Incorporation of human decision making preference into evolutionary multi objective optimization

Incorporating Decision Maker Preference in Multi-Objective Evolutionary Algorithms Lily Rachmawati B Eng Hons., NUS A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2008 Acknowledgements There are many people whom I wish to thank for the support they have rendered me throughout the course of the Doctoral program I gratefully acknowledge the financial support National University of Singapore during the course of the program I sincerely thank my supervisor, Dr Dipti Srinivasan, for the suggestions and encouragements that helped shape the research direction I also thank the many anonymous referees, whose valuable feedback have contributed greatly to the research work accomplished I am also thankful to the family and friends that have supported me in direct and indirect means from the beginning of the course I would like to extend my especial gratitude to my parents, sister and brother, for the countless advice and encouragement they have given Many thanks as well to Ms Jessica Kusuma, Ms Riyanti Teresa, Mr Lisman Komaladi, Mr Arief Adhitya, Mr Steven Halim, and many others who have offered prayers, encouragement and support during the writing and revision of this thesis Finally, and most importantly, I would like to thank the almighty God for His enduring grace and love i Contents Introduction 1.1 Motivation of Research 1.2 Objectives and Scope of Research 1.3 Contribution of the Research 1.4 Outline of the Thesis 11 Concepts and Terminology 12 2.1 Multi-objective Optimization Problem 12 2.2 Pareto Optimality 13 2.3 Pareto Dominance 14 2.4 Decision Maker Preference 16 2.5 Conclusion 17 Decision Making Preference in Multi Objective Optimization 18 3.1 Features of Human Decision Making 20 3.2 Desirable Properties of Preference-based Evolutionary Search 23 3.3 Approaches based on the Relative Importance of Objectives 28 3.3.1 MAUT-based Algorithms 29 3.3.2 Lexicographic Ordering 37 3.3.3 Outranking-based Algorithms 38 3.4 Approaches based on a Goal/Reference Vector 41 3.5 Approaches based on Optimality of Trade Off 45 ii 3.6 50 3.6.1 Population-based Indicator 50 3.6.2 3.7 Other Approaches Fuzzy Logic 50 51 3.7.1 a priori and Interactive Preference Incorporation 52 3.7.2 3.8 Issues in Preference Incorporation into Evolutionary Algorithms Coevolutionary and Fitness-based Preference Integration 56 Conclusion 58 Fitness Functions in Multi objective Evolutionary Algorithms 60 4.1 Deriving Total Order from Partial Order 62 4.2 Aggregative Fitness Functions 66 4.2.1 Random and Dynamic Weighted Aggregation 66 4.2.2 Maximin Function 67 Dominance-based ranking with niching 68 4.3.1 NSGA-II 70 4.3.2 SPEA2 71 4.3.3 PAES 71 4.3.4 NPGA 72 4.3.5 Distributed Pairwise Comparison 73 4.3.6 SEAMO 74 4.3.7 ε-MOEAs 74 4.3.8 Steady State Replacement 76 4.3 4.4 Conclusion Imprecise Goal Vectors 78 79 5.1 Fuzzy Sets 80 5.2 Preference Representation 81 5.2.1 Goal Vectors 82 5.2.2 Degree of Imprecision 83 5.2.3 Invalid Imprecise Goal Vectors 83 Preference Elicitation 84 5.3 iii 5.4 Preference Integration 85 5.5 Empirical Evaluation 89 5.6 Conclusion 95 Knee Solutions 97 6.1 Preference Representation 99 6.2 Weighted Sum Niching 100 6.3 Parallel Local Weighted-Sum Based Optimization 103 6.3.1 Stage-One Optimization: Information Gathering 104 6.3.2 Estimation of potential knee solutions 105 6.3.3 Stage-Two Optimization: Integration of Preference into MOEA 6.3.4 108 Interactive Preference Articulation 110 6.4 Computational complexity 111 6.5 Empirical Evaluation 113 6.5.1 Performance Metric 114 6.5.2 Simulation Details 118 6.5.3 Covergence, Accuracy and Diversity: A Comparative Analysis 124 6.5.4 Weighted-sum Niching and Parallel Local Weighted-Sum Optimization: Further Remarks 130 6.6 Conclusion 134 Relative Importance of Objectives 7.1 137 Preference Representation 139 7.1.1 Preference structure 139 7.1.2 Mathematical Interpretation 141 7.2 Elicitation of Preference 146 7.3 Integration of Preference information into MOEA 149 7.4 Empirical Evaluation 151 7.4.1 Simulation Details 152 iv 7.4.2 7.5 Result and Discussion 157 Conclusion 161 Conclusion 8.1 163 Contributions 163 8.1.1 8.1.2 Optimality of Trade-Off 164 8.1.3 8.2 Imprecise Goal Vector 164 Relative Importance of Objectives 165 Future Work 166 A Conditions of MAUT 168 B Preference Functions 169 C Steady State MOEA Results 171 D Elicitation Algorithm 180 v Abstract The objective of the majority of work in Evolutionary Multi Objective Optimization (EMOO) is to develop Multi-Objective Evolutionary Algorithms (MOEA) able to find a well-distributed approximation of the Pareto optimal front An evenly distributed set of non dominated solutions equips the decision maker with the trade-off behavior associated with the problem at hand so that (s)he could select from the set the most suitable solutions The aposteriori manual application of human preference to the selection of solutions is rendered impractical in higher dimensional problems as a very large number of solutions would be required to approximate the entire extent of the Pareto front meaningfully An apriori/interactive incorporation of preference information into a MOEA averts the problem by concentrating on a subset of the Pareto front The approach avails the decision maker with a higher resolution in the region of interest in the objective space and aids the progression of the elite population towards the true optima Human preference in multi-objective decision making contains uncertainties and anomalies that are to be taken into account in a formal model of preference The uniqueness of the evolutionary computation approach renders the direct adoption of modelling and implementation techniques developed for classical optimization approaches unsuitable This thesis documents research effort into the articulation and incorporation of preference information into EMOO Models of preference formulated in terms of the importance ranking of objectives, an imprecisely specified reference vector, and objective trade off and their implementations in MOEAs are reviewed Three preference incorporation schemes encompassing the representation, elicitation and implementation of preference information are also proposed The approaches are designed for easy adoption into major state-of-the-art general-purpose MOEAs The first guides the population of solutions towards an imprecisely specified goal vector The second directs the search to regions of optimum trade-off in the Pareto front The third incorporates importance ranking of objective functions into MOEAs The proposed mathematical model of preference caters to incomparability and features a functional correspondence between explicated importance ranking of objectives and a specific subset of the Pareto front The preference elicitation algorithm devised facilitates scalable explication of preference The preference incorporation techniques are validated in an empirical study that involves difficult test problems and comparison with similar preference-based algorithms as well as baseline MOEAs Preferencebased performance metrics are also proposed where applicable to measure the concord between obtained solutions and explicated preference ii List of Figures 2.1 Ideal and Nadir objective vectors 14 3.1 Selection of best solution 25 3.2 Guided dominance (Shaded: area dominated by Xi ) 31 3.3 Dotted: Region dominated by solution, Stripes: Region nondominated with respect to solution 3.4 33 Dotted: Region dominated by solution, Stripes: Region nondominated with respect to solution 34 3.5 Knee Solution 45 3.6 Knee solution: a bulge of the Pareto front 47 3.7 Knee in Concave Region 49 4.1 Block Diagram: Evolutionary algorithm 61 4.2 Pareto dominance imposes partial order on the objective space 64 4.3 Bi-objective example with Maximin Fitness (given in brackets) 68 5.1 Case 1: Vector T lying in dominating region 87 5.2 Case : Vector T lying in non-dominated region 87 5.3 Case : Vector T lying in dominated region 88 5.4 Pareto front of problem ZDT1 91 5.5 Pareto front of problem ZDT2 91 5.6 Pareto front of problem ZDT3 92 5.7 Results for problem ZDT1 94 i 5.8 Results for problem ZDT2 95 5.9 Results for problem ZDT3 95 5.10 Results for problem ZDT4 96 6.1 Knee Region 101 6.2 Multiple Knee Regions 102 6.3 Block Diagram 104 6.4 Test Problem DEB2DK-2 with K = to 121 6.5 Captures of the population at different instants in stage (DO2DK, K=2, s=1) 131 6.6 Result obtained for test problem DO2DK (K=1,s=0) with parallel local optimization (δ = 0.1, 0.2, 0.5) 132 6.7 Result obtained for test problem DO2DK (K=2,s=1) with proposed approach (δ = 0.1, 0.2, 0.5) 133 6.8 Result obtained for test problem DO2DK (K=4,s=1) with proposed approach (δ = 0.1, 0.2, 0.5) 133 6.9 Result obtained for test problem DO2DK (K=2,s=1) with weighted sum niching in [92] 134 6.10 Result obtained for test problem DO2DK (K=4,s=1) with weighted sum niching in [92] 134 7.1 Left: f1 P f2 , Middle: f1 If2 , Right: f2 P f1 142 7.2 List and graphs of c1 and c2 143 7.3 Example 1: Three dimensional view (target subset highlighted) 144 7.4 Example 1: View from f1 -f2 plane (target subset highlighted) 144 7.5 Example 1: View from f1 -f3 plane (target subset highlighted) 145 7.6 Example 1: View from f2 -f3 plane (target subset highlighted) 145 7.7 List and graphs of c1 , c2 and c3 146 7.8 Pareto front of problem DTLZ2 154 7.9 Pareto front of problem DTLZ5 157 7.10 Generational Distance in Six Objective Problems 160 ii prefmat[i][index2] = prefmat[i][index1]; if (relation==1)/*case: index1 P index2, insert below index1*/ /*if maximum then just insert*/ maxi = -1; j=1; for (j=1;j¡(size+1);j++) if (prefmat[i][index1]¡prefmat[i][j]) maxi = -100; if (prefmat[i][j]==(prefmat[i][index1]+1)) maxi = j; if (maxi==-1) prefmat[i][index2]=prefmat[i][index1]+1; else /* not so easy case */ dummy = infix(prefmat,size, i, maxi,index2, 1); if ((all).chains2[i] == 1) /*insert obj index1 into the chains*/ if (relation==2)/*case: indifference*/ prefmat[i][index1] = prefmat[i][index2]; if (relation==1)/*case: preferred*/ if(prefmat[i][index2]==1) /*easy case, insert at head of chain*/ for(j=1;j¡(size+1);j++) if (prefmat[i][j]¿0) prefmat[i][j]=prefmat[i][j]+1; prefmat[i][index1]=1; else /* infix insert, not so easy*/ j=1; while(prefmat[i][j]!=(prefmat[i][index2]-1) AND j¡(size+1)) j+=1; dummy = infix(prefmat, size, i,j,index1,2); return 1; void removeduplicate(int** prefmat, int size) int i,j,k, count[size]; for(i=0;i¡size;i++) if (prefmat[i][0]¿0) for(k=0;k¡size;k++) if (prefmat[k][0]¿0 AND k!=i) j=1; 183 while ((prefmat[i][j]-prefmat[k][j]) ¿=0 AND (j¡size+1)) j+=1; if (j==(size+1)) for (j=0;j¡(size+1);j++) prefmat[k][j]=0; int main () int obj1,obj2,i,j,nobj, quit, rel, counter; char binrel; int **S; FILE *fptinputs; FILE *fptcomp; quit = ’n’; fptinputs = fopen(”userinputs.out”,”w”); fptcomp = fopen(”completepref.out”,”w”); fprintf(fptinputs,”# This file contains the user-input preference”); fprintf(fptcomp,”# This file contains the complete preference information”); printf(” No of objective functions : ”); scanf (” while (nobj ¡2) printf(” You entered printf(” No of objective functions : ”); scanf (” /* initialize matrix S*/ S = (int**) malloc(nobj*sizeof(int*)); for (i=0;i¡nobj;i++) S[i] = malloc((nobj+1)*sizeof(int)); for (j=0;j¡(nobj+1);j++) if ((i+1)==j —— j==0) S[i][j]=1; else S[i][j] = 0; printf(” Start entering user preference information in the format: m R n ”); printf(” m, n : integers indices of objective functions defined from to printf(” R : the binary relation between the two objectives m and n ”); printf(” Preference ”); printf(” Indifference ”); quit = 2; counter = 1; while (quit!=1) /*input binary relations */ 184 printf(” Enter index of the first objectives: ”); scanf(” while (obj1¡1 —— obj1¿nobj) printf(” Value of objective index should be between and printf(” Enter index of the first objectives: ”); scanf(” printf(” Enter index of the second objectives: ”); scanf(” while (obj2¡1 —— obj2¿nobj) printf(” Value of objective index should be between and printf(” Enter index of the second objectives: ”); scanf(” printf(” Enter the binary relation: ”); scanf(” /*analyze if the input is valid, insert into pref matrix*/ i = insert(S,nobj, obj1, obj2, rel); /* remove all substrings */ removeduplicate(S, nobj); /* interpret the relation */ if (rel==1) binrel = ’P’; else if (rel==2) binrel = ’I’; /* write into the input list*/ fprintf(fptinputs,”input # /*print out the partial ordering list */ for (i=0;i¡nobj;i++) if (S[i][0]==1)/*active chain, print out content*/ printf(””); for (j=1;j¡(nobj+1);j++) printf(” /*further inputs*/ printf(” Quit (1 yes /2 no) ?”); scanf(” counter += 1; 185 Bibliography [1] Arrow, J., ”Social Choice and Individual Values”, Cowles Commission Monograph, vol 12, Wiley, New York, 1951 [2] Balling, R., ”The Maximin Fitness Function; Multiobjective City and Regional Planning,” in C.M Fonseca et al (Eds), Proceedings of EMO 2003, Lecture Notes in Computer Science, Springer Verlag, pp 1-15, 2003 [3] Bandyopadhyay S., Saha S., Maulik U., and Deb K, ”A Simulated AnnealingBased Multiobjective Optimization Algorithm: AMOSA”, in IEEE Transactions on Evolutionary Computation, pp 269-283, 2008 [4] Belegundu, A.D et al ”Multi-objective Optimization of Laminated Ceramic Composites Using Genetic Algorithms,” in Proceedings of the 5th AIAA/NASA/USAF/ISSMO Symposium on Multidiscipinary Analysis and Optimization, Washington, D.C AIAA, pp 1015-1022, 1994 [5] Benayoun R., de Montgolfier J., Tergny J and Laritchev O., ”Linear Programming with Multiple Objective Functions: Step Method (STEM)”, in Mathematical Programming, vol 1, no 3, pp 366-375, 1971 [6] Berry A and Vamplew P., ”The Combative Accretion Model - Multiobjective Optimization Without Explicit Pareto Ranking,” in C A Coello Coello et al (Eds), Proceedings of EMO 2005, Berlin: Springer-Verlag, pp 77-91, 2005 [7] Bestle D., and Eberhard P., ”Dynamic System Design via Multicriteria Optimizationf”, in Multiple Criteria Decision Making: Proceedings of the Twelfth International Conference, Lecture Notes in Economics and Mathematical Systems, Hagen (Germany), vol 448, Springer-Verlag, Berlin, Heidelberg, pp 467-478, 1997 [8] Branke J., Kauβlerand J., Schmeck H., ”Guidance in evolutionary multi-objective optimization”, in Advances in Engineering Software, no 32, pp 499-507, 2001 186 [9] Branke J., Deb K., Dierolf H and Osswald M., ”Finding Knees in Multi-objective Optimization,” in The Eighth Conference on Parallel Problem Solving from Nature VIII, pp 722-731, 2004 [10] Branke, J and Deb, K., ”Integrating User Preferences into Evolutionary MultiObjective Optimization”, KanGAL Report No 2004004, May 2004 [11] Brans, J.P., Mareschal, B., and Vincke Ph., ”A New Family of outranking methods in culticriteria analysis”, in Brans, J (Ed.), Operational Research’84, Elsevier Science Publishers B.V., pp 408-421, 1984 [12] T.M Chan, K.F Man, K.S Tang and S Kwong, ”A Jumping Gene Paradigm for Evolutionary Multiobjective Optimization”, in IEEE Transactions on Evolutionary Computation, vol 12, no 2, pp 143-159, 2008 [13] Chankong V and Haimes Y Y, ”The Interactive Surrogate Worth Trade-off Method for Multiobjective Decision Making”, S Zionts (Ed.), Multiple Criteria Problem Solving,Lecture Notes in Economics and Mathematical Systems, vol 155, Springer-Verlag, Berlin, Heidelberg, pp 42-67, 1978 [14] Chankong V and Haimes Y Y, Multiobjective Dceision Making Theory and Methodology, Elsevier Science Publishing Co., Inc., New York, 1983 [15] Coelho R.F., Bersini H., and Bouillard P, ”Parametrical mechanical design with constraints and preferences: application to a purge valve”, in Computer Methods in Applied Mechanics and Engineering, vol 192, no 39, pp 4355-4378, 26 September 2003 [16] Coello C.A.C., ”Handling Preferences in Evolutionary Multiobjective Optimization: A Survey”, in Proc of the 2000 Congress on Evolutionary Computation, 2000 [17] Coello C.A.C., Van Veldhuizen D A and Lamont G.B., Evolutionary algorithms for solving multi-objective problems, New York : Kluwer Academic, 2002 [18] Coello C.A.C., ”20 Years of Evolutionary Multi-Objective Optimization: What Has Been Done and What Remains to be Done”, in Yen, G.Y., and Fogel, D.B., Computational Intelligence: Principles and Practice, pp 73-88, 2006 [19] Corne D W., Jerram N.R , Knowles J.D , and Oates M.J , ”PESA-II: Regionbased Selection in Evolutionary Multiobjective Optimization”, in Proceedings of the Genetic and Evolutionary Computation Conference-GECCO01, Morgan Kaufmann Publishers, pp 283-290, 2001 [20] Cvetkovic D., and Coello C.A.C., ”Human Preferences and their Applications in Evolutionary Multi-Objective Optimization”, unpublished 187 [21] Cvetkovic, D., and Parmee I.C., ”Use of preferences for GA-based multi-objective optimization”, in Banzhaf, W., and Daida J (eds), Proceedings of the Genetic and Evolutionary Computation Conference-GECCO99, San Mateo, CA: Morgan Kaufmann, pp 1504-1509, July 1999 [22] Cvetkovic, D., and Parmee I.C., ”Genetic-algorithm-based multi-objective optimization and conceptual engineering design,” in Proceedings of the 1999 Congress on Evolutionary Computation-CEC99, NJ: IEEE Press, pp 29-36, 1999 [23] Cvetkovic, D., Evolutionary Multi-Objective Decision Support Systems for Conceptual Design, PhD thesis, School of Computing, University of Plymouth, Plymouth, UK, 2000 [24] Cvetkovic, D and Parmee, I.C., ”Preferences and Their Application in Evolutionary Multiobjective Optimization”, in IEEE Transactions on Evolutionary Computation, vol 6, no.1, pp 42-57, February 2002 [25] D Corne, J.D Knowles ”No Free Lunch and Free Leftovers Theorems for Multiobjective Optimisation Problems”, in Proceedings of the Evolutionary MultiObjective Optimization Conference-EMO03, pp 327-341, 2003 [26] Das, I., Dennis J.E., ”Normal-boundary intersection: an new method for generating the Pareto surface in multicriteria optimization problems”, SIAM Journal on Optimization, vol 8, 631-657, 1998 [27] Das, I., ”On characterizing the ”knee” of the Pareto curve based on NormalBoundary Intersection”, in Structural Optimization, vol 18, pp 107-115, 1999 [28] K Deb and S Agrawal, ”Simulated binary crossover for continuous search space,” in Complex Systems, vol 9, pp 115-148, 1995 [29] Deb K ”Solving goal programming problems using multi-objective genetic algorithms”, in Proceedings of the Congress on Evolutionary Computation-CEC99, IEEE, pp 77-84, 1999 [30] Deb, K and Chaudhuri, S., ”I-EMO: An Interactive Evolutionary Multi-Objective Optimization Tool”, Kanpur Genetic Algorithms Laboratory Report Number 2005003 [31] Deb K., Multiobjective Optimization using Evolutionary Algorithm, Chichester, England, John Wiley and Sons, 2001 [32] Deb K., Pratap A., Agarwal S., and Meyarivan T., ”A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II”, in IEEE Transactions on Evolutionary Computation, vol 6, no 2, pp 182-197, April 2002 188 [33] Deb K., ”Multi-objective evolutionary algorithms: Introducing bias among Pareto-optimal solutions,” in Advances in Evolutionary Computing: Theory and Applications, London, Springer-Verlag, pp 263-292, 2003 [34] Deb K., Mohan M., and Mishra S., ”Towards a Quick Computation of WellSpread Pareto-optimal Solutions”, in Proceedings of the Second Evolutionary Multi-Criterion Optimization Conference-EMO03, Lecture Notes in Computer Science, vol 2632, pp 222-236, 2003 [35] Deb K., Mohan M., and Mishra S., ”Evaluating the -Domination Based Multiobjective Evolutionary Algorithm for a Quick Computation of Pareto-Optimal Solutions”, in Evolutionary Computation, vol 13, no 4, , pp 501-525, 2005 [36] Deb K., Thiele L., Laumanns M and Zitzler E., ”Scalable Test Problems for Evolutionary Multiobjective Optimization,” in A Abraham, L Jain and R Goldberg (Eds), Evolutionary Multiobjective Optimization: Theoretical Advances and Applications, Springer, USA, pp 105-145, 2005 [37] Deb K and Sundar J., ”Reference Point Based Multi-Objective Optimization Using Evolutionary Algorithms”, in Proceedings of GECCO 2006, Seattle, Washington, pp 635 - 642, 2006 [38] De Weck, O.L., ”Multi-objective Optimization: History and Promise,” Invited Keynote Paper, GL2-2, The Third China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems, Kanazawa, Japan, October 30-November 2, 2004 [39] Dyer J.S and Sarin R.K., ”Multicriteria Decision Making”, in Holzman, A.G and Dekker M (Eds.), Mathematical Programming for Operations Researchers and Computer Scientists, pp 123-148, 1981 [40] Ericsson, K.A and Kintsch, W., ”Long term working memory”, in Psychological Review, vol 102, pp 211-245, 1995 [41] Fieldsend J.E and Singh S., ”A Multi-Objective Algorithm based upon Particle Swarm Optimisation, an Efficient Data Structure and Turbulence”, in Proceedings of the 2002 U.K Workshop on Computational Intelligence, Birmingham, UK , pages 37-44, 2-4 Sept 2002 [42] Fieldsend J.E., Everson R.M., and Singh S., ”Using unconstrained elite archives for multiobjective optimization”, in IEEE Transactions on Evolutionary Computation, vol 7, no.3, pages 305-323, 2003 189 [43] Fonseca C.M and Fleming P.J, ”Genetic Algorithms for Multiobjective Optimization: Formulation, Discussion and Generalization,” in S Forrest (Ed), Proceedings of the FiBh International Conference on Genetic Algorithms,San Mateo, California, pp 416-423, 1993 [44] Fonseca C.M and Fleming P.J., ”An Overview of Evolutionary Algorithms in Multiobjective Optimization,” in Evolutionary Computation, no vol 1, pp 1-16, 1995 [45] Fonseca C.M and Fleming P.J.,”Multiobjective Optimization and Multiple Constraint Handling with Evolutionary Algorithms-Part I: A Unified Formulation”, in IEEE Transactions on Systems, Man and Cybernetics - Part A: Systems and Humans, vol 28, no 1, pp.26-37, January 1998 [46] Frank M, and Wolfe P., ”An Algorithm for Quadratic Programming, in Naval Research Logistics Quarterly, vol 3, no 1-2 ,pp 95-110, 1956 [47] Gardiner, L.R., and Vanderpooten, D., ”Interactive Multiple Criteria Procedures: Some Reflections”, in Climaco, J (ed), Multicriteria Analysis, Springer Verlag, Heidelberg, pp 290-301, 1997 [48] Goh C.K, and Tan K.C., ”A Competitive-Cooperative Coevolutionary Paradigm for Dynamic Multiobjective Optimization”, to appear in IEEE Transactionson Evolutionary Computation [49] Goldberg D.E., Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, 1989 [50] Geiger M.J., ”The Interactive Pareto Iterated Local Search Metaheuristic and its Application to the Biobjective Portfolio Optimization Problem”, in Proceedings of the 2007 IEEE Symposium on Computational Intelligence in Multicriteria Decision Making, pp 193-199, 2007 [51] Geiger, M.J, Wenger, W., and Habenicht, W., ”Interactive Utility Maximization in Multi-Objective Vechicle Routing Problems: A Decision Maker in the Loop Approach”, in Proceedings of the 2007 IEEE Symposium on Computational Intelligence in Multicriteria Decision Making, pp 178-184, 2007 [52] Geoffrion, A.M, Dyer, J.S and Feinberg, A., ”An Interactive Approach for MultiCriterion Optimization, with an Application to the Operation of an Academic Department”, in Management Science, vol 19, no 4, pp.357-368, 1972 [53] Greenwood, G.W., Hu, X, and D’Ambrosio, J.G, ”Fitness functions for Multiple Objective Optimization Problems: Combining Preferences with Pareto Rankings,” in Foundations of Genetic Algorithms, pp.437-455, 1996 190 [54] Haimes, Y.Y., and Hall, W.A ”Multiobjectives in Water Resources Systems Analysis: The Surrogate Worth Trade-off Method,” in Water Resources Research, vol 10, no 4, pp.615-624, 1974 [55] Haimes, Y.Y., Hall, W.A., and Freedman, H.T., Multi-objective Optimization in Water Resources Systems: The Surrogate Worth Trade-off Method, Elsevier, The Netherlands, 1975 [56] Hallam, N., Blanchfield, P., and Kendall, G., ”Handling diversity in Evolutionary Multiobjective Optimization”, in Proceedings of the Congress on Evolutionary Computation-CEC05, pp 2233 - 2240, 2005 [57] Horn J., Nafpliotis N., and Goldberg D.E., ”A niched Pareto Genetic algorithm for multiobjective optimization,” in Z Michalewicz (Ed.) Proceedings of the First IEEE Conference on Evolutionary Computation, Piscataway, NJ: IEEE Press, pp 82-87, 1994 [58] M.T Jensen, Reducing the Run-time Complexity of Multi- Objective EAs: The NSGA-II and Other Algorithms, in IEEE Transactions on Evolutionary Computation, vol 7, no 5, pp.503-515, 2003 [59] Jin Y., Okabe T., and Sendhoff B., ”Adapting weighted aggregation for multiobjective evolution strategies,” in Proceedings of First International Conference on Evolutionary Multi-Criterion Optimization, Lecture Notes in Computer Science, pages 96110, Zurich, March 2001, Springer [60] Jin, H and Wong, M-L, ”Adaptive Diversity Maintenance and Convergence Guarantee in Multiobjective Evolutionary Algorithm”, in Proceedings of the IEEE Congress on Evolutionary Computation-CEC03, pp 2498 - 2505, 2003 [61] Y Jin and B Sendhoff, ”Incorporation of Fuzzy Preferences into Evolutionary Multiobjective Optimization”, in Proceedings of the 4th Asia Pacific Conference on Simulated Evolution and Learning, vol 1, pp 26-30, Singapore, November 2002 [62] Keeney, RL and Raiffa, H., Decision with Multiple Objectives : Preferences and Value Tradeoffs, New York: John Wiley and Sons, 1976 [63] Knowles J and Corne D., ”The Pareto archived evolution strategy: A new baseline algorithm for multiobjective optimization,” in Proceedings of the 1999 Congress on Evolutionary Computation-CEC99, Piscataway, NJ: IEEE Press, pp 98-105, 1999 [64] Knowles, J and Corne, D., ”Properties of an Adaptive Archiving Algorithm for Storing Nondominated Vectors”, in IEEE Transactions on Evolutionary Computation, vol 7, no 1, pp 100-116, April, 2003 191 [65] Knowles, J., and Corne, D., ”How to Maintain at Most N Well-Distributed, Nodominated Solutions to a Pareto Optimization Problem,” unpublished [66] Korhonen, P., Larichev O, Moshkovich H., Mechitov A and Wallenius J., ”Choice behavior in a computer-aided multiattribute decision task”, in Journal of Multicriteria Decision Analysis, vol 6, pp 233-246, 1997 [67] Kumar R and Rockett P., ”Improved sampling of the pareto-front in multiobjective genetic optimizations by steady-state evolution: a Pareto converging genetic algorithm,” in Evolutionary Computation, vol 10, no 3, pp 283-314, 2002 [68] Kursawe,F., ”A variant of evolution strategies for vector optimization,” in Schwefel, H.P and Manner, R (Eds.), Proceedings of Parallel Problem Solving from Nature, Berlin, Germany: Springer-Verlag, pp.193-197, 1990 [69] Larichev, O.I., Moshkovich, H.M., and Rebrik, S.B., ”Systematic Research into human behavior in multiattribute object classification problems”, Acta Psychologica, vol 68, pp 171-182, 1988 [70] M Laumanns, G Rudolph, and H.-P Schwefel, ”A Spatial Predator-Prey Approach to Multiobjective Optimization: a Preliminary Study”, in A.E Eiben, T Back, M Schonauer, and H.-P Schwefel (Eds.), Proceedings of the Fifth Int Conf on Parallel Problem Solving from Nature(PPSN V), Berlin: Springer-Verlag, pp 241-249, 1998 [71] Laumanns, M., Thiele, L., Deb, K and Zitzler, E., ”Combining convergence and diversity in evolutionary multiobjective optimization”, in Evolutionary Computation, vol 10, no 3, pp 263-282, 2002 [72] Li X., ”Better Spread and Convergence: Particle Swarm Multiobjective Optimization Using the Maximin Fitness Function”, in Proceedings of GECCO 2004, Berlin: Springer Verlag, pp 117-128, 2004 [73] De Lit P., Latinne P., Rekiek B., and Delchambre A., ”Assembly planning with an ordering evolutionary algorithm”, in International Journal of Production Research, vol 39, no 16, pp 36233640, 2001 [74] Luce, R.D., ”Semiorders and a theory of utility discrimination”, in Econometrica, vol 24, pp 178-191, 1956 [75] Markowitz, H.M., Portfolio Selection, Wiley, New York, 1959 [76] Mattson, C A., Mullur, A A., and Messac, A., ”Smart Pareto Filter: Obtaining a Minimal Representation of Multi-objective Design Space,” in Engineering Optimization, vol 36, no 6,pp 721.740, 2004 192 [77] Miettinen, K M., Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, USA, 1999 [78] Munda, G., ”Multiple-criteria decision aid: Some epistemological considerations”, in Journal of Multi-Criteria Decision Analysis, vol 2, pp.41-55, 1993 [79] Mumford C L., ”Simple Population Replacement Strategies for a Steady-State Multiobjective Evolutionary Algorithm”, in K Deb et al (Eds), Proceedings of the Genetic and Evolutionary Computation Conference-GECCO04, Berlin: Springer Verlag, pp 1389-1400, 2004 [80] Mumford-Valenzuela C L ”A Simple Approach to Evolutionary Multiobjective Optimization”, in Ajith Abraham, Lakhmi Jain and Robert Goldberg, (Eds.), Evolutionary Computation Based Multi-Criteria Optimization: Theoretical Advances and Applications, Springer-Verlag, 2004 [81] Nakayama H., Kaneshige K, Takemoto S., Watada Y., ”An Application of a Multi-Objective Programming Technique to Construction Accuracy Control of Cable-Stayed Bridges”, in European Journal of Operational Research, vol 87, no 3, pp 731-738, 1995 [82] Nebro, A J.; Luna, F.; Alba, E.; Dorronsoro, B.; Durillo, J J.; Beham, A,” AbYSS: Adapting Scatter Search to Multiobjective Optimization”, in IEEE Transactions on Evolutionary Computation, vol., no., pp., 2008 [83] Pareto V., Manual of Political Economy, Augustus M Kelley, 1969 [84] Parmee I C and Watson A H., Preliminary airframe design using co-evolutionary multiobjective genetic algorithms, in W Banzhaf and J Daida (Eds.), Proceedings of the Genetic and Evolutionary Computation Conference-GECCO99,San Mateo, CA: Morgan Kaufmann, pp 16571665, 1999 [85] Parmee, I.C., Cvetkovic, D., Watson, A.H., and Bonham, C.R., ”Multi-objective satisfaction within an interactive evolutionary design environment”, in Evolutionary Computation, vol 8, no 2, pp 197-222, 2000 [86] Poloni, C., ”Hybrid GA for multiobjective aerodynamic shape optimization,” in Genetic Algorithms in Engineering and Computer Science, G Winter, J Periaux, M Galan, and P Cuesta, Eds., New York:Wiley, pp 397-414, 1997 [87] Rachmawati, L and Srinivasan D., ”A Hybrid Fuzzy Evolutionary Algorithm for A Multi-Objective Resource Allocation Problem”, in Proceedings of the 2005 Hybrid Intelligent Systems Conference, pp 55-60, 2005 193 [88] Rachmawati, L and D Srinivasan, ”A Fuzzy Evolutionary Algorithm for Combinatorial Optimization of Soft Objectives”, in Third International Conference on Computational Intelligence, Robotics and Autonomous Systems, 13 - 16 Dec 2005, Singapore [89] Srinivasan D and Rachmawati L, ”An efficient multi-objective evolutionary algorithm with steady-state replacement model,” in Proceedings the Genetic and Evolutionary Computation Conference-GECCO06, pp 715-722, 2006 [90] Rachmawati, L and Srinivasan D., ”A multi-objective evolutionary algorithm with weighted-sum niching for convergence on knee regions”, Proceedings the Genetic and Evolutionary Computation Conference-GECCO06, pp 749-750, 2006 [91] Rachmawati, L and Srinivasan D., ”Preference Incorporation in Multi-objective Evolutionary Algorithms: A Survey”.in Proceedings of Congress on Evolutionary Computation, 2006 [92] Rachmawati, L and Srinivasan D., ”A Multi-objective Genetic Algorithm with Controllable Convergence on Knee Regions”, in Proceedings of Congress on Evolutionary Computation, 2006 [93] Rachmawati, L and Srinivasan D., ”Dynamic resizing for grid-based archiving in evolutionary multi objective optimization”, in Proceedings of Congress on Evolutionary Computation, pp 3975 - 3982, 2007 [94] Srinivasan, D and Rachmawati, L., ”Efficient Fuzzy Evolutionary AlgorithmBased Approach for Solving the Student Project Allocation Problem”, in , IEEE Transactions on Education, accepted for future publication [95] Rachmawati, L and Srinivasan D., ”Steady-state Replacement Strategy for the Selection of Elite Solutions in Evolutionary Multiobjective Optimization”, inIEEE Transactions on Evolutionary Computation, conditionally accepted [96] Rachmawati, L and Srinivasan D., ”A Multi-objective Evolutionary Algorithm with Controllable Focus on the Knees of the Pareto Front”, inIEEE Transactions on Evolutionary Computation, accepted [97] Rachmawati, L and Srinivasan D., ”The Systematic Representation, Elicitation and Incorporation of Objective Importance Ranking in Multi-Objective Evolutionary Algorithm” [98] B Rekiek, Assembly line design: multiple objective grouping evolutionary algorithm and the balancing of mixed-model hybrid assembly line, Ph.D thesis, Universite Libre de Bruxelles, 2001 194 [99] Roy B., ”Classement et choix en presence de points de vue multiples(la methode Electre)”, in Revue Francaise d’Informatique et de Recherche Operationnelle, vol 8, pp 57-75, 1968 [100] Roy, B and Skalka, J.M., ”Electre IS - aspects methodologiques et guide d’utilisation”, Document du LAMSADE, vol 30, p 125, 1984 [101] Roy, B and Bertier P., ”La methode Electre II - une application eu media planning”, in Ross M (Ed.), OR’72, North Holland Publishing Company, pp 291-302, 1973 [102] Roy, B., ”Electre III: un algorithme declassement fonde sur une representation floue des preferences en presence des criteres multiples”, in Cahiers du CERO, vol 20, pp 3-24, 1978 [103] Roy, B., and Hugonnard, J.-Chr, ”Ranking of suburban line extension projects on the Paris metro system by a multicriteria method”, in Transportation Research, vol 16 A, pp301-312, 1982 [104] Roy, B., ”How outranking relations helps multiple criteria decision making”, in Cochrane J, Zeleny M (eds) Multicriteria Decision Making, University of South Carolina, pp.179-201, 1973 [105] Roy, B., in Methodologie Multicritere d’Aide a la Decision, Paris: Economica, 1985 [106] Roy, B and Mousseau, V., ”A Theoretical Framework for Analysing the Notion of Relative Importance of Criteria”, in Journal of Multi-criteria Decision Analysis, vol 5, pp.145-159, 1996 [107] G Rudolph and A Agapie, ”Convergence Properties of Some Multiobjective Evolutionary Algorithms,” Proceedings of the 2000 Conference on Evolutionary Computation, vol 2, Piscataway, New Jersey: IEEE Press, pp 1010-1016, July 2000 [108] Saaty, T.L., The Analytic Hierarchy Process: Planning, priority setting, resource allocation McGraw Hill, New York, 1980 [109] Sakawa, ”Interactive Multiobjective Decisionmaking by Sequantial Proxy Optimizatrion Technique: SPOT”, in European Journal of Operational Research, vol 9, no 4, pp.386-396, 1982 [110] Schott J., ”Fault tolerant design using single and multicriteria genetic algorithm optimization”, Master’s thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 1995 195 [111] Scott MJ and Antonsson EK, ”Arrow’s theorem and engineering design decision making”, in Research in Engineering Design, vol 11, no 4, pp 218-228, 1999 [112] Smith K.I., Everson R.M., Fieldsend J.E., Murphy C., and Misra R., ”Dominance-Based Multiobjective Simulated Annealing”, in IEEE Transactions on Evolutionary Computation, vol., no., pp 323-342, 2008 [113] Solso, R.L., , Covnitive Psychology, Allyn and Bacon, Inc., Boston, 1988 [114] Srinivas N and Deb K., ”Multiobjective optimization using nondominated sorting in genetic algorithms,” in Evolutionary Computation, vol 2, no 3, pp 221248, 1994 [115] Szidarovszky, F ”Notes on Multiobjective Dynamic Programming”, Working paper 379-1, Dept of Systems and Industrial Engineering, University of Arizona, Tucson, 1979 [116] K.C Tan, E.F Khor, T.H Lee and R Sathikannan, ”An Evolutionary Algorithm with Advanced Goal and Priority Specification for Multi-objective Optimization”, in Journal of Artificial Intelligence Research, vol 18, pp 183-215, 2003 [117] Tversky, A, ”Intransitivity of Preferences”, in Psychological Review, vol 76, pp 31-48, 1969 [118] D A Van Veldhuizen, Multiobjective evolutionary algorithms: Classifica- tions,analyzes, and new innovations, Ph.D dissertation, Graduate School of Eng of the Air Force Inst of Technol., Air Univ., June 1999 [119] von Winterfeldt, D, Fischer GW, ”Multiattribute utility theory: Models and assessment procedures”, in D Wendt, C Vlek (Eds), Utility, probability and human decision making, Reidel, Dordrecht, 1975 [120] Vincke Ph., ”Preferences and numbers” in A Colorni, M Paruccini, and B Roy (Eds), A-MCD-A - Aide Multi Critre la Dcision - Multiple Criteria Decision Aiding, Joint Research Center, The European Commission, pp 343354, 2001 [121] Wallenius J., ”Comparative Evaluation of Some Interactive Approaches to Multicriterion Optimization,” in Management Science, vol 21, no 12, pp 1387-1396, 1975 [122] Wierzbicki, A.P., ”A Methodological Guide to Multiobjective Optimization, Optimization Techniques Part 1, Lecture Notes in Control and Information Sciences, vol 22, pp 99-123, 1980 196 [123] Zadeh, L ”Optimality and Non-Scalar-Valued Performance Criteria,” IEEE Transactions on Automatic Control, No 59, 1963 [124] Zadeh, L., ”The concept of a linguistic variable and its application to approximate reasoning”, in Information Sciences, Part 1:vol 8, pp 199-249, Part 2: vol 8, pp 301-357, Part 3: vol 9,pp 43-80, 1975 [125] Zadeh, L., ”A theory of approximate reasoning”, in Machine Intelligence 9, Wiley, New York, pp 149-194, 1979 [126] Zitzler E and Thiele L., Multiobjective optimization using evolutionary algorithmsA comparative case study, in A E Eiben et al (Eds), Proceedings of Parallel Problem Solving from Nature V, Berlin, Germany: Springer, pp 292301, 1998 [127] Zitzler E and Thiele L., ”Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach,” IEEE Transactions on Evolutionary Computation, vol 3, no 4, pp 257-271, 1999 [128] Zitzler E., Deb K and Thiele L ”Comparison of Multiobjective Evolutionary Algorithms: Empirical Results,” in Evolutionary Computation, vol 8, no 2, pp 173-195, 2000 [129] Zitzler E., Laumanns M., Thiele L., ”SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization”, in Evolutionary Methods for Design, Optimization and Control, 2002 [130] Zitzler E., Thiele L., Laumanns M., Fonseca C M., da Fonseca V G., ”Performance Assessment of Multiobjective Optimizers: An Analysis and Review,” in IEEE Transactions on Evolutionary Computation, vol 7, no 2, pp 117-131, 2003 [131] Zitzler, E., and Kunzli, S, ”Indicator-Based Selection in Multiobjective Search”, in Proceedings of the eighth conference on Parallel Problem Solving in Nature, Lecture Notes on Computer Science, pp 832-842, 2004 [132] Q Zhang; A Zhou; Y Jin , ”RM-MEDA: A Regularity Model-Based Multiobjective Estimation of Distribution Algorithm”, in IEEE Transactions on Evolutionary Computation, pp 41-63, 2008 197 ... 17 Chapter Decision Making Preference in Multi Objective Optimization This chapter presents a review of the a priori / interactive incorporation of preference into Multi Objective Optimization. .. Decision Making Preference in Multi Objective Optimization 18 3.1 Features of Human Decision Making 20 3.2 Desirable Properties of Preference- based Evolutionary Search... Importance of Objectives The relative importance of objectives is an operationally intuitive notion of decision making preference in multi objective optimization The importance of objectives