated until the number of iterations reaches the specified number T. Then the searching is stopped and the set of A at that time is regarded as a set of optimal solutions.
4 Numerical Experiments and Analysis Two scenarios with different destinations, vehicles and two tasks requirements are studied. The format of the ‘Scenarios’ is [Destination, Vehicle], which stands for the number of destination and vehicle separately. The minimum of the total mission time cost matrix includes flight time between nodes and the task execution time. Tij, a N×(N+M) matrix, represents the time cost of the flight time between nodes, while Tsk, a K×N×M matrix, represents the time cost of the task execution. 4.1 Experimental 1. Scenario [2, 3] For the Scenario [2, 3] with two destinations, three vehicles and two tasks, the preferred value of targets is shown in Table 4. The profits that vehicles perform different tasks are listed in the Table 5, A set of Pareto solutions can be obtained by MOPSO, and the Pareto front fitness value is shown in Table 6. Table 4. Targets prefered value of Scenario [2,3] Destination Destination1 Destination2
Preferred value V 5.0 8.0
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Table 5. The UAV received value on performing the different tasks Vehicles
Received value P Task-1 Task-2 4 7 9 5 6 9
Vehicle1 Vehicle2 Vehicle3
Table 6. The fitness of Scenario [2,3] Objectives J1 J2 J3
11.5131 185 0
Fitness 16.88552 126.6441 195 209 0 0
213.18341 210 0
4.2 Experimental 2. Scenario [3, 4] For the Scenario [3, 4] with three destinations, four vehicles and two tasks, the preferred value and profits are shown in the Table 7 and Table 8. The Pareto front fitness value is shown in Table 9. Table 7. Targets prefered value of Scenario [3,4] Destination Destination1 Destination2 Destination3
Preferred value V 5.0 15.0 10.0
Table 8. The UAV received value on performing the different tasks Vehicles Vehicle1 Vehicle2 Vehicle3 Vehicle4
Received value P Task-1 Task-2 18 10 10 20 5 12 10 4
Table 9. The fitness of Scenario [3,4] Objectives J1 J2 J3
Fitness 10 475 0
11.51671 525 0
12.98741 645 0
25.84051 670 0
219.42261 710 0
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5 Conclusion Many of the experiments with different scenarios have been tested, each scenario, a set of Pareto optimal can be achieved. On the other hand, the Pareto front are not continue, since the existence of constrain requirements which is the third fitness J3 which determines the feasible solutions are limited. In summary, the proposed multi-objective vehicle assignment model can reduce the dimension of the solution space and be easily adapted by MOPSO algorithms. Furthermore, the constrain treatment strategy, which considers the violations as an objective, is an effective method. The future work is going to refine the model for more complicate scenarios and improve algorithm’s flexibility, stability and distribution uniformity for more tasks. Acknowledgments The authors gratefully acknowledge the support of NNSF(Grant No.60903005). Special thanks go to Dr. Russ Eberhart, Xiaohui HU and Yaobin Chen at Indiana University-Purdue University Indianapolis for their assistance and collaboration.
References 1. Chandler, P., Pachter, M., Swaroop, D., Fowler, J.: Complexity in UAV cooperative control. In: American Control Conference, ACC, pp. 1831–1836 (2002) 2. Schumacher, C., Chandler, P.R., Pachter, M., Pachter, L.S.: Optimization of Air Vehicle Operations Using Mixed-Integer Linear Programming. Air Force Research Lab (AFRL/VACA) Wright-Patterson AFB, OH Control Theory Optimization Branch (2006) 3. Guo, W., Nygard, K.E., Kamel, A.: Combinatorial Trading Mechanism for Task Allocation. In: Proceedings of the 14th International Conference on Computer Applications in Industry and Engineering, Las Vegas, Nevada, USA (2001) 4. Arulselvan, A., Commander, C.W., Pardalos, P.M.: A hybrid genetic algorithm for the target visitation problem. Naval Research Logistics (2007) 5. Vijay, K.S., Moises, S., Rakesh, N.: Priority-based assignment and routing of a fleet of unmanned combat aerial vehicles. Elsevier Science Ltd. 35, 1813–1828 (2008) 6. Pan, F., Hu, X., Eberhart, R., Chen, Y.: A New UAV Assignment Model Based on PSO. In: IEEE Swarm Intelligence Symposium (SIS 2008), St. Louis, USA (2008) 7. Pan, F., Chen, J., Tu, X.-Y., Cai, T.: A multiobjective-based vehicle assignment model for constraints handling in computational intelligence algorithms. In: International Conference Humanized Systems 2008, Beijing, P.R.China (2008)
Correlative Particle Swarm Optimization for Multi-objective Problems Yuanxia Shen1,2, Guoyin Wang1, and Qun Liu1 1
Institute of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2 Anhui University of Technology, Maanshan 243002, China [email protected], [email protected]
Abstract. Particle swarm optimization (PSO) has been applied to multiobjective problems. However, PSO may easily get trapped in the local optima when solving complex problems. In order to improve convergence and diversity of solutions, a correlative particle swarm optimization (CPSO) with disturbance operation is proposed, named MO-CPSO, for dealing with multi-objective problems. MO-CPSO adopts the correlative processing strategy to maintain population diversity, and introduces a disturbance operation to the nondominated particles for improving convergence accuracy of solutions. Experiments were conducted on multi-objective benchmark problems. The experimental results showed that MO-CPSO operates better in convergence metric and diversity metric than three other related works. Keywords: Multi-objective problems, Correlative particle swarm optimization, Population diversity.
1 Introduction Optimization plays a major role in the modern-day design and decision-making activities. The multi-objective optimization (MOO) is a challenging problem due to the inherent confliction nature of objective to be optimized. As evolutionary algorithm can deal simultaneously with a set of possible solutions in a single run, it is especially suitable to solve MOO problems. Since Schaffer proposed a Vector Evaluated Genetic algorithm (VEGA) in 1984[1], many evolutionary MOO algorithms have been developed in the past decades. The most of studies published on multi-objective genetic algorithm (MOGA) and multiobjective evolutionary algorithm (MOEA) in different fashion, such as, SPEA2 [2], NSGA [3]. Particle swarm optimization (PSO) is a class of stochastic optimization technique that is inspired by the behavior of bird flocks [4]. As PSO has the fast convergence speed, Coello Coello employed PSO to solve MOO problems, and proposed MOPSO in 2004 [5]. The main difference between the original single-objective PSO and the
multi-objective PSO (MOPSO) is the local guide (gbest) distribution must be redefined in order to obtain a set of non-dominated solutions (Pareto front). To maintain population diversity is also important for solving MOO, and then several techniques [5-7] are introduced to PSO, e.g. an adaptive-grid mechanism, an adaptive mutation operation. However, this results in desired diversity and convergence still not close enough to the Pareto front. In order to improve convergence and diversity of solutions, a correlative particle swarm optimization (CPSO) with disturbance operation is proposed, named MO-CPSO. MO-CPSO adopts the correlative processing strategy to maintain population diversity, and introduces the disturbance operation to the optimal particles have found for improving convergence accuracy of solutions. The remainder of this paper is organized as follows. Basic concepts of MOO are described in section 2. CPSO is introduced in section 3. MO-CPSO is presented in section 4. Experimental results on some benchmark optimization problems are discussed in section 5. Conclusions are drawn in section 6.
2 Basic Concepts of MOO In general, many real-world applications involve complex optimization problems with various competing specifications and constraints. Without loss of generality, we consider a minimization problem with decision space Y which is a subset of real numbers. For the minimization problem, it tends to find a parameter set y for Min F ( y ) y∈Y
y ∈ RD
(1)
where y = [y1, y2, . . . , yD] is a vector with D decision variables and F = [f1, f2, . . . , fM] are M objectives to be minimized. In the absence of any preference information, a set of solutions is obtained, where each solution is equally significant. Pareto dominance and Pareto optimality are defined as follows: Definition 1 (Pareto dominance). A solution y= [y1, y2, . . . , yD] is said to dominate the other solution z = [z1, z2, . . . , zD], if both statement below are satisfied. 1. The solution y is no worse than z in all objectives, or fi(y)≤fi(z) for all i {1,2, . . ., M}. 2. The solution y is strictly better than z in at least one objective, or fi(y)
∈
∈
∈
Definition 2 (Pareto optimality). For a general MO problem, a given solution y F (where F is the feasible solution space) is the Pareto optimality if, and only if there is no z F that dominates y.
∈
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Definition 3 (Pareto front). The front obtained by mapping the Pareto optimal set (OS) into the objective space is called POF. JK POF = { f = ( f1 ( x)," , f M ( x)) | x ∈ OS }
(2)
The determination of a complete POF is a very difficult task, owing to the presence of a large number of suboptimal Pareto fronts. By considering the existing memory constraints, the determination of the complete Pareto front becomes infeasible and requires the solutions to be diverse covering its maximum possible regions.
3 Correlative Particle Swarm Optimization (CPSO) 3.1 Standard PSO (SPSO) A swarm in PSO consists of a number of particles. Each particle represents a potential solution of the optimization task. Each particle adjusts its velocity according to the past best position pbest and the global best position gbest in such a way that it accelerates towards positions that have had high objective (fitness) values in previous iterations. The position vector and velocity vector of the particle i in N-dimensional space can be indicated as xi =(xi1,…, xin , …, xiN) and vi=(vi1,…, vin , … ,viN) respectively. The updating velocity and position of the particles are calculated using the following two equations: Vi , j (t + 1) = wVi , j (t ) + c1r1i , j ( pbesti , j (t ) − X i , j (t )) + c2 r2i , j ( gbest j (t ) − X i , j (t )). X i , j ( t + 1) = X i , j ( t ) + V i , j ( t + 1).
(3) (4)
where, w is the inertia weight; c1 and c2 are positive constants known as acceleration coefficients; Random factors r1 and r2 are independent uniform random numbers in the range [0,1]. The value of velocity vector vi can be restricted to the range [-vmax, vmax] to prevent particles from moving out of the search range. vmax represents the maximal magnitude of the element of velocity vector vi,j. 3.2 Correlative Particle Swarm Optimization (CPSO) In SPSO model, the strategy with independent random coefficients is used to process gbest and pbest. This strategy makes no difference to exploit gbest and pbest, and lets the cognitive and the social components of the whole swarm contribute randomly to the position of each particle in the next iteration. In CPSO, correlative factors are used to process gbest and pbest, and create the relationship between gbest and pbest, where correlative factors are correlated random factors.
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In [8], Shen and Wang pointed out the positive correlation between random factors can maintain population diversity. In order to improve diversity of solutions, the random factors are positive correlation in this paper. The Gaussian Copula is used to describe correlated random factors. The Gaussian copulas is a member of the Elliptical copulas family, which is by far the most popular copula used in the framework of intensity or structural models because that it is easy to simulate. The updating velocity of the particle is calculated using the following equation: ⎧⎪Vi , j (t + 1) = wVi , j (t ) + c1r1 i, j (t )( pbesti , j (t ) − X i , j (t )) + c2 r2 i, j ( gbest j (t ) − X i , j (t )) ⎨ −1 −1 ⎪⎩ H ( r1 i, j (t ), r2 i , j (t )) = Φ ρ (Φ ( r1 i, j (t )), Φ ( r2 i , j (t ))) ρ > 0
(5)
where, H is the joint distribution function of correlative factors. Φρ denotes the joint distribution function of a standard 2-dimensional normal random vector with correlation matrix, and Φ is the univariate standard normal distribution function. Φ-1 is the inverses function of Φ. ρ denotes the correlation coefficient between correlated random factors r1 and r2, where 0<ρ<1.
4 MO-CPSO In single-objective problems, the term gbest represents the best solution obtained by the whole swarm. In MO problems, more than one conflicting objectives must all need be optimized. The number of non-dominated solutions which are located on/near the Pareto front will be more than one. To resolve this problem, the concept of nondominance is used and an archive of non-dominance solutions is maintained, from which a solution is picked up as the gbest in MO-CPSO. The historical archive stores non-dominance solutions to prevent the loss of good particles. The archive is updated at each cycle, e.g., if the candidate solution is not dominated by any members in the archive, it will be added to the archive. Likewise, any archive members dominated by this solution will be removed form the archive. In MO problems, there are many non-dominated solutions which are located on the Pareto front. This paper introduces a disturbance operation to non-dominated solutions in the archive for trying to find better solutions or other non-dominated solutions. The disturbance operation will randomly select m non-dominated solutions from the archive to put noise into their positions, and is shown as (6) X i , j (t ) = X i , j (t ) + b *η * X i , j (t )
(6)
Where, b is a positive constant; η is a Gaussian distribution noise with mean value of 0, and the variance is 1. MO-CPSO is described in Fig.1.
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/Ns: size of the swarm; MaxIter: maximum member of iterations; d: the dimensions of the search space./ (1) t = 0, randomly initialize S0; /St: swarm at iteration t / Ь initialize xi,j, ię{1, . . . ,Ns} and ję{1, . . . ,d}/ xi,j: the j-th coordinate of the i-th particle / Ь initialize vi,j, ię{1, . . . ,Ns} and ję{1, . . . ,d}/vi,j: the velocity of i-th particle in j-th dimension / Ь pbestiĸ xi, ię{1, . . . ,Ns} / pbesti: the coordinate of the personal best of the i-th particle / (2)Evaluate each of the particles in S0. (3)A0ĸnon_dominated(S0) /returns the non-dominated solutions from the swarm; At: archive at iteration t / (4) for t = 1 to t = MaxIter:, Ь for i = 1 to i = Ns / update the swarm / / generating correlative factors / Set correlation coefficient U of correlative factors to be 1, generate the correlative factors are uniform in the range [0,1] (a) Given two independent random variables t1,t2 from U(0,1) (b) Calculate k1=)-1(t1), k2=)-1(t2), where ) (.) is standard Normal distribution; (c) Perform Cholesky transformation: k1= k1, k2=U k1 +(1-U2)0.5 k2˗ (d) Calculate r1=) (k1), r2=) (k2), r1 and r2 are thus simulated from the elliptical copula with correlation coefficient U / updating the velocity of each particle / Ь vi= w vi + c1r1(pbesti -xi) + c2r2(Regb-xi) / Regb is the non-dominant solution that is randomly taken form the archive/ /updating coordinates / Ь xi = xi + vi (5) Evaluate each of the particles in St. (6) /updating the archive / Atĸnon_dominated(St). (7)/disturbance operation to the randomly selected solutions in At/ AtĸSelected (non_dominated(St))(1+b*Ș); (8) END Return At Fig. 1. Pseudo code of MO-CPSO
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5 Experimental Results 5.1 Test Functions and Parameters Setting In the context of MOO, the benchmark problems must pose sufficient difficulty to impede searching for the Pareto optimal solutions. In this paper, four benchmark problems are selected to test the performance of the proposed MO-CPSO. The definition of these test functions is summarized in Table 1. In this experiment, the maximum fitness evaluation (FE) is set at 10000. The population size is set at 100 for all problems. The correlation coefficient ρ is set at 0.95. The parameter b in the disturbance operation is set at 0.05. Table 1. Test functions Test functions Schaffer’s study (SCH) Fonseca,s and Fleming’s study (FON)
Poloni’s study (POL)
Kursawe’s study (KUR)
Definition Minimize F=(f1(x),f2(x)), where f (x) = x2, f (x) = (x − 2)2, x ∈[−103,103] 1 2 Minimize F=(f1(x),f2(x)), where
(
)
(
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f1 ( x) = ∑ i =1[−10 exp( −0.2 xi2 + xi2+1 )], f 2 ( x) = ∑ i =1 [| xi |0.8 +5sin( xi3 )], 2
3
xi ∈ [−5,5], i = 1, 2,3
5.2 Performance Metrics The knowledge of Pareto front of a problem provides an alternative for selection from a list of efficient solutions. It thus helps in taking decisions, and also, the knowledge gained can be used in situations where the requirements are continually changing. In order to provide a quantitative assessment for the performance of MO optimizer, two issues are taken into consideration, i.e. the convergence to the Pareto-optimal set and the maintenance of diversity in solutions of the Pareto-optimal set. In this paper, convergence metric γ [7] and diversity metric δ [7] have as qualitative measures. Convergence metric is used to measure the extent of convergence of the obtained set of solutions. The smaller is the value of γ, the better is the convergence toward the POF. Diversity metric is used to measure the spread of solutions lying on the POF. For the most widely and uniformly spread out set of non-dominated solutions, diversity metric δ would be very small.
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5.3 Experimental Results and Discussions Results for the convergence metric and the diversity metric obtained using MO-CPSO, are given in Table 2 and 3, where results of NSGA- , MOPSO and MOIPSO come form Ref. [7]. From the results, they are evident that MO-CPSO converges better than the other three algorithms. In order to clearly visualize the quality of solutions obtained, figures have been plotted for the obtained Pareto fronts with POF. As can been seen form Fig. 2, the front obtained from MO-CPSO has the high extent of coverage and uniform diversity for all test problems. In a word, the performance of MO-CPSO is better in converges metric and diversity metric. It must be noted that MOPSO adopts an adaptive mutation operator and an adaptive-grid division strategy to improve its search potential, while MOIPSO adopts search methods including an adaptive-grid mechanism, a self-adaptive mutation operator, and a novel decision-making strategy to enhance balance between the exploration and exploitation capabilities. MO-CPSO only adopts disturbance operation to solve MOO problems, and no other parameters are introduced.
Ⅱ
Table 2. Results of the convergence metric for test problems Test function SCH FON POL KUR
Fig. 2. Pareto solutions of MOPSO and MO-CPSO. (a) SCH, (b) FON, (c) POL, (d)KUR.
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6 Conclusion In this paper, we propose MO-CPSO for multi-objective problems, where the correlative processing strategy is used to maintain population diversity, and the disturbance operation is adopted to improve convergence accuracy of solutions. Experimental results show that the proposed algorithm can find solutions with good diversity and convergence, and is an efficient approach for complex multi-objective optimization problems. Acknowledgments. This paper is supported by Chongqing Key Lab of Computer Network and Communication Technology No.CY-CNCL-2009-03.
References 1. Schaffer, J.D.: Multiple objective optimization with vector evaluated genetic algorithms. PhD thesis, Vanderbilt University (1984) 2. Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach. Transactions on Evolutionary Computation 3(4), 257– 271 (2000) 3. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002) 4. Kennedy, J., Eberhart, R.C.: Particle Swarm Optimization. In: Proceeding of International Conference on Neural Networks, pp. 1942–1948. IEEE Press, Perth (1995) 5. Coello, C.A.C., Pulido, G.T., Lechuga, M.S.: Handling multiple objectives with particle swarm optimization. IEEE Transactions on Evolutionary Computation 3(3), 256–280 (2004) 6. Liu, D.S., Tan, K.C., Goh, C.K., Ho, W.K.: A multi-objective memetic algorithm based on particle swarm optimization. IEEE Transaction on Systems, Man and Cybernetics, Part b: Cybernetics 37(1), 42–61 (2007) 7. Agrawal, S., Dashora, Y., Tiwari, M.K., Son, Y.J.: Interactive particle swarm: a paretoadaptive metaheuristic to multiobjective optimization. IEEE Transaction on Systems, Man and Cybernetics, Part a: Systems and Humans 38(2), 258–278 (2008) 8. Shen, Y.X., Wang, G.Y., Tao, C.M.: Particle swarm optimization with novel processing strategy and its application. International Journal of Computational Intelligence Systems 4(1), 100–111 (2011)
A PSO-Based Hybrid Multi-Objective Algorithm for Multi-Objective Optimization Problems Xianpeng Wang and Lixin Tang Liaoning Key Laboratory of Manufacturing System and Logistics, The Logistics Institute, Northeastern University, Shenyang, 110004, China [email protected], [email protected]
Abstract. This paper proposes a PSO-based hybrid multi-objective algorithm (HMOPSO) with the following three main features. First, the HMOPSO takes the crossover operator of the genetic algorithm as the particle updating strategy. Second, a propagating mechanism is adopted to propagate the non-dominated archive. Third, a local search heuristic based on scatter search is applied to improve the non-dominated solutions. Computational study shows that the HMOPSO is competitive with previous multi-objective algorithms in literature. Keywords: Multi-objective optimization, hybrid particle swarm optimization.
The third category combines or incorporates the advantages of other EA algorithms or local search algorithms into MOPSO to improve its exploration and exploitation abilities. Li [10] incorporated the main mechanisms of the NSGA-II into PSO, and developed a hybrid MOPSO. Srinivasan and Seow [11] developed the particle swarm inspired evolutionary algorithm (PS-EA) that is a hybrid between PSO and EA. Liu et al. [12] adopted the local search and proposed a memetic algorithm based on PSO, in which a new particle updating strategy is adopted based on the concept of fuzzy global-best to avoid premature convergence and maintain diversity. The fourth category focuses on the influence of variations of parameters on the performance of MOPSO and adopts adaptive parameter control strategy. Tripathi et al. [13] proposed the time variant MOPSO, in which the vital parameters (i.e., w, c1, c2) change adaptively with iterations so as to help the algorithm to explore the search space more efficiently. In this paper, we propose a new multi-objective optimization algorithm based on PSO and denote it as HMOPSO. The rest of this paper is organized as follows. Section 2 gives the introduction of MO optimization. The details of the HMOPSO are introduced in Section 3. Section 4 reports and analyzes the computational results on benchmark problems. Finally, the paper is concluded in Section 5.
2 Proposed HMOPSO Algorithm 2.1 Algorithm Overview The implementation of the HMOPSO can be presented in Fig. 1. In the following of the paper, we use EXA to represent the external archive and PBA[i] to represent the personal best archive of each particle i. 2.2 EXA Propagating Mechanism As mentioned in the literature review of MOPSO, many previous researches focused on the selection of gbest (or guiding solutions) from the EXA, but few of them took into account the quality of the EXA (i.e., the number and diversity of non-dominated solutions stored in the EXA). In the experiment, we found that at the beginning iterations (or beginning search process) of canonical MOPSO algorithms, the EXA generally has very few non-dominated solutions stored in it. Since there are only very few non-dominated solutions to be selected as gbest that has significant influence on the flight of particles, particles in the swarm tend to be attracted by the same nondominated solution or very close non-dominated solutions and thus the swarm may converge quickly and be stuck to a local optimal front (especially for MO problems with many local optimal fronts). Therefore, in this paper we propose a propagating mechanism to propagate the EXA whenever the number of non-dominated solutions in it is too small so as to avoid premature convergence and improve the exploration ability of MOPSO. Let nEXA denote the maximum size of the EXA and |EXA| denote the current size of the EXA, and then the propagating mechanism can be described as follows.
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Begin: Initialization: 1. Set the termination criterion, and initialize the values of parameters such as the size of the population, the size of EXA, the size of PBA[i] (note that all particles have the same size of PBA[i]), the mutation probability. 2. Set EXA and PBA[i] to be empty. 3. Randomly initialize all the particles in the swarm. 4. Evaluate each particle in the swarm, and store each particle i in PBA[i]. 5. Store the non-dominated particles in the swarm in EXA. while (the termination criterion is not reached) do 1. EXA-propagating-mechanism () % extend EXA when necessary % 2. Particle-flight-mechanism () % particle flight using crossover % 3. Particle-mutation () 4. Evaluate each particle in the swarm. 5. for each particle i in the swarm PBA[i]-update-strategy () End for 6. for each non-dominated particle i in the swarm EXA-update-strategy () using the non-dominated particle i End for 7. EXA-improvement () % local search on EXA % End while Report the obtained non-dominated solutions in EXA. End
Fig. 1. The main procedure of HMOPSO
Step 1. If |EXA| = 1, go to Step 2; otherwise, go to Step 3. Step 2. Perturb the single solution in EXA for nEXA times to generate other nEXA new solutions. Step 3. Randomly select two solutions in EXA, and use the simulated binary crossover (SBX) operator to generate two offspring solutions. Select the best non-dominated one as the new solution. Repeat this procedure until nEXA new solutions are obtained. Step 4. Use the EXA-update-strategy described in the following section 3.7 to update the EXA with the nEXA new solutions. 2.3 Particle Flight Mechanism The canonical MOPSO algorithm updates particles using the flight equations, which consists of three parameters, i.e., w, c1, and c2. When combined with other algorithms, the hybrid MOPSO will have more parameters, which will cause great difficulty in the parameter tuning. Therefore, in this paper we do not follow the canonical flight equations, but prefer to adopt a new strategy based on the SBX operator of GA. Since there have been many research report with respect to the parameters of the SBX operator, it is reasonable to follow the suggested setting, and thus there is no parameters to be tuned in the adopted update mechanism. For each particle i, this strategy has two simple steps: (1) randomly select a pbest from PBA[i], use the crossover operator to generate two offspring solutions from the particle i and its
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selected pbest, and then randomly select a gbest from the EXA; and (2) use the SBX crossover operator to generate two offspring solutions from the selected pbest and gbest, and then select the best one based on Pareto dominance as the new particle. 2.4 Particle Mutation To improve the search diversity, our HMOPSO algorithm also use the mutation operation. For each dimension of each particle in the swarm, we first generate a random number rnd in [0, 1]. If rnd nEXA (the maximum size of the EXA), calculate the crowding distance of all solutions in the EXA, and then remove the most crowded solution (i.e., the solution with the least crowding distance) from the EXA. Repeat this step until |EXA| = nEXA. 2.7 EXA Improvement Since the selection of gbest from the EXA has significant influence on the performance of MOPSO, it is then clear that the improvement on the EXA can improve the MOPSO because this will help to provide better candidate solutions to be selected as gbest. Motivated by this idea, we develop a local search heuristic named the EXAimprovement to further improve the quality of the EXA, i.e., the distance of the EXA to the true Pareto front and the diversity of solutions in the EXA. This local heuristic can be viewed as a simplified version of scatter search (SS) because it adopts the concept of reference set (denoted as REF) of SS in [14].
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To give the EXA-improvement method, we first define the distance between k
max solutions x and y of a MOP as d ( x, y ) = ∑ fi ( x) − fi ( y) ( fi max − fi min ) , in which fi i =1
and f i min are the maximum and minimum values of the ith objective function in the EXA, respectively. Let nREF denote the maximum size of the REF, then the EXAimprovement method can be described as follows. Step 1. Construct REF Step 1.1 If |EXA| = nREF, store all solutions in EXA in the REF. Step 1.2 If |EXA| < nREF, store all solutions in EXA to REF, and then use the non-dominated sorting method to classify the particles in the swarm into different levels. Starting from front 1, randomly select a particle and store it in REF, until |REF| = nREF. Step 1.3 If |EXA| > nREF, then perform the following procedures. Step 1.3.1 Calculate the crowding distance of each solution in the EXA and then store them in the non-ascending order of their crowding distances in a list L. Step 1.3.2 Select the first nREF / 2 solutions in L and add them to REF, and then delete them from L. Step 1.3.3 Select the solution p with the maximum value of the minimum distance to REF from L, add it to REF and then delete it from L. Repeat this step until another nREF/2 solutions are added to REF. Step 2. Generate new solutions from REF. Select two solutions from REF, use the SBX operator to generate two offspring solutions, and then select the best one as the new solution. Note that there are a total of nREF(nREF-1)/2 new solutions generated. Step 3. Update the EXA. Use the obtained nREF(nREF-1)/2 new solutions to update the EXA based on the EXA-update-strategy.
3 Computational Experiments We adopt the General Distance (GD), the Spacing (SP), and the Maximum Spread (MS) to evaluate the algorithm’s performance. Based on the experimental results, the following parameter setting is adopted: npop = 100, nEXA =100, nREF =10, nPBA =5, and nprop =0.3. The HMOPSO is compared with other powerful or state-of-the-art algorithms such as the NSGA-II [2], the MOPSO [7] (denoted as cMOPSO), and the MOPSO with crowding distance [15] (denoted as MOPSO-CD). These three algorithms are selected because they are proven to be very effective and often used by many researchers. In this experiment, the maximum runtime is used as the stopping criterion because all the algorithms are written in C++ and run on the same computer. In addition, for each problem 30 independent duplications were carried out and we select the best one. The computational results of each test problem are given in Figures 2-5. Based on these results, it is clear that the proposed HMOPSO outperforms the other algorithms. In addition, the proposed HMOPSO can reach the true Pareto fronts for all test MOPs, and it shows a very robust performance. Among the rival algorithms, NSGA-II can also reach the true Pareto fronts of all test MOPs, but its performance on the
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distribution of the non-dominated solutions is not as good as that obtained by the proposed HMOPSO. The cMOPSO and MOPSO-CD cannot reach the true Pareto fronts of all ZDT series of problems. Specially, they get trapped in local optimum for ZDT4. For the other test MOPs, they show a bad performance on the distribution of the obtained non-dominated solutions. With the incorporation of the crowding distance of NSGA-II, the MOPSO-CD obtains significant improvements with comparison to the cMOPSO. 1.2
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4 Conclusion In this paper, we investigated the improvement to the canonical MOPSO algorithm and proposed three main strategies. First, the traditional update equations for particles’ positions are replaced by a new particle flight mechanism that is based on the crossover operator in GA. Second, motivated by observations that there are few non-dominated solutions for some problems in the starting process of MOPSO, we proposed a propagating mechanism to improve the quality and diversity of the external archive. Third, a modified version of scatter search was adopted as the local search to improve the external archive. In addition, we adopted the DOEs method to analyze the influences of each parameter and their interactions on the performance of our HMOPSO algorithm. In the comparative study, HMOPSO is compared against existing state-of-the-art multi-objective algorithms through the use of benchmark test problems. The results indicate that our HMOPSO algorithm is competitive or superior to the NSGA-II, and much better than two MOPSO algorithms in the literature for all benchmark problems.
Acknowledgements This research is supported by Key Program of National Natural Science Foundation of China (71032004), National Natural Science Foundation of China (70902065), National Science Foundation for Post-doctoral Scientists of China (20100481197), and the Fundamental Research Funds for the Central Universities (N090404018).
References 1. Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002) 2. Knowles, J.D., Corne, D.W.: Approximating the nondominated front using the Pareto archived evolution strategy. Evolutionary Computation 8, 149–172 (2000)
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3. Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength Pareto evolutionary algorithm. Computer Engineering Networks Lab (TIK), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, Technical Report, 103 (2001) 4. Nebro, A.J., Luna, F., Alba, E., Dorronsoro, B., Durillo, J.J., Beham, A.: AbYSS Adapting scatter search to multiobjective optimization. IEEE Transactions on Evolutionary Computation 12(4), 439–457 (2008) 5. Hu, X., Eberhart, R.C.: Multiobjective optimization dynamic neighborhood particle swarm optimization. In: Proceedings of Congress on Evolutionary Computation, pp. 1677–1681 (2002) 6. Mostaghim, S., Teich, J.: Strategies for finding local guides in multi-objective particle swarm optimization (MOPSO). In: Proceedings of IEEE Swarm Intelligence Symposium, pp. 26–33 (2003) 7. Coello, C.A.C., Pulido, G.T., Lechuga, M.S.: Handling multiple objectives with particle swarm optimization. IEEE Transactions on Evolutionary Computation 8(3), 256–279 (2004) 8. Chow, C.K., Tsui, H.T.: Autonomous agent response learning by a multi-species particle swarm optimization. In: Proceedings of Congress on Evolutionary Computation, pp. 778– 785 (2004) 9. Yen, G.G., Leong, W.F.: Dynamic multiple swarms in multiobjective particle swarm optimization. IEEE Transactions on Systems, Man, and Cybernetics – Part A 39(4), 890– 911 (2009) 10. Goh, C.K., Tan, K.C., Liu, D.S., Chiam, S.C.: A competitive and cooperative coevolutionary approach to multi-objective particle swarm optimization algorithm design. European Journal of Operational Research 202(1), 42–54 (2010) 11. Li, X.D.: A non-dominated sorting particle swarm optimizer for multiobjective optimization. In: Cantú-Paz, E., Foster, J.A., Deb, K., Davis, L., Roy, R., O’Reilly, U.-M., Beyer, H.-G., Kendall, G., Wilson, S.W., Harman, M., Wegener, J., Dasgupta, D., Potter, M.A., Schultz, A., Dowsland, K.A., Jonoska, N., Miller, J., Standish, R.K. (eds.) GECCO 2003. LNCS, vol. 2723, pp. 37–48. Springer, Heidelberg (2003) 12. Srinivasan, D., Seow, T.H.: Particle swarm inspired evolutionary algorithm (PS-EA) for multiobjective optimization problem. In: Proceedings of Congress on Evolutionary Computation, pp. 2292–2297 (2003) 13. Tripathi, P.K., Bandyopadhyay, S., Pal, S.K.: Multi-Objective Particle Swarm Optimization with time variant inertia and acceleration coefficients. Information Science 177(22), 5033–5049 (2007) 14. Martí, R., Laguna, M., Glover, F.: Principles of scatter search. European Journal of Operational Research 169(2), 359–372 (2006) 15. Raquel, C.R., Naval Jr., P.C.: An effective use of crowding distance in multiobjective particle swarm optimization. In: Proceedings of Conference on Genetic Evolutionary Computation, pp. 257–264 (2005)
The Properties of Birandom Multiobjective Programming Problems Yongguo Zhang1 , Yayi Xu2 , Mingfa Zheng2 , and Liu Ningning1 1
College of Elementary Education, Xingtai University Xingtai, Hebei, 054001, China 2 College of Science, Air Force Engineering University, Xi’an, Shanxi, 710051, China {yongguo924,mingfa103}@163.com, {mingfazheng,lnn}@126.com
Abstract. This paper is devoted to the multiobjective programming problem based on the birandom theory. We first propose the birandom multiobjective programming (BRMOP) problem and its expected value model. Then we present the concepts of non-inferior solution, called expected-value efficient solutions and expected-value wake efficient solutions, and their properties are also discussed. The results obtained in this paper can provide theoretical basis for designing algorithms to solve the BRMOP problem. Keywords: birandom variable, multiobjective programming , expectedvalue efficient solution.
1
Introduction
The multiobjective programming problems are studied by many researchers such as [2], [7], [8]. For given multiobjective problem, its absolute optimal solutions which optimize each objective functions simultaneously usually dons’t exist, so we consider their non-inferior solutions in a sense, which are Pareto optimal solutions in common use. There are various types of uncertainties in the real-world problem. As we known, random phenomena is one class of uncertain phenomena which has been well studied. Based on the probability, stochastic multiobjective programming problems have been presented such as [1], [10]. In a practical decision-making process, we often face a hybrid uncertain environment where linguistic and frequent nature coexist. For the examples of two fold uncertainty, we may refer to [9], Liu [3], [4], Liu[5], Liu and Liu [6], Yazenin [11]. To deal with this two fold uncertainty, it is required to employ birandom theory [9]. The multiobjective programming in birandom environment have not been developed well, therefore, following the idea of stochastic multiobjective programming, this paper devotes the birandom multiobjective programming (BRMOP) problems based on the birandom theory. For the birandom parameters, we consider their expectation which convert the BRMOP problem into the expected-value model of random fuzzy multiobjective (EVBRMOP) which Y. Tan et al. (Eds.): ICSI 2011, Part II, LNCS 6729, pp. 34–40, 2011. c Springer-Verlag Berlin Heidelberg 2011
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is a deterministic multiobjective problem. By the deterministic problem above, we can obtain the expected-value efficient solutions or expected-value wake efficient solutions to the BRMOP problem, and their relations are also discussed, whose results can provide theoretical basis for designing algorithm to solve the proposed problem. This paper is organized as follows. The next section provides a brief review on the related concepts and results in birandom theory. Section 3 presents the BRMOP problem and its expected value model. Furthermore, based on the expected value model, the expected-value efficient solution and expected-value wake efficient solution to the BRMOP is proposed, and their properties are discussed. Finally, Section 4 provides a summary of the main results of this paper.
2
Preliminaries
Let ξ be a random variable defined on the probability space(Ω, Σ, Pr), where Ω is a universe and Σ is the σ-algebra of subsets of Ω and Pr is a probability measure defined on (Ω, Σ, Pr). Definition 2.1.[9] A birandom variable is a function ξ from the probability space (Ω, Σ, Pr) to the set of random variables such that Pr{ξω ∈ B} is a measurable function of ω for any Borel set B of . Proposition 2.1.[9] Assume that ξ is a birandom variable defined on the probability space (Ω, Σ, Pr). Then for ω ∈ Ω, we have (1) Pr{ξω ∈ B} a random variable for any B ∈ B(); (2) E[ξω ] is a random variable provided that E[ξω ] is finite for fixed ω ∈ Ω. Definition 2.2.[9] Let be a birandom variable defined on the probability space (Ω, Σ, Pr). The expected value E[ξ] is defined by[9] ∞ 0 E[ξ] = E[E[ξω ]] = Pr{E[ξω ] ≥ r}dr − Pr{E[ξω ] ≤ r}dr (1) 0
−∞
provided that at least one of the two integrals is finite. From Eq.(1), we can obtain the following expectation of birandom variable i.e., (2) E[ξ] = Eω [Eω [ξω (ω)]], where ω × ω ∈ Ω × Ω .
3 3.1
Birandom Multiobjective Programming Expected Value Model of Birandom Multiobjective Programming
If y = (y1 , y2 , · · · , yn )T , z = (z1 , z2 , · · · , zn )T ∈ Rn , then we define: y = z ⇐⇒ yi = zi , i = 1, 2, · · · , n y > z ⇐⇒ yi > zi , i = 1, 2, · · · , n y >= z ⇐⇒ yi >= zi , i = 1, 2, · · · , n, y≥z⇐⇒yi >= zi , i = 1, 2, · · ·, n; there exist j0 at least such that yj0 > zj0 , 1 =< j0 <= n, i.e., y = z.
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Similarly, we can define y < z, y <= z, y ≤ z. Considering the birandom multiobjective programming (BRMOP) problem as follows: ⎫ ⎪ min f(x, ξ) = (f1 (x, ξ), f2 (x, ξ), · · · , fp (x, ξ)T ) ⎬ x∈R (3) (BRMOP) s.t. g(x, ξ) = (g1 (x, ξ), g2 (x, ξ), · · · , gm (x, ξ))T <= 0 ⎪ h(x, ξ) = (h1 (x, ξ), h2 (x, ξ), · · · , hl (x, ξ))T = 0, ⎭ where decision-making variable x ∈ Rn , ξ is a continuous birandom variable. For the BRMOP problem, we assume the condition that fj (x, ξω (ω)), j = 1, 2, · · · , p, is borel measure function defined on measure space (Ω, Σ, Pr), then, by the definition of random fuzzy variable, we can easily obtain that fj (x, ξω ) = Eω [fj (x, ξω (ω)] is a random variable for given x ∈ Rn and ω ∈ Ω . To solve the BRMOP problem, based on birandom theory, we present the expected value model of birandom multiobjective programming (EVBRMOP) problem which is a deterministic multiobjective programming problem as follows: min E[f (x, ξ)] = (E[f1 (x, ξ)], E[f2 (x, ξ)], · · · , E[fp (x, ξ)])T ,
x∈D
(4)
where D={x∈Rn|E[g(x, ξ)] = (E[g1 (x, ξ)], E[g2 (x, ξ)], · · · , E[gm (x, ξ)])T<= 0, E[h(x, ξ)] = (E[h1 (x, ξ)], E[h2 (x, ξ)], · · · , E[hl (x, ξ)])T = 0}. Theorem 3.1.1. Let ξ be a birandom variable, f(x, t) and g(x, t) be convex vector function on x for any given t, then the EVBRMOP problem is a convex programming. Proof. To prove the theorem, it is sufficient to illuminate that E[f (x, ξ)] is a convex vector function and feasible region D is a convex set. By the assumed conditions, for any given t, we can obtain: f(λx1 + (1 − λ)x2 , t) ≤ λf(x1 , t) + (1 − λ)f(x2 , t), for any λ ∈ [0, 1] and x1 , x2 ∈ n . It is evidence that the following inequality f(λx1 + (1 − λ)x2 , ξ) ≤ λf(x1 , ξ) + (1 − λ)f(x2 , ξ)
(5)
holds for ω ×ω ∈ Γ ×Ω. Taking the expectation of random variable to inequality (5), by the linear properties of random variable, we can obtain: Eω [f (λx1 + (1 − λ)x2 , ξω (ω))] ≤ λEω [f (x1 , ξω (ω))] + (1 − λ)Eω [f (x2 , ξω (ω))]. Using the similar method, we know that the random variable Eω [f (x, ξω (ω))] have linear properties, so we have Eω [Eω [f (λx1+(1−λ)x2 , ξω(ω))]]≤λEω [Eω [f (x1 , ξω(ω))]]+(1−λ)Eω [Eω [f (x2 , ξω(ω))]],
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which shows E[f (x, ξ)] is a convex vector function. Next we will illuminate that feasible region D is a convex set. If x1 , x2 ∈ D, it follows from the convexity of vector function g that g(λx1 + (1 − λ)x2 , ξ) ≤ λg(x1 , ξ) + (1 − λ)g(x2 , ξ), for any t and λ ∈ [0, 1]. Similarly, by the linear properties of random variable, we can obtain: Eω [g(λx1 +(1−λ)x2 , ξω (ω))] ≤ λEω [g(x1 , ξω (ω))]+(1−λ)Eω [g(x2 , ξω (ω))] <= 0. (6) It follows from the linear properties that Eω [Eω [g(λx1+(1−λ)x2 , ξω(ω))]]≤λEω [Eω[g(x1 , ξω(ω))]]+(1−λ)Eω[Eω [g(x2 , ξω(ω))]]<=0.
On the other hand, because h(ξ, t) is linear vector function, we can obtain h(λx1 + (1 − λ)x2 , ξ) = λh(x1 , ξ) + (1 − λ)h(x2 , ξ). Similarly, it follows from the linear properties that E[h(λx1 + (1 − λ)x2 , ξ)] = λE[h(x1 , ξ)] + (1 − λ)E[h(x2 , ξ)].
(8)
Obviously, by Eq.(7)and Eq.(8),we know the feasible region D is a convex set. Hence, the EVBRMOP problem is a convex programming. The proof is complete. 3.2
The Expected-Value Non-inferior Solutions and Their Relations
Definitions 3.2.1. For the EVBRMOP problem, if x∗ ∈ D, we say that x∗ is the expected-value absolute optimal solution to the BRMOP problem whose solution set is denoted Dab if it satisfies the following conditions: E[f (x∗ , ξ)] <= E[f (x, ξ)], namely,
E[fj (x∗ , ξ)] <= E[fj (x, ξ)], for all j = 1, 2, · · · , p.
Definitions 3.2.2. For the EVBRMOP problem, if x∗ ∈ D, we say that x∗ is the expected-value efficient solutions to the BRMOP problem whose solution set is denoted Dpa if it satisfies the following conditions: there does’t exist x ∈ D such that E[f (x, ξ)] ≤ E[f (x∗ , ξ)]. namely, E[fj (x∗ , ξ)] <= E[fj (x, ξ)], for all j = 1, 2, · · · , p,
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and there must exist some j0 at least such that E[fj0 (x∗ , ξ)] < E[fj0 (x, ξ)]. Definitions 3.2.3. For the EVBRMOP problem, if x∗ ∈ D, we say that x∗ is the expected-value wake efficient solutions to the BRMOP problem whose solution set is denoted Dwpa if it satisfies the following conditions: there does’t exist x ∈ D such that E[f (x, ξ)] < E[f (x∗ , ξ)]. Theorem 3.2.1. Dab ⊂ Dpa ⊂ Dwpa ⊂ D. Proof. We first prove that Dab ⊂ Dpa . If Dab = φ, then the result is immediate. If not, suppose that x∗ ∈ Dab , and x∗ ∈ / Dpa , then, by the definition of the expected-value efficient solution, their must exist x ∈ D such that E[f (x, ξ)] ≤ E[f (x∗ , ξ)], namely,
E[fj (x, ξ)] ≤ E[fj (x∗ , ξ)],
for all j = 1, 2, . . . , p, and their exists j0 at least such that E[fj (x, ξ)] < E[fj (x∗ , ξ)], 1 <= j0 <= p, which implies the contradiction with x∗ ∈ Dab . Hence, Dab ⊂ Dpa . Then we prove that Dpa ⊂ Dwpa . If x∗ ∈ Dpa , and x∗ ∈ / Dwpa, then, by the definition of expected-value wake efficient solution, their must exist x ∈ D such that E[f (x, ξ)] < E[f (x∗ , ξ)], namely, E[fj (x, ξ)] < E[fj (x∗ , ξ)] holds for all j = 1, 2, . . . , p. Thus, we can obtain: E[f (x, ξ)] ≤ E[f (x∗ , ξ)], which implies x∗ ∈ / Dpa . By the previous assumption, we obtain the contradiction with x∗ ∈ Dpa . Hence, Dpa ⊂ Dwpa . It follows from the definition of the expected-value wake efficient solution that Dwpa ⊂ D, which proves the desired theorem. Theorem 3.2.2. (1) If Dab = φ, then Dab = Dpa . (2)If h(x, ξ) is linear vector function, f(x, t) and g(x, t) are strict convex vector function on x, then we can obtain Dpa = Dwpa . Proof. It follows from Theorem 3.2.1 that we need only to prove Dab ⊃ Dpa . If x∗ ∈ Dpa , and x∗ ∈ / Dab , since Dab = φ, their must exist x ∈ Dab , by the definition of expected-value absolute optimal solution, we can obtain E[f (x, ξ)] <= E[f (x∗ , ξ)]. Since x∗ ∈ / Dab , we have E[f (x, ξ)] = E[f (x∗ , ξ)]. It follows from
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the inequality above that E[f (x, ξ)] ≤ E[f (x∗ , ξ)], which is a contradiction with x∗ ∈ Rpa . Hence, Dab ⊃ Dpa , which implies the required conclusion. (2) It follows from Theorem 3.2.1 that we need only to prove Dwpa ⊂ Dpa . If x∗ ∈ Dwpa , and x∗ ∈ / Dpa , we know that their must exist x ∈ D, and x = x∗ , such that E[f (x, ξ)] ≤ E[f (x∗ , ξ)]. By the assumed conditions and Theorem 3.1, we can obtain that D is a convex set, hence, αx + (1 − α)x∗ ∈ D for any given α ∈ (0, 1). Since f(x, ξ) is strict convex vector function on D, and f(x, ξ) is also conmonotonic, by noting the inequality just given, it easy to know that E[f (αx + (1 − α)x∗ , ξ)] < αE[f (x, ξ)] + (1 − α)E[f (x∗ , ξ)] < E[f (x∗ , ξ)], which is a contradiction with x∗ ∈ Dwpa . Thus, Dwpa ⊂ Dpa , which proves the required theorem.
4
Conclusions
Based on birandom theory, the BRMOP problem and its expected value model has been introduced in this paper. Since the non-inferior solutions play important role to multiobjective problem, the expected-value efficient solutions and expected-value wake efficient solutions of the BRMOP problem are presented and their relations are also studied. The results in this paper which can be as theoretical tool to design algorithm for solving BRMOP problem. Acknowledgments The authors Mingfa Zheng and Yayi Xu were supported by National Natural Science Foundation of China under Grant 70571021, and the Shanxi Province Science Foundation under Grant SJ08A02.
References 1. Benabdelaziz, F., Lang, P., Nadeau, R.: Pointwise efficiency in multiobjective stochastic linear Prograaming. Jourmal of Operational Research Sociaty 45, 11–18 (2000) 2. Hu, Y.D.: The efficient theory of multiobjective programming. Shanghai Since and Technology Press, China (1994) 3. Liu, B.: Fuzzy random dependent-chance programming. IEEE Trans. Fuzzy Syst. 9, 721–726 (2001) 4. Liu, B.: Uncertain programming. Wiley, New York (1999) 5. Liu, B.: Random fuzzy dependent-chance programming and its hybrid intelligent algorithm. Information Sciences 141, 259–271 (2002) 6. Liu, Y.K., Liu, B.: Expected value operator of random fuzzy variable operator. International Journal of Uncertainty, Fuzziness, Knowlledge-Based Systems 11, 195–215 (2003) 7. Lin, C.Y., Dong, J.L.: The efficient theory and method of multiobjective programming. Jilin Educational Press, China (2006)
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8. Ma, B.J.: The efficient rate of efficient solution to linear multiobjective programming. Jounal of Systems Engineering and Electronic Techology 2, 98–106 (2000) 9. Peng, J., Liu, B.: Birandom variables and birandom programming. Technical (2003) 10. Stancu-Minasian, I.M.: Stochastic programming with multiple objective functions. Buckarest (1984) 11. Yager, R.R.: A foundation for a theory of possibility. Journal of Cybernetics 10, 177–204 (1980)
A Modified Multi-objective Binary Particle Swarm Optimization Algorithm Ling Wang, Wei Ye, Xiping Fu, and Muhammad Ilyas Menhas Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200072, China [email protected]
Abstract. In recent years a number of works have been done to extend Particle Swarm Optimization (PSO) to solve multi-objective optimization problems, but a few of them can be used to tackle binary-coded problems. In this paper, a novel modified multi-objective binary PSO (MMBPSO) algorithm is proposed for the better multi-objective optimization performance. A modified updating strategy is developed which is simpler and easier to implement compared with standard discrete binary PSO. The mutation operator and dissipation operator are introduced to improve the search ability and keep the diversity of algorithm. The experimental results on a set of multi-objective benchmark functions demonstrate that the proposed MBBPSO is a competitive multi-objective optimizer and outperforms the standard binary PSO algorithm in terms of convergence and diversity. Keywords: Binary PSO, Multi-objective optimization, Pareto.
Particle Swarm Optimization (PSO) was firstly developed by Kennedy and Eberhart in 1995[6]. It is originated by imitating the behavior of a swarm of birds trying to search for food in an unknown area [5]. Owing to its simple arithmetic structure, high convergence speed and excellent global optimization ability, PSOs have been researched and improved to solve various multi-objective optimization problems. However, standard PSO and most of its improved versions work in continuous space, which mean they cannot tackle the binary-coded problems directly. To make up for it, Kennedy extended the PSO and proposed a novel discrete binary PSO (DBPSO) [6]. Based on DBPSO, researchers have introduced binary PSO to solve multi-objective problems. Abdul [8] proposed a multi-objective DBPSO, called BMPSO, to select the cluster head for lengthening the network lifetime and preventing network connectivity degradation. Peng and Xu [7] proposed a modified multi-objective binary PSO combining DBPSO with immune system to optimize the placement of the phasor measurement unit. These works prove that DBPSO-based multi-objective optimizers are efficient in solving MOPs. Nevertheless, the previous works on single objective optimization problems show that the optimization ability of DBPSO is not ideal [9], [10]. So we propose a novel modified multi-objective binary PSO (MMBPSO) in this paper to achieve the better multiobjective search ability and simplified the implementation of algorithm. The rest of the paper is organized as follows. In Section 2, the brief introduction on DBPSO and a modified binary PSO algorithm is given first, and then the proposed MMBPSO algorithm are described in detail. Section 3 validates the MMBPSO with several benchmark problems, and the optimization performance and comparison are also illustrated. Finally, some concluding remarks are given in Section 4.
2 Modified Multi-objective Binary Particle Swarm Optimization 2.1 Standard Modified Binary Particle Swarm Optimization Shen and Jiang [11] developed a modified binary PSO (MBPSO) algorithm for feature selection. In MBPSO, the updating formulas are demonstrated as Eq. (1-3). X
X
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The parameter α, called static probability, should be set properly. A small value of α can improve the convergent speed of the algorithm but makes MBPSO be trapped in the local optimum easily; while MBPSO with a big α may be ineffective as it cannot utilize the knowledge gained before well [9]. Although the update formulas of MBPSO and DBPSO are different, the updating strategy is still the same. In MBPSO, each particle still flies through the search space according to its past optimal experience and the global optimal information of the group. The Eq. (1) is an exhibition of inertia which represents the information that a particle inherited from its previous generation. The Eq. (2) represents particle’s cognitive capability which draws the particle to its own best position. The Eq. (3) is the particle’s social capacity which leads the particle to move to the best position found by the swarm [12].
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2.2 Modified Multi-objective Binary Particle Swarm Optimization Although MBPSO has been successfully adopted to solve various problems such as numerical optimization problem, feature selection and multidimensional knapsack problem, it is obvious that standard MBPSO cannot tackle Pareto-based multiobjective optimization problems. So we extend MBPSO and propose a novel modified multi-objective binary PSO. 2.2.1 Updating Operator To achieve the good convergence and diversity performance simultaneously in multiobjective problem, it should very carefully counterpoise the global and local search capability. In original MBPSO, the control parameter α has not the function of adjusting the global search ability which spoils the performance of algorithm in MOPs. To make up of this drawback, we modified the updating operator as Eq. (4-6).
X
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(4)
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(5)
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(6)
Here the parameter β can adjust the probability of tracking the two different best solutions. According to the Eq. (4-6), MMBPSO is easy to stick in the local optimal. For instance, if X , P and P are all equal to “1”, X will be “1” forever and vice versa. So the dissipation operator and the mutation operator are introduced to keep the diversity and enhance the local search ability. 2.2.2 Dissipation Operator The dissipation operator, as Eq. (7), is defined as randomly re-initializing a particle with some probability which brings the new particle into the swarm and retains the diversity effectively. Due to the characteristic of randomness, the probability of dissipation operating pd should not be a big value which may destroy the basic updating mechanism of algorithm. Usually pd is set between 0.05 and 0.15 to prevent the loss of optimal information. X
Reini Xi
(7)
rand
2.2.3 Mutation Operator The dissipation operator can greatly improves the diversity, but it may destroy the useful information at the same time as it operates on the level of individual. To further enhance the optimization ability of algorithm, the mutation operation is also introduced which is defined as Eq. (8). Different from the dissipation operator, the mutation operator works on the level of bit with the probability pm. So mutation can improve the local search ability as well as keeping the diversity of algorithm. if rand
X
1, 0,
if X if X
0 1
(8)
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2.2.4 The Updating of Personal Best and Global Best Each particle is guided by its two best individuals, i.e., the personal best solution (P ) and the global best solution of swarm (P ) to perform the search. So the updating of P and P is very important for the optimization performance of algorithm. In Paretobased MOPs, the goal of algorithm is to find the diverse non-dominated solutions laid in the Pareto front, which means that attention should be paid to the diversity as well as convergence when we design the algorithm. To realize this goal, the niche count [13] as a density measure is adopted in MMBPSO to select P . Niche count is defined as the number of the particles in the niche. For example, σ is the niche which indicates the radius of the neighborhood in Figure 1. From Figure 1, we can find that particle B has 4 neighbors while particle A has 8 neighbors, that is, particle B has a less crowded niche than particle A. So particle B is superior to particle A in terms of diversity. In this work, the σ is calculated as Eq. (9). σ
max
max N
min 1
min
(9)
where max and min are the maximum and minimum values of the two objective functions, and N is the population size.
A
B
Fig. 1. An example of niche count
During each iteration process, the non-dominated solution set is sorted according to the niche count. P for each generation is randomly chosen among top 10%“less crowded” non-dominated particles in the set. To encourage MMBPSO to search the whole space and find more non-dominated solutions, P is replaced by the nondominated current particle.
3 Experiments 3.1 Benchmark Functions and Performance Metrics To test the performance of the proposed MMBPSO, five well-known benchmark functions, i.e., ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6 [15] are adopted in the paper. All problems have two objective functions and without any constraint. Multiobjective optimizer is designed to achieve two goals: 1) convergence to the Paretooptimal set and 2) maintenance of diversity in solutions of the Pareto-optimal set. These two tasks cannot be measured adequately with one performance metric. So the
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convergence metric Υ proposed in [15] and the diversity metric S proposed in [14] are adopted to evaluate the performance of MMBPSO. 3.2 Experimental Results MMBPSO was applied to optimize the 5 benchmark functions, and each function was run 30 times independently. For a comparison, BMPSO [8] and DBPSO with the same P and P updating strategy are also used to solve these benchmarks. The population size and the maximum generation of all algorithms are 200 and 250 respectively, and each decision variable is coded to 30 bits. The other parameter settings of 3 algorithms are shown in Table 1. The optimization results are listed in the Table 2-3 and are drawn in Fig.2 -3 as well. The experimental results of convergence metric in Fig. 2 demonstrates that the proposed MMBPSO finds better solutions which are more closely related to the real Pareto front. For functions ZDT1, ZDT2, ZDT3 and ZDT6, MMBPSO has no difficulty to reach P ; but for function ZDT4, the performance of MMBPSO is not ideal due to the 21 different local Pareto-optimal fronts in the search space. However, MMBPSO has much better convergence values than BMPSO, MDBPSO in all 5 benchmark functions. The metrics of diversity drawn in Fig. 3 also show that MMBPSO is better than the other two algorithms on all functions. To evaluate the performances of 3 algorithms more exactly and clearly, Fig. 4 plots the founded Pareto front of MMBPSO, MDBPSO and BMPSO on all functions, which displays that MMBPSO outperforms MDBPSO and BMPSO. Table 1. Parameters settings of MMBPSO, MDBPSO and BMPSO Algorithm MMBPSO MDBPSO BMPSO [10]
Parameters α=0.55, β=0.775, pd=0.1, pm=0.001; 2.0, c 2.0 , ω 0.8, v 5,5 ; c c 2.0, c 2.0 , ω 0.8, v 5,5 . ZDT1
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Fig. 2. Box plots of the convergence metric obtained by MMBPSO, MDBPSO and BMPSO
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Fig. 3. Box plots of the distance metric obtained by MMBPSO, MDBPSO and BMPSO
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Fig. 4. The founded Pareto front of MMBPSO, MDBPSO and BMPSO on ZDT series functions Table 2. The results of convergence metric Υ Algorithm ZDT1 ZDT2 ZDT3 ZDT4 ZDT6
Mean Variance Mean Variance Mean Variance Mean Variance Mean Variance
4 Conclusion In this paper, a novel multi-objective modified binary particle swarm optimization is proposed. Compared with DBPSO, the proposed MMBPSO developed an improving updating strategy which is simpler and more easily to implement. The mutation operator and dissipation operator are introduced to improve its search ability and keep the diversity of algorithm. The modified global best and local best solutions updating strategy help MMBPSO converge to the Pareto front better. Five well-known benchmark functions were adopted for testing the proposed algorithm. The experimental results proved that the proposed MMBPSO can find better solutions than MDBPSO and BMPSO. Especially, the superior of MMBPSO to MDBPSO demonstrated the advantages of the developed updating strategy in terms of convergence and diversity. Acknowledgments. This work is supported by Research Fund for the Doctoral Program of Higher Education of China (20103108120008), the Projects of Shanghai Science and Technology Community (10ZR1411800 & 08160512100), Mechatronics Engineering Innovation Group project from Shanghai Education Commission, Shanghai University “11th Five-Year Plan” 211 Construction Project and the Graduate Innovation Fund of Shanghai University (SHUCX102218).
References 1. Xiang, Y., Sykes, J.F., Thomson, N.R.: Alternative formulations for ptimal groundwater remediation design. J. Water Resource Plan Manage 121(2), 171–181 (1995) 2. Das, D., Datta, B.: Development of multi-objective management models for coastal aquifers. J. Water Resource Plan Manage 125(2), 76–87 (1999) 3. Erickson, M., Mayer, A., Horn, J.: Multi-objective optimal design of groundwater remediation systems: application of the niched Pareto genetic algorithm (NPGA). Advances in Water Resources 25(1), 51–65 (2002) 4. Sharaf, A.M., El-Gammal, A.: A novel discrete multi-objective Particle Swarm Optimization (MOPSO) of optimal shunt power filter. In: Power Systems Conference and Exposition, pp. 1–7 (2009)
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5. Clerc, M., Kennedy, J.: The particle swarm—explosion, stabilityand convergence in a multidimensional complex space. IEEE Trans. Evol. Comput. 6(1), 58–73 (2002) 6. Kennedy, J., Eberhart, R.C.: A discrete binary version of the particle swarm algorithm, Systems, Man, and Cybernetics. In: IEEE International Conference on Computational Cybernetics and Simulation, vol. 5, pp. 4104–4108 (1997) 7. Peng, C., Xu, X.: A hybrid algorithm based on immune BPSO and N-1 principle for PMU multi-objective optimization placement. In: Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, pp. 610–614 (2008) 8. Abdul Latiff, N.M., Tsimenidis, C.C., Sharif, B.S., Ladha, C.: Dynamic clustering using binary multi-objective Particle Swarm Optimization for wireless sensor networks. In: IEEE 19th International Symposium on Personal, Indoor and Mobile Radio Communications, pp. 1–5 (2008) 9. Wang, L., Wang, X.T., Fei, M.R.: An adaptive mutation-dissipation binary particle swarm optimisation for multidimensional knapsack problem. International Journal of Modelling, Identification and Control 8(4), 259–269 (2009) 10. Wang, L., Wang, X.T., Fu, J.Q., Zhen, L.L.: A Novel Probability Binary Particle Swarm Optimization Algorithm and Its Application. Journal of Software 9(3), 28–35 (2008) 11. Qi, S., Jian, H.J., Chen, X.J., Guo, L.S., Ru, Q.Y.: Modified particle swarm optimization algorithm for variable selection in MLR and PLS modeling: QSAR studies of antagonism of angiotensin II antagonists. European Journal of Pharmaceutical Sciences 22(2-3), 145– 152 (2004) 12. Jahanbani Ardakani, A., Fattahi Ardakani, F., Hosseinian, S.H.: A novel approach for optimal chiller loading using particle swarm optimization. Energy and Buildings 40, 2177– 2187 (2008) 13. Li, X.: A non-dominated sorting particle swarm optimizer for multiobjective optimization. In: The Genetic and Evolutionary Computation Conference, pp. 37–48 (2003) 14. Gong, M., Liu, C., Cheng, G.: Hybrid immune algorithm with Lamarckian local search for multi-objective optimization. Memetic Computing 2(1), 47–67 (2010) 15. Deb, K., Jain, S.: Running performance metrics for evolutionary multi-objective optimization. Technical Report, no. 2002004 (2002)
Improved Multiobjective Particle Swarm Optimization for Environmental/Economic Dispatch Problem in Power System* Yali Wu, Liqing Xu, and Jingqian Xue School of Automation and Information Engineering, Xi’an University of Technology, shaanxi, China [email protected], [email protected], [email protected]
Abstract. An improved particle swarm optimization based on cultural algorithm is proposed to solve environmental/economic dispatch (EED) problem in power system. Population space evolves with the improved particle swarm optimization strategy. Three kinds of knowledge in belief space, named situational, normative and history knowledge are redefined respectively to accordance with the solution of multi-objective problem. The results of standard test systems demonstrate the superiority of the proposed algorithm in terms of the diversity and uniformity of the Pareto-optimal solutions obtained. Keywords: Environmental/economic dispatch, Cultural algorithm, Particle swarm optimization, Multi-objective optimization.
1 Introduction With the increasing concern of environmental pollution, operating at absolute minimum cost can no longer be the only criterion for economic dispatch of electric power generation. Environmental/economic dispatch (EED) is becoming more and more desirable for not only resulting in great economical benefit, but also reducing the pollutants emission [1]. However, minimizing the total fuel cost and total emission are conflicting in nature and they cannot be minimized simultaneously. Hence, the EED problem is a large-scale highly constrained nonlinear multi-objective optimization problem. Over the past decade, the meta-heuristic optimization methods have been significantly used in EED primarily due to their nice feature of population-based search [2]. Many multi-objective evolutionary algorithms such as niched Pareto genetic algorithm (NPGA) [3], non-dominated sorting genetic algorithm (NSGA) [4], strength Pareto evolutionary algorithm (SPEA) [5] and NSGA-II [6, 7] have been introduced to solve the EED problem with impressive success. *
Manuscript received January 2, 2011. This work was supported by Natural Science Foundation of Shaanxi Province (Grant No.2010JQ8006) and Science Research Programs of Education Department of Shaanxi Province (Grant No.2010JK711).
As a new population-based algorithm, particle swarm optimization (PSO) has several key advantages over the other existing optimization techniques in terms of simplicity, convergence speed and robustness. Several PSO-based approaches [8-11] have been proposed to solve the EED problem, but all the improved algorithms cannot consider the effective utilization of evolution knowledge. In this paper, an improved particle swarm optimization based on cultural algorithm (CA-IMOPSO) is proposed to solve EED problem. Circular crowded sorting approach is used to generate a set of well-distributed Pareto-optimal solutions, and the global best individual in multi-objective optimization domain was redefined through a new multi-objective fitness roulette technique. The evolutionary process of MOPSO in population space is controlled by adaptive adjustment policy.
2 Problem Statement The typical EED problem can be formulated as a bi-criteria optimization model. The two conflicting objectives, i.e., fuel cost and pollutants emission, should be minimized simultaneously while fulfilling certain system constraints. This problem is formulated as follows. 2.1 Problem Objectives Objective 1: Minimization of fuel cost. The total fuel cost F ( PG ) can be represented as follows:
Fi ( PGi ) = ai + bi PGi + ci PGi 2
(1)
Where M is the number of generators committed to the operating system, ai , bi , ci are fuel cost coefficients of i -th generator, and PG is the real power output of the i -th i
generator. Objective 2: Minimization of pollutants emission. The emissions can be modeled through a combination of polynomial and exponential terms [12].
Ei ( PGi ) = α i + β i PGi + γ i PGi 2 + ξi exp(λi PGi )
(2)
Where αi , βi , γ i ,ξi , λi are coefficients of the i -th generator emission characteristics. 2.2 Problem Constraints Constraint 1: Generation capacity constraint. For normal system operations, real power output of each generator is restricted by lower and upper bounds as follows:
PGmin ≤ PGi ≤ PGmax i i
(3)
where PGimin and PGimax are the minimum and maximum power generated by generator i , respectively.
Constraint 2: Power balance constraint. The total power generation must cover the total demand PD and the real power loss in transmission lines PLoss. NG
PD + PLOSS − ∑ PGi = 0
(4)
i =1
NG NG
NG
i =1 j =1
i =1
PLOSS = ∑∑ PGi Bij PGi + ∑ PGi B0i + B00
(5)
where Bij is the transmission loss coefficient, B0i is the i -th element of the loss coefficient vector and B00 is the loss coefficient constant.
3 Improved Algorithm (CA-IMOPSO) The structure of CA-IMPSO algorithm is shown in fig.1 [13], in which population space and belief space are linked through acception function and influence function. Adjust Belief space Acceptance Selection
Influence
Population space
Performance Function
Variation
Fig. 1. Spaces of a cultural algorithm
3.1 Particle Swarm Optimization (PSO) in the Population Space
A particle status on the population space is characterized by two factors: its position and velocity, which are updated by following equations [14]: vid (t + 1) = wvid (t ) + c1r1d (φid (t ) − xid (t )) + c2 r2 d (φ gd (t ) − xid (t ))
(6)
xid (t + 1) = xid (t ) + vid (t + 1)
(7)
where vid represents the d -th dimensional velocity of particle i ; xid represents the d -th dimensional position values of particle i ; φid represents the best previous position of particle i ; φgd represents the best position among all particles in the population. r1d and r2 d are two independently uniformly distributed random variables with range [0, 1]; c1 and c2 are acceleration coefficients; w is the inertia weight. 3.2 The Knowledge of the Belief Space
Three knowledge sources, named situational knowledge, normative knowledge and history knowledge, are considered in belief space.
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Situational Knowledge. Situational knowledge is a set of exemplary individuals useful for the experiences of all individuals. So the initial situational knowledge is choosen from the nondominated set of the population space depending on the diversity and uniformity. The variation operator of differential evolution techniques is used to update the situational knowledge, that is: xi' = xi + F ⋅ ( pi , r1 + pi , r 2 )
(8)
where xi is the i -th individual in the situational knowledge, and pi , r1 , pi , r 2 are different particles in the nondominated set. F ∈ [ 0,1] is the ratio factor of differential evolution. If xi' dominates xi , then xi' replaces xi . If neither of them dominates each other, select the new individual at random. Normative Knowledge. Normative knowledge consists of a set of promising ranges. It provides a standard guiding principle within which individual adjustments can be made. Normative knowledge is updated in two means:
1) For the i -th dimension of each nondominated individual, if the fitness is located in the iteration, we generate a child around it. Else, we generate a child randomly on homogeneous distribution in the iteration; 2) Replace the i-th dimension of a particle with its minimum and maximum velocities to generate two children. If one child dominates the particle and the other child, then adopt it as the new non-dominated individual; Else if both of the children dominates the particle but non-dominates each other, then choose one of them as the new nondominated individual. History Knowledge. History knowledge keeps track of the history of the search process and records key events in the search. It might be either a considerable move in the search space or a discovery of a landscape change. Individuals use the history knowledge for guidance in selecting a moving direction [11]. The history knowledge will be used later to adapt the distribution of the individuals after finding the Pareto-front. 3.3 Communication Protocol Acceptance Function. The global worst one of the belief space is replaced by the global best of the population space every Acc generation. Acc = Bnum + t / Tmax × Dnum
(9)
Where Bnum and Dnum are two constants. The global best of population space is the least number of the individual. And the global worst of the belief space is the individual with the shortest crowding distance in Pareto-front.
Influence Function. After each Inf generation, the global worst one of the population space is replaced by the global best of the belief achieve. Inf = Bnum + (Tmax − t ) / Tmax × Dnum
(10)
The global best individual of the belief space is the one with the longest crowding distance in Pareto-front. And the global worst individual of the population space is the one with the largest number. 3.4 Archiving Mechanism
The non-dominated solutions of the archive are composed of two parts. Some of them are new non-dominated solutions in population space; the others are new nondominated solutions in belief space. A circular crowding sorting algorithm is adopted in this paper to improve the uniformity of Pareto optimal front.
4 Implement of CA-IMOPSO for EED Problem In this section, the proposed algorithm was applied to the standard IEEE 30-bus sixgenerator test system [15]. This power system is connected through 41 transmission lines and the total system demand amounts to 2.834 p.u. 4.1 Encoding of Particles
The first step is the encoding of the decision variables. The power output of each generator is selected as the gene, and many genes comprise a particle which represents a candidate solution for the EED problems. That is, every particle j consists of N real coded string such as x j = {PG1, j , PG 2, j ,......PGM , j } , where PGi , j , i = 1, 2,", M means the power output of the i -th generator with respect to the j -th particle. 4.2 Parameter Setting of the Proposed Algorithm
The inertia weight ω , acceleration coefficient c1 and c2 is defined as follows. ω = ωmax − ((ωmax − ωmin ) / Tmax )t
(11)
c1 = (c1 f − c1i )t / Tmax + c1i
(12)
c2 = (c2 f − c2i )t / Tmax + c2i
(13)
Where t is the current iteration number and Tmax is the maximum iteration number. The other parameters are set in the following. The size of swarm and archive set are both fixed at 50. ω min = 0.4 , ω max = 0.9 , c1i = 2.5, c1 f = 0.5, c2i = 0.5, c2 f = 2.5 , F = 0.5 , Tmax = 7000 .
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5 Simulation Results and Discussions Two cases have been considered. In Case 1, the generation capacity and the power balance constraints with neglecting Ploss are considered; in Case 2, the generation capacity and the power balance constraints with considering Ploss are considered. 5.1 Multiobjective Optimization Using CA-IMOPSO
In next page, the Pareto-optimal sets are shown in Figs.2(a) for case 1 and in Figs.2(b) for case 2. It can be seen that the CA-IMOPSO technique preserves the diversity and uniformity of the Pareto-optimal front and solve effectively the problem in both cases considered. The non-dominated solutions with CA-IMOPSO for case 1 and case 2 are compared to those reported in the literature [10], [4], [3], [5]. And the best two nondominated solutions with the proposed approach and those reported for case 1 and case 2 are given in Table 1 and 2 respectively. 0.225
0.22
0.22
0.215
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Fig. 2. The pareto-optimal front of EED problem Table 1. The result of minimum cost in case 1 with different algorithms FCPSO [10]
Table 2. The result of minimum cost in case 2 with different algorithms FCPSO [10]
NSGA [4]
NPGA [3]
SPEA [5]
MOPSO
CA-IMOPSO
P1
0.1130
0.1168
0.1245
0.1279
0.1139
0.1209
P2
0.3145
0.3165
0.2792
0.3163
0.3252
0.2894
P3
0.5826
0.5441
0.6284
0.5803
0.6161
0.5814
P4
0.9860
0.9447
1.0264
0.9580
0.9689
0.9894
P5
0.5264
0.5498
0.4693
0.5258
0.4998
0.5220
P6
0.3450
0.3964
0.3993
0.3589
0.3464
0.3557
Cost
607.7862
608.245
608.147
607.86
608.790
605.881
Emission
0.2201
0.2166
0.2236
0.2176
0.2191
0.2203
From the table we can conclude that the proposed CA-IMOPSO technique is superior to all reported techniques. And it demonstrates the potential and effectiveness of the proposed technique to solve EED problem.
6 Conclusion In this paper, a novel multiobjective particle swarm optimization technique based on cultural algorithm has been proposed and applied to environmental economic dispatch optimization problem. The results of the EED problem show the potential and efficiency of the proposed algorithm. In addition, the simulation results also reveal the superiority of the proposed algorithm in terms of the diversity and quality of the obtained Pareto-optimal solutions.
References 1. Talaq, J.H., EI-Hawary, F., EI-Hawary, M.E.: A summary of environmental/economic dispatch algorithms. J. IEEE Trans. Power Syst. 9(3), 1508–1516 (1994) 2. Lingfeng, W., Chanan, S.: Environmental/economic power dispatch using a fuzzified multi-objective particle swarm optimization algorithm. J. Electr. Power Syst. Research 77, 1654–1664 (2007) 3. Abido, M.A.: A niched Pareto genetic algorithm for multiobjective environmental/economic dispatch. J. Electr. Power Energy Syst. 25(2), 97–105 (2003) 4. Abido, M.A.: A novel multiobjective evolutionary algorithm for environmental/ economic power dispatch. J. Electr. Power Syst. Research 65, 71–91 (2003) 5. Abido, M.A.: Multiobjective evolutionary algorithms for electric power dispatch problem. J. IEEE Trans. Evolut. Comput. 10(3), 315–329 (2006) 6. King, R.T.F., Rughooputh, H.C.S., Deb, K.: Evolutionary multi-objective environmental/Economic dispatch: Stochastic versus deterministic approaches. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 677–691. Springer, Heidelberg (2005)
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7. Basu, M.: Dynamic economic emission dispatch using nondominated sorting genetic algorithm-II. J. Electr. Power. Energy Syst. 30(2), 140–210 (2008) 8. Wang, L.F., Singh, C.: Environmental/economic power dispatch using a fuzzified multiobjective particle swarm optimization algorithm. J. Electr. Power Syst. Res. 77(12), 1654– 1664 (2007) 9. Cai, J.J., Ma, X.Q., Li, Q., Li, L.X., Peng, H.P.: A multi-objective chaotic particle swarm optimization for environmental/economic dispatch. J. Energy Convers Manage. 50(5), 1318–1325 (2009) 10. Agrawal, S., Panigrahi, B.K., Tiwari, M.K.: Multiobjective particle swarm algorithm with fuzzy clustering for electrical power dispatch. J. IEEE Trans. Evolut. Comput. 12(5), 529–541 (2008) 11. Daneshyari, W., Yen, G.G.: Cultural MOPSO: A cultural framework to adapt parameters of multiobjective particle swarm optimization. In: C. IEEE Congress. on Evolut. Comput., pp. 1325–1332 (2009) 12. Farag, A., Al-Baiyat, S., Cheng, T.C.: Economic load dispatch multiobjective optimization procedures using linear programming techniques. J. IEEE Trans. Power Syst. 10(2), 731–738 (1995) 13. Landa, B., Carlos, A., Coello, C.: Cultured differential evolution for constrained optimization. J. Comput Methods in Applied Mechanics and Engine 195, 4303–4322 (2006) 14. Yunhe, H., Lijuan, L., Yaowu, W.: Enhanced particle swarm optimization algorithm and its application on economic dispatch of power systems. J. Proc. of CSEE 24(7), 95–100 (2004) 15. Hemamalini, S., Simon, S.P.: Emission Constrained Economic Dispatch with Valve-Point Effect using Particle Swarm Optimization. In: C. IEEE Region. 10 Confer., pp. 1–6 (2008)
A New Multi-Objective Particle Swarm Optimization Algorithm for Strategic Planning of Equipment Maintenance Haifeng Ling1,2 , Yujun Zheng3 , Ziqiu Zhang4 , and Xianzhong Zhou1 1
School of Management & Engineering, Nanjing University, Nanjing 210093, China 2 PLA University of Science & Technology, Nanjing 210007, China 3 College of Computer Science & Technology, Zhejiang University of Technology, Hangzhou 310014, China 4 Armament Demonstration & Research Center, Beijing 100034, China ling [email protected], [email protected], [email protected], [email protected]
Abstract. Maintenance planning plays a key role in equipment operational management, and strategic equipment maintenance planning (SEML) is an integrated and complicated optimization problem consisting of more than one objectives and constraints. In this paper we present a new multi-objective particle swarm optimization (PSO) algorithm for effectively solving the SEML problem model whose objectives include minimizing maintenance cost and maximizing expected mission capability of military equipment systems. Our algorithm employs an objective leverage function for global best selection, and preserves the diversity of non-dominated solutions based on the measurement of minimum pairwise distance. Experimental results show that our approach can achieve good solution quality with low computational costs to support effective decision-making.
1
Introduction
Maintenance planning plays a key role in equipment operational management. In general, strategic equipment maintenance planning (SEMP) involves various kinds of equipment, different maintenance policies and quality evaluation measures, and complex sets of constraints, and thus is typically modeled as a complicated multi-objective optimization problem. Despite this, most researches on equipment maintenance planning (e.g., [4,7,15]) still focus on the narrow use of single approaches such as expert conversation, mathematical programming, proration, case based reasoning, etc, which have serious limitations in expressing relationships and tradeoffs between the objectives. In recent years, some nature-inspired heuristic methods such genetic algorithm, simulated annealing, and ant colony optimization have been applied
This work was supported in part by grants from National Natural Science Foundation (No. 60773054, 61020106009, 90718036) of China.
Y. Tan et al. (Eds.): ICSI 2011, Part II, LNCS 6729, pp. 57–65, 2011. c Springer-Verlag Berlin Heidelberg 2011
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in the domain of equipment maintenance and have shown their advantages both in problem-solving effectiveness and solution quality. For example, Kleeman and Lemont [9] designed a multi-objective genetic algorithm to solve the aircraft engine maintenance scheduling problem, which is a combination of a modified job shop problem and a flow shop problem. Verma and Ramesh [14] viewed the initial scheduling of preventive maintenance as a constrained non linear multi-objective decision making problem, and proposed a genetic algorithm that simultaneously optimizes the objectives of reliability, cost and newly introduced criteria, nonconcurrence of maintenance periods and maintenance start time factor. Yang and Huang [16] also proposed a genetic algorithm for multi-objective equipment maintenance planning, but the model used a simplified function that evaluates equipment capability based on equipment cost and thus limited its practicality. Ai and Wu [1] used a hybird approach based on simulated annealing and genetic algorithm for communication equipment maintenance planning, the they did not consider multiple objectives. Recently the authors presented in [19] an efficient multi-objective tabu search algorithm, which was capable of solving large problems with more than 45000 equipments of 500 kinds. Particle swarm optimization (PSO) [8] is a population-based global optimization technique that enables a number of individual solutions, called particles, to move through a hyper dimensional search space in a methodical way to search for optimal solution(s). Each particle represents a feasible solution which has a position vector x and a velocity vector v, which are adjusted at iteration by learning from a local best pbest found by the particle itself and a current global best g best found by the whole swarm. PSO is conceptually simple and easy to implement, and has demonstrated its efficiency in a wide range of continuous and combinatorial optimization problems [2]. Since 2002, multi-objective PSO (MOPSO) has attracted much attention among researchers and has shown promising results for solving multi-objective optimization problems (e.g., [3,13,11,6]). In this paper we define a multi-objective integer programming model for MESP which considers objectives include minimizing maintenance costs (including costs of maintenance materiel and workers) and maximizing expected mission capability of equipment systems (via layered quadratic functions). We then propose a MOPSO algorithm for the problem model, which uses an objective leverage function for global best selection and preserves the diversity of non-dominated solutions based on the measurement of minimum pairwise distance. Experimental results show that our approach can achieve good solution quality with low computational costs to support effective decision- making.
2 2.1
Problem Model Problem Description and Hypothesis
SEMP needs to determine the numbers of different kinds of equipment to be maintained at different levels according to the overall mission requirements and the current conditions of all equipment [18]. There are two key aspects to assess
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an SEMP solution: the overall maintenance cost and the overall mission capability after maintenance. Thus SEMP is typically a multi-objective optimization problem, for which the improvement of one objective may cause the degradation of another. On of the basic principles of equipment maintenance is to assign each equipment to an appropriate maintenance level according to the quality of the equipment. In this paper, we roughly suppose there are three quality levels of equipment, namely A, B, and C, and two maintenance levels, namely I and II; Typically, equipment of quality level A does not need to be maintained, and equipment of quality level C and B should be maintained at the level I and II respectively1 . 2.2
Cost Evaluation
Suppose there are m kinds of equipment to be maintained and n kinds of materiel to be consumed for maintenance, then we can define two materiel to be materiel consuming matrices, namely Z = (zij )m×n and Z = (zij )m×n , where zij and zij denote materiel j consumed for equipment i at the maintenance level I and II respectively. Let cj be the price of material j, xi be the number of equipment i to be maintained at level I and xi be that at level II, then the total cost of maintenance material can be calculated as follows: CE =
m n
cj (zij xi + zij xi )
(1)
i=1 j=1
Moreover, let ti be the number of (average) maintenance hours of equipment i to be maintained at level I, si be the relevant cost (in RMB Yuan) per hour, and ti and si be that at level II, then the total working hour and working cost can be respectively calculated by (2) and (3): MT =
m
(ti xi + ti xi )
(2)
i=1
CT =
m (si ti xi + si ti xi )
(3)
i=1
And thus the overall maintenance cost C = CE + CT . 2.3
Mission Capability Evaluation
B C Let xA i , xi , and xi be the current numbers of equipment i at the quality level A, B and C respectively, and αi , βi , and γi be the wearing coefficients during a given period respectively2 . Given the maintenance number xi and xi , after the 1 2
The model based on this hypothesis can be easily extended to include more quality and maintenance levels. In detail, αi is the probability of degradation from quality A to B, βi is that from B to C, and γi is that from A to C. All coefficients are in range (0,1).
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given period the expected numbers of equipment i of the quality level A, B, and C are respectively calculated as follows: A x A i = (1 − αi − γi )(xi + xi + xi )
A B x B i = αi (xi + xi + xi ) + (1 − βi )xi C B A x C i = xi + βi xi + γi (xi + xi + xi )
(4) (5) (6)
Now for equipment i, its mission capability can be evaluated based on the weighted sum x i of the numbers of equipment at different quality level as follows (For most equipment the weight wiA can be set to 1 and the weight wiC is very small): B B x i = wiA x A i + wiC x C (7) i + wi x i And the mission capability I of the whole equipment system can be evaluated using the quadratic function as follows: I=
m m
aij x i x j +
i=1 j=1
m
bi x i + c
(8)
i=1
where aij is the correlation coefficient, bi is the covariance coefficient, and c is the constant coefficient. In real-world applications, we usually have aij = 0 for most i and j, and thus the number of coefficients is typically far less than m2 . 2.4
The Multi-Objective Optimization Problem Model
Based on above analysis, we get the following multi-objective optimization problem model for SEMP: m m
max I =
min C =
aij x i x j +
i=1 j=1 m n
m
bi x i + c
(9)
i=1 cj (zij xi + zij xi ) +
i=1 j=1
s.t. I ≥ I
m
(si ti xi + si ti xi )
(10)
i=1
(11)
C≤C
(12)
MT ≤ M T m xi ≤ X
(13)
i=1 m
xi ≤ X
(14) (15)
i=1
where I is the lower limit of the overall mission capability, C is the upper limit of the overall cost, M T is the upper limit of the total working hour, and X and X are the upper limits of the numbers of equipment can be maintained at level I and II respectively.
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3 3.1
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MOPSO Algorithm The Algorithm Framework
The SEMP model described above is a multi-objective integer programming problem. Although the standard PSO algorithm works on continuous variables, the truncation of real values to integers will not affect significantly the performance of the method when the range of decision variables is large [10]. The following presents our MOPSO algorithm for the SEMP that searches for the Pareto-optimal front rather than a single optimal solution: 1. Set the basic algorithm parameters, and randomly generate a swarm P of p feasible solutions. 2. For each particle η in the swarm, initialize its velocity v η = 0, and set pηbest be its initial position xη . 3. Select all non-dominated solutions from P and save them in the archive N P . 4. Choose a solution gbest from N P such that: gbest = max{θ ∈ N P |w1 I(θ) − w2 C(θ)}
(16)
where and w1 and w2 are two preference weights satisfying w1 , w2 ∈ (0, 1) and that w1 + w2 = 1. 5. Update the velocity and position of each η in P according to the following movement equations: v η = wv η + c1 r1 (pηbest − xη ) + c2 r2 (g best − xη )
(17)
x =x +v
(18)
η
6. 7. 8. 9. 10.
η
η
where w is the inertia weight, c1 and c2 are learning factors, and r1 and r2 are random values between (0, 1). If the position xη violates the problem constraints (11)∼(15), reset xη = pηbest and reset v η = 0. η Update each local best solution pbest . Compute SI = θ∈N P I(θ) and SC = θ∈N P C(θ). Update the non-dominated solution set N P based on the new swarm, and then compute ΔSI = θ∈N P I(θ) − SI and ΔSC = SC − θ∈N P C(θ). If the termination condition is satisfied, then the algorithm stops; else update the inertia weight according to the following equations and then goto step 4: k w = wmax − max (wmax − wmin ) k min(w1 + 0.1, w1max ) if ΔSI < ΔSC w1 = max(w1 − 0.1, w1min ) else w2 = 1 − w1
(19) (20) (21)
where k is the current iteration number, k max is the maximum iteration number of the algorithm, wmax and wmin are the maximum and minimum inertia weights respectively, and w1max and w1min are the maximum and minimum first-preference weights respectively.
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In the algorithm we use an “objective leverage function” L(θ) = w1 I(θ)−w2 C(θ) for global best selection, and a solution θ whose value of L(θ) is the maximum among all solutions in N P is selected as g best . Typically, w1 and w2 can both be initialized to 0.5, or be manually set based on the preference of the decisionmaker. After each iteration, we compute ΔSI , the increasing of summarized value objective function I of N P , and ΔSC , the decreasing of summarized value objective function C of N P . If ΔSI is less than ΔSC , we increase the preference weight w1 and decrease w2 , and vice versa. This strategy significantly decreases the computational cost for global best selection since at each time it uses only one non-dominated solution as the g best for all particles; on the other hand, the changing of values of w1 and w2 according to the ΔSI and ΔSC helps to simultaneously evolve the two objectives and thus to improve the diversity of the solutions. However, to ensure the strategy works effectively, we should scale those coefficients in (9) and (10) such that I and C are of the same order of magnitude. 3.2
Size and Diversity of the Solutions Set
During the search procedure, the size of approximating non-dominated solution set N P may increase rapidly and the performance of search will decrease significantly. Therefore, a reasonable approach is to limit the size of the solution set, which will cause us to decide whether to insert a new non-dominated solution η when the size of N P reaches the limit and, if so, which archive solution θ should be removed. In our MOPSO algorithm, we use the diversity of solutions in N P as the criterion, i.e., a θ ∈ N P should be replaced with η if the diversity can be (potentially) improved by doing so, since the preservation and improvement of diversity of which is crucial not only to avoid loosing potentially efficient solutions but also avoid premature convergence. Here we employ a simple approach based on the minimum pairwise distance [5] which is with low computational costs. In detail, whenever N P contains more than one member, it records two solutions θa and θb , the Euclidean distance between which is the minimum among all pairs in N P : dis(θa , θb ) = min dis(x, y) (22) x,y∈N P ∧x=y
When the size of N P reaches the size limit |N P |max , the following procedure is applied for possible inclusion of a new solution η: If η is dominated by any θ ∈ N P , then η is discarded. Else if η dominates some θ ∈ N P , then remove those θ and insert η. Else if minθ∈N P ∧θ=θa dis(η, θ) > dis(θa , θb ), then remove θa and insert η. Else if minθ∈N P ∧θ=θb dis(η, θ) > dis(θa , θb ), then remove θb and insert η. Else choose a closet z ∈ N P to η; if minθ∈N P ∧θ=z dis(η, θ) > dis(η, z), then remove z and insert η. 6. Else discard η.
1. 2. 3. 4. 5.
A New Multi-Objective Particle Swarm Optimization Algorithm Table 1. Parameter setting in the algorithms, where M = total number of equipment
m
i=1
63
B C xA i + xi + xi is the
|N P |max kmax p c1 c2 wmax w min w1max w1min MOPSO-A max( M/10, 50) M/2 m/10 0.8 1.0 0.9 0.1 0.9 0.1 MOPSO-B max( M/10, 50) M/2 m/10 0.8 1.0 0.9 0.1 MOTS max( M/10, 50) M Algorithm
Table 2. Computational experiments conducted on the test problem instances m
M
MOPSO-A t |N P |
4
I
∗
MOPSO-B C
∗
t |N P |
I
∗
MOTS C
∗
t |N P |
I∗
C∗
50
300
0.1
5 30.6 11.4
0.1
5 30.6 11.4
0.1
5 30.6 11.4
50
900
0.2
9 75.2 35.7
0.2
9 75.2 35.7
0.1
9 75.2 35.7
100 1800
0.7
13 24.3 89.0
0.7
13 24.3 89.0
0.6
13 24.3 89.0
100 9750
2.4
28 45.9 12.6
2.7
26 40.1 12.6
2.2
30 45.9 14.3
200 3800
2.6
18 14.9 12.1
2.7
17 14.6 12.1
2.6
18 14.9 12.1
200 17200
21.8
36 92.5 42.2
26.1
39 79.0 42.2
42.2
40 73.6 42.2
300 4900
22.2
20 80.4 89.6
26.6
23 72.1 93.2
40.5
23 72.1 87.5
300 23600
63.6
39 54.2 44.7
60.5
33 52.3 43.8 181.7
37 53.9 46.7
500 9500
72.1
27 36.7 25.0
74.1
29 36.3 25.2 208.0
30 36.9 28.1
500 45800 292.4
46 50.0 12.5 329.2
50 48.7 12.8 1480.9
48 49.9 12.7
800 13650 283.3
36 34.9 22.2 280.7
36 34.9 22.2 1675.8
35 29.6 22.0
800 64000 2791.2
49 75.3 48.8 3039.4
50 70.1 48.8
1000 15000 2846.6
38 29.4 14.2 3022.3
38 27.3 14.2
1000 82000 6558.9
50 69.3 60.0
Computational Experiments
The presented MOPSO algorithm (denoted by MOPSO-A) has been tested on a set of SEMP problem instances and compared with two other algorithms: – A basic MOPSO algorithm (denoted by MOPSO-B) where the gbest is randomly selected from N P . – A multi-objective tabu search algorithm (denoted by MOTS) proposed in [19].
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The experiments are conducted on a computer of 2 × 2.66GHz AMD Athlon64 X2 processor and 8GB memory. The basic parameter values are given in Table 1. The performance measures include the CPU time t (in seconds), the number of non-dominated solutions |N P |, the result maximum mission capability I ∗ and minimum maintenance cost C ∗ . In order to improve the clarity of comparison, all the values of I ∗ and C ∗ are scaled into the range (0, 100). The summary of experimental results are presented in Table 2 (the maximum running time on every instance is 2 hours, and that “ ” denotes the algorithm fails to stop within the time). As we can see from the computational results, for small-size problem instances where m ≤ 100 and M ≤ 1800, all the three algorithms reach the same Paretooptimal front; but with the increasing of instance size, two PSO algorithms exhibit significant performance advantage over the tabu search algorithm; for large-size problems, MOPSO-A also exhibits certain performance advantage over MOPSO-B. On the other hand, the result I ∗ obtained by MOPSO-A is always no less than that obtained by MOPSO-B, and C ∗ obtained by MOPSO-A is always no more than that obtained by MOPSO-B except for one case (which is italicized in Table 2). This demonstrate that our strategy from global best selection plays an important role for improving the quality of result solutions.
5
Conclusion
The paper presents an effective multi-objective particle swarm optimization (PSO) algorithm for solving the SEML problem model. Our algorithm employs an objective leverage function for global best selection and preserves the diversity of non-dominated solutions based on the measurement of minimum pairwise distance, and thus decreases the computational cost and improve the quality of result solution set. As demonstrated by the experimental results, the proposed algorithm are quite efficient even for large-size problem instances. We are now extending the algorithm by introducing non-dominated sorting method [11] which will increase the computational cost but can evolve the swarm more close to the true Pareto front, and thus is more appropriate for medium-size problem instances. Further research will also include the fuzziness of maintenance costs and mission capability to decrease the sensitivity of the model and improve the adaptivity of the algorithm.
References 1. Ai, B., Wu, C.: Genetic and simulated annealing algorithm and its application toequipment maintenace resource optimization. Fire Control & Command Control 35(1), 144–145 (2010) 2. Clerc, M.: Particle Swarm Optimization. ISTE, London (2006) 3. Coello, C.A.C., Lechuga, M.S.: MOPSO: A proposal for multiple objective particle swarm optimization. In: Proceedings of Congress on Evolutionary Computation, vol. 2, pp. 1051–1056. IEEE Press, Los Alamitos (2002)
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4. Fletcher, J.D., Johnston, R.: Effectiveness and cost benefits of computer-based decision aids for equipment maintenance. Comput. Human Behav. 18, 717–728 (2002) 5. Hajek, J., Szollos, A., Sistek, J.: A new mechanism for maintaining diversity of Pareto archive in multi-objective optimization. Adv. Eng. Softw. 41, 1031–1057 (2010) 6. Ho, S.-J., Ku, W.-Y., Jou, J.-W., Hung, M.-H., Ho, S.-Y.: Intelligent particle swarm optimization in multi-objective problems. In: Ng, W.-K., Kitsuregawa, M., Li, J., Chang, K. (eds.) PAKDD 2006. LNCS (LNAI), vol. 3918, pp. 790–800. Springer, Heidelberg (2006) 7. Jayakumar, A., Asgarpoor, S.: Maintenance optimization of equipment by linear programming. Prob. Engineer. Inform. Sci. 20, 183–193 (2006) 8. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks, Perth WA, Australia, pp. 1942–1948 (1995) 9. Kleeman, M.P., Lamont, G.B.: Solving the aircraft engine maintenance scheduling problem using a multi-objective evolutionary algorithm. In: Coello, C.C., Aguirre, A.H., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 782–796. Springer, Heidelberg (2005) 10. Laskari, E.C., Parsopoulos, K.E., Vrahatis, M.N.: Particle swarm optimization for integer programming. In: Proceedings of Congress on Evolutionary Computing, pp. 1582–1587. IEEE Press, Los Alamitos (2002) 11. Li, X.: A non-dominated sorting particle swarm optimizer for multiobjective optimization. In: Cant´ u-Paz, E., Foster, J.A., Deb, K., Davis, L., Roy, R., O’Reilly, U.-M., Beyer, H.-G., Kendall, G., Wilson, S.W., Harman, M., Wegener, J., Dasgupta, D., Potter, M.A., Schultz, A., Dowsland, K.A., Jonoska, N., Miller, J., Standish, R.K. (eds.) GECCO 2003. LNCS, vol. 2723, pp. 37–48. Springer, Heidelberg (2003) 12. Liu, D., Tan, K., Goh, C., Ho, W.: A multiobjective memetic algorithm based on particle swarm optimization. IEEE Trans. Syst. Man. Cybern. B 37, 42–50 (2007) 13. Parsopoulos, K.E., Vrahatis, M.N.: Particle dwarm optimization method in multiobjective problems. In: Proceedings of the 2002 ACM Symposium on Applied Computing, pp. 603–607. ACM Press, New York (2002) 14. Verma, A.K., Ramesh, P.G.: Multi-objective initial preventive maintenance scheduling for large engineering plants. Int. J. Reliability Quality & Safety Engineering 14, 241–250 (2007) 15. Xu, L., Han, J., Xiao, J.: A combinational forecasting model for aircraft equipment maintenance cost. Fire Control & Command Control 33, 102–105 (2008) 16. Yang, Y., Huang, X.: Genetic algorithms based the optimizing theory and approaches to the distribution of the maintenance cost of weapon system. Math. Prac. Theory 24, 74–84 (2002) 17. Yu, G., Li, P., He, Z., Sun, Y.: Advanced evolutionary algorithm used in multiobjective constrained optimization problem. Comput. Integ. Manufact. Sys. 15, 1172–1178 (2009) 18. Zhang, Z., Wang, J., Duan, X., et al.: Introduction to Equipment Technical Support. Military Science Press, Beijing (2001) 19. Zheng, Y., Zhang, Z.: Multi-objective optimization model and algorithm for equipment maintenance palnning. Comput. Inter. Manufact. Sys. 16, 2174–2180 (2010)
Multiobjective Optimization for Nurse Scheduling Peng-Yeng Yin*, Chih-Chiang Chao, and Ya-Tzu Chiang Department of Information Management, National Chi-Nan University Nantou 54561, Taiwan [email protected]
Abstract. It is laborious to determine nurse scheduling using human-involved manner in order to account for administrative operations, business benefits, and nurse requests. To solve this problem, a mathematical formulation is proposed where the hospital administrators can set multiple objectives and stipulate a set of scheduling constraints. We then present a multiobjective optimization method based on the cyber swarm algorithm (CSA) to solve the nurse scheduling problem. The proposed method incorporates salient features from particle swarm optimization, adaptive memory programming, and scatter search to create benefit from synergy. Two simulation problems are used to evaluate the performance of the proposed method. The experimental results manifest that the proposed method outperforms NSGA II and MOPSO in terms of convergence and diversity performance measures of the produced results. Keywords: cyber swarm algorithm, adaptive memory programming, scatter search, multiobjective optimization, nurse scheduling.
1 Introduction Nurse scheduling, which is among many other types of staff scheduling, intends to automatically allot working shifts to available nurses in order to maximize hospital value/benefit subject to relevant constraints including governmental regulations, nurse skill requirement, minimal on-duty hours, etc. There are several solution methods proposed in the last decade for dealing with the nurse scheduling problem. These methods can be divided into three categories: mathematical programming, heuristics, and metaheuristics. Most of the methods aimed to solve a single-objective formulation, only few of them [1-4] addressed a more complete description of real-world hospital administration and attempted multiobjective formulation of nurse scheduling. Nevertheless, due to the high complexity of multiobjective context, the authors of [1-3] converted the multiobjective formulation into a single-objective program by the weighting-sum technique. The weighting-sum technique fails to identify optimal solutions if the Pareto front is non-convex and the value of the weights used to combine multiple objectives is hard to determine. This paper proposes a cyber swarm algorithm (CSA) for the Multi-Objective Nurse Scheduling Problem (MONSP). The CSA is a new metaheuristic approach which marries the major features of particle swarm optimization (PSO) and scatter search. *
The CSA has been shown to be more effective than several state-of-the-art methods for single-objective optimization [5]. The contribution of this paper includes the following. (1) We devise a multiobjective version for the CSA. The proposed method, named MOCSA, is general and can be employed to solve many classes of problems with multiobjective context; (2) we show the effectiveness of MOCSA in tackling the generic multiobjective nurse scheduling problem. The non-dominated solutions obtained by MOCSA are superior to those produced by other competing methods in terms of the dominance strength and the diversity measure on the solution front; and (3) the multi-dimensional asymptotic Pareto front is shown in the objective space to illustrate the comparative performances of competing methods. The remainder of this paper is organized as follows. Section 2 presents a literature review of existing methods for the nurse scheduling problem and introduces the central concepts of multiobjective optimization. Section 3 describes the problem formally and articulates the proposed method. Section 4 presents experimental results together with an analysis of their implications. Finally, concluding remarks and discussions are given in Section 5.
2 Related Works To assist various operations performed in a hospital, a work day is normally divided into two to four shifts (for example, a three-shift day may include day, night, and late shift). Each nurse is allocated to a number of shifts during the scheduling period with a set of constraints. A shift is fulfilled by a specified number of nurses with different medical skills depending on the operations to be performed in the shift. The adherent constraints with nurse scheduling are necessary hospital regulations when taking into account the wage cost, execution of operations, nurses’ requests, etc. The constraints can be classified as hard constraints and soft constraints. Hard constraints should be strictly satisfied and a schedule violating any hard constraints will not be acceptable. Soft constraints are desired to be satisfied as much as possible and a schedule violating soft constraints is still considered feasible. The objective could involve the reduction of the human resource cost, satisfaction of nurses’ request, or minimization of violations to any soft constraints. Most existing works seek to optimize one objective, only few consider multiple objectives when search for solutions. Berrada et al. [1] proposed the first attempt to find a nurse schedule optimizing several soft constraints simultaneously. The lexico-dominance technique is applied where the priority order of the soft constraints is pre-specified and is used to determine the quality of solutions. Burke et al. [3] applied the weighting sum technique but the weight values are determined by the priority order of objectives obtained after close consultation with hospitals. Burke et al. [4] proposed a simulated annealing multiobjective method which generates the non-dominated solutions to obtain an approximate Pareto front. A widely accepted notion in decision science field for multiobjective optimization is to search the Pareto-optimal solutions which are not dominated by any other solutions. A solution x dominates another solution y, denoted x ; y , if x is strictly better than y in at least one objective and x is no worse than y in the others. The plots of objective values for all Pareto-optimal solutions form a Pareto front in the objective space. It is usually hard to find the true Pareto front due to the high complexity of the problem nature. Alternatively, an approximate Pareto front is searched for. The
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quality of this front is evaluated by two measures: (1) The convergence measure indicates how close the approximate front is converging to the true front, and (2) the diversity measure favors the approximate front whose plots are evenly spread on the front. Classical multiobjective optimization methods include lexico-dominance, weighting sum, and goal programming. However, multiple runs of the applied method are needed to obtain a set of non-dominated solutions. Recently, metaheuristic algorithms have been introduced as a viable technique for multiobjective optimization. Notable applications have been proposed using Strength Pareto Evolutionary Algorithm (SPEA II) [6], Non-dominated Sorting Genetic Algorithm (NSGA II) [7], and Multi-Objective Particle Swarm Optimization (MOPSO) [8].
3 Proposed Method This paper deals with the MONSP on a shift-by-shift basis. Each working day is divided to three shifts (day, night, and late shift), and the total shifts in a scheduling period are numbered from 1 to S (1 indicates the day-shift of the first day, 2 indicates the night-shift of the first day, etc). Assume that there are M types of nurse skills, and skill type m is owned by Tm nurses. The aim of the MONSP is to optimize multiple objectives simultaneously by allotting appropriate nurses to the shifts subject to a set of hard constraints. By using the notations introduced in Table 1, we present the mathematical formulation of the addressed MONSP as follows. Table 1. Notations used in the addressed MONSP formulation Lmj Umj Wm Rm Cmj Pmij xmij
Min. number of nurses having skill m required to fulfill shift j Max. number of nurses having skill m required to fulfill shift j Min. number of shifts a nurse having skill m should serve in a scheduling period Max. number of consecutive working days that a nurse having skill m can serve Cost incurred by allotting a nurse having skill m to shift j Pmij = 1 if nurse i having skill m is satisfied with shift j assignment; Pmij = 1 if unsatisfied; and Pmij = 0 if no special preference xmij = 1 if nurse i having skill m is allotted to shift j; otherwise, xmij = 0
－
Minimize
M
Tm
S
∑∑∑ x
f1 =
m =1 i =1 j =1
f2 =
Minimize
f3 =
(1)
(2)
∑ ∑ ∑ x (1 − P )
(3)
S
Tm
∑ ∑ ⎜⎜ ∑ x m =1 j =1
Minimize
C mj ⎞ − L mj ⎟⎟ ⎠
M
⎛
mij
M
Tm
⎝
mij
i =1
S
m =1 i =1 j =1
mij
mij
Subject to S
∑x j =1
mij
≥ W m ∀m, i
(4)
mij
≥ L mj
∀m, j
(5)
Tm
∑x i =1
Multiobjective Optimization for Nurse Scheduling Tm
∑x i =1
≤ U mj
∀m, j
(6)
≤1
r = 1, 4, 7, …, S-2 ∀m, i
(7)
r = 1, 4, 7, …, S-2 ∀m, i
(8)
mij
r+2
∑x
mij
69
j =r
r + 3 ( Rm +1 )−1
∑x
mij
≤ Rm
j=r
x mij ∈ {0, 1}
∀m, i, j
(9)
The first objective (Eq. (1)) intends to minimize the cost incurred by performing the nurse schedule. The second objective (Eq. (2)) tries to minimize the deviation between the minimum number of required nurses for a shift and the number of nurses really allotted to that shift. The third objective originally intends to maximize the total nurses’ preference Pmij about the schedule, it is converted to a minimization objective by using 1 Pmij (Eq. (3)). The first constraint (Eq. (4)) stipulates that the number of shifts fulfilled by a nurse having skill m should be greater than or equal to a minimum threshold Wm. Eq. (5) and Eq. (6) describe that the number of nurses having skill m which are allotted to shift j should be a value between Lmj and Umj. The fourth constraint (Eq. (7)) indicates any nurse can only work for at most one shift during any working day. Finally, the fifth constraint (Eq. (8)) requests that the nurse having skill m can serve for at most Rm consecutive working days.
－
Fig. 1. The conception diagram of the MOCSA
One of the notable PSO variants is the Cyber Swarm Algorithm (CSA) [5] which facilitates the reference set, a notion from scatter search [9], keeping the most influential solutions. To seek the approximate Pareto optimal solutions for the MONSP problem, we propose the multiobjective version of the CSA, named MOCSA. Fig. 1 shows the conception diagram of the MOCSA which consists of four memory components. The swarm memory component is the working memory where a population of swarm particles evolve to improve their solution quality. The individual memory reserves a separate space for each particle and stores the pseudo non-dominated solutions by reference to all the solutions found by this designated particle only. Note that the
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pseudo non-dominated solutions could be dominated by the solutions found by other particles, but we propose to store the pseudo non-dominated solutions because our preliminary results show that these solutions contain important diversity information along the individual search trajectory and they assist in finding influential solutions that are overlooked by just using global non-dominated solutions. The global memory tallies the non-dominated solutions that are not dominated by any other solutions found by all the particles. The solutions stored in the global memory will be output as the approximate Pareto optimal solutions as the program terminates. Finally, the reference memory taking the notion of reference set from scatter search [9] selects the most influential solutions based on objective values and diversity measures. The MOCSA exploits the guiding information by the manipulations on different types of adaptive memory. The details of the features of MOCSA are presented as follows. Particle Representation and Fitness Evaluation. Given S working shifts to be fulfilled, there are at most 2 S possible allocations (without considering scheduling constraints) for assigning a nurse to available shifts. Hence, a nurse schedule can be encoded as a value between [0, 2 S 1]. Assume a population of U particles is used, where a particle Pi = {pij}, indicating the schedule for all the nurses. The fitness of the ith particle is a four-value vector (f1, f2, f3, f4). The objective values evaluated using Eqs. (1)-(3) are referred to as the first three fitness values (f1, f2, f3). The fourth fitness value f4 serves as a penalty which computes the amount of total violations incurred by any constraint (Eqs. (4)-(8)). We assume that a feasible solution always dominates any infeasible solution.
－
Exploiting guiding information. The CSA extends the learning form using pbest and gbest by additionally including another solution guide which is systematically selected from the reference set, storing a small number of reference solutions, denoted RefSol[m], m = 1, 2, …, RS, observed by all particles by reference to fitness values and solution diversity. For implementing the MOCSA, the selecting of solution guides is more complex because multiple non-dominated solutions can play the role of pbest, gbest and RefSol[m]. Once the three solution guides were selected, particle Pi updates its positional vector in the swarm memory by the guided moving using Eqs. (10) and (11) as follows. ⎛ ⎛ ω1ϕ1 pbestij + ω 2ϕ 2 gbest j + ω3ϕ3 RefSol[m] j ⎞ ⎞ ,1≤m≤RS (10) vijm ← K ⎜⎜ vij + (ϕ1 + ϕ 2 + ϕ 3 )⎜⎜ − pij ⎟⎟ ⎟⎟ ω ϕ + ω ϕ + ω ϕ 1 1 2 2 3 3 ⎝ ⎠⎠ ⎝
Pi←non-dominated
{ ( f (P + v ) 1 ≤ k ≤ 4) k
i
m i
m ∈ [1, RS ]
}
(11)
where K is the constriction factor, ω and ϕ are the weighting value and the cognition coefficient for the three solution guides pbest, gbest and RefSol[m]. As RefSol[m], 1≤m≤RS is selected in turn from the reference set, the process will generate RS candidate particles for replacing Pi. We choose the non-dominated solution from the RS candidate particles. If there exist more than one non-dominated solutions, the tie is broken at random. Nevertheless, all the non-dominated solutions found in the guided moving are used for experience memory update as noted in the following. Experience memory update. As shown in Fig. 1, experience memory consists of individual memory, global memory and reference memory, where the rewarded
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experience pbest, gbest and RefSol[m] are stored and updated. The individual memory tallies the personal rewarded experience pbest for each individual particle. Because there may exist more than one non-dominated solution in the search course of a particle (here, the non-dominance only refers to all the solutions found by this particle), we save all these solutions in the individual memory. Any solutions in the individual memory can serve as pbest in the guided moving, and we’ll present the Diversity strategy [10] for selecting pbest from the individual memory. By contrast to individual memory, the global memory stores all the non-dominated solutions found by the entire swarm. Hence, the content of the global memory is used for the final output of the approximate Pareto-optimal solutions. During the evolution, the solutions in the global memory are also helpful in assisting the guided moving of particles by serving as gbest. The Sigma strategy [11] is employed in our method for selecting gbest from the global memory. The reference memory stores a small number of reference solutions selected from individual and global memory. According to the original scatter search template [9], we facilitate the 2-tier reference memory update by reference to the fitness values and diversity of the solutions. Selecting solution guides. First, the Diversity strategy for selecting pbest is employed where each particle selects from its individual memory a non-dominated solution as pbest that is the farthest away from the other particles in the objective space. Thus, the particle is likely to produce a plot of objective values equally-distanced to those of other particles, improving the diversity property of the solution front. Second, we apply the Sigma strategy for selecting gbest from the global memory. For a given particle, the Sigma strategy selects from the global memory a non-dominated solution as gbest which is the closest to the line connecting the plot of the particle’s objective values to the origin in the objective space, improving the convergence property of the solution front. Finally, the third solution guide, RefSol[m], m = 1, 2, …, RS, is systematically selected from the reference memory. These reference solutions have good properties of convergence and diversity, so their features should be fully explored in the guided moving for a particle.
4 Result and Discussion We have intensively consulted administrators and senior staffs at the Puli Christian Hospital (http://www.pch.org.tw/english/e_index.html). A dataset consisting of two problem instances was thus created for assessing the objective values of the nurse schedules produced by various algorithms. The first problem instance (Problem I) requires to determine the optimal scheduling of 10 nurses with two levels of skills in a planning period of one week, while the second problem instance (Problem II) consists of 25 nurses with three different skills to be scheduled in a period of four weeks. Among others, NSGA II and MOPSO are two notable methods and are broadly used as performance benchmarks. We thus choose these two methods for performance comparison. All the algorithms were coded using C# language, and the following experiments were conducted on a 2.4GHz PC with 1.25GB RAM. The quality of the final solution front is evaluated in two aspects: the convergence of the produced front to the true Pareto front and the diversity of the produced front manifesting the plots of objective values that are evenly spaced on the front. The convergence measure named Hypervolume calculates the size of the fitness space covered by the produced front.
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To prevent the bias preferred to a less number of efficient points, the Hypervolume is normalized by the final number of solutions produced. The solutions with a smaller Hypervolume value is more desired because they are closer to the true Pareto front. The diversity measure named Spacing which estimates the variance of the distance between adjacent fitness plots. The solutions with a smaller Spacing value are more desired because these solutions exhibit a better representation of a front. As all the competing algorithms are stochastic, we report the average performance index values over 10 independent runs. Each run of a given algorithm is allowed with a period of duration of 80,000 fitness evaluations. Table 2 lists the values of the performance indexes for the solution fronts produced by the competing algorithms. For Problem I, the MOCSA gives the smallest Hypervolume value indicating the produced solution front converges closer to the true Pareto front than the other two algorithms. The Spacing value for the MOCSA is also the smallest among all which discloses that the non-dominated solutions produced by MOCSA spread more evenly on the front. On the other hand, the NSGA II produces the greatest values (worst performance) for both Hypervolume and Spacing, while the MOPSO generates the intermediate values. The experimental outcome for Problem II is slightly different with the previous case. The NSGA II gives the smallest Hypervolume value (best performance) although its spacing value indicates that the produced solutions are not well distributed on the front. The MOCSA produces the second smallest Hypervolume value and the smallest Spacing value among all competitors, supporting the claim that the MOCSA is superior to the other two algorithms. The MOPSO generates the worst Hypervolume value and a median Spacing value. Fig. 2(a) shows the plots of the multiobjective values of all the solutions for Problem I obtained by different algorithms. It is seen that the front produced by MOCSA is closer to the origin. We can also observe that the spread of the solutions are better Table 2. The values of performance indexes obtained by competing algorithms
MOCSA NSGA II MOPSO
Problem I Hypervolume Spacing 2.42E+07 1.41 9.45E+07 2.37 6.17E+07 2.01
(a)
Problem II Hypervolume Spacing 9.86E+07 3.40 8.37E+07 7.82 1.35E+08 4.24
(b)
Fig. 2. The multiobjective-valued front for simulation problems
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distributed on the front than those produced by the other two methods. The front generated by the MOPSO is next to that produced by the MOCSA by reference to the visual distance to the origin. The front generated by the NSGA II is the farthest to the origin and the obtained solutions are not evenly distributed on the front. For Problem II as shown in Fig. 2(b), we can see the front produced by the NSGA II is the closest to the origin although the obtained solutions are still not evenly distributed on the front. The MOCSA produces the front next to that of NSGA II, but better spacing is observed. Finally, MOPSO front is the furthest to the origin where the distribution of the obtained solutions on the front is also better than that produced by the NSGA II.
5 Conclusions In this paper, we have presented a multiobjective cyber swarm algorithm (MOCSA) for solving the nurse scheduling problem. Based on a literature survey, we propose a mathematical formulation containing three objectives and five hard constraints. In contrast to most existing methods which transform multiple objectives into an integrated one, the proposed MOCSA method tackles the generic multiobjective setting and is able to produce approximate Pareto front. The experimental results on two simulation problems manifest that the MOCSA outperforms NSGA II and MOPSO in terms of convergence and diversity measures of the produced fronts.
References 1. Berrada, I., Ferland, J., Michelon, P.: A multi-objective approach to nurse scheduling with both hard and soft constraints. Socio-Economic Planning Sciences 30, 183–193 (1996) 2. Azaiez, M.N., Al Sharif, S.S.: A 0-1 goal programming model for nurse scheduling. Computers & Operations Research 32, 491–507 (2005) 3. Burke, E.K., Li, J., Qu, R.: A Hybrid Model of Integer Programming and Variable Neighbourhood Search for Highy-Constrained Nurse Rostering Problems. European Journal of Operational Research 203, 484–493 (2010) 4. Burke, E.K., Li, J., Qu, R.: A Pareto-based search methodology for multi-objective nurse scheduling. Annals of Operations Research (2010) 5. Yin, P.Y., Glover, F., Laguna, M., Zhu, J.X.: Cyber swarm algorithms – improving particle swarm optimization using adaptive memory strategies. European Journal of Operational Research 201, 377–389 (2010) 6. Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength pareto evolutionary algorithm. Technical Report 103, ETH, Switzerland (2001) 7. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transaction on Evolutionary Computation 6, 42–50 (2002) 8. Coello Coello, A.C., Pulido, G.T., Lechuga, M.S.: Handling multiple objectives with particle swarm optimization. IEEE Trans. on Evolutionary Computation 8, 256–279 (2004) 9. Laguna, M., Marti, R.: Scatter Search: Methodology and Implementation in C. Kluwer Academic Publishers, London (2003) 10. Branke, J., Mostaghim, S.: About selecting the personal best in multi-objective particle swarm optimization. In: Runarsson, T.P., Beyer, H.-G., Burke, E.K., Merelo-Guervós, J.J., Whitley, L.D., Yao, X. (eds.) PPSN 2006. LNCS, vol. 4193, pp. 523–532. Springer, Heidelberg (2006) 11. Mostaghim, S., Teich, J.: Strategies for finding local guides in multi-objective particle swarm optimization (MOPSO). In: Proceedings of the IEEE Swarm Intelligence Symposium 2003 (SIS 2003), Indianapolis, Indiana, USA, pp. 26–33 (2003)
A Multi-Objective Binary Harmony Search Algorithm Ling Wang, Yunfei Mao, Qun Niu, and Minrui Fei Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronics and Automation, Shanghai University, Shanghai, 200072 [email protected]
Abstract. Harmony Search (HS) is an emerging meta-heuristic optimization method and has been used to tackle various optimization problems successfully. However, the research of multi-objectives HS just begins and no work on binary multi-objectives HS has been reported. This paper presents a multi-objective binary harmony search algorithm (MBHS) for tackling binary-coded multiobjective optimization problems. A modified pitch adjustment operator is used to improve the search ability of MBHS. In addition, the non-dominated sorting based crowding distance is adopted to evaluate the solution and update the harmony memory to maintain the diversity of algorithm. Finally the performance of the proposed MBHS was compared with NSGA-II on multi-objective benchmark functions. The experimental results show that MBHS outperform NSGA-II in terms of the convergence metric and the diversity metric. Keywords: binary harmony search, multi-objective optimization, harmony search.
Now HSs have been successfully applied in a wide range of optimization problems in the scientific and engineering fields. However, most of these works focused on the single-objective optimization problems in the continuous or discrete space; and so far just several researches are concerned with the binary-coded problems or multiobjective optimization problems. On binary-coded optimization problems, Geem [10] firstly used HS to solve the water pump switching problem where the candidate value for each decision variable is “0” or “1”. Then Greblicki and Kotowski [11] analyzed the properties of HS on the one dimensional binary knapsack problem and the optimization performance of HS is unsatisfactory. Afterwards, Wang et al[12] pointed out that the pitch adjustment rule of HS cannot performs its function for binary-coded problems which is the root of the poor performance. To make up for it, Wang proposed a binary HS algorithm in which a new pitch adjustment operator was developed to ameliorate the optimization ability. On the multi-objective optimization problems, Geem and Hwangbo [13] studied the satellite heat pipe design problem which need simultaneously consider two objectives, i.e., the thermal conductance and the heat pipe mass. However, the authors transformed this multi-objective problem into a single objective function by minimizing the sum the individual error between current function value and optimal value. And Geem [14] later used HS to tackle the multi-objective time-cost optimization problem for scheduling a project. In this work, the dominance-based comparison for selection was adopted to achieve the trade-off of the time and cost. As far as we know, there is no work reported on the multi-objective binary HS (MBHS). To extend HS to tackle the multi-objective binary-coded problems, a new Pareto-based multi-objective binary HS is proposed in the work. This paper is organized as follow. Section 2 briefly introduces the standard HS algorithm. Then the proposed MBHS is described in Section 3 in details. Section 4 presents the experimental results of MBHS on the benchmark functions and the comparisons with NSGA-II are also given. Finally, some conclusions are drawn in Section 5.
2 Harmony Search Algorithm Harmony Search Algorithm is inspired by the improvising process of musicians. HS mimics this process by keeping a matrix of the best solution vectors named the Harmony Memory (HM). The number of vectors that can be simultaneously remembered in the HM called the Harmony Memory Size (HMS). These memory vectors are initialized with HMS solutions randomly generated for each decision variable. The search procedure after initialization is called improvisation which includes three operators, i.e., harmony memory considering operator, pitch adjusting operator and random selection operator. The harmony memory considering rate (HMCR), which is between 0 and 1, controls the balance between exploration and exploitation during improvisation. A random number is generated and compared with HMCR during search process for each decision variable. If it is smaller than HMCR, the memory vector in HM is taken into consideration for generating the new value; otherwise a value is randomly selected from the possible ranges of the decision variable. Each decision variable of the new solution vector obtained from the HM is examined to determine whether it should be pitch adjusted. The pitch adjusting rate (PAR) decides the ratio of
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pitch-adjustment. Another random number between 0 and 1 is generated and the pitch-adjustment operating as Eq. (1) is executed if it is not bigger than PAR. new ⎪⎧ xi + rand ().BW in continuous space xi new = ⎨ new in discrete space ⎪⎩ xn
(1)
Here xi new is the i-th element of the new harmony solution vector; rand() is the random number; BW is an arbitrary distance band width; and xn new is a neighboring value of xi new . If the new harmony vector is better than the worst solution vector in HM in terms of the fitness value, it will be included in the HM and the existing worst harmony solution vector is excluded from HM. This process runs iteratively till the terminated rules are satisfied.
3 Multi-Objective Binary Harmony Search Algorithm The standard multi-objective HS can be used to deal with binary-coded multiobjective optimization problems, but the disfunction of the pitch adjustment operator in binary space will spoil the performance greatly. So the multi-objective binary harmony search algorithm (MBHS) is proposed in this paper to achieve the satisfactory optimization ability. In MBHS, the harmony vector is formed by the binary-string. For a N-dimension problem, the HM with the size of HMS can be represented as Eq. (2) and initialized randomly,
⎤ ⎥ x2, N ⎥ ⎥ ... ⎥ ... ⎥ xHMS −1, N ⎥ ⎥ xHMS , N ⎥⎦
x1, N
(2)
where xi , j ∈ {0,1} is the j-th element of i-th harmony memory vector. Like the standard HS, MBHS also uses three updating operators, that is, harmony memory consideration operator, pitch adjustment operator and random selection to generate the new solutions. 3.1 Harmony Memory Consideration Operator and Radom Selection Operator
In MBHS, harmony memory consideration operator (HMCO) and random selection operator (RSO) are used to perform the global search. MBHS performs HMCO with the probability of HMCR, i.e., picking a value in HM; while it runs RSO with the rate of (1-HMCR), i.e., choosing a feasible value not limited to HM, which means that the bit is re-initialized stochastically to be “0” or “1”. The process of HMCO and RSO can be described as Eq. (3-4)
where x j is the j-th bit of the new harmony solution vector; r1 and r2 are two independent random number between 0 and 1. 3.2 Pitch Adjustment Operator
If the element of the new harmony comes from the HM, it need to be adjusted by pitch adjustment operator (PAO) with the probability PAR. However, in binary space, the value of the each element in HM is bound to be “0” or “1”, so the standard definition of PAO in HS will be degraded to mutation operation [12]. If we simply abandon the PAO, the algorithm will lack the operator to perform local search. To remedy it, the pitch adjustment operator as Eq. (5) is used in MBHS. ⎧⎪ B j xj = ⎨ ⎪⎩ x j
r ≤ PAR otherwise
(5)
where r is a random number; B j is the j-th element of the best harmony solution vector in HM. The PAO executes a local search based on the current solution and the optimal solution which will help MBHS find the global optima effectively and efficiently. 3.3 Updating of HM
The new generated harmony vector is added into the HM. Then all the solutions in HM are sorted according to the fitness values and the solution with the worst fitness is removed from HM. In the multi-objective optimization problems, the two major goals of Pareto-based optimizer are to pursue the convergence to the Pareto-optimal set as well as maintain the diversity. To achieve it, the non-domination sort strategy based on crowding distance is adopted to sort the HM vectors.
4 Result and Discussion Following the previous work, five multi-objective optimization functions, i.e., SCH*[15], FON[16] and Deb*[17], are chosen as benchmark problems. 4.1 Performance Measures
In this work, the convergence metric γ and the diversity metric Δ proposed in [18] are adopted to evaluate the performance.
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(1) convergence metric γ The convergence metric γ is used to measure the closeness of solutions in obtained Pareto-optimal set to true Pareto-optimal set and it is calculated as Eq.(6): | p*|
di = min j =1
⎛ f m (hi ) − f m ( p j ) ⎞ ⎟ f m max − f m min ⎠ m =1 ⎝ k
∑⎜
(6)
|H |
γ =
∑d i =1
i
(7)
|H |
where p* = ( p1 , p2 ,...... p| p* | ) is the true Pareto-optimal set, H = ( h1 , h2 ,......h| A| ) is the obtained Pareto-optimal set, f m max is the maximum of the m-th objective function and f m min is the minimum of the m-th objective function. In this work, a set of | p* |= 400 uniformly distributed Pareto-optimal solutions is used to calculate the convergence metric γ . (2) diversity metric Δ The diversity metric is computed as Eq. (8): Δ=
d f + dl +
HMS −1
∑ i =1
−
di − d
(8)
d f + dl + ( HMS − 1)
where di is the distance between two successive solutions in the obtained Paretooptimal set; d is the mean value of all the di ; d f and d l are the two Euclidean distances between the extreme solutions and the boundary solutions of the obtained non-dominated set. 4.2 Result and Discussion
For MBHS, a reasonable set of parameter values are adopted, i.e., HMCR=0.9, and PAR=0.03; each decision variable are coded with 30 bits. For a comparison, NSGA-II [18] with the default parameters is used to solve these problems as well. MBHS and NSGA-II both ran with 50000 function evaluations. Table 1-2 list the optimization results of MBHS and NSGA-II and box plots of γ and Δ are given in Fig.1 and Fig.2. According to the results in Table 1-2, it is reasonable to claim that the proposed MBHS is superior to the NSGA-II. Fig.1 indicated that MBHS generally achieved solutions with higher quality in comparison with NSGA-II in terms of convergence metric. And in Fig.2, the comparison of the diversity metric indicated that MBHS is able to find a better spread of solutions and obviously outperforms NSGA-II in all problems.
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-3
x 10
-3
FON
x 10
DEB2
DEB1
1.25
6 4
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1.2
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Convergence
8
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1
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0
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Fig. 1. Box plot of the convergence metrics γ obtained by MBHS and NSGA-II
FON 0.65
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Diversity
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Fig. 2. Box plot of the diversity metrics
MBHS
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△ obtained by MBHS and NSGA-II
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FON SCH1 SCH2 DEB1 DEB2
NSGA-II Variance
Mean
Variance
1.9534481E-003
2.5725898E-003
1.9009196E-003
1.9787263E-004
9.7508949E-004
5.9029912E-005
9.7769396E-004
6.9622480E-005
7.3687049E-004
5.6615711E-005
7.4402367E-004
5.2879053E-005
1.0286786E-003
5.8010990E-005
1.0697121E-003
6.6791139E-005
8.3743810E-003
1.5211841E-002
9.8603419E-002
1.0217030E-001
Table 2. Mean and Variance of the diversity metric Δ NSGA-II
MBHS Mean FON SCH1
9.6845154E-002 1.1668542E-001
Variance
Mean
Variance
6.2345711E-002
7.8416829E-001
2.9294262E-002
1.0841259E-002
4.2701519E-001
3.5264364E-002
SCH2
9.4714113E-001
1.6775193E-003
1.0347253E+000
2.7413411E-002
DEB1
4.7516338E-001
5.5063477E-003
6.3378683E-001
1.9689019E-002
DEB2
6.6037039E-001
1.8871529E-001
6.8131960E-001
1.1085579E-001
5 Conclusion This paper presented a new multi-objective binary harmony search algorithm for tackling the multi-objective optimization problems in binary space. A modified pitch adjustment operator is used to perform a local search and improve the search ability of algorithm. In addition, the non-dominated sorting based on crowding distance is adopted to evaluate the solution and update the HM which insures a better diversity performance as well as convergence of MBHS. Finally the performance of the proposed MBHS was compared with NSGA-II on five well-known multi-objective benchmark functions. The experimental results show that MBHS outperforms NSGAII in terms of the convergence metric and the diversity metric.
Acknowledge This work is supported by Research Fund for the Doctoral Program of Higher Education of China (20103108120008), the Projects of Shanghai Science and Technology Community (10ZR1411800 & 08160512100), ChenGuang Plan (2008CG48), Mechatronics Engineering Innovation Group project from Shanghai Education Commission, Shanghai University “11th Five-Year Plan” 211 Construction Project and the Graduate Innovation Fund of Shanghai University.
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References 1. Geem, Z., Kim, J., Loganathan, J.: A new heuristic optimization algorithm: harmony search. J. Simulations 76, 60–68 (2001) 2. Pan, Q., Suganthan, P., Tasgetiren, M., Liang, J.: A self-adaptive global best harmony search algorithm for continuous optimization problems. Applied Mathematics and Computation 216, 830–848 (2010) 3. Wang, C., Huang, Y.: Self-adaptive harmony search algorithm for optimization. Expert Systems with Applications 37, 2826–2837 (2010) 4. Li, H., Li, L.: A novel hybrid real-valued genetic algorithm for optimization problems. In: International Conference on Computational Intelligence and Security, pp. 91–95 (2008) 5. Omran, M., Mahdavi, M.: Global-best harmony search. Applied Mathematics and Computation 198, 643–656 (2008) 6. Li, L., Huang, Z., Liu, F., Wu, Q.: A heuristic particle swarm optimizer for optimization of pin connected structures. Computers & Structures 85, 340–349 (2007) 7. Geem, Z.: Particle-swarm harmony search for water network design. Engineering Optimization 41, 297–311 (2009) 8. Jang, W., Kang, H., Lee, B.: Hybrid simplex-harmony search method for optimization problems. In: IEEE Congress on Evolutionary Computation, pp. 4157–4164 (2008) 9. Wang, X., Gao, X.Z., Ovaska, S.J.: A hybrid optimization method for fuzzy classification systems. In: 8th International Conference on Hybrid Intelligent Systems, pp. 264–271 (2008) 10. Geem, Z.: Harmony search in water pump switching problem. In: Wang, L., Chen, K., S. Ong, Y. (eds.) ICNC 2005. LNCS, vol. 3612, pp. 751–760. Springer, Heidelberg (2005) 11. Greblicki, J., Kotowski, J.: Analysis of the Properties of the Harmony Search Algorithm Carried Out on the One Dimensional Binary Knapsack Problem. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) EUROCAST 2009. LNCS, vol. 5717, pp. 697– 704. Springer, Heidelberg (2009) 12. Wang, L., Xu, Y., Mao, Y., Fei, M.: A Discrete Harmony Search Algorithm. Communications in Computer and Information Science 98, 37–43 (2010) 13. Geem, Z., Hwangbo, H.: Application of harmony search to multi-objective optimization for satellite heat pipe design. Citeseer, pp. 1–3 (2006) 14. Geem, Z.: Multiobjective Optimization of Time Cost Trade off Using Harmony Search. Journal of Construction Engineering and Management 136, 711–716 (2010) 15. Schaffer, J.: Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of the 1st International Conference on Genetic Algorithms, pp. 93–100 (1985) 16. Fonseca, C., Fleming, P.: Multiobjective optimization and multiple constraint handling with evolutionary algorithms. II. Application example. IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans 28, 38–47 (2002) 17. Deb, K.: Multi-objective genetic algorithms: Problem difficulties and construction of test problems. Evolutionary Computation 7, 205–230 (1999) 18. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Trans. on Evolutionary Computation 6, 182–197 (2002)
A Self-organized Approach to Collaborative Handling of Multi-robot Systems Tian-yun Huang1,2, Xue-bo Chen2, Wang-bao Xu1,2, and Wei Wang1 1
Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, 116024 Liaoning, China 2 School of Electronics and Information Engineering, Liaoning University of Science and Technology, 114051 Liaoning, China [email protected], [email protected], [email protected], [email protected]
Abstract. The purpose of this paper is to develop a general self-organized approach to multi-robot’s collaborative handling problem. Firstly, an autonomous motion planning graph (AMP-graph) is described for individual movement representations. An individual autonomous motion rule (IAM-rule) based on “free-loose” and “well-distributed load-bearing” preferences is presented. By establishing the simple and effective individual rule model, an ideal handling formation can be formed by each robot moving autonomously under their respective preferences. Finally, the simulations show that both the AMP-graph and the IAMrule are valid and feasible. On this basis, the self-organized approach to collaborative hunting and handling with obstacle avoidance of multi-robot systems can be further analyzed effectively. Keywords: Self-organized, Collaborative handling, Formation control.
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a good train of thought for collaborative handling of multi-robot systems, even swarm systems [8][9][10]. The main objective of this paper is to initiate a study on selforganized approach to multi-robot’s collaborative handling problem. For individual movement representations, an autonomous motion planning graph (AMP-graph) is described. An individual autonomous motion rule (IAM-rule) including two kind of “free-loose” and “well-distributed load-bearing” preferences is presented. By establishing the simple and effective individual rule model, an ideal handling formation can be formed by each robot moving autonomously under their respective preferences. The simulations show that both the AMP-graph and the IAM-rule are valid and feasible. Considering many uncertain factors in the handling process, before continuing any further we will make three necessary assumptions: First, the handling process happen in the ideal plane. Second, the rim of object exist any solid handling points which hardly produce deformation. Lastly, handling robots with strong bearing capacity don’t sideslip and deflect in the handling process. Based on these assumptions, a selforganized approach will be design.
2 Autonomous Motion Planning Model Definition 1. Based on local sensing, each robot can complete collaborative handling task only through their own simple rules, we call it multi-robot self-organized handling (MSH). Based on Definition 1, we will make three assumptions [11], which are the base of autonomous motion planning model of handling robots: Assumption 1. Each handling robot can obtain location information of object. Assumption 2. Each handling robot has a local sensor, by which the robot can obtain the position information of a finite number of its neighboring robots. Assumption 3. There are always some simple rules, by which each robot can autonomously move under the respective preferences to form an ideal handling formation. Next, the details of autonomous motion planning model will be described. Definition 2. In the absolute coordinates XaOaYa, the robot Ri can obtain four location information denoted by T0=(xt0,yt0), Ri=(xi,yi), Rpi=(xpi,ypi), Rqi=(xqi,yqi), which are of any target point T0 within the object, the robot Ri and two neighboring robots Rpi, Rqi of the robot Ri. A target constraint line clti is the position vector from the robot Ri to the target T0, denoted by clti=(xt0-xi)+i(yt0-yi). A target constraint angle θ ti is the angle from X-axis to clti, denoted by θ ti =arctan((yt0-yi)/(xt0-xi)). Two interconnected constraint lines clpi and clqi is the position vector from the robot Ri to its neighboring robot Rpi and Rqi, denoted by clpi=(xpi-xi)+i(ypi-yi), clqi=(xqi-xi)+i(yqi-yi). Two interconnected constraint angles θ pi and θ qi is the deflection angle from X-axis to clpi and clqi, denoted by θ pi =arctan((ypi-yi)/(xpi-xi)), θ qi =arctan((yqi-yi)/(xqi-xi)). And two interconnected deflection angle θpti and θqti are the angle from clti to clpi and clqi, denoted by θ pti = θ pi - θ ti , θ qti = θ qi - θ ti . The desired linear velocity vei is decomposed into a vertical component vtdi in the direction of clti and a horizontal component vtpi in the direction perpendicular to clti. The desired deflection angle θ eti is the angle from vtdi to vei, denoted by θ eti = θ ti - θ ei .
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T0 clti
vei vtdi
Rpi clpi
θ ei
Rqi θ ti θ pi vtpi clqi θ tpi θ qi Ri
Fig. 1. The AMP-graph
Then, an autonomous motion planning graph (AMP-graph) is formed by location information of any target point T0 and two neighboring robots Rpi, Rqi, shown in Fig. 1.
3 Individual Autonomous Motion Rule Model From the above discussion, we note that the key of MSH problem is how to design some simple rules by which each robot can autonomously determine the direction of motion at every moment, and by which all the handling robots can be distributed evenly to various points around the edge of the target within a finite time. There are two parts in the moving process: (1) aimed at collision avoidance when the robots move from initial points to target; (2) aimed at well-distributed load-bearing when all the robots reach the edge of the target. Considering the parameters in definition 2 and two different motion processes, an individual autonomous motion rule (IAM-rule) with the “freeloose” and “well-distributed load-bearing” preferences will be designed. 3.1 The IAM-Rule Based on the “Free-Loose” Preference Given in Definition 2, the desired linear velocity vei is a vector sum of vtdi in the direction of clti and vtpi in the direction perpendicular to clti. For the sake of simplicity, only the target T0, two neighbors Rpi, Rqi of the ith robot Ri are taken into account in the IAM-rule based on the “free-loose” preference. Ensure that vtpi points to the “freeloose” space while vtdi always point in the direction of cl0i and two constraint conditions vtdi= fd(|clti|,|clpi|,|clqi|) and vtpi=fp(|clpi|,|clqi|) are satisfied, where fd, fp are the vertical and horizontal potential functions. In the process of moving to the target, the rule will make all the robots coordinate into the ideal formation and all the robots tend towards scatter each other and towards gather relative to the target, therefore we call it the IAM-rule based on the “free-loose” preference. The “free-loose” space modeling. Let us consider the robot Ri in the relative coordinates in which Y-axis always point to the target T0, the first and forth quadrants are defined as the positive quadrants since θ pti , θ qti are positive within them and the second and forth quadrants are defined as the negative quadrants since θ tpi , l
negative within them, then the “free-loose” space can be described:
θ qti are
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The direction of the “free-loose” space points to 1) the opposite direction of the space two neighbors belong to, when two neighbors lay together in the positive or negative quadrant. 2) the direction of the space the interconnected constraint line with greater X-axis component belong to, when two neighbors lay respectively in the positive or negative quadrant. Thus, the description can be expressed mathematically as follows:
⎧⎪Cli = cl pi sin θ pti + clqi sin θ qti ⎨ ⎪⎩Cli = cl pi sgn(θ pti ) + clqi sgn(θ qti ) l ⎧θ tpi = θ ti ⎪ sgn(θ pti )+ sgn(θ qti ) ⎨ π l 2 ⎪θ tpi = θ ti + (−1) ⋅ sgn(Cli ) ⋅ 2 ⎩
clti ≠ 0 clti = 0
(3.1)
Cli ≤ ε Cli > ε
(3.2)
where ε is a permissible error. Because Cli covers all of the information to determine autonomous motion of Ri, we call it interconnected characteristics parameter with “free-loose” feature. Cli denotes the vector sum of the X-axis components of two interconnected direction lines clpi and clqi if the robot Ri has not reach the edge of the target, or the vector sum of clpi and clqi if the robot Ri has reach. Similarly, because
θ tpil covers all the possible direction of “free-loose” space of Ri, therefore we call it
autonomous motion direction angle with “free-loose” feature. Specially, the desired linear velocity vei point in the direction of θ ti if “free-loose” space do not exist, that is, ∃ε , θ tpi = θ ti , if |Cli| ≤ ε . l
We know that the arrow of the Y-axis represents the direction that all the robots tend towards gather relative to the target and the arrow of the X-axis represents the direction that all the robots tend towards scatter each other on the edge of the target. Therefore, the desired angle θ ei at every moment of autonomous motion with the IAM-rule based on the “free-loose” preference can be obtained as follow:
⎧θ ei ⎪ ⎪θ ei ⎨ ⎪θ ei ⎪ * ⎩θ ei
= θ ti
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Cli ≤ ε and clti = 0
Eq. (3.3) describes every process of multi-robot self-organized handling. According to Definition 2, the desired angle θ ei is the deflection angle of vei and xa if two interconnected constraint lines exist and the robot does not reach the edge of the
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target, that is, |clti| ≠ 0 and |Cli| > ε . Specially, when two interconnected constraint lines do not exist, that is |Cli|=0, the desired angle θ ei coincides with the target constraint angle θ ti . When the robot reaches the edge of the target and the interconnected constraint lines exist, that is, |clti|=0 and |Cli| > ε , the desired angle
θ ei coincides
with θ . When the robot reaches the edge of the target and the interconnection can be l ti
negligible, that is, |clti|=0 and |Cli| angel θ coinciding with θ ti .
≤ ε , the robot obtain a stable desired
* ei
Now, we turn to the second motion process of multi-robot self-organized handling. 3.2 The IAM-Rule Based on the “Well-Distributed Load-Bearing” Preference After a uniform dispersed formation is formed by autonomous motion with the IAMrule based on the “free-loose” preference, that is, |clti|=0 and |Cli| ≤ ε , all the handling robots smoothly lift up the object together to measure the load bearing data which are used as the parameter of the IAM-rule based on the “well-distributed loadbearing” preference. Similar to the “free-loose” preference, only the load-bearings of the two nearest neighbors at both left and right sides of Ri are taken into account. Ensure that Ri always move along the edge of the object and in the direction of neighbor with larger load-bearing, the IAM-rule will make the load-bearing of all the robots tending towards average, therefore we call it the IAM-rule based on the “welldistributed load-bearing” preference. The “well-distributed load-bearing” space modeling. Similar to the “free-loose” preference, the robot Ri in the relative coordinates in which Y-axis always point to the target T0, the first and forth quadrants are defined as the positive quadrants since θ pti , θ qti are positive in them and the second and forth quadrants are defined as the negative quadrants since θ pti ,
θ qti are negative in them, then the “well-distributed
load-bearing” space can be described: The direction of the “well-distributed load-bearing” space points to the direction of the space the neighbor with larger load-bearing belong to. Corresponds to the "free-loose" preference model, the description can be expressed mathematically as follows:
Cbi = G pi sgn(θ pdi ) + Gqi sgn(θ qdi )
(3.4)
b ⎧θ ei = θ tpi = θ ti Cbi ≤ ε ⎪ π ⎪ b Cbi > ε ⎨θ ei = θ tpi = θ ti + sgn(Cb ) ⋅ 2 ⎪ ⎪Gei* = G0 / n all Cbi ≤ ε ⎩
(3.5)
where Gpi and Gqi are the load-bearing of the two nearest neighbors at both left and right sides of Ri. Because Cbi covers all of the information to determine the direction
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of autonomous motion of Ri, we call it interconnected characteristics parameter with
“well-distributed load-bearing” feature. Similarly, because θ tpi covers all the possible b
direction of “well-distributed load-bearing” space of Ri, therefore we call it autonomous motion direction angle with “well-distributed load-bearing” feature. Specially, if |Cbi| ≤ ε , i=1,2…n, then all the robots receive the weight equally denoted by G*i=G0/n. 3.3 General Remarks on Multi-robot Self-organized Handling Remark 1. Effective sensing range is the maximum range within which the omnidirection sensor of each handling robot can detect a target, denoted by Rs. If the minimum distance between the robot and the object beyond effective sensing range Rs, the robot follows any given point T0=(xt0,yt0) within the object, or the robot follows the point T0i=(xt0i,yt0i) located nearest from the edge of the object. Remark 2. By setting the parameters of the potential field function, the IAM-rule can maintain collision avoidance between any two robots. When the distance between the robots is smaller, the potential field function make the interconnected deflection angle increasing rapidly to produce greater repulsive interaction. Specially, when the “freeloose” spaces in all directions do not exist, the robot is forced to remain stationary and wait for a chance to autonomous motion. Remark 3. Effective interconnected radius δ is the maximum value within which the interaction between any two robots Rpi, Rqi exists, that is, ∃ δ i , |clpq|=|clpq| if |clpq| ≤ δ i , or |clpq|=0 if |clpq| > δ i ,p ≠ q ∈ {1,2,…,n}.
4 Simulations and Analysis In order to test the validity and feasibility of the IAM-rule based on the “free-loose” preference, two simulations are carried out, in which 8 handling robots are present. The group of robots is required to start from a disordered state and to then form the relatively well-distributed formation around the edge of the object described as an ellipse, where the parametric equations of the ellipse are x=3cos(t)-3, y=1.5sin(t)+1. Each robot obtains the same motion parameters, denoted by r the radius of robot, by Rs effective sensing radius, by δ effective interconnected radius, by ε the minimum distance difference between two robots, by T0 any given point within the object and by λ the step factor. The parameter values of the trajectory control are shown in Table 1 and the initial position information of all the handling robots are shown in Table 2. Table 1. The parameter values of the trajectory control Parameter Value
r 0.2
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Fig. 2. The moving process of 8 handling robots with IAM-rule (36 steps)
From Fig. 2, we observe that after 36 steps all the handling robots distribute uniformly around the edge of the target, so the IAM-rule based on the “free-loose” preference can effectively make multi-robot systems form the ideal handling formation corresponding to formation control [12][13][14]. In the initial period R7 follows the known point T0 within the object, since the object from which the initial position of R7 farther away can not be perceived, coincides with Remark 1. Due to the smaller distance between R1 and R4 in the initial period, R1 and R4 obtain two larger desired deflection angles θ et1 and θ et 4 , coincide with Remark 2. In addition, although the robot R2, R7 and R8 are neighbors each other, the interaction between them are negligible in the initial period because of the greater distances each other, and then the re-establishment of the interaction makes their trajectories deflected during the autonomous motion, coincides with Remark 3. It is to be noted that because each robot is always fond of moving in the direction of “free-loose” space, the robots in the periphery of the group possess more dispersity and make the ones within the group pulled by the “free-loose” space to spread to the periphery gradually, thus the relatively dispersed characteristics for the group are formed finally. If each robot satisfies local collision avoidance conditions under the special circumstances of Remark 2, we might as well call it “strict collision avoidance”.
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5 Conclusion and Future Work The self-organized approach with IAM-rules has the following advantages over other methods. Firstly, a simple and effective individual autonomous motion rule(IAM-rule) model is established, by which an ideal handling formation can be formed by each robot moving autonomously under their respective preferences. Compared with the centralized control used for multi-robot collaborative handling, the self-organized approach with IAM-rule is simple and has fewer bottlenecks in communication and computing caused by the centralized data processing and leader guiding. For the robot itself, if the information of the target and two neighbors are obtained by local perception, it will determine their own desired speed, thus less information processing are beneficial to making rapid judgment. Secondly, the self-organized approach with IAM-rule has the good characteristics for strict collision avoidance which provides a solution for coordination problem of swarm systems. The IAM-rule can be applied to explanation and resolution of group behaviors since the “free-loose” preference coincides with individual behavior of real system. Thirdly, it may provide a novel train of thought for emergence control modeling, which is verified by the simulation that the system can be controlled to produce certain characteristic and function of emergence by constructing simple individual rules. The paper is the basement of research on emergence of multi-robot collective behavior. Future work include: 1.On this basis, self-organized approach to multi-robot systems’ collaborative hunting and handling with obstacle avoidance can be further analyzed effectively. 2. More rules with certain preferences can be designed to jointly complete more complex function of swarm systems. 3. Based on the IAM-rule, leader emergence can be further discussed. Acknowledgments. Supported by the National Natural Science Foundation of China (Grant No. 60874017).
References 1. Kim, K.I., Zheng, Y.F.: Two Strategies of Position and Force Control for Two Industrial Robots Handling a Single Object. Robotics and Autonomous Systems 5, 395–403 (1989) 2. Kosuge, K., Oosumi, T.: Decentralized Control of Multiple Robots Handling an Object. In: IEEE/ RJS Int.Conf. on Intelligent Robots and Systems, vol. 1, pp. 318–323 (1996) 3. Yamashita, A., Arai, T., et al.: Motion Planning of Multiple Mobile Robots for Cooperative Manipulation and Transportation. IEEE Transactions on Robotics and Automation 19(2) (2003) 4. Koga, M., Kosuge, K., Furuta, K., Nosaki, K.: Coordinated Motion Control of Robot Arms Based on the Virtual International Model. IEEE Transactions on Robotics and Autonomous Systems 8 (1992) 5. Wang, Z., Nakano, E., Matsukawa, T.: Cooperating Multiple Behavior-Based Robots for Object Manipulation. In: IEEE /RSJ/GI International Conference on Intelligent Robots and Systems IROS 1994, vol. 3, pp. 1524–1531 (1994)
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6. Huang, T.-y., Wang, X.-n., Chen, X.-b.: Multirobot Time-optimal Handling Method Based on Formation Control. Journal of System Simulation 22, 1442–1465 (2010) 7. Kosuge, K., Taguchi, D., Fukuda, T., Sakai, M., Kanitani, K.: Decentralized Coordinated Motion Control of Manipulators with Vision and Force Sensors. In: Proc. of 1995 IEEE Int. Conf. on Robotics and Automation, vol. 3, pp. 2456–24162 (1995) 8. Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules. IEEE Transactions on Automatic Control 48, 988–1001 (2003) 9. Turgut, A.E., Çelikkanat, H., Gökçe, F., Şahin, E.: Self-organized Flocking in Mobile Robot Swarms. Swarm Intelligence 2, 97–120 (2008) 10. Gregoire, G., Tu, H.C.Y.: Moving and Staying Together Without a Leader. Physica D 181, 157–170 (2003) 11. Xu, W.B., Chen, X.B.: Artificial Moment Method for Swarm Robot Formation Control. Science in China Series F: Information Sciences 51(10), 1521–1531 (2008) 12. Balcht, T., Arkin, R.C.: Behavior-based Formation Control for Multi-robot Teams. IEEE Transactions on Robotics and Automation 14, 926–939 (1998) 13. Lawton, J.R., Beard, R.W., Young, B.J.: A Decentralized Approach to Formation Maneuvers. IEEE Transactions on Robotics and Automation 19, 933–941 (2003) 14. Das, A.K., Fierro, R., et al.: A vision-based formation control framework. IEEE Transactions on Robotics and Automation 18, 813–825 (2002)
An Enhanced Formation of Multi-robot Based on A* Algorithm for Data Relay Transmission Zhiguang Xu1, Kyung-Sik Choi1, Yoon-Gu Kim2, Jinung An2, and Suk-Gyu Lee1 1
Department of Electrical Eng. Yeugnam Univ., Gyongsan, Gyongbuk, Korea 2 Daegu Gyeongbuk Institute of Science & Technology, Daegu, Korea [email protected], {robotics,sglee}@ynu.ac.kr, {ryankim9,robot}@dgist.ac.kr
Abstract. This paper presents a formation control method of multi-robot based on A* algorithm for data relay transmission. In our system, we choose Nanotron sensor and compass sensor to execute the tasks of distance measurement, communication and obtaining moving direction. Since there exists data disturbance from Nanotron sensor when there is an obstacle between two robots. Therefore, we embed path planning algorithm information control. The leader robot (LR) knows the whole information of environment, and sends its moving information and corner information as a node to FRs. The FRs regard the node information which received from LR as temporary target to increase the efficiency of multi-robot formation by optimal path. From the simulations and experiments, we will show desirable results of our method. Keywords: multi-robot, formation, path planning, data relay transmission.
techniques from omni-directional vision systems, automatic control, and dynamic programming. This strategy is avoiding static obstacles and dynamic objects from reestablishing positions of each robot. In our system, we assume LR knows whole information of environment. FRs follow LR or forward FRs. We consider that the FRs use Nanotron sensor to obtain distance information. Robots keep in certain distance with each other to avoid collision between each robot. FRs follow the frontal robot in given distance range and plan the path based on the knowledge of nodes which are received from LR. To get shortest trajectory, FRs also apply A* algorithm. The paper is organized as follows. Section 2 derives the mathematical descriptions for embedding path planning algorithm in our system. In section 3, simulation results which coded by Matlab show good performance of our proposed method. In section 4, experiment result validates proposed method.
2 Related Works In multiple robots system, there are three main control approaches, leader-follower based approach [12], behavior-based approach [13] and virtual structure approach [14]. Several control algorithms, such as EKF [15], I/O linearization [16], sliding mode control method [17] are common used to control each robot. In practical, computer process should be loaded if we use the control approaches and control methods above to do multi-robot formation. However, the MCU of our system is AVR, so it is difficult to make our system commercialize and industrialize. In our robots’ system, each robot just utilizes on-board sensors to localization, but redundant sensor data will make a great burden on the aim controller. Consequently, we adopt a more realistic practical application to achieve our control goal which implant path planning algorithm to each robot so as to reduce the computational burden and satisfy our control system requirements. 2.1
Algorithm Description State Function and System Flow Chart
The motion of each robot is described in terms of P = (x, y, θ)T, where x, y and θ are the x coordinate, the y coordinate and the bearing respectively. The trajectory of each robot has the form of (x, y) with the velocity v and the angular velocity ω. The model of robot Ri as the form of:
Fig. 2 shows flow chart of LR process which knows the whole information of environment. If LR does not reach its destination, LR will send MI to rear robots at each time step, such as moving distance, heading angle and node information. When LR arrives at a corner, it will turn 90 degree and regard the next step position of LR as a node. Fig. 3 describes flow chart of FR maintaining a given distance range. To reduce the steps from start point to goal point and maintain the communication with LR, the FRs use A* algorithm to plan the path, where the FRs make use of the information nodes received from LR.
Fig. 2. Flow chart of LR
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Fig. 3. Flow chart of FRs for maintaining a given distance with LR
For the FRs, the node is target which received from the LR. And when LR moves in the environment, there is not only just one node. So the FRs must reach every node as target. However, if two nodes are very close, to increase the efficiency of navigation, the FRs will use A* algorithm to obtain a shortest path and eliminate useless nodes. 3.2
Path Planning
A* algorithm is widely used in graph searching algorithm which includes heuristic function and evaluation function to sort the nodes. This algorithm expressed as f(n) which consists of two functions as follows: g(n) is defined as the cost to go, and h(n) is the cost remaining after having gone, and it is chosen as the Euclidean distance from the other end of the new candidate edge to the destination. The searching process of our system is as follows: (1) Marking the initial node and expanding the unmarked subsequent nodes or the child nodes; (2) Calculating the evaluation function value for each subsequent node, and sorting by evaluation function. Then identifying and marking the node of the minimum evaluation function value; (3) Iterating upper steps to recording the shortest path until the current node same as the goal node [2] [8]. f(n) = g(n) + h(n).
(2)
Fig. 4 is pseudo code of A* in the simulation. In the pseudo code, EXP means horizontal and vertical position on nodes, evolution function, cost function, and heuristic value function, respectively. OPEN and CLOSE sets are to store available path information and disable path information, respectively. To search evolution function with minimum value, there is the comparison step between EXP and OPEN.
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If Have a new node(goal) A*(current position, goal) Closed set = the empty set Open set = includes current position(CP) Store node(start) in OPEN While Open set != empty & path movable Do calculate N’s G, H, F and save in EXP Compare EXP and OPEN IF F is minimum then flag=1 Else flag=0 and Add node(n) in OPEN Update path cost and CLOSE End while
Fig. 4. Pseudo code of A* algorithm
4 4.1
Simulation and Experiment Simulations
We use Matlab 2008b to do the simulation of the multi-robot formation. The simulation describes the moving trajectory of one LR and several FRs. We give the whole map information and path to LR. FRs follow the LR, and planning the path to keep the formation at the same time. The distance range is constrained from 1m to 3m. The environment is separated by cells; one cell size is 1m by 1m. Fig. 5 (a) describes the trajectories of one LR and three FRs which without A* algorithm. The three FRs follow the leader robot by the same trajectory of leader robot. In order to obtain accurate distance measurement data, the LR have to wait the rear FRs until the distance between these two robots is in the minimize value when the LR arrive a corner. It may also reduce the efficiency of formation. However, in Fig. 5 (b), the black circles are denoted as the information nodes sending by leader robot. When the distance measurement data is disturbed, especially the robots go through a corner, LR will send the node information to the robots which moving behind it. Then the FRs use A* algorithm to plan the shortest path to reach its node (as target) within minimize steps. This method not only increases the efficiency of formation but also prevent the effectiveness from data disturbance in Nanotron sensor. Fig. 6 shows the step comparison histogram of one leader robot with different number of follower robots from start point to goal point using A* algorithm or not. If the leader robot just has one follower robot, the step of follower robot using A* algorithm is 39, it is 12 steps less than the follower robot without A* algorithm. If the leader robot has two follower robots using A* algorithm, the step of follower robots is 48, however, if the follower robots do not use A star algorithm, the step is 57. In three robots case, the three robots use 48 steps from start point to goal point using A* algorithm. If without A* algorithm, the three followers use 57 steps. From the comparison, we get the result that the steps using A* algorithm is much less than the steps of follower robots without A* algorithm. The follower robots are able to reach their goal points using A* algorithm more efficiently.
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Fig. 5. Trajectories of one LR and three FRs, (a) without A* algorithm, (b) using A* algorithm
Fig. 6. Step comparison histogram of one leader robot with different number of follower robots using A* algorithm or not
4.2 Experiments In the experiment, we embed whole map information to LR, such as the distance between target and corner information. FR navigates autonomously and follows the LR within a given distance to keep required form. When LR arrive at the corner, it will send the corner information to FR for executing A* algorithm. Each robot in the team executes localization using motor encoder and Nanotron sensor data based on Kalman Filter. From some paper, the researchers obtain θ calculated from the relationship between motor encoder and the length of robot’s two wheels. However, heading angle data from compass sensor is more accurate than the data calculated from encoder, we get θ value from XG1010 and let robot go straight using XG1010. Robots are in indoor environment and there is just one corner. The initial distance between LR and FR is 3 meter. When LR robot move 3 meter, it will
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experiment result:one leader robot and one follower robot
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Fig. 7. Experiment result about LR and FR moving trajectories
turn left and send the node information to rear FR by Nanotron sensor. At this time, FR plans optimal path to the temporary target based on A* algorithm to keep the required form with LR. We get the robot’s position values and orientation value at each step. And when robots go straight, from its real trajectory, we measure the error in x axis and y axis is less than 1 centimeter. Then we use Matlab draw the trajectories of each robot as Fig. 7.
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Conclusion
In multiple mobile robots system, it is so important to share the moving information of every robot to increase the efficiency of cooperation. The FRs move to its node (as target) with A* path planning algorithm using the information node which is achieved from LR. Basically, the proposed method obtains a respectable result as we want. The steps of FRs using A* path planning algorithm are much less than the steps of FRs without A* algorithm. The simulation and experiment results show that the robots embedded A* algorithm could obtain better performance on efficiency. For future research, we are going to realize this multi-robot formation control among more number of robots. And we will consider more complex environment, such as exist some obstacles.
Acknowledgment This research was carried out under the General R/D Program of the Daegu Gyeongbuk Institute of Science and Technology (DGIST), funded by the Ministry of Education, Science and Technology (MEST) of the Republic of Korea.
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References 1. Farinelli, A., Iocchi, L., Nardi, D.: Multi-robot Systems: A Classification Focused on Coordination. IEEE Transactions on Systems, Man, and Cybernetics, Part-B: Cybernetics 34(5), 2015–2028 (2004) 2. Wang, K.H.C., Botea, A.: Tractable Multi-Agent Path Planning on Grid Maps. In: Int. Joint Conf. on Artificial Intelligence, pp. 1870–1875 (2009) 3. Seo, W.J., Ok, W.J., Ahn, J.H., Kang, S., Moom, B.: An Efficient Hardware Architeture of the A-star Algorithm for the Shortest Path Search Engine. In: Fifth Int. Joint Conf. INC, IMS and IDC, pp. 1499–1502 (2009) 4. Scrapper, C., Madhavan, R., Balakirsky, S.: Stable Navigation Solutions for Robots in Complex Environments. In: Proc. IEEE Int. Workshop on Safety, Security and Rescue Robotics (2007) 5. Cai, C., Yang, C., Zhu, Q., Liang, Y.: Collision Avoidance in Multi-Robot Systems. In: Proc. IEEE Int. Conf. on Mechatronics and Automation, pp. 2795–2800 (2007) 6. Castillo, O., Trujillo, L., Melin, P.: Multiple objective optimization genetic algorithms for path planning in autonomous mobile robots. Int. Journal of Computers, Systems and Signals 6(1), 48–63 (2005) 7. Li, W., Zhang, W.: Path Planning of UAVs Swarm using Ant Colony System. In: Fifth Int. Conf. on Natural Computation, vol. 5, pp. 288–292 (2009) 8. Yao, J., Lin, C., Xie, X., Wang, A.J., Hung, C.C.: Path planning for virtual human motion using improved a star algorithm. In: Seventh Int. Conf. on Information Technology, pp. 1154–1158 (2010) 9. Li, H., Yang, S.X., Biletskiy, Y.: Neural Network Based Path Planning for A Multi-Robot System with Moving Obstacles. In: Fourth IEEE Conf. on Automation Science and Engineering (2008) 10. Otte, M.W., Richardson, S.G., Mulligan, J., Grudic, G.: Local Path Planning in Image Space for Autonomous Robot Navigation in Unstructured Environments. Technical Report CU-CS-1030-07, University of Colorado at Boulder (2007) 11. Sibo, Y., Gechter, F., Koukam, A.: Application of Reactive Multi-agent System to Vehicle Collision Avoidance. In: Twentieth IEEE Int. Conf. on Tools with Artificial Intelligence, pp. 197–204 (2008) 12. Consolini, L., Morbidi, F., Prattichizzo, D., Tosques, D.: A Geometric Characterization of Leader-Follower Formation Control. In: IEEE International Conf. on Robotics and Automation, pp. 2397–2402 (2007) 13. Balch, T., Arkin, R.C.: Behavior-based Formation Control for Multi-robot Teams. IEEE Trans. on Robotics and Automation 14, 926–939 (1998) 14. Lalish, E., Morgansen, K.A., Tsukamaki, T.: Formation Tracking Control using Virtual Structures and Deconfliction. In: Proc. of the 2006 IEEE Conf. on Decision and Control (2006) 15. Schneider, F.E., Wildermuth, D.: Using an Extended Kalman Filter for Relative Localisation in a Moving Robot Formation. In: Fourth Int. Workshop on Robot Motion and Control, pp. 85–90 (2004) 16. Desai, J.P., Ostrowski, J., Kumar, R.V.: Modeling formation of multiple mobile robots. In: Proc. of the 1998 IEEE Int. Conf. on Robotics and Automation, Leuven, Belgium (1998) 17. Sánchez, J., Fierro, R.: Sliding Mode Control for Robot Formations. In: Proc. of the 2003 IEEE Int. Symposium on Intelligent Control, Houston, Texas (2003) 18. Hanback Electronics, http://www.hanback.co.kr/ 19. Atmel Corporation, http://www.atmel.com/ 20. MicroInfinity, http://www.minfinity.com/
WPAN Communication Distance Expansion Method Based on Multi-robot Cooperation Navigation Yoon-Gu Kim1, Jinung An1, Kyoung-Dong Kim2, Zhi-Guang Xu2, and Suk-Gyu Lee2 1
Daegu Gyeongbuk Institute of Science and Technology, 50-1, Sang-ri, Hyeonpung-myeon, Dalseong-gun, Daegu, Republic of Korea 2 Department of Electrical Engineering, Yeungnam University, 214-1, Dae-dong, Gyongsan, Gyongbuk, Republic of Korea {ryankim9,robot}@dgist.ac.kr, [email protected], [email protected], [email protected]
Abstract. Over the past decade, an increasing number of researches and developments for personal or professional service robots are attracting considerable attention and interest in industry and academia. Furthermore, the development of intelligent robots is strongly promoted as a strategic industry. To date, most of the practical and commercial service robots are controlled remotely. The most important technical issue of remote control is wireless communication, especially in indoor and unstructured environments where communication infrastructure may be hampered. Therefore, we propose a multi-robot cooperation navigation method for securing the communication distance extension of the remote control based on wireless personal area networks (WPANs). The concept and implementation of following navigation are introduced, and performance verification is carried out through navigation experiments in real or test-bed environments. Keywords: WPAN, Communication distance expansion, Multi-robots, Remote control.
The Kalman filter, a well-known algorithm widely applied in the robotics field, is based on the linear mean square error filtering for state estimation. A set of mathematical equations in the Kalman filter is implemented adequately as a compensator and an optimal estimator for some types of noises. Therefore, it has been used for stochastic estimation of measurements with noisy sensors. This filter can minimize the estimated error covariance when the robot is placed under presumed conditions. For the given spectral characteristics of an additive combination of signal and noise, the linear operation based on these input yields the best results with minimum square error of the signal from the noise. The distinctive feature of the Kalman filter, described in its mathematical formulation in terms of a state space analysis, is that its solution is computed recursively. Park [1] approached the recognition of position and orientation of a mobile robot using encoders and ubiquitous sensor networks (USNs). For this, the USNs are consisted of four fixed nodes and a mobile node. The robot is based on the fuzzy algorithm using information from the encoder and the USNs. Incidentally, this proposal has errors in the recognition of a USN when considering the exploration of each robot without fixed nodes. In addition, the noises caused by the friction between the road surface and the wheels and the control error of the motor affect the localization estimation acquired from the encoders. In addition, the measured errors accumulate while a robot navigates. In order to solve these problems, we proposed a localization and navigation system, which is based on the IEEE 802.15.4a protocol, to measure the distance between the robots and a compass sensor to obtain the heading angle of each robot. The IEEE 802.15.4a protocol allows for high aggregate throughput communication with a precision ranging capability. Nanotron techniques developed their first Chirp spread spectrum (CSS) smart RF module—smart nanoLOC RF with ranging capabilities. The proposed method is based on a modified Kalman filter, which is adapted in our system to improve the measurement quality of the wireless communication module, and the compass sensor for reducing the error in the localization and navigation process. This paper is organized as follows. Section 2 introduces related works and discusses localization approaches and the application of the IEEE 802.15.4a protocol to our system. Section 3 presents the proposed multi-robot-based localization and navigation. Section 4 explains and analyzes the experimental results. Finally, Section 5 presents the conclusion of this research and discusses future research directions.
2 Related Works 2.1 Localization Approaches In general, localization is divided into relative localization and absolute localization. Relative localization is the process of estimating a mobile robot’s state or pose (location and orientation) relative to its initial one in the environment; it also called dead reckoning (DR). Generally, an encoder, a gyroscope, and an inertial measurement unit (IMU) are used for localization by DR. It is easy and economically efficient to implement DR localization; however, DR has a critical drawback in that it is easily affected by external noises, resulting in error accumulation.
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Absolute localization is based on telemetric or distance sensors and may avoid the error accumulation of relative localization. Absolute localization is a global localization using which it is possible to estimate the current pose of the mobile robot even if the conditions of the initial pose are unknown and the robot is kidnapped and teleported to a different location [2]. The basic principle of absolute localization is based on probabilistic methods and the robot’s belief or Bayes’ rule. The former is a probability density function of the possible poses. The latter updates the belief according to the information. Taking into account the problem of approximating the belief, we can classify localization into Gaussian filter-based localization and non-parametric filter-based localization. The extended Kalman filter (EKF) [4] and the unscented Kalman filter (UKF) [3] are included in the former. Markov localization [5] and Monte Carlo localization [2] are included in the latter. EKF localization represents the state or pose of the robot as Gaussian density to estimate the pose using EKF. UKF localization addresses the approximation issues of the EKF. The basic difference between the EKF and the UKF stems from the manner in which Gaussian random variables (GRV) are represented for propagating through system dynamics [3]. In the EKF, state distribution is approximated by GRV, which is then propagated analytically through the first-order linearization of a nonlinear system. This can introduce large errors in the true posterior mean and the covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF addresses this problem by using a deterministic sampling approach. The state distribution is also approximated by GRV. In contrast, it is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, which are propagated through the true nonlinear system. The EKF achieves only first-order accuracy. Neither the explicit Jacobian nor the Hessian calculation is necessary for the UKF. Remarkably, the computational complexity of the UKF is of the same order as that of the EKF [3]. Markov localization approximates the posterior pose of a robot using a histogram filter over a grid decomposition of the pose space. Hence, it is called grid localization. Monte Carlo localization approximates the posterior pose of a robot using a particle filter that represents the pose of the robot by a set of particles with important weight. This non-parametric filter-based localization can resolve the global localization and kidnap problem through multi-modal distribution. 2.2 IEEE 802.15.4A IEEE 802.15 related to the wireless personal area network (WPAN) is the standard protocol developed by many task groups (TG) in IEEE. In particular, IEEE 802.15.4 is the standard of the low power for driving devices, the low cost for establishment, and the available industrial, scientific, and medical (ISM) band. In addition, IEEE 802.15.4a provides enhanced ranging information among nodes through its adaptation of wireless communication. As a result, we decided to use this protocol for sensor networking. IEEE 802.15.4a was standardized in August 2007 based on the low complexity, low cost, and low energy in a WPAN environment and its capability to simultaneously allow for correspondence and distance measurement. IEEE 802.15.4a chooses two PHY techniques, namely, the ultra-wide band (UWB) method and the
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chirp spread spectrum (CSS) method with the centre of Samsung and Nanotron[6, 7]. UWB is a technique used for local distance communication and it is used for communicating signals with shorter pulse width in the baseband without a carrier. Owing to the extremely short pulse width, the applied frequency bandwidth is long. Therefore, it appears as though normal noise exists in channels of low output power. It does not affect the wireless device. However, it is difficult in long distance communication because it is a baseband communication and the output has low voltage. Its frequency range is 3.4 GHz~10 GHz. CSS was developed in the 1940s, and it is referred to as the dolphin and bat communication. It has been typically used in radars because it has some advantages such as strong interference and availability to long distance communication. After 1960, it was expanding into industrialization, and grafted linear sweep into chirp signal to get the significant information. CSS uses its entire allocated bandwidth to broadcast a signal, making it robust to channel noise. Moreover, even in the low voltage case, multi-path fading will not be much affected. The frequency of the CSS method is the 2.4 GHz ISM band.
Fig. 1. Wireless communication expansion scenario based on a multi-robot system
Fig. 2. System architecture
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Fig. 3. Operation flowcharts of leader robot and follower robot
4 Experimental Results For this experiment, we placed two robots at certain distances in the linear corridor. Figure 4 shows the measured distance errors with keeping each distance interval between the FR and the LR. This experiment shows that the error for maintaining a specific interval is decreased when the Kalman filter is applied to the distance measurement. The Kalman filter estimates a more correct distance between the predicted encoder distance information and the measured WPAN distance information, as summarized in equations (1)–(7). We tried to simulate how well the FR follows LR by the leader following operation, which is based on the WPAN distance measurement. Figure 5 shows the simulation results of the leader following navigation of a follower robot. The simulation is the navigation results of the FR following the LR and navigating in a 10 m × 10 m area. The RF sensor data and compass sensor data have uncertainty error factors. Therefore, the objective of the proposed system is to achieve accuracy in the WPAN sensor network system by using the Kalman filter. However, the Kalman filter requires a considerable amount of data for the estimation. Even in this case, the system cannot move perfectly when the measurement data are dispersed. To solve this problem, we have to ignore the dispersed data. Therefore, the system cannot avoid resulting in errors. Figure 6 shows an experiment of the multirobot cooperation navigation for valid wireless communication distance expansion.
xˆ k−+1 = xˆ k + u k + wk
(1)
,
d k +1 = d k − ( Δt vk cosθ k )2 + ( Δt vk sinθ k )2
,
(2)
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θ k +1 = θ k +
Δt vk tanφk L ,
Pk−+1 = Pk + σ w2 k K =
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(3)
(4)
,
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2 Pk−+1 + σ RF k +1
(5) ,
xˆ k +1 = xˆ k−+1 + K ( z k +1 − xˆ k−+1 ) , Pk +1 = Pk−+1 (1 − K )
(6)
(7)
.
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Fig. 4. Measured distance errors with keeping each interval b/t FR and LR
Fig. 5. Simulation of the leader robot following navigation by a follower robot
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Fig. 6. Multi-robot cooperation navigation for wireless communication distance expansion
5 Conclusion We proposed a multi-robot cooperation navigation method for securing a valid communication distance extension of the remote control based on the WPAN. The concept and implementation of the LR following navigation were introduced, and performance verification was carried out through navigation experiments in real or test-bed environments. The proposed multi-robot cooperation navigation method verified the effect and reliability of securing valid wireless communication and expanding the valid communication distance in an indoor and special-purpose service robot. Acknowledgments. This research was carried out under the General R/D Program sponsored by the Ministry of Education, Science and Technology(MEST) of the Republic of Korea and the partial financial support by the Ministry of Knowledge Economy(MKE), Korea Institute for Advancement of Technology(KIAT) and DaeguGyeongbuk Leading Industry Office through the Leading Industry Development for Economic Region.
References 1. Jong-Jin, P.: Position Estimation of a Mobile Robot Based on USN and Encoder and Development of Tele-operation System using the Internet. The Institute of Webcasting Internet and Telecommunication (2009) 2. Sebastian, T., Dieter, F., Wolfram, B., Frank, D.: Robust Monte Carlo Localization for Mobile Robots. Artificial Intelligence 128, 99–141 (2001) 3. Wan, E.A., van der Merwe, R.: Kalman Filtering and Neural Networks. In: The Unscented Kalman Filter, ch. 7. Wiley, Chichester (2001) 4. Greg, W., Gary, B.: An Introduction to the Kalman Filter. Technical Report: TR 95-041, University of North Carolina at Chapel Hill (July 2006) 5. Dieter, F., Wolfram, B., Sebastian, T.: Active Markov Localization for Mobile Robots in Dynamic Environments. Journal of Artificial Intelligence Research 11(128), 391–427 (1999) 6. Jeon, H.S., Woo, S.H.: Adaptive Indoor Location Tracking System based on IEEE 802.15.4a. Korea Information and Communications Society 31, 526–536 (2006)
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7. Lee, J.Y., Scholtz, R.A.: Ranging in a Dense Multipath Environment using an UWB Radio Link. IEEE Journal on Selected Areas in Comm. 20(9) (2002) 8. http://www.hanback.co.kr/ 9. http://www.aichi-steel.co.jp/ 10. http://www.minfinity.com/
Relative State Modeling Based Distributed Receding Horizon Formation Control of Multiple Robot Systems* Wang Zheng1,2, He Yuqing2, and Han Jianda2 1 2
Graduate School of Chinese Academy of Sciences, Beijing, 100049, P.R. China State Key Laboratory of Robotics, Shenyang Institute of Automation, Shenyang, 110016, P.R. China {wzheng,heyuqing,jdhan}@sia.cn
Abstract. Receding horizon control has been shown as a good method in multiple robot formation control problem. However, there are still two disadvantages in almost all receding horizon formation control (RHFC) algorithms. One of them is the huge computational burden due to the complicated nonlinear dynamical optimization, and the other is that most RHFC algorithms use the absolute states directly while relative states between two robots are more accurate and easier to be measured in many applications. Thus, in this paper, a new relative state modeling based distributed RHFC algorithm is designed to solve the two problems referred to above. Firstly, a simple strategy to modeling the dynamical process of the relative states is given; Subsequently, the distributed RHFC algorithm is introduced and the convergence is ensured by some extra constraints; Finally, formation control simulation with respect to three ground robots is conducted and the results show the improvement of the new given algorithm in the real time capability and the insensitiveness to the measurement noise. Keywords: multiple robot system, formation control, distributed receding horizon control, relative state model.
1 Introduction Formation control, multiple robot systems working together as a fixed geometry configuration, has been widely researched in the past decades. And a great deal of strategies have been introduced and presents their great validity in both theory and reality, such as leader-following[1], behavior based[2], and virtual structure [3], etc. Receding horizon control (RHC), also called model predictive control (MPC), with the abilities of handling constraints and optimization, has been paid more and more attentions in the field of formation control in most recent. However, there are. One of the huge disadvantages in almost all existing receding horizon formation control (RHFC) is the huge computational burden due to the required online optimization algorithm. In order to solve this problem, distributed RHFC (DRHFC) seems a good method and some researching works have been published [4-9]. *
This work is supported by the Chinese National Natural Science Foundation: 61005078 and 61035005.
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However, there are some problems in DRHFC algorithms in most practical applications: 1) Absolute states of each individual robot are difficult to be obtained by other robots since intercommunication is lack of reliability in poor environment. 2) Most DRHFC algorithms use the absolute states directly while relative states between two robots are more accurate and easier to be measured in many applications[16]. Relative state model, i.e., which determines the relative motion law between two robot systems considering each individual model simultaneously in detail, is a new concept originated from the multiple satellite formation control[10]. Both relative kinematics model[11] and relative dynamics model[12] described this kind of relative motion. And, these relative state models have been applied to many distributed formation problems recently. In this paper, a new DRHFC strategy is proposed by introducing relative state model to deal with the above disadvantages. And the remainder of this paper is organized as follows: First, in section 2, the relative state model between two robot systems and a whole formation model are derived. Second, the formation strategy and distributed control law is realized in section 3. Subsequently, in section 4, a simulation results are presented to verify the validity of the proposed algorithm. Finally, the conclusions are given in section 5.
2 System Modeling 2.1 Relative Model We consider the formation control problem of N (N≥2) robot systems, and each individual robot’s dynamical model can be denoted as follows,
xi0 = fi 0 ( xi0 , ui )
(1)
where xi0 ∈ \ n (i=1,2,…,N) and ui ∈ \ m are the state vector and control input vector of the ith robot, respectively; f i 0 (⋅) are some nonlinear smooth functions with predefined structure. Generally, Eq.(1) describes the motion of the robot system in the global coordinate fixed with the earth [14-15]. Thus, xi0 is often called absolute states. Actually, for most member robot system in a formation, only relative states information are necessary to keep a high precise formation, so it is necessary to obtain the dynamical equation considering relative states between two interested robot systems. In this paper, we denote the relative model of robot i and robot j as follows,
x ij = f ji ( xij , ui , u j )
(2)
where xij ∈ \n is the relative state vector with the same dimensions of individual state xi and x j . ui , u j ∈ \ m are the control input of robot i and j, respectively. The methods for modeling relative state equations can be founded in [11] and [12].
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2.2 Formation Model In a formation control problem, suppose that every robot i has ni neighbor robots (neighbors of the ith robot mean the robots which can exchange information with robot i), and all the neighbors of robot i consist of a set Ni. There are two roles in our formation architecture, Na (Na≤N) leaders and N-Na followers. Leaders mean these robots which know their own desired states profile. While Followers denote these robots have no a priori-knowledge about their own desired states profile, and they can only follow their neighbor robots to keep the formation. Thus, the leader robot can be modeled using absolutely state equation and the follower robot can be modeled as several relative state equations with his neighbor robots. Thus, each robot’s state equation combined to its neighbors can be denoted as follows, ⎡ xi0 ⎤ ⎡ fi 0 ( xi0 , ui ) ⎤ ⎢ ⎥ ⎢ ⎥ # ⎢# ⎥=⎢ ⎥ ⎢ x ij ⎥ ⎢ f ji ( xij , ui , u j ) ⎥ ⎢ ⎥ ⎢ ⎥ # ⎣⎢ # ⎦⎥ ⎣⎢ ⎦⎥
(3.a)
# ⎡#⎤ ⎡ ⎤ ⎢ x i ⎥ = ⎢ f i ( x i , u , u ) ⎥ ⎢ j⎥ ⎢ j j i j ⎥ ⎢⎣ # ⎥⎦ ⎢⎣ ⎥⎦ #
(3.b)
where vector xi = [ xi " xij "]T and xi = [" xij "]T denote the leader and follower’s states, respectively. For the purpose of simplification, Eq.(3.a) and Eq.(3.b) can be transformed uniformly as xi = fi ( xi , ui , u−i )
(4)
where u−i = ["u j "]T is all the neighbors’ control inputs. Combining all the system states and models, the whole formation system’s formation model should be expressed as follow, x = f ( x, u )
(5)
where x = [ x1 ," xN ]T is the total states of all robots, and u = [u1 ,", uN ]T the total control input. f ( x, u ) = [" fi ( xi , ui , u−i )"]T is the summation of all the individual robots’ model (4).
3 Distributed Receding Horizon Formation Control 3.1 Cost Function
Before introducing the distributed receding horizon formation control algorithm, we first give some notations used in the following sections. For any vector x ∈ \ n , x
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denotes the vector norm. x
2 P
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= xT Px is P-weight 2-norm of vector x, where P is an
arbitrary positive-definite real symmetric matrix. Also, λmax ( P ) and λmin ( P) denote the largest and smallest eigenvalue of P, respectively. xijc , xi0c , xic and
x c = [ x1c ," xNc ]T are the desired states. In general, the following cost function is used in RHFC algorithm, N N ⎧ ⎪ L( x, u ) = ∑ L ( xi , ui ) = ∑ ⎨γ xi0 − xi0 c i =1 i =1 ⎪ ⎩
2 Qi0
1 + (1 − γ ) ∑ x ij − xij c 2 j ∈N i
2 Q ij
+ ui
2 Ri
⎫⎪ ⎬ ⎪⎭
(6)
where
for i ∈ {1,… , Na} (robot i is a leader) for i ∈ {N − Na,… , N } (robot i is a follower)
⎧1 ⎩0
γ =⎨
is a positive constant for distinguishing leader and follower. Weighted matrix Qi0 , Qij and Ri are all positive definite matrixes, and Qij = Qi j . Let Q = diag ("Qi0 "Qij ") and R = diag (" Ri ") , the integrated cost function can be equivalently rewritten as
L ( x, u ) = x − x c
2 Q
+ u
2
(7)
R
Splitting the cost function (7) as following distributed cost function for each individual robot,
Li ( xi , ui ) = xi − xic
2 Qi
+ ui
2 Ri
= γ xi0 − xi0 c
2 Qi0
1 + (1 − γ ) ∑ xij − x ij c 2 j ∈Ni
2 Qij
+ ui
2 Ri
(8)
Then, the distributed formation control problem can be described as: Design some distributed controllers ui = ki ( xi ) by solving a optimal control problem with respect to the distributed const function (8) for each individual robot i to make the formation system (5) converge to the desired formation state x c . 3.2 Algorithm
Since some cost Li(xi,ui) depends upon the relative states xij , which is subject to dynamics model (2), robot i must predict some relative trajectories according to ui and ui over each prediction horizon. That means, during each update, robot i will receive an assumed control trajectories uˆi (⋅; tk ) from its neighbors[9]. Then, by solving the optimal control problem using model (2), the assumed relative state trajectories can be computed. Likewise, robot i should transmit an assumed control to all neighbors for their own behavior optimization. Thus, the optimal control problem for each individual robot system can be denoted as
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Problem 1. For every robot i ∈ {1,… , N } and at any update time tk, give initial conditions xi(tk), and assumed controls uˆ−i (⋅; tk ) , for all s ∈ [tk , tk + T ] find J i∗ ( xi (tk )) = min J i ( xi (tk ), ui (⋅; tk )) ui ( ⋅ )
(9)
where J i ( xi (tk ), ui (⋅; tk )) = ∫
tk + T
tk
Li ( xi ( s; tk ), ui (s; tk ))ds + M i ( xi (tk + T ; tk ))
subject to dynamics constrains xi (s; tk ) = f i ( xi (s; tk ), ui (s; tk ), u− i (s; tk )) input constrains ui ( s; tk ) ∈ U terminal constrains xi (tk + T ; tk ) ∈ Ωi (ε i )
(10)
and compatibility input constrains
ui ( s; tk ) − uˆi ( s; tk ) ≤ δ 2κ where terminal set is defined as Ω i (ε i ) = { xi | xi − xic
(11) 2
≤ ε i } , given the constants
κ , ε i ∈ (0, ∞) . Constraint (11) is used to reduce the prediction error due to the difference between what a robot plans to do and what neighbors believe that robot will plan to do. Details about defining constraint (11) can be found in [9]. Terminal function Mi(.) should be chosen to drive the terminal state enter the terminal set (10) so that system close-loop stability can be guaranteed. By solving Problem 1, the optimal control solution, we can obtain the optimal control profile ui∗ (τ ; tk ) , τ ∈ [tk , tk + T ] . And, the close-loop system for which stability is to be guaranteed is x(τ ) = f ( x(τ ), u∗ (τ )), τ ≥ t0
(12)
with the applied distributed receding horizon control law
u∗ (τ ; tk ) = (u1∗ (τ ; tk ),", uN∗ (τ ; tk )) for τ ∈ [tk , tk + 1) , and the receding horizon control law is updated when each new initial state update x(tk ) ← x(tk +1 ) is available. Following the succinct presentation in [9], we state the control algorithm. Algorithm 1. At time t0 with initial state xi(t0), the distributed receding horizon controller for any robot i ∈ {1,… , N } is as follows,
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Data: xi(t0), T ∈ (0, ∞) , δ ∈ (0, T ] . Initialization: At time t0, solve Problem 1 for robot i, setting uˆi (τ ; t0 ) = 0 and uˆ− i (τ ; t0 ) = 0 for all τ ∈ [t0 , t0 + T ] and removing constraint (11). At every update interval, (1) Over any interval [tk, tk+1): a) Apply ui∗ (τ ; tk ) , τ ∈ [tk , tk +1 ) , b) Compute uˆi (τ ; tk +1 ) = uˆi (τ ) as ⎧⎪u ∗ (τ ; tk ) τ ∈ [tk +1 , tk + T ) uˆi (τ ; t k +1 ) = ⎨ i τ ∈ [tk + T , tk +1 + T ] ⎪⎩0
c) Transmit uˆi (τ ; tk +1 ) to neighbors and receive uˆ− i (τ ; tk +1 ) from neighbors. (2) At any time tk: a) Measure current state xi(tk), b) Solve Problem 1 for robot i, yielding ui∗ (τ ; tk ) , τ ∈ [tk , tk + T ] . 3.3 Stability Analysis
In this section, the stability analysis of algorithm 1 is given and the main result is somewhat similar to the work in reference [9]. So, the primary lemmas and theorems will be given with a simple explanation. Lemma 1. For a given fixed horizon time T>0, and for the positive constant ξ defined by
ξ = 2ρmax λmax (Q)ANT δ 2κ The function J*(.) satisfies N
tk +1
i =1
tk
J ∗ ( x(tk +1 )) − J ∗ ( x(tk )) ≤ − ∑ ∫
Li ( xi∗ ( s; tk ), ui∗ ( s; tk )) ds + ξδ 2
(13)
for any δ ∈ (0, T ] . In (13), ρ max ∈ (0, ∞) is a positive constant for restricting the state boundary, such that xi ( s; tk ) − xic ≤ ρ max for all s ∈ [tk , tk + T ] . Constant A ∈ (0, ∞) , restricting the boundary of uncontrollable input, satisfies x ij1 − xij 2 ≤ A u j1 − u j 2
at invariant ui
*
subject to relative model (2). Lemma 1 shows that J (.), the optimal value function, decreases from one update to the next along the actual closed-loop trajectories by properly choosing the update interval δ . That is, sufficient small δ will ensure the monotonically decreasing characteristic of objective function J*(.), satisfies J ∗ ( x(τ )) − J ∗ ( x(tk )) ≤ − x(tk ) − x c
2 Q
(14)
Theorem 1. For a given fixed horizon time T>0 and for any state x (t0 ) ⊂ X at initialization, if there exist an proper update time δ satisfies (14), then the formation can converge to x c asymptotically.
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A small fixed upper bound on δ is provided that guarantees all robots have reached their terminal constraint sets via the distributed receding horizon control. After applying the previous lemmas, J*(.) is shown to be a Lyapunov function for the closed-loop system and the remainder of the proof follows closely along the lines of the proof of Theorem 1 in [13].
4 Simulation In this section, we will conduct some simulations to verify the supposed algorithm. Considering two dimensional bicycle-style robot system, shown an Fig.1, and its absolute and relative state model are stated as, ⎡ xi ⎤ ⎡υi cos θi ⎤ ⎢ y ⎥ ⎢υ sin θ ⎥ i ⎥ ⎢ i⎥ =⎢ i ⎢θi ⎥ ⎢ ui1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣υi ⎦ ⎣ ui 2 ⎦
(15.a)
⎡ x ij ⎤ ⎡υ j cos θ ij − υi + y ij ui1 ⎤ ⎢ i⎥ ⎢ ⎥ i i ⎢ y j ⎥ = ⎢ υ j sin θ j − x j ui1 ⎥ ⎢θij ⎥ ⎢ ⎥ −ui1 + u j1 ⎢ ⎥ ⎢ ⎥ ui 2 ⎢⎣ υi ⎥⎦ ⎢⎣ ⎥⎦
(15.b)
Simulation of three robots formation is presented. Robot-1 is a leader robot. As two followers, robot-2 follows robot-1 by the measured relative states, and robot-3 simultaneously follows robot-1 and robot-2. Set update interval δ =0.2s and predict horizon T=1s. At initial time, there robots are located at (2, 2), (1, 3) and (3, 1) in the global coordinate, respectively, and the desired formation is at initial time instant. In order to do some comparisons, we conducts the simulations using both absolute state based DRHFC method (DRHFC-A) and the supposed algorithm in this paper (DRHFC-B). The simulations are conducted using Matlab Optimization Toolbox solver at a PC (Intel(R) Core(TM)i5, M450 @ 2.40GHz). υj
υj
θ ij
θj
( x ij , y ij )
(x j , y j )
υi ( xi , yi )
θi
υi ( xi , yi )
Fig. 1. Absolute and relative modeling of robots
θi
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Simulation 1: Time consuming There robots, keeping initial formation geometry, move along x axis with velocity 1m/s during the first 5 seconds. At the time 5s, an abrupt change of the leader robot’s desired position in the direction of y axis happens, i.e., the desired trajectory of the leader robot is as follows, ⎧[2 + t , 2], t ∈ [0,5] ⎨ ⎩[2 + t ,3], t ∈ [5,10] The whole simulation takes 10 second, and the trajectories of the robots are shown in Fig.2 where the five dashed circles of each individual robot denote their five different predictive states in every time interval. The relative position between robot-1 and robot-2 is shown at Fig. 3, where the dashed line denote the simulation results of DRHFC-A (algorithm in reference [9]) and the solid line describe the results of DRHFC-B (the proposed algorithm in this paper). From Fig.3, it can be seen that the precision of these two algorithms is similar. Time=5.4s
Time=9.6s Leader Follower1 Follower2
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Fig. 2. Trajectories of three robots formation at 5.4s and 9.6s respectively
Since DRHFC-B takes one relative model instead of two absolute models while solving the optimal problem at every interval. The computing time will be naturally reduced. Computing time of the two algorithms is shown in Fig.4, with average cost time Time(DRHFC-A)=3.18 and Time(DRHFC-B)=1.81. That means DRHFC-B is more effective than DRHFC-A. Also, comparisons are conducted in different simulation environment as shown in Table 1, and the similar results can be concluded. Relative Positions 14
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Fig. 4. Computing time at every update interval
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Saving time 42.20% 23.78% 41.90% 24.24% 37.82% 36.79% 41.87% 42.78%
Simulation 2: Insensitiveness to measurement noise There robots keep initial formation geometry stationary for 10 seconds, this time desired trajectory of the leader robot changes to be [2, 2], t ∈ [0,10] Since there is no filter in controllers, and σ2=0.01m2 Gaussian white noise contained in every measured absolute and relative states, robots’ formation are interfered dramatically as shown in Fig.5. We chose object function J*(.)>0 to measure the stationary noise disturbance, larger J*(.) representing stronger disturbance. Fig.6 displays the compared cost function with average J*(A)=0.01196 and J*(B)=0.00351. That means DRHFC-B has less disturbance than DRHFC-A. Relative Positions 0.035 1.2 x12(m)
1.1
J(A)=0.011959 J(B)=0.0035076
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Fig. 5. Relative positions of robot 1 and 2
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Fig. 6. Effect of noise disturbance
5 Conclusion In this paper, a new decentralized receding horizon formation control based on relative state model was proposed. The new designed algorithm has the following advantages: 1) The relative states, instead of the absolute states are used, since the latter is the only requirement for most member robots in a formation and easier to be measured; 2) Computation burden and influence from measurement noise is reduced. However, as a classical leader-follower scheme, some disadvantages will still exist in the proposed algorithm, which is common in most DRHFC algorithm. Such as, how to select proper parameters as the receding horizon time T and update period δ .
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References 1. Das, A.K., Fierro, R., Kumar, V.: A vision-based formation control framework. J. IEEE Transactions on Robotics and Automation 18(5), 813–825 (2002) 2. Balch, T., Arkin, R.C.: Behavior-based formation control for multi-robot teams. J. IEEE Transactions on Robotics and Automation 14(6), 926–939 (1998) 3. Lewis, M.A., Tan, K.H.: High precision formation control of mobile robots using virtual structures. J. Autonomous Robots 4(4), 387–403 (1997) 4. Camponogara, E., Jia, D., Krogh, B.H., Talukdar, S.: Distributed model predictive control. J. IEEE Control Systems Magazine 22(1), 44–52 (2002) 5. Motee, N., Sayyar-Rodsari, B.: Optimal partitioning in distributed model predictive control. In: Proceedings of the American Control Conference, pp. 5300–5305 (2003) 6. Jia, D., Krogh, B.H.: Min-max feedback model predictive control for distributed control with communication. In: Proceedings of the American Control Conference, pp. 4507–4512 (2002) 7. Richards, A., How, J.: A decentralized algorithm for robust constrained model predictive control. In: Proceedings of the American Control Conference, pp. 4261–4266 (2004) 8. Keviczy, T., Borrelli, F., Balas, G.J.: Decentralized receding horizon control for large scale dynamically decoupled systems. J. Automatica 42(12), 2105–2115 (2006) 9. Dunbar, W.B., Murray, R.M.: Distributed receding horizon control for multi-vehicle formation stabilization. J. Automatica 42(4), 549–558 (2006) 10. Inalhan, G., Tillerson, M., How, J.P.: Relative dynamics and control of spacecraft formations in eccentric orbits. J. Guidance, Control, and Dynamics 25(1), 48–59 (2002) 11. Chen, X.P., Serrani, A., Ozbay, H.: Control of leader-follower formations of terrestrial UAVs. In: Proceedings of Decision and Control, pp. 498–503 (2003) 12. Wang, Z., He, Y.Q., Han, J.D.: Multi-unmanned helicopter formation control on relative dynamics. In: IEEE International Conference on Mechatronics and Automation, pp. 4381– 4386 (2009) 13. Chen, H., Allgower, F.: Quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. J. Automatica 34(10), 1205–1217 (1998) 14. Fukao, T., Nakagawa, H., Adachi, N.: Adaptive tracking control of a nonholonomic mobile robot. J. IEEE Transactions on Robotics and Automation 16(5), 609–615 (2002) 15. Béjar, M., Ollero, A., Cuesta, F.: Modeling and control of autonomous helicopters. J. Advances in Control Theory and Applications 353, 1–29 (2007) 16. Leitner, J.: Formation flying system design for a planer-finding telescope-occulter system. In: Proceedings of SPIE the International Society for Optical Engineering, pp. 66871D-10 (2007)
Simulation and Experiments of the Simultaneous Self-assembly for Modular Swarm Robots Hongxing Wei1, Yizhou Huang1, Haiyuan Li1, and Jindong Tan2 1
School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, 100191, Beijing, China [email protected] 2 Electrical Engineering Department, Michigan Technological University, 49931, Houghton, USA [email protected]
Abstract. In our previous work, we have proposed a distributed self-assembly method based on Sambot platform. But there have interference of the infrared sensors between multiple Sambots. In this paper, two interference problems with multiple DSAs are solved and a novel simultaneous self-assembly method is proposed to enhance the efficiency of the self-assembly of modular swarm robots. Meanwhile, the simulation platform is established; some simulation experiments for various configurations are made and the results are analyzed for finding out evidence for further improvement. The simulation and physical experiment results verify the effectiveness and scalability of the simultaneous self-assembly algorithm which is more effective to shorten the assembly time. Keywords: swarm, self-assembly, modular robot.
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docking interfaces. It can move fully autonomously and dock with another Sambot from four directions. Through docking with each other, multiple Sambots can organize into a collective robot [7]. The algorithm for self-assembly is complex and because of high cost of hardware experiment, a simulation platform for Sambot robot is required. Using Microsoft Robotics Studio (MSRS), we design a simulation platform according to physical Sabmot system and some simulation experiments of autonomous construction for various configurations are conducted. In our previous work [7], [8], we have proposed a distributed self-assembly method based on Sambot platform. There have three types of Sambots, including Docking Sambots (DSA), SEED and Connected Sambots (CSA). Single DSA experiments for some configurations have been conducted. But because there have interference of the infrared sensor between multiple Sambots, the simultaneous self-assembly have not been realized. In this paper, two interference problems in Wandering and Locking phase are found out and solved. A simultaneous self-assembly method is designed to enhance the efficiency of the self-assembly of modular swarm robots. Meanwhile, the simultaneous docking of multiple Sambots in Locking phase has been realized. The simulation and physical experiment results show that the simultaneous self-assembly control method is more effective for the autonomous construction of swarm robots. The paper is organized as follows. In section 2, the overall structure of the Sambot robot is described and simulation platform of Sambot is introduced. In section 3, two interference problems in Wandering and Locking phase are analyzed and a simultaneous self-assembly control method is proposed. In section 4, based on the Sambot simulation platform, some simulation experiments are demonstrated to verify the selfassembly algorithm suitable for autonomous construction of various configurations. The simulation results are provided and analyzed. In section 5, physical experiments are implemented and the results are discussed. Finally, conclusions are given and the ongoing work is pointed out.
2 The Sambot Robot and Simulation Platform 2.1 Overall Structure of Sambot The Sambot is an autonomous mobile and self-assembly modular robot including a power supply, microprocessors, sensors, actuators, and a wireless communication unit, which is composed of a cubic main body and an active docking interface, as shown in Fig. 1 (a). The control system of each Sambot is composed of a main microcontroller and four salve microprocessors. The Sambot has two types of communications: the ZigBee wireless communication and the CAN bus communication. The former can be used to achieve global wireless communication among multiple Sambots but here it is not used. The latter takes effect only after two or multiple Sambots finish docking. In autonomous construction phase, CAN bus is adopted to communicate the information and command between Sambots and share the parameters. The bus can support 128 nodes at the most which is large enough for most engineering application.
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Fig. 1. The structure of Sambot. (a) a Sambot robot ; (b) simulated Sambot module; (c) simulated cross quadruped configuration; (d) simulated parallel quadruped configuration.
2.2 Simulation Platform While some researches are being performed, we use Microsoft Robotics Studio (MSRS) to build our simulation platform for more complex strcuture and large quantity of swarms. The simulation model is shown in Fig. 1(b). To realize physics-based simulation, we should design a class which contains inspection module, control module and execution module (as shown in Fig. 2). The inspection module contains gyroscope, infrared sensor and bumper sensor. Control module works as ports in simulation environment which receives message from inspection module and makes decision according to the information. Then robot carries out performance according to these decisions. In Fig. 1 (c) and (d), the simulated cross quadruped configuration and simulated parallel quadruped configuration have been demonstrated. Analysis of Configuration
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3 The Simultaneous Self-assembly Algorithms This section presents three types of roles of different Sambot robots in the selfassembly control model and a newly improved simultaneous self-assembly algorithm. In our previous work [8], a control model consisiting of SEED, CSA, DSA, and CCST (configuration connection state table) is proposed. However in phase of selfassembly, only a single docking Sambot can enter the platform and finish docking once, which is obviously efficient for large quantity of swarms. But, here, a simultaneous self-assembly algorithm for various configuration is designed for improving the assembly. These are carried out based on a bounded experimental platform.
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3.1 Interferences of Self-assembly with Multiple DSAs In our previous work [8], in order to avoid the collision of simultaneous docking, the DSA Sambots are added onto the experimental platform one by one. The DSA is an “isolated” wander and does not have the information of the target configuration and the global coordination. Its controller works according to a series of behaviors of the DSA, including Wandering, Navigation, Docking and Locking. Obviously, if simultaneous docking is available, efficiency of self-assembly would be improved. Self-assembly with multiple DSAs has two interference problems to solve. One appears in Wandering phase, and the other in Locking phase.
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1. In the phase of Wandering, when there is only one DSA to dock with the current configuration, the DSA searches for the Docking_Direction (infrared emitters) without another DSA’s interference. However, if there are multiple DSAs wandering simultaneously, interference would occur from other Sambots’ infrared emitters. In such cases, the DSAs might mistake anther DSA as the current configuration and then miss the target. As shown in Fig. 3 (a), in the process of searching SEED or CSA, detecting sensors of DSA (2) detect DSA (1) before find SEED and DSA (1) is mistaken as current configuration. Then it will navigate around DSA (1). Although DSA (2) still can get away from DSA (1) after DSA (1) is not within the perception of DSA (2), this process is unprofitable. So it is necessary to distinguish current configuration and DSA. 2. In the Locking phase, for simultaneous docking of multiple Sambots, information transmitting conflict can cause the deadlock. Because of CAN bus characteristics and sensors’ limitation, the bus is shared simultaneously by two or more Docking Sambots. When two docking interfaces of current configuration are docked with Sambot A and B meanwhile, Sambot A waits for the record end of Sambot B while Sambot B waits for record end of Sambot A. For example, in Fig. 3 (b) DSA (1) and DSA (2) are docking simultaneously with the SEED, the SEED needs to communicate with them. However, in previous self-assembly algorithm, docking time difference is used to recognize which interface is docked with and further define DSA’ node number in connection state table, here which is unavailable and need to be improved. 3.2 Solution of Interference and Simultaneous Self-assembly Algorithm To achieve simultaneous self-assembly with multiple DSAs, We proposed the improved algorithm to solve the interference problems for obtain better efficiency of self-assembly with multiple DSAs.
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1. In order to avoid infrared sensors’ interference in the phase of Wandering, noting that the DSAs are wandering but the current configuration always remains static. Therefore, when detecting infrared sensors of a DSA receive signals, the object detected by the DSA may be the current configuration or another DSA. At that moment, the DSA moves forward a short step and then rotates around it by certain angle. After this moment, if the signal disappears, the object in the front must be another DSA. Otherwise, it might be the current configuration. However, a possible exception is that two or more DSAs might be interfered simultaneously, which may lead to a wrong judgments. In this situation, the DSA would be in endless deadlock. A function is designed to monitor this sutiation periodically to terminate the deadlock. The Wandering algorithm for multiple DSAs is improved as the following example. If the object detected by DSA is the current configuration, operation scenario is as shown in Fig. 4. First, detecting infrared sensors of DSA have input signal reflected by SEED (a). DSA rotates to the right by a certain angle (b). After that, DSA takes a certain distance forward (c). Then, DSA rotates turns left by a fixed angle and detects SEED again (d). However, as shown in Fig. 5, if the detected object is another DSA, the object will move away from the original place in movement process of DSA and at last is not within perception of DSA (Fig. 5 (d)). Therefore, this method can be used to distinguish current configuration and DSA and solve sensors’ interference problem. 2. Referring to the ordered resource allocation policy to prevet the deadlock by Havender, solution to avoid information conflict is designed. To solve the problem, an ordered conmunication process is introduced. Here, the four interfaces of the same Sambot belongs a group and the smaller the Samot’s node number in connection state table the smaller group number. Meanwhile, the four interfaces in the same group as the suquence of the front, left, back and right are respectively defined as 1, 2, 3 and 4(shown in Fig. 6). Fig 3 (b) gives a possible deadlock. Once deadlock happens,
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interface of a lower number (here back) is delayed, until the information of high number has been transmitted and deadlock is removed, that is, comunication is running as an ordered allocation. Two improved algorithms to solve the corresponding interference problems are added to self-assembly control method. Multiple DSAs are able to simultaneously self-assembly into the target configuration according to design requirement. Obviously, it will shorten the assembly time, which would be analyzed in next sections through simulation and physical experiments.
the ordered resource allocation Left front right back Group 1
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Fig. 6. Solution to avoid the information conflict using ordered resource allocation policy
4 Simulation and Analysis 4.1 Simulation of Snake-Like and Quadruped Configuration On the simulation platform, we try to construct a snake-like configuration and a cross quadruped configuration. We will take experiments with 5, 6 and 7 robots as examples to show the distribution of completion time. Fig. 7 shows the process of the self-assembly experiments of snake-like and cross quadruped configuration. Fig. 8 shows the distribution of completion time. As shown in these graphs, as the number of robots increases, the completion time grows quickly. To show the variation trend of completion time, we expand the number of robots to 11 robots. As shown in Fig. 9, the curvilinear path of completion time is almost like quadratic curve whose slope grows. When the number of robots grows to a certain value, the completion time becomes unacceptable. However, the slope of expected curve should stay the same or even goes down. To explain the phenomenon, we can focus on a single robot. For each robot, the time of docking with other robot stays the same. Then we should pay attention to wandering and navigating state. In wandering state, as the number of robots grows, the probability of interference from other robots increases. It becomes more difficult to find the SEED or CSA. In navigating state, there are two main reasons: one is the distance for a robot to navigate increases as the configuration grows. The second is the interference from other robots when one robot is in navigating state, it can be brought back to wandering state by other robot. So to reduce the completion time, we should improve the wandering and navigating algorithms.
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(a) snake-like configuration
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Fig. 7. The self-assembly experiments of the snake-like and cross quadruped configuration on simulation platform
(a) Snake-like configuration
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Fig. 8. Completion time of snake-like and cross quadruped configuration on simulation platform
Fig. 9. Average time of snake-like and cross quadruped configuration
4.2 The Simulation of Complex Configuration Fig. 10 shows the process of the self-assembly experiments of H-form and parallel quadruped configuration on simulation platform. The Fig.11 shows the distribution of completion time.
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Fig. 10. The self-assembly experiments on the H-form and quadruped configuration on simulation platform
Fig. 11. Distribution of completion time of the H-form and quadruped configurations on simulation platform
5 Physical Experiments
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Based on Sambot modules, on a platform of 1000 mm 1000 mm, we conduct the simultaneous self-assembly experiments with multiple DSAs for both the snake-like and the quadruped configurations. The SEED is also located at the platform center, but the DSAs are put randomly at the four corners. 1. The simultaneous self-assembly of the snake-like configuration with multiple DSAs is shown in Fig. 12. As for linear configuration, in simultaneous self-assembly process, simultaneous docking conflict doesn’t exit but DSA’s sensors are possible to be interfered by another DSA. 2. The simultaneous self-assembly of the quadruped configuration with multiple DSAs. As indicated by the red arrows in Fig. 13, all the four lateral interfaces of the SEED are Docking-Directions which remarkably enhance the experimental efficiency. Transmitting information conflict to the deadlock and sensor interference are possible to happen. However, the simultaneous self-assembly algorithm can deal with the problems. The experimental results verify the effectiveness of the algorithm.
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Fig. 12. The self-assembly experiment of the snake-like configuration with multiple DSAs
Fig. 13. The self-assembly experiment of the quadruped configuration with multiple DSAs
6 Conclusions and Future Work This paper proposed a simultaneous self-assembly control algorithm based on our novel self-assembly modular robot, Sambot. That can be used to realize reconfiguration by autonomous construction. Each module of Sambot is a fully self-contained, mobile robot that has the characteristics of both the chain-type and the mobile swarm robots. In distributed state, each DSA is an autonomous mobile robot; the control model has distributed characteristics. A simultaneous self-assembly algorithm is proposed to enhance the docking efficiency by solving transmitting information conflict and sensor interference. On simulation platform, we make simultaneous self-assembly experiments of various configuration and analyze the efficiency. We succeed in autonomously constructing the snake-like and the quadruped configurations with five Sambots on physical platform which verify simultaneous self-assembly control algorithm comparing the previous researches.
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Some ongoing researches still deserve studying. It is significant that wandering and navigating algorithm still needs further improvement using evolutionary algorithm. Moreover, it is necessary to establish an autonomous control system for the selfassembly of some given configurations, the movement of the whole configuration, the evolutionary reconfiguration to another arbitrary robotic structure and so on.
Acknowledgments This work was supported by the 863 Program of China (Grant No. 2009AA043901 and 2009AA043903), National Natural Science Foundation of China (Grant No. 60525314), Beijing technological new star project (Grant No. 2008A018).
References 1. Whitesides, G.M., Grzybowski, B.: Self-Assembly at All Scales. J. Science 295, 2418– 2421 (2002) 2. Christensen, A.L., Grady, R.O., Dorigo, M.: Morphology Control in a Multirobot System. J. IEEE Robotics & Automation Magzine 14, 18–25 (2007) 3. Anderson, C., Theraulaz, G., Deneubourg, J.L.: Self-assemblages in Insect Societies. J. Insectes Sociaux 49, 99–110 (2002) 4. Bojinov, H., Casal, A., Hogg, T.: Multiagent Control of Self-reconfigurable Robots. J. Artificial Intelligence 142, 99–120 (2002) 5. Klavins, E.: Programmable Self-assembly. J. IEEE Control Systems Magazine 27, 43–56 (2007) 6. Christensen, A.L., O’Grady, R., Dorigo, M.: Morphology Control in a Multirobot System. J. IEEE Robotics & Automation Magazine 14(4), 18–25 (2007) 7. Hongxing, W., Yingpeng, C., Haiyuan, L., Tianmiao, W.: Sambot: a Self-assembly Modular Robot for Swarm Robot. In: The 2010 IEEE Conference on Robotics and Automation, pp. 66–71. IEEE Press, Anchorage (2010) 8. Hongxing, W., Dezhong, L., Jiandong, T., Tianmiao, W.: The Distributed Control and Experiments of Directional Self-assembly for Modular Swarm Robot. In: The 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 4169–4174. IEEE Press, Taipei (2010)
Impulsive Consensus in Networks of Multi-agent Systems with Any Communication Delays Quanjun Wu1, , Li Xu1 , Hua Zhang2 , and Jin Zhou2 1
Department of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 200090, China [email protected] 2 Shanghai Institute of Applied Mathematics and Mechanics and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China
Abstract. This paper considers consensus problem in directed networks of dynamic agents having communication delays. Based on impulsive control theory on delayed dynamical systems, a simple impulsive consensus protocol for such networks is proposed, and a generic criterion for solving the average consensus problem is analytically derived. Compared with some existing works, a distinctive feature of this work is to address average consensus problem for networks with any communication delays. It is shown that the impulsive gain matrix in the proposed protocol play a key role in seeking average consensus problems. Simulations are presented that are consistent with our theoretical results. Keywords: average consensus; impulsive consensus; directed delayed networked multi-agent system; fixed topology; time-delay.
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Recently, the distributed coordination in dynamic networks of multi-agents has emerged as a challenging new research area. The applications of multi-agent systems are diverse, ranging from cooperative control of unmanned air vehicles, formation control of mobile robots, control of communication networks, design of sensor-network, to flocking of social insects, swarm-based computing, etc., [1,2,3,4]. Agreement and consensus protocol design is one of the important problems encountered in decentralized control of communicating-agent systems. To achieve cooperative consensus, a series of works have been performed recently [1,2,3,4,5,6]. Jadbabaie et al. provided a theoretical explanation for the consensus behavior of the Vicsek model using graph theory [1]. Fax et al. emphasized the role of information flow and graph Laplacians and derived Nyquist-like criterion for stabilizing vehicle formations [2]. Olfati-Saber et al. investigated a systematical framework of consensus problem in networksof agents. Three consensus problems were discussed: directed networks with fixed topology, directed
Corresponding author.
Y. Tan et al. (Eds.): ICSI 2011, Part II, LNCS 6729, pp. 128–135, 2011. c Springer-Verlag Berlin Heidelberg 2011
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networks with switching topology, as well as undirected networks with communication time-delays and fixed topology [3]. Moreau used a set-valued Lyapunov approach to study consensus problems with unidirectional time-dependent communication links [4]. Ren et al. extended the results to unidirectional communication and relaxed the connectivity assumption to the assumption that the union of the communication graphs has a spanning tree [5]. Time-delays often occur in such systems as transportation and communication systems, chemical and metallurgical processes, environmental models and power networks [7,8,9]. In many scenarios, networked systems can possess a dynamic topology that is time-varying due to node and link failures/creations, packet-loss, asynchronous consensus, state-dependence, formation reconfiguration, evolution, and flocking. There has been increasing interest in the study of consensus problem in dynamic networks of multi-agents with time-delays in the last several years [3,7,8,9]. It has been noticed the existing studies on consensus problem are predominantly to give some consensus protocols for networks of dynamic agents having communication delays with various network topology. However, these consensus protocols are only valid for some specific small communication delays [3,7,8,9]. For example, Olfati-Saber et al. discussed average consensus problems in undirected networks having a common constant communication delay with fixed topology and switching topology. They presented the following main result (See Theorem 10 in [3]): A sufficient and necessary condition for seeking average consensus in an undirected connected network is that the communication delays are less than a positive threshold. Therefore, this motivates the present investigation of average consensus problems in networks of dynamic agents for any communication delays particularly regarding practical engineering applications. This present paper considers consensus problem in directed networks of dynamic agents with fixed topology for any communication delays. It can generalize to the case of switching topology. The primary contribution of this work is to propose a novel yet simple impulsive consensus protocol for such networks, which is the generalization of corresponding results existing in the literature. A generic criterion for solving the average consensus problem is derived based on impulsive control theory on delayed dynamical systems. It is demonstrated that average consensus in the networks is heavily dependent on impulsive gain matrix in the proposed consensus protocol. Finally, simulations are presented that are consistent with our theoretical results. The paper is organized as follows. A simple impulsive consensus protocol is proposed in Section 2. In Section 3, we focus on the average consensus problem in directed delayed networks of dynamic agents with fixed topology. Some simulation results are provided in Section 4. Finally, we summarize the main conclusions in Section 5.
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Let R = (−∞, +∞) be the set of real numbers, R+ = [0, +∞) be the set of nonnegative real numbers, and Z + = {1, 2, · · · } be the set of positive integer
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numbers. For the vector x = [x1 , · · · , xn ] ∈ Rn , x denotes its transpose. Rn×n stands for the set of n × n real matrixes, for the matrix A = [aij ]n×n ∈ Rn×n , A denotes its transpose, As = (A + A )/2 stands for the symmetric part of A. The spectral norm of A is defined as A = [λmax (AA )]1/2 . E is the identity matrix of order n. In this paper, we are interested in discussing average consensus problem in directed delayed networks of dynamic agents with fixed topology, where the information (from vj to vi ) passes through edge (vi , vj ) with the coupling timedelays 0 < τ (t) ≤ τ . Here we assume that the communication topology of G is balanced and has a spanning tree. Moreover, each agent updates its current state based upon the information received from its neighbors. As L is a balanced matrix, an average consensus is asymptotically reached and α = ( i xi (0))/n = Ave(x). The invariance of Ave(x) allows decomposition of x according to the following equation: x = α1 + η, (1) where η = (η1 , · · · , ηn )T ∈ Rn satisfies 1T η = 0. Here, we refer to η as the (group) disagreement vector. The vector η is orthogonal to 1 and belongs to an (n − 1)-dimensional subspace. Let xi be the state of the ith agent. Suppose each node of a graph is a dynamic integrator agent with dynamics: x˙ i (t) = ui (t), i = 1, 2, · · · , n. (2) where ui (t) is the control input (or protocol) at time t. In [3], Olfati-Saber and Murray presented the following linear time-delayed consensus protocol: ui (t) = aij xj (t − τ ) − xi (t − τ ) . (3) vj ∈Ni
They presented the following main result [3]: Proposition 1. Assume the network topology G is fixed, undirected, and connected. Then, the protocol (3) globally asymptotically solves the average-consensus problem if and only if the following condition is satisfied: (i) τ ∈ (0, τ ∗ ) with τ ∗ = π/2λn , λn = λmax (L). Obviously, the consensus protocol (3) is invalid for any τ ≥ τ ∗ . The main objective of this section is to design and implement an appropriate protocol such that (2) uniformly asymptotically solves the average consensus problem for any communication delays. This is to say, limt→+∞ xi (t) − xj (t) = 0, for τ ∈ (0, +∞) and all i, j ∈ Z + . Based on impulsive control theory on delayed dynamical systems, we propose the following impulsive consensus protocol: aij xj (t − τ (t)) − xi (t − τ (t)) ui (t) = vj ∈Ni
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where bij ≥ 0 are constants called as the control gain, δ(t) is the Dirac function [9,10]. Remark 1. If bij = 0 for all i, j ∈ n, then the protocol (4) becomes a linear consensus protocol (3) corresponding to the neighbors of node vi . Clearly, consensus protocol (4) is the generalization of corresponding results existing in the literature [3,7,8,9]. It should be noted that the latter part of the impulsive consensus protocol (4) has two aims. On one hand, if τ (t) < τ ∗ , we can utilize it to accelerate the average consensus of such systems. On the other hand, if τ (t) ≥ τ ∗ , it can solve average consensus for any communication time-delays. This point will be further illustrated through the numerical simulations. Under the consensus protocol (4), the system (2) has the following form x(t) ˙ = −Lx(t − τ (t)), t = tm , t ≥ t0 , Δx(t) = x(t) − x(t− ) = −M x(t), t = tm , m ∈ Z + ,
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where n M = (mij )n×n is a Laplacian defined by mij =s −bij , j = i, and mij = k=1,k=i bik , j = i. The eigenvalues of the matrix M can be ordered as 0 = λ1 (M s ) < λ2 (M s ) ≤ · · · ≤ λn (M s ). Moreover, η evolves according to the (group) disagreement dynamics given by η(t) ˙ = −Lη(t − τ (t)), t = tm , t ≥ t0 , (6) (E + M )η(t) = η(t− ), t = tm , m ∈ Z + In what follows, we will consider the average consensus problem of (5) with fixed topology. We will prove that under appropriate conditions the system achieves average consensus uniformly asymptotically.
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Based on stability theory on impulsive delayed differential equations, the following sufficient condition for average consensus of the system (5) is established. Theorem 1. Consider the delayed dynamical network (5). Assume there exist positive constants α, β > 0, such that for all m ∈ N , the following conditions are satisfied: s (A1 ) 2 + 2λ2 (M ) + λ2 (M M ) · L ≤ α; (A2 ) ln 1 + 2λ2 (M s ) + λ2 (M M ) − α(tm − tm−1 ) ≥ β > 0. Then the delayed dynamical network (5) achieve average consensus uniformly asymptotically. Proof. Since the graph G has a spanning tree, by using Lemma 3.3 in [5], then its Laplacian M has exactly one zero eigenvalue and the rest n− 1 eigenvalues all have positive real-parts. Furthermore, M s is a symmetric matrix and has zero row sums. Thus, the eigenvalues of matrices M s and M M can be ordered as 0 = λ1 (M s ) < λ2 (M s ) ≤ · · · ≤ λn (M s ),
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and
0 = λ1 (M M ) < λ2 (M M ) ≤ · · · ≤ λn (M M ).
On the other hand, since M s and M M are symmetric, by the basic theory of Linear Algebra, we know η (t)M s η(t) ≥ λ2 (M s )η (t)η(t),
1 η = 0.
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Let us construct a Lyapunov function of the form V (t, η(t)) =
1 η (t)η(t). 2
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When t = tm , for all η(t) ∈ S(ρ1 ), 0 < ρ1 ≤ ρ, we have − η (tm )(E + M )(E + M )η(tm ) = η (t− m )η(tm ),
By (7) and (8), we get − 1 + 2λ2 (M s ) + λ2 (M M ) η (tm )η(tm ) ≤ η (t− m )η(tm ), that is V (tm , η(tm )) ≤
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(10)
1 t, then ψ(t) is strictly increasing and 1 + 2λ2 (M s ) + λ2 (M M ) ψ(0) = 0 with ψ(t) < t for all t > 0. Hence, the condition (ii) of Theorem 1 in [10] is satisfied. For any solutions of Eqs. (6), if Let ψ(t) =
V (t − τ (t), η(t − τ (t))) ≤ ψ −1 (V (t, η(t)).
(11)
Calculating the upper Dini derivative of V (t) along the solutions of Eqs. (6), and by using the inequality x y + y x ≤ εx x + ε−1 y y, we can get that D + V (t) = −η Lη(t − τ (t)) ≤ L · V (t, η(t)) + sup V (s, η(s)) t−τ ≤s≤t
≤ 2 + 2λ2 (M s ) + λ2 (M M ) · LV (t, δ(t)) ≤ αV (t, η(t)). Letting g(t) ≡ 1 and H(t) = αt. Thus, the condition (iii) of Theorem 1 in [10] is satisfied. The condition (A2 ) of Theorem 1 implies that μ tm ds − g(s) ds ψ(μ) H(s) tm−1 1
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The condition (iv) of Theorem 1 in [10] is satisfied. Let w1 (|x|) = w2 (|x|) = |x2 |/2, so the condition (i) of Theorem 1 in [10] is satisfied. Therefore, all the conditions of Theorem 1 in [10] are satisfied. This completes the proof of Theorem 1. Remark 2. Theorem 1 shows that, average consensus of the delayed dynamical network (5) not only depends on the topology structures of the entire network, but also is heavily determined by the impulsive gain matrix M and the impulsive interval tm − tm−1 . In addition, the conditions of Theorem 1 are all sufficient conditions but not necessary, i.e., the dynamical networks can achieve average consensus uniformly asymptotically, although one of the conditions of Theorem 1 may fail.
4
Simulations
As an application of the above theoretical results, average consensus problem for delayed dynamical networks is worked out in this section. Meanwhile, simulations with various impulsive gains matrices are given to verify the effectiveness of the proposed impulsive consensus protocol, and also visualize the impulsive gain effects on average consensus problem of the delayed dynamical networks. Here we consider a directed network with fixed topology G having 100 agents as in Fig. 1. It is easy to see that G has a spanning tree. Matrix L is given by ⎞ ⎛ 2 −1 0 · · · −1 ⎜ −1 2 −1 · · · 0 ⎟ ⎟ ⎜ ⎟ ⎜ L = ⎜ 0 −1 2 · · · 0 ⎟ . ⎜ .. .. .. . . .. ⎟ ⎝ . . . . . ⎠ −1 0
0 ··· 2
100×100
Fig. 1. A directed network with fixed topology having 100 agents
For simplicity, we consider the equidistant impulsive interval tm − tm−1 ≡ Δt. It is easy to verify that if the following conditions hold, 2 + 2λ2 (M s ) + λ2 (M M ) × 4 ≤ α; and ln 1 + 2λ2 (M s ) + λ2 (M M ) − α(tm − tm−1 ) ≥ β > 0.
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xi(t),(i=1,2,...,100)
10
8
6
4
2
0
0
5
10 t
15
20
10
10
9
9
8
8
7
7 x (t),(i=1,2,...,100)
6 5 4
6 5 4
i
xi(t),(i=1,2,...,100)
Fig. 2. The change process of the state variables of the delayed dynamical network (5) without impulsive gain in case τ (t) = τ ∗ = π/8
3
3
2
2
1 0
1
0
0.5
1 (a) t
1.5
2
0
0
0.5
1 (b) t
1.5
2
Fig. 3. Average consensus process of the agents state of the delayed dynamical network (5) with different impulsive gains matrices in case τ (t) = 1.0
then all the conditions of Theorem 1 are satisfied, which means the delayed dynamical network (5) achieve average consensus uniformly asymptotically. Let the equidistant impulsive interval be taken as Δt = 0.02. Fig. 2 is the simulation result corresponding to change process of the state variables of the delayed dynamical network (5) having the communication delay τ (t) = τ ∗ = π/2λn = π/8 with the impulsive gain matrix M = 0 in time interval [0, 20]. This clearly shows that average consensus is not asymptotically reached, which is consistent with the result of Proposition 1. Fig. 3 demonstrates the change process of the state variables of the delayed dynamical network (5) having the communication delay τ (t) = 1 with different impulsive gain mij = −0.015, i = j, mij = 1.485, i = j, α = 30, β = 2.7322 and mij = −0.018, i = j, mij = 1.782, i = j, α = 36, β = 2.9169 in time interval [0, 2], respectively, which satisfy the conditions of Theorem 1. It can be shown that impulsive average consensus is finally achieved, and the impulsive gain matrix heavily affect consensus of the delayed dynamical network.
5
Conclusions
This paper has developed a distributed algorithm for average consensus in directed delayed networks of dynamic agents. We have proposed a simple impulsive consensus protocol for such networks for any communication delays, and some generic sufficient conditions under which all the nodes in the network achieve
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average consensus uniformly asymptotically have been established. It has been indicated that average consensus in the networks is heavily dependent on communication topology of the networks and impulsive gain. Finally, numerical results have been used to show the robustness and effectiveness of the proposed impulsive consensus protocol.
Acknowledgment This work was supported by the National Science Foundation of China (Grant Nos. 10972129 and 10832006), the Specialized Research Foundation for the Doctoral Program of Higher Education (Grant No. 200802800015), the Innovation Program of Shanghai Municipal Education Commission (Grant No. 10ZZ61), the Shanghai Leading Academic Discipline Project (Project No. S30106), and the Scientific Research Foundation of Tongren College (Nos. TS10016 and TR051).
References 1. Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules. IEEE Trans. Autom. Contr. 48, 988–1001 (2003) 2. Fax, J.A., Murray, R.M.: Information Flow and Cooperative Control of Vehicle Formations. IEEE Trans. Autom. Contr. 49, 1465–1476 (2004) 3. Olfati-Saber, R., Murray, R.M.: Consensus Problems in Networks of Agents with Switching Topology and Time-Delays. IEEE Trans. Autom. Contr. 49, 1520–1533 (2004) 4. Moreau, L.: Stability of Multiagent Systems with Time-Dependent Communication Links. IEEE Trans. Autom. Contr. 50, 169–182 (2005) 5. Ren, W., Beard, R.W.: Consensus Seeking in Multiagent Systems Under Dynamically Changing Interaction Topologies. IEEE Trans. Autom. Contr. 50, 655–661 (2005) 6. Hong, Y.G., Hu, J.P., Gao, L.X.: Tracking Control for Multi-Agent Consensus with an Active Leader and Variable Topology. Automatica 42, 1177–1182 (2006) 7. Sun, Y.G., Wang, L., Xie, G.M.: Average Consensus in Networks of Dynamic Agents with Switching Topologies and Multiple Time-Varying Delays. Syst. Contr. Lett. 57, 175–183 (2008) 8. Lin, P., Jia, Y.M.: Average Consensus in Networks of Multi-Agents with both Switching Topology and Coupling Time-Delay. Physica A 387, 303–313 (2008) 9. Wu, Q.J., Zhou, J., Xiang, L.: Impulsive Consensus Seeking in Directed Networks of Multi-Agent Systems with Communication Time-Delays. International Journal of Systems Science (2011) (in press), doi:10.1080/00207721.2010.547630 10. Yan, J., Shen, J.H.: Impulsive Stabilization of Functional Differential Equations by Lyapunov-Razumikhin Functions. Nonlinear Anal. 37, 245–255 (1999)
FDClust: A New Bio-inspired Divisive Clustering Algorithm Besma Khereddine1,2 and Mariem Gzara1,2 1
Multimedia InforRmation systems and Advanced Computing Laboratory (MIRACL) Sfax, Tunisia 2 Institut supérieur d’informatique et de mathématique de Monastir [email protected], [email protected]
Abstract. Clustering with bio-inspired algorithms is emerging as an alternative to more conventional clustering techniques. In this paper, we propose a new bio-inspired divisive clustering algorithm FDClust (Artificial Fish based Divisive Clustering algorithm). FDClust takes inspiration from the social organization and the encounters of fish shoals. In this algorithm, each artificial fish (agents) is identified with one object to be clustered. Agents move randomly on the clustering environment and interact with neighboring agents in order to adjust their movement directions. Two Groups of similar objects will appear through the movement of agents in the same direction. The algorithm is tested and evaluated on several real benchmark databases. The obtained results are very interesting in comparison with Kmeans, Slink, Alink, Clink and Diana algorithms. Keywords: Clustering, data mining, hierarchical clustering, divisive clustering, swarm intelligence, fish shoals.
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merging them into parental nodes (ii) divisive clustering that builds a top-down hierarchy by splitting greater clusters into smaller ones starting from the entire data set. Researchers seek to invent new approaches to enhance the resolution of the clustering problem and to achieve better results. Recently, research on and with the bio-inspired clustering algorithms has reached a very promising state. The basic motivation of these approaches stems from the incredible ability of social animals and other organisms (ants, bees, termites, birds, fish, etc) to solve complex problems collectively. These algorithms use a set of similar and rather simple artificial agents (ant, bee, individual, etc) to solve the clustering problem. These algorithms can be divided into three main categories according to data representation [1]: (i) an agent represents a potential solution to the clustering problem to be optimized such as genetic [2,3] and particle swarm optimization clustering algorithms [4,5], (ii) data points which are objects in the universe, are moved by agents in order to form clusters. Examples of such approaches are ant-based clustering algorithms [6] [7], (iii) each artificial agent represents one data set. These agents move on the universe to form groups of similar entities, for example Antree [8] and AntClust [9]. In this work, we propose a new bio-inspired divisive clustering algorithm: artificial Fish based Divisive Clustering algorithm (FDClust). This algorithm takes inspiration from the social organization and the encounters of fish shoals phenomena. Several studies have shown that fish shoals are assorted according to several characteristics [10][11]. During fish shoals encounters, an individual fish decides to join or to leave a group according to its common characteristics with the already existing group members [12][13]. Shoals encounters may result in the fission of the group into two homogenous shoals. Thus real fish are able to solve the sorting problem. These phenomena can be easily adapted to solve the clustering problem. In FDClust, an artificial fish represents an object to be clustered. The encounters of two artificial shoals results in the fission of the group into two clusters of similar objects. FDClust builds a binary tree of clusters. It applies recursively this process to split each node into two homogenous clusters. The reminder of the paper is organized as follows. Section 2 first describes the social organization of fish species and then the encounter phenomenon of fish shoals. In section 3 we present the FDClust algorithm in details. Experimental results are presented and discussed in section 4. Section 5 concludes the paper and gives suggestions for future work.
2 Social Organization and Encounters of Fish Shoals Fish are strikingly social organisms [14]. Several biological studies have observed and developed theoretical models to understand the fish shoals structures. In [13] the authors have stated that fish shoals are not random aggregations of individuals but they are instead assorted with respect to several factors including species and size. Croft et al [14] provided evidence that juveniles display assortative shoaling based on color pattern. Shoaling by color has also been reported in mollies [12]. Shoaling by species and body length was observed in several species [13][11]. The homogeneity of group composition has associated benefits such as anti-predator defense and foraging efficiency.
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Fig. 1. Diagram showing the two forms of fission events that were recorded a) a rear fission event, b) a lateral fission event [14]
Shoals membership is not necessarily stable over time. Individuals are exchanged between groups [14]. Fish shoals are thus open groups (groups where individuals are free to leave and join). Theoretical models of open groups assert that socials animals make adaptive decisions about joining groups on the basis of a number of different phenotypic traits of existing group members. Hence, individuals prefer to associate with similar conspecifics, those of similar body length and those free of parasite [13]. Active choice of shoal mates has been documented for many fish species. During shoals encouters individuals may actively choose neighboring fish that are of a similar phenotype. Fish have limited vision and then cannot interact with the total group members but only with perceived ones. Thus, shoals encounters provide an individual based mechanism for shoal assortment. Since individuals can make decisions based on the composition of available shoals, other group members are a source of information about the most adaptive decisions [15]. Group living is likely to be based on a continuous decision-making process, with individuals constantly evaluating the profitability of joining, leaving or staying with others, in each encounter with other groups. The encounters of fish shoals result in shoal fission or fusion. Fission (but not fusion) events are shown to be an important mechanism in generating phenotypic assortment [14]. Shoal fission events are divided into two categories (figure 1): (i) rear fission events where the two resulting shoals maintained the same direction of travel and fission occur due to differential swimming speeds, (ii) lateral fission events where the two resulting shoals are separated due to different directions of travel [14]. The social organization of fish shoals is based on the phenotypic similarity. The continuous decision-making process is based on the maintenance of social organization with neighboring group members. The behavior of real fish during shoals encounters makes them able to solve collectively the sorting problem. Our study of these phenomena (particularly the fission events) from a clustering perspective results in the development of a clustering model for solving the divisive clustering problem. The core task in such a problem is to split a candidate cluster into two distant parts. In our model, this task is achieved by the simulation of the encounters of two groups of artificial fish. The model is described in the next section.
3 FDClust: A New Bio-inspired Divisive Clustering Algorithm The FDClust algorithm constructs a binary tree of clusters by starting from all objects in the same cluster and splitting recursively each candidate cluster into two
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sub-clusters until each object form one cluster. At each step the cluster with the highest diameter among those not yet splitted is partitioned into two sub-clusters. To achieve the partitioning of a group of objects into two homogenous groups, FDClust applies a bipartitioning procedure that takes inspiration from the shoals encounters phenomenon. During shoals encounters, real fish are able to evaluate dynamically the profitability of joining, leaving or staying with neighboring agents. This decision making process is based on the maintenance of social organization of the entire group. Fish shoals are phenotypically assorted by color, size and species. Shoals encounters may result in the fission of the group into two well-organized groups (assorted groups). In lateral fission, groups are separated due to two different directions of swimming. To achieve the division of the candidate cluster into two sub-clusters, we use two artificial fish shoals. The encounter of these two groups of artificial fish results in a lateral fission of the group into two homogenous groups. Artificial fish (agents) are initially randomly scattered on the clustering environment. Each agent is an object to be clustered. Each agent is randomly associated a direction left or right. Since real fish have only local vision, artificial agents interact only with neighboring agents to make adaptive decisions about joining or leaving a group. Each agent has to make a binary decision whether to move to the left or to the right. Agents take the same direction as most similar agents in their neighborhood. Artificial fish join finally their appropriate group composed with similar agents. The initial group is then separated into two sub-groups of similar objects due to the two directions of travel left and right. Two groups of agents are formed those having the left direction and those having the right direction. 3.1 Clustering Environment and Agents Vision The clustering environment is a rectangular 2D grid G. Its width is w= ⎡ n ⎤ and length is L = ⎡ n + 2 A ⎤ , where A is a positive parameter and n is the number of objects. Objects are initially scattered randomly in the central square of the grid of size w×w (figure 2). Two objects cannot occupy initially the same cell of the grid. Each agent has initially a random direction left (←) or right (→). Artificial agents have a limited vision. An agent can perceive only s×s neighboring cells (figure 2). Agents are allowed to move to a cell still occupied by other agents. Let n i be the
(a)
(b)
Fig. 2. FDClust: a) Clustering environment b) Agents vision
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number of agents in the neighborhood v ( p i ) of the agent p i . If n ≤ v ( p ) = s × s then the agent p interacts with all his neighbors, else it interacts only with n v = s × s neighbors chosen randomly among those situated in his neighborhood. We note pv ( p i ) the set of agents with which the agent p i can interact. i
i
i
3.2 Agents Movements Each agent has an initial preferable direction left (←) or right (→). This direction is initially fixed randomly. Agents move with identical speed. In one step, an agent can move to one of its neighboring cells whatever the left one or the right one. It chooses actively its travel direction through the interactions with its neighboring agents. An agent interacts with at most n v nearest neighbors among those situated in his local neighborhood. In fact, agents can occupy the same cell as other agents. To take the decision about the next travel direction, the agent p i evaluates its similarity with agents from
pv ( p i )
that have the direction left (→) (respectively right (←)). These
two similarities are calculated as follow: p
sim
( pi,→ ) = 1 −
j
p
m *
j
pv
∈ pv ( p ∈ pv ( p j
i
) / dir
∑
i
( p 2
j
d
) / dir
) =→
( pi, p
( p
j
) =→
∈ v ( p i ) / dir ( p
p j ∈ pv ( p i ) / dir ( p j ) =← 2
sim ( p i , ← ) = 1 −
∑
d
p j ∈ pv ( p i ) / dir ( p
m * p
j
( pi, p
j
) =←
∈ pv ( p i ) / dir ( p
j
)
j
) =→
)
j
j
(1)
(2)
) =←
Where m is the number of attribute considered to characterize the data. An agent has the tendency to have the same direction of travel as its most similar neighbors. If agents in pv ( p ) that have the left direction are more similar to p than those having the right direction, than p will move to the cell at its left and vice versa. An agent will apply the following rules: i
i
i
• •
If | If |
•
If pv ( p ) ≠ to the right. If pv ( p ) ≠ to the left.
•
pv ( p
pv ( p
i
)
i
)
i
i
|=0 then the agent will stand by. |≠0 and sim ( p i , → ) = sim ( p i , ← ) then the agent will stand by. and
0
0
and
sim
( p
i
, →
) >
sim
( p
i
, ←
)
sim ( p i , → ) < sim ( p i , ← )
then the agent will move then the agent will move
3.3 The Algorithm FDClust FDClust starts by all objects gathered in the same cluster. At each step it applies the bi-partitioning algorithm to the cluster to be splitted until each object constitutes one cluster. It is a hierarchical divisive clustering algorithm (figure 3).
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1. 2. 3. 4. 5.
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Initially, the universal cluster C containing all objects is to be splitted. Bi-partitioning(C). Eliminate C from the list of clusters to be splitted, and add Cr and Cl to this list. Select the cluster C with the highest diameter among those not yet splitted. If |C|=1 stop else go to step 2. Fig. 3. The algorithm FDClust
Input: number of objects N, the size of the perception zone s*s, the movement step p and the number of iterations T. Output: Cl and Cr 1. Scatter objects of cluster C in the central square of the grid 2. Associate random direction (ĺ or ĸ) to each object 3. For t=1 to T do 4. For i=1 to N do 5. If(| pv ( p ) |=0) then stand by else 6. compute sim ( p i , → ) and sim ( p i , ← ) i
7.
If
8.
direction ( p )= ĺ and move to the right. If sim ( p , → ) < sim ( p , ← ) then
9.
direction ( else stand by
sim
( p
i
, →
) >
sim
( p
i
, ←
)
then
i
i
10. for i=1 to N do 11. if direction ( p )= ĺ then 12. Else p i ∈ Cl 13. end 14. end 15. Return Cl et Cr i
i
p
i
p
i
)=ĸ and move to the left.
∈ Cr
Fig. 4. Bi-partitioning algorithm
The bi-partitioning algorithm (figure 4) receives as a parameter a cluster C composed of n objects, the size of the perception zone s×s and the number of iterations T. The output of the algorithm is two clusters Cl and Cr. It assigns randomly to each object its corresponding coordinate on the grid and its initial direction left or right. The algorithm consists of T iterations. At each iteration, each agent evaluates its similarity with neighboring ones having the left direction (respectively the right direction), takes the decision on its following direction and computes its new coordinates on the grid. After T iterations, two clusters will be formed Cl and Cr, where Cl (respectively Cr) is the set of objects having the left direction (respectively right direction). The computational complexity of the bi-partitioning procedure is Ο(T n v n) with T the number of iteration, n v the maximum number of neighboring agents and n the number of objects in the candidate cluster.
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4 Tests and Results To evaluate our algorithm, we have used real databases issued from the machine learning repository [16]. Table 1. Real data bases Data base Iris Glass Thyroid Soybean Wine Yeast
N 150 214 215 47 178 1484
M 4 9 5 35 13 8
K 3 6 3 4 3 10
The main features of the databases are summarized in Table 1. In each case the number of attributes (M), the number of classes (K) and the total number of objects (N) are specified. To evaluate our algorithm we have used the following measures:
The intra clusters inertia: used to determine how homogonous the objects in clusters are with each others (where, Gi is the center of the cluster I, d is the Euclidean distance): I =
1 K
¦
K
i =1
¦
xi ∈ C
d ( x i , Gi ) 2
(3)
The recall, the precision and the F-measure: are based on the idea of comparing a resulting partition with a real or a reference partition. The relative recall (respectively precision and F-measure) of the reference class Ci to the resulting class Cj are defined as follows: n ij
recall ( i , j ) =
N
precision (i , j ) =
i
n ij Nj
F (i, j ) = 2
precision precision
( i , j ) * recall ( i , j ) + recall
(i, j ) (i, j )
Where n ij is the number of objects or individuals present in the reference class Ci and in the resulting class Cj. Ni and Nj represent respectively the total number of objects in the class Ci and Cj. To evaluate the entire class Ci, we just choose the maximum of values obtained within Ci: recall ( i ) = max ( recall ( i , j )) precision ( i ) = max ( precision ( i , j )) F ( i ) = max ( F ( i , j )) j
j
j
The global value of the recall (r) , the precision (p) and F-measure (F) for all classes will be respectively ( p i is the weight of the class Ci): r =
∑ i
p i × recall (i )
p = ∑ pi × précision(i) i
F =
∑ i
p i × F (i )
where p i =
Ni (4) ∑ Nk k
In table 2, we present the obtained results for FDClust, kmeans, Alink, Clink, Slink and Diana algorithms. Since FDClust and kmeans are stochastic, we give the min, the max the mean and the standard deviation of 100 runs.
FDClust: A New Bio-inspired Divisive Clustering Algorithm Table 2. FDClust: experimental results Iris I R P F Glass I r p F Thyroid I r p F Soybean I r p F Wine I r p F Yeast I r p F
FDClust min 0 .047 max 0.051 mean 0.05 sd 0.01 min 0.86 max 0.92 mean 0.88 sd 0.01 min 0.85 max 0.92 mean 0.89 sd 0.01 min 0.85 max 0.92 mean 0.88 sd 0.01 FDClust min 0,09 max 0.14 mean 0.1 sd 0.03 min 0,32 max 0.75 mean 0.49 sd 0.05 min 0.41 max 0.7 mean 0.57 sd 0.07 min 0.37 max 0.61 mean 0.48 sd 0.05 FDClust min 0.02 max 0.11 mean 0.034 sd 0.01 min 0.54 max 0.72 mean 0.66 sd 0.03 min 0.59 max 0.74 mean 0.71 sd 0.01 min 0.49 max 0.68 mean 0.64 sd 0.03 FDClust
Kmeans min 0.05 max 0.052 mean 0.051 sd 0.01 min 0.89 max 0.92 mean 0.92 sd 0.009 min 0.66 max 0.89 mean 0.86 sd 0.08 min 0.7 max 0.88 mean 0.85 sd 0.06 Kmeans min 0,08 max 0.1 mean 0.83 sd 0.04 min 0,52 max 0.85 mean 0.63 sd 0.06 min 0.52 max 0.78 mean 0.63 sd 0.05 min 0.53 max 0.74 mean 0.6 sd 0.04 Kmeans min 0.03 max 0.056 mean 0.049 sd 0.03 min 0.86 max 0.78 mean 0.78 sd 0.01 min 0.71 max 0.9 mean 0.9 sd 0.02 min 0.77 max 0.87 mean 0.87 sd 0.01 Kmeans
min 1.42 max 1.57 mean 1.43 sd 0.02 min 0.62 max 0.8 mean 0.79 sd 0.03 min 0.71 max 1 mean 0.94 sd 0.04 min 0.75 max 0.97 mean 0.93 sd 0.04 FDClust min 0.32 max 0.34 mean 0.32 sd 0.03 min 0.86 max 0.98 mean 0.94 sd 0.05 min 0.68 max 0.93 mean 0.87 sd 0.05 min 0.73 max 0.93 mean 0.87 sd 0.04 FDClust min 0.011 max 0.012 mean 0.011 sd 0.001 min 0.21 max 0.64 mean 0.37 sd 0.09 min 0.29 max 0.47 mean 0.34 sd 0.03 min 0.22 max 0.52 mean 0.34 sd 0.05
min 1.57 max 1.64 mean 1.61 sd 0.02 min 0.53 max 1 mean 0.93 sd 0.07 min 0.51 max 1 mean 0.9 sd 0.11 min 0.41 max 1 mean 0.89 sd 0.11 Kmeans min 0.27 max 0.32 mean 0.28 sd 0.05 min 0.8 max 0.96 mean 0.95 sd 0.01 min 0.56 max 0.96 mean 0.95 sd 0.03 min 0.65 max 0.96 mean 0.94 sd 0.02 Kmeans min 0.032 max 0.035 mean 0.033 sd 0.0001 min 0.32 max 0.45 mean 0.39 sd 0.02 min 0.57 max 0.82 mean 0.65 sd 0.03 min 0.35 max 0.53 mean 0.48 sd 0.02
Slink 0.07
Alink 0.047
Clink 0.047
Diana 0.046
0.99
0.88
0.88
0.88
0.66
0.9
0.91
0.89
0,77
0,87
0,88
0,88
Slink 0,2
Alink 0,091
Clink 0,1
Diana 0,082
0,97
0,86
0,87
0,84
0.53
0.56
0.56
0.56
0.62
0.62
0.62
0.62
Slink 0.09
Alink 0.054
Clink 0.054
Diana 0.049
0.97
0.92
0.96
0.88
0.8
0.87
0.67
0.9
0.87
0.88
0.75
0.87
Slink
Alink
Clink
Diana
1.57
1.39
1.57
1.39
0.95
1
0.95
1
0.88
1
0.88
1
0.9
1
0.9
1
Slink 0.52
Alink 0.29
Clink 0.52
Diana 0.32
0.98
0.93
0.98
0.87
0.58
0.93
0.58
0.82
0.67
0.93
0.67
0.81
Slink 0.07
Alink 0.07
Clink 0.042
Diana 0.035
0.98
0.98
0.5
0.47
0.67
0.67
0.72
0.54
0.72
0.72
0.54
0.53
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For the database Iris, our algorithm generates the best results according to all considering measures in comparison with other algorithms. For the data bases Glass and thyroid, FDClust encounters some difficulty in the determination of real cluster structure, but the obtained clusters are homogenous. For the data base soybean, all algorithms generate good partitions and results are nearby. For the data base Wine FDClust generates a partition of a good quality in term of inertia, recall, precision and F_measure in comparison with those obtained by the other algorithms. For the data base yeast, FDClust generates the best partition in term of intraclusters inertia but like Kmeans it has a difficulty in detecting real clusters structures. Comparing with other algorithms, we note that for all data bases FDClust has recorded good performances. Moreover FDClust has the advantage of having lower complexity than the other hierarchical algorithms.
5 Conclusion Bio-inspired clustering algorithms are an appropriate alternative to traditional clustering algorithms. Research on bio-inspired clustering algorithms is still an on-going field of research. In this paper we have presented a new approach for divisive clustering with artificial fish. It is based on the shoal encounters and social organization of fish shoals phenomena. The obtained results are encouraging. As prospects, we attempt to extend our algorithm by considering more than two directions of travels. A candidate cluster may be divided into more than two sub-clusters.
References 1. Bock., H., Gaul, W., Vichi, M.: Studies in classification, data analuysis, and knowldge organization (2005) 2. Falkenauer, E.: A new representation and operators for genetic algorithms applied to grouping problems. Evolutionary Computation 2(2), 123–144 (1994) 3. Maulik, U., Bandyopadhyay, S.: Genetic algorithm-based clustering technique. Pattern Recognition 33, 1455–1465 (2000) 4. Sandra Cohen, C.M., Leandro de Castro, N.: Data Clustering with Particle Swarms. In: IEEE Congress on Evolutionary Computations 2006 (2006) 5. Chen, C.-Y., Ye, F.: Particle swarm optimization algorithm and its application to clustering analysis. In: Proceedings of IEEE International Conference on Networking, Sensing and Control, pp. 789–794 (2004) 6. Lumer, E., Faieta, B.: Diversity and adaptation in populations of clustering ants. In: Cliff, D., Husbands, P., Meyer, J., W., S. (eds.) Proceedings of the Third International Conference on Simulation of Adaptive Behavior, pp. 501–508. MIT Press, Cambridge (1994) 7. Gzara., M., Jamoussi., S., et Elkamel, A., Ben Abdallah, H.: L’algorithme CAC: des fourmis artificielles pour la classification automatique. Accepté à paraitre dans la revue d’intelligence artificielle (2011) 8. Azzag, H., Guinot, C., Oliver, A., Venturini, G.: A hierarchical ant based clustering algorithm and its use in three real-world applications. In: Dullaert, W., Marc Sevaux, K.S., Springael, J. (eds.) European Journal of Operational Research (EJOR). Special Issue on Applications of Metaheuristics (2005)
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9. Labroche, N., Monmarché, N., Venturini, G.: A new clustering algorithm based on the chemical recognition system of ants. In: van Harmelen, F. (ed.) Proceedings of the 15th European Conference on Artificial Intelligence, pp. 345–349 (2002) 10. Krause, J., Butlin, R.K., Peuhkuru, N., Prichard, V.: The social organization of fish shoals: a test of the predictive power of laboratory experiments for the field. Biol. Rev. 75, 477– 501 (2000a) 11. McCann, L.I., Koehn, D.J., Kline, N.J.: The effects of body size and body markings on nonpolarized schooling behaviour of zebra fish (Brachydanio rerio). J. Psychol. 79, 71–75 (1971) 12. Krause, J., Godin, J.G.: Shoal choice in the banded killifish (Fundulus diapha-nus, Teleostei, Cyprinodontidae) – Effects of predation risk, fish size, species compo-sition and size of shoals. Ethology 98, 128–136 (1994) 13. Crook, A.C.: Quantitative evidence for assortative schooling in a coral reef. Mar. Ecol. Prog. Ser. 179, 17–23 (1999) 14. Theodorakis, C.W.: Size segragation and effects of oddity on predation risk in minnow schools. Anim. Behav. 38, 496–502 (1989) 15. Croft, D.P., Arrowsmith, B.J., Bielby, J., Skinner, K., White, E., Couzin, I.D., Margurran, I., Ramnarine, I., Krausse, J.: Mechanisms underlying shoal composition in Trinidadian guppy (Poecilia). Oikos 100, 429–438 (2003) 16. Blake, C.L., Merz, C.J.: UCI repository of machine learning databases (1998)
Mining Class Association Rules from Dynamic Class Coupling Data to Measure Class Reusability Pattern Anshu Parashar1 and Jitender Kumar Chhabra2 1
2
Haryana College of technology & Management Kaithal,136027, India Department of Computer Engineering, National Institute of Technology, Kurukshetra, Kurukshetra 136 119, India [email protected]
Abstract. The increasing use of reusable components during the process of software development in the recent times has motivated the researchers to pay more attention to the measurement of reusability. There is a tremendous scope of using various data mining techniques in identifying set of software components having more dependency amongst each other, making each of them less reusable in isolation. For object-oriented development paradigm, class coupling has been already identified as the most important parameter affecting reusability. In this paper an attempt has been made to identify the group of classes having dependency amongst each other and also being independent from rest of the classes existing in the same repository. The concepts of data mining have been used to discover patterns of reusable classes in a particular application. The paper proposes a three step approach to discover class associations rules for Java applications to identify set of classes that should be reused in combination. Firstly dynamic analysis of the Java application under consideration is performed using UML diagrams to compute class import coupling measure. Then in the second step, for each class these collected measures are represented as Class_Set & binary Class_Vector. Finally the third step uses apriori (association rule mining) algorithm to generate Class Associations Rules (CAR’s) between classes. The proposed approach has been applied on sample Java programs and our study indicates that these CAR’s can assist the developers in the proper identification of reusable classes by discovering frequent class association patterns. Keywords: Coupling, Data Mining, Software Reusability.
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some of the already existing quality components and development of new highly reusable component. The reuse of software components in software development leads to increased productivity, quality, maintainability etc [3,23]. The success of reusability is highly dependent on proper identification of whether a particular component is really reusable or not. These measures help to develop, store and identify reusable components [21]. Reuse of Class code has been frequent in practice. It is essential & tricky to identify a set of needed classes to reuse together or alone. Hence it is always desirable to find out the classes along with their associated classes [17]. Class coupling plays a vital role to measure reusability and selecting classes for reuse in combination because the highly coupled classes are required to be reused as a group [7]. One can define a class Ca related to class Cb if Ca must use Cb in all future reuse. So group of dependent classes should be reused together for ensuring the proper functioning of the application [22]. There is a thrust of software metrics especially reusability metric as an active research area in the field of software measurement. Software metric is a quantitative indicator of an attribute of a software product or process. There are some reuse related metric models like cost productivity, return on investment, maturity assessment, failure modes and reusability assessment etc [20]. For developer who wants to reuse components, reusability is one of the important characteristic. It is necessary to measure the reusability of components in order to recognize the reuse of components effectively. So classes must be developed as Reusable to effectively reuse them later. Developers should be trained or facilitated to use reusable components e.g. classes because it is hard to understand the structure of classes developed by others [24]. If developers do not have any prior knowledge about the coupling of classes they want to reuse, then they need to spend few time to understand the association pattern of classes. So there is a need to develop some mechanism that helps to know what combination of classes to reuse. By viewing class association rules and patterns, developer can predict required set of classes and can avoid unnecessary, partial class reuse. So for reuse, issues like maintaining class code repository, deciding what group of classes should be incorporated into repository & their association patterns and identifying exact set of classes to reuse, need to be addressed. It will reduce some reuse efforts. To discover the class association rules data mining can be used. By using data mining technology, one can find frequently used classes and their coupling pattern in a particular java application. 1.1 Data Mining and Its Usage in Reusability Data mining is the process of extracting new and useful knowledge from large amount of data. Mining is widely used to solve many business problems such as customer profiling, customer behavior modeling, product recommendation, fraud detection etc [25]. Data mining techniques can be used to analyze software engineering data to better understand the software and assist software engineering tasks. It also helps in programming, defect detection, testing, debugging, maintenance etc. In component reuse, mining helps in numerous ways such as to decide which components we should reuse, what is the right way to reuse, which components may often be reused in combinations etc [25]. The general approach of mining software engineering data consists of following steps:
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a) b) c) d)
Identify the software engineering problem to be solved by mining Identify and study data source Extract & preprocess data Mine the data e.g discover association rules
Due to the popularity of open source concept large amount of source code of classes is available on Internet as software repositories. Some also exists in large software companies where developer in one group may reuse classes written by other groups. For this reason, it is desirable to have mining tools that tell explicitly the class association patterns. Finding associations provides a distinct advantage in highly reusable environment. By searching for class patterns with high probability of repetitions we can correlate one set of classes with other set of classes. Also class associations will help developer to know which classes are likely to be reused together. The process of selecting required set of classes to reuse is complicated and requires some fundamental knowledge about the class structure, relationship or interaction with other classes. For this either software developer learns the reuse pattern of classes by their continuous experience or reading documentations/ manuals or by browsing such mined class association rules. The latter being feasible practically. In this paper, we explore market basket analysis technique to mine class association rules (CAR’s) from vast collection of class coupling data for particular project/program. This can be helpful in reusing classes by capturing association rules between classes. By querying or browsing such association rules a developer can discover patterns for reusing classes. For this purpose firstly dynamic analysis of java application is done using UML diagrams to collect class import coupling data. Then in second step, these collected data are represented as Class Set & Binary Class Vector. Then finally in the third step market basket analysis (apriori) technique is applied on Class Set representation to find frequently used classes and association rules between them. Further Class Vector representation is used to measure cosine similarity between classes. The measured values are analyzed to compare import coupling pattern between classes. The rest of the paper is organized as follows. Section 2 discusses the related works. Section 3 describes the proposed methodology to mine class association rules and class coupling behavior. Section 4 shows example case study to illustrate our approach .Section 5 presents results and discussion. Finally, Section 6 concludes this paper.
2 Related Works For object-oriented development paradigm, class coupling has been used as an important parameter effecting reusability. Li et.al. [19], Yacoup et. al [18] , Arisholm et. al.[2] proposed some measures for coupling. Efforts have been made by the researchers to measure reusability through coupling and cohesion of components [5]. Gui et al [6,7] and Choi et al [4] provided some reusability measures based on coupling and cohesion. ISA [8] methodology has been proposed to identify data cohesive subsystems. Gui et al [10] proposed a new static measure of coupling to assess and rank the reusability of java components. Arisholm et. al.[2] have provided a method for identifying import coupled classes with each class at design time using UML
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diagrams. Data mining is focused on developing efficient techniques to extract relevant information from very large volumes of data that may be exploited, for example, in decision making, to improve software reliability and productivity [9]. Association rule discovery from large databases is an important data mining task. Few algorithms have been proposed for mining the association rules like market basket analysis [1],apriori[11],Ranked Multilabel Rule (RMR) [12], CAR[13], CMAR [14],ARMC[13]. Michail[28] considered the problem of discovering association rules that identify library components that are often reused in combination in the ET++ application framework . Yin et al[15] proposed a Classification approach CPAR based on Predictive Association Rules, which combines the advantages of both associative classification and traditional rule-based classification. Cosine similarity (between two vectors) and Jaccard similarity coefficient are often used to compare documents in text mining[30]. We find the association mining approach proposed by Agrawal et al[1,11,3] and cosine similarity measure very simple and go well with our idea. So to predict class reusability pattern of a particular java application, we are using cosine similarity measure and association mining approach.
3 Proposed Methodology The concepts of data mining have been used to discover patterns of reusable classes in a particular application. These patterns are further helpful in reusing the classes. Association rules between classes and class coupling behaviour are used to identify the class reusability patterns. For this purpose, association mining algorithm [1, 11] is used to mine class association rules (CAR) from class import coupling data. To know the class coupling behaviour the cosine similarity measure can be applied on class import coupling data. Our approach to mine class association rules and class coupling behavior consists of three steps: 1. 2. 3.
Collection of Class import coupling data through UML. Representation of Collected Data. Mining of Class Association Rules (CAR) & Prediction of class import coupling behavior. The steps are described in section 3.1 to 3.3. 3.1 Collection of Class Import Coupling Data through UML Dynamic analysis of a program is a precondition for finding the association rules between classes. Dynamic analysis of programs can be done through UML diagrams [27]. Significant advantages of using UML are its language independence and computation of dynamic metrics based on early design artifacts. Erik Arisholm[2] referred UML models to describe dynamic coupling measures as a way to collect for each class its import coupled classes. They used following formula for calculating class import coupling IC_OC (Ci). I C _ O C ( c 1)
{ ( m 1 , c 1 , c 2 ) | ( ( o 1 , c 1 ) R o c ) ( ( o 1 , c 2 ) R o c | N ) c 1 z c 2 ( o 1 , m 1 | o 2 , m 2 ) M E }
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IC_OC (Ci) counts the number of distinct classes that a method in a given object uses. This formula can be used to measure dependency of one class to other classes in terms of its import coupling. 3.2 Representation of Collected Data Data thus collected in step one should be represented in some suitable intermediate representation, so that mining algorithm can be applied easily to find the class associations. In this we propose to represent data in two forms: 3.2.1 Class Set Representation Class import coupling data of each class can be represented by class set representation. IC_Class_Set represents classes coupled (import) with a class. For example , Let C= {C1,C2,C3,C4,C5} is set of classes of an application , IC_Class_Set(C1)= {C1,C3,C4 } means that class C1 is coupled(import) with classes C3 and C4.Class C1 itself is included in its import coupled class set to have complete set of classes used . For an application, the collection of IC_Class_Set of all classes is called as IC_SET (application). 3.2.2 Class Vector Representation Class vector of a class also represents the import coupled classes with a given class but it is in vector form and is w.r.t. all classes of the application. Suppose C is the ordered set of classed in a particular application , then for a class Ci , class vector is represented as C_V(Ci)=[1,0,1,1,0] . Here 1 at place j indicates that Ci class is coupled (import) with class j and 0 at place k indicates no coupling (import) of class Ci with class Ck . From above two representations, IC_SET (application) is used to mine class association rules through apriori approach and C_V(Ci) is used for measuring class coupling behavior by cosine similarity measure. 3.3 Mining of Class Association Rules and Prediction of Class Import Coupling Behavior 3.3.1 Mining Class Association Rules To mine class association rules for java programs basic apriori approach proposed by Agrawal et al[1,11,3] has been used. They have used the apriori approach (Market basket analysis) to analyze marketing transaction data for determining which products customers purchase together. The concept of support and confidence has been used to find out association rules for the purchased products. The support of an itemset is defined as the proportion of transactions in the transaction data set which contain the itemset and confidence of a rule X→Y is defined as an estimate of the probability P(Y | X), the probability of finding the RHS of the rule in transactions under the condition that these transactions also contain the LHS[29].Based on that we are considering a collection of class import coupling data for an application i.e. IC_SET (application) to find the set of Class Association Rules (CAR). Association rules are required to satisfy a user-specified minimum support and minimum confidence at the same time. Association rule generation is usually split up into two separate phases. In First phase, minimum support min_sup is applied to mine all frequent classes called as Frequent Class Combination Set (FCCS). The process to find FCCS is as follows:
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1: i=1 2: Create candidate class set CSi having all classes and their Support. (The Support for a class Ci is the frequency of occurrence of that class in IC_SET (application). 3: Create large class set Li by eliminating Class Set from CSi having Support sup<min_sup 4: Repeat 5: i=i+1 6: Create candidate class set CSi having Cartesian product of sets in Li-1 and calculate their support from IC_SET (application). 7: Create Large set Li by eliminating Class Set from CSi having sup <min_sup 8: Until (no scope to built large class set) So Class Set in CSi give frequent Class Combination Set (FCCS).After this in Second phase, FCCS and minimum confidence min_conf constraint are used to form CAR. The support and confidence values for each pair of classes in FCCS are calculated using below mentioned formulas 1 &2 [1,11,3]: support(Ci → Cj) =
confidence(Ci → Cj) =
no of tuples containing both Ci & C j total no of tuples no of tuples containing both Ci & C j no of tuples containing Ci
(1)
(2)
So a rule Ci → Cj holds in the IC_SET with confidence cf% if cf% tuples of ICOUP_SET contain Ci also contain Cj . The rule Ci → Cj has support sp% if sp% tuples of IC_SET contain Ci U Cj. So the association rule Ci → Cj on IC_SET is worth considering if set of tuples in IC_SET holds this rule with sup (Ci → Cj) >min_sup & conf (Ci → Cj) >min_conf. As a result of this, the distinction can be made between classes that are being used together often and classes that are not. These rules suggest which classes can be reused as a group and repository designer can then use these rules to put frequently used classes in the repository. 3.3.2 Measuring Class Coupling Behavior The class vector representation C_V(C) of classes is used to compute Cosine Similarity between classes on a scale of [0, 1]. The Cosine Similarity [16] of two class vectors C_V (Ci) & C_V (Cj) is defined as:
Cos_Sim(Ci,Cj) =
Ci .C j Ci C j
The value 1 means that the coupling pattern of classes Ci & Cj is identical, 0 means completely different[16,26].So using Cosine similarity one can analyze which classes have similar, nearly similar or completely different coupling pattern. In next section, we demonstrate our approach of Mining Class Association Rules and Measuring Class Coupling Behavior of a sample application.
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4 Example Case Study We are using a small example to illustrate our approach for Mining Class Association Rules and measuring class coupling behavior. We consider example java application MYSHAPES. Application MYSHAPES has class set C(MYSHAPES)= { myshape, circle, square, shape }. In the first step import coupling data are collected for MYSHAPES using UML approach. In second step these collected values are represented asIC_SET(MYSHAPES). The next sections 4.1 & 4.2 show the third step of our approach. We assume min_sup>25%. 4.1 Mining Class Association Rules As a first part of third step of methodology, the method given in section 3.3.1 is applied on IC_SET (MYSHAPES) to find Frequent Class Combination Set (FCCS) and is shown in figure 1.Then the output FCCS is used to form Class Association Rules(CAR) having min_conf ≥90%. Table 1 lists the CAR with confidence more than 90% for the application MYSHAPES. 4.2 Measuring Class Coupling Behavior To find out the behavior of each class in terms of class coupling pattern we use class vector representation (C_V) of MYSHAPES (table 2) and compute Cosine similarity measure between classes as mentioned in section 3.3.2. Following table 3 shows the computed Cos_Sim(Class1,Class2).
5 Results and Discussion We can measure the reusability pattern of classes by analyzing their association rules and import coupling patterns. CAR’s of application MYSHAPES (figure 2) suggest whenever a class on the left hand side of rule is to be reused, there is strong probability with 100% confidence that classes on right side of the rule will also be reused. From figure 2 it is observed that whenever class square is to be reused class shape will also be reused. From figure 2 it is observed that the cosine similarity between classes circle and shape is 1 and myshape and square is.71. It suggests that import coupling behavior of classes circle & shape are exactly similar i.e. they are always used together while classes myshape and square are sometimes import coupled to some common classes. Our study shows that FCCS, CAR’s and Cos_sim between classes can be helpful for a repository designer/user to predict which classes are required to be reused in combination and what is the coupling pattern of classes. The effectiveness of Class association rule is dependent on type of coupling attributes used to know import coupling between classes, ways to represent coupling data and accuracy of association mining algorithm applied to it.
Mining Class Association Rules from Dynamic Class Coupling Data
IC_SET (MYSHAPES)
CS1
Class_Id
IC_Class_Set
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sup
myshapes
myshape, circle, square, shape
myshapes
01
circle
circle, square, shape
circle
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square , shape
shape
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shape
circle , square , shape
L1
CS2 Class_Set
sup
{ circle ,square }
03
{ circle ,shape }
03
{ square shape }
03
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sup
circle
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square
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CS3 L2 Class_Set
sup
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{ circle ,shape }
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{ square shape }
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Frequent Class Combination Set(FCCS) { circle ,square, shape }
Fig. 1. Frequent Class Combination Mining Steps
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Application
Frequent Class Combinations Set (FCCS)
MYSHAPES
{circle , square, shape}
CAR
1.
circleĺsquare
(sup=75% , conf=100%)
2.
circleĺshape
(sup=75% , conf=100%)
3.
shapeĺsquare
(sup=100% , conf=100%)
4.
squareĺshape
(sup=100% , conf=100%)
5.
circleĺshape, square
Table 2. Class vector of MYSHAPES
(sup=75% , conf=100%)
Table 3. Cosine similarity between of MYSHAPES
myshape
circle
square
shape
C_V(myshape)
1
1
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1
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0
1
1
1
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Scale
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.87
Cos_Sim(myshape,square)
.71
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.87
Cos_Sim(circle,square)
.81
Cos_Sim(circle,shape)
1
Cos_Sim(square,shape)
.87
1 0.8
squareĺshape
SCALE
circleĺshape, square
0.6 0.4
CAR’s
0.2
shapeĺsquare
0
circleĺshape circleĺsquare 0% 20% 40% 60% 80%100% SUPPORT
Fig. 2. CAR and their Support
Cos Sim(Class1,Class2)
Fig. 3. Cosine Similarities between Classes
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6 Conclusions In this paper, an attempt has been made to determine class reusability pattern from dynamically collected class import coupling data of java application. Our initial study indicates that basic technique of market basket analysis (apriori) and cosine similarity measure can be constructive to find out class association rules (CAR’s) and class import coupling behaviour. Currently, we have deduced CAR’s for a sample java application. However, the approach can also be applied on larger java applications. Moreover, other association mining and clustering algorithms can be explored to apply on class coupling data for finding class reusability patterns.
References 1. Agrawal, R., Imielinski, T., Swami, A.: Mining Association Rules between Sets of Items in Large Databases. In: ACM, SIGMOD, pp. 207–216 (1993) 2. Arisholm, E.: Dynamic Coupling Measurement for Object-Oriented Software. IEEE Transactions on Software Engineering 30(8), 491–506 (2004) 3. Negandhi, G.: Apriori Algorithm Review for Finals, http://www.cs.sjsu.edu 4. Choi, M., Lee, J.: A Dynamic Coupling for Reusable and Efficient Software System. In: 5th IEEE International Conference on Software Engineering Research, Management and Applications, pp. 720–726 (2007) 5. Mitchell, A., Power, F.: Using Object Level Run Time Metrics to Study Coupling Between Objects. In: ACM Symposium on Applied Computing, pp. 1456–1462 (2005) 6. Gui, G., Scott, P.D.: Coupling and Cohesion Measures for Evaluation of Component Reusability. In: ACM International Workshop on Mining Software Repository, pp. 18–21 (2006) 7. Taha, W., Crosby, S., Swadi, K.: A New Approach to Data Mining for Software Design. In: 3rd International Conference on Computer Science, Software Engineering, Information Technology, e-Business, and Applications (2004) 8. Montes, C., Carver, D.L.: Identification of Data Cohesive Subsystems Using Data Mining Techniques. In: IEEE International Conference on Software Maintenance, pp. 16–23 (1998) 9. Xie, T., Acharya, M., Thummalapenta, S., Taneja, K.: Improving Software Reliability and Productivity via Mining Program Source Code. In: IEEE International Symposium on Parallel and Distributed Processing, pp. 1–5 (2008) 10. Gui, G., Scott, P.D.: Ranking reusability of software components using coupling metrics. Elsevier Journal of Systems and Software 80, 1450–1459 (2007) 11. Agrawal, R., Srikant, R.: Fast Algorithms for Mining Association Rules. In: 20th International Conference on Very Large Data Bases, pp. 487–499 (1994) 12. Thabtah, F.A., Cowling, P.I.: A greedy classification algorithm based on association rule. Elsevier journal of Applied Soft Computing 07, 1102–1111 (2007) 13. Zemirline, A., Lecornu, L., Solaiman, B., Ech-Cherif, A.: An Efficient Association Rule Mining Algorithm for Classification. In: Rutkowski, L., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2008. LNCS (LNAI), vol. 5097, pp. 717–728. Springer, Heidelberg (2008) 14. Li, W., Han, J., Pei, J.: CMAR: Accurate and Efficient Classification Based on Multiple Class-Association Rules. In: International Conference on Data Mining, pp. 369–376 (2001)
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15. Yin, X., Han, J.: CPAR: Classification based on Predictive Association Rules 16. Cosine Similarity Measure, http://www.appliedsoftwaredesign.com/cosineSimilarityCalculat or.php 17. Lee, Y., Chang, K.H.: Reusability and. Maintainability Metrics for Object-Oriented Software. In: ACM 38th Annual Southeast Regional Conference, pp. 88–94 (2000) 18. Yacoub, S., Ammar, H., Robinson, T.: Dynamic Metrics for Object-Oriented Designs. In: IEEE 6th International Symposium Software Metrics, pp. 50–61 (1999) 19. Li, W., Henry, S.: Object Oriented Metrics that predict Maintainability. In: Technical Repot, Virginia Polytechnic Institute and State University (1993) 20. Shiva, S.J., Shala, L.A.: Software Reuse: Research and Practice. In: Proceedings of the IEEE International Conference on Information Technology, pp. 603–609 (2007) 21. Bhatia, P.K., Mann, R.: An Approach to Measure Software Reusability of OO Design. In: Proceedings of the 2nd National Conference on Challenges & Opportunities in Information Technology, pp. 26–30 (2008) 22. Eickhoff, F., Ellis, J., Demurjian, S., Needham, D.: A Reuse Definition, Assessment, and Analysis Framework for UML. In: International Conference on Software Engineering (2003), http://www.engr.uconn.edu/~steve/Cse298300/eickhofficse2003s ubmit.pdf 23. Caldiera, G., Basili, V.R.: Identifying and Qualifying Reusable Software Components. IEEE Journal of Computer 24(2), 61–70 (1991) 24. Henry, S., Lattanzi, M.: Measurement of Software Maintainability and Reusability in the Object Oriented Paradigm. In: ACM Technical Report (1994) 25. Xie, T., Pei, J.: Data mining for Software Engineering, http://ase.csc.ncsu.edu/dmse/dmse.pdf 26. Cosine Similarity, http://en.wikipedia.org/wiki/Cosine_similarity 27. Gupta, V., Chhabra, J.K.: Measurement of Dynamic Metrics Using Dynamic Analysis of Programs. In: Proceedings of the Applied Computing Conference, pp. 81–86 (2008) 28. Michail, A.: Data Mining Library Reuse Patterns in User-Selected Applications. In: 14th IEEE International Conference on Automated Software Engineering, pp. 24–33 (1999) 29. Associations Rule, http://en.wikipedia.org/wiki/Association_rule_learning 30. Jaccard Index, http://en.wikipedia.org/wiki/Jaccard_index
An Algorithm of Constraint Frequent Neighboring Class Sets Mining Based on Separating Support Items Gang Fang, Jiang Xiong, Hong Ying, and Yong-jian Zhao College of Mathematics and Computer Science, Chongqing Three Gorges University Chongqing 404000, P.R. China [email protected], [email protected], [email protected], [email protected]
Abstract. For the reasons that present constraint frequent neighboring class sets mining algorithms need generate candidate frequent neighboring class sets and have a lot of repeated computing, and so this paper proposes an algorithm of constraint frequent neighboring class sets mining based on separating support items, which is suitable for mining frequent neighboring class sets with constraint class set in large spatial database. The algorithm uses the method of separating support items to gain support of neighboring class sets, and uses up search to extract frequent neighboring class sets with constraint class set. In the course of mining frequent neighboring class sets, the algorithm only need scan once database, and it need not generate candidate frequent neighboring class sets with constraint class set. By these methods the algorithm reduces more repeated computing to improve mining efficiency. The result of experiment indicates that the algorithm is faster and more efficient than present mining algorithms when extracting frequent neighboring class sets with constraint class set in large spatial database. Keywords: neighboring class set; constraint class set; separating support items; up search; spatial data mining.
class set with constraint class set, and MFNCSWCC written in [6] need generate many candidates and have a lot of repeated computing when it uses iterative search to generate frequent neighboring class set with constraint class set. Hence, this paper proposes an algorithm of constraint frequent neighboring class sets mining based on separating support items, denoted by CMBSSI, which need not generate candidate when mining frequent neighboring class sets with constraint class set.
2 Definition and Problem Description Spatial data set is made up of each spatial object in spatial domain. We use this data structure as
