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[19] S. Phelps and M. Koksalan, “An interactive evolutionary metaheuristic for multiobjective combinatorial optimization,” Management Science, vol. 49, no. 12, pp. 1726–1738, December 2003. [20] J. W. Fowler, E. S. Gel, M. Koksalan, P. Korhonen, J. L. Marquis, and J. Wallenius, “Interactive evolutionary multi-objective optimization for quasi-concave preference functions,” 2009, submitted to European Journal of Operational Research. [21] A. Jaszkiewicz, “Interactive multiobjective optimization with the pareto memetic algorithm,” Foundations of Computing and Decision Sciences, vol. 32, no. 1, pp. 15–32, 2007. [22] J. Figueira, S. Greco, and R. Slowinski, “Building a set of additive value functions reprsenting a reference preorder and intensities of preference: GRIP method,” European Journal of Operational Research, vol. 195, no. 2, pp. 460–486, 2009. [23] E. Zitzler, M. Laumanns, and L. Thiele, “SPEA2: Improving the strength pareto evolutionary algorithm for multiobjective optimization,” in Evolutionary Methods for Design Optimization and Control with Applications to Industrial Problems, K. C. Giannakoglou, D. T. Tsahalis, J. Périaux, K. D. Papailiou, and T. Fogarty, Eds. Athens, Greece: International Center for Numerical Methods in Engineering (CIMNE), 2001, pp. 95–100. [24] A. M. Geoffrion, “Proper efficiency and theory of vector maximization,” Journal of Mathematical Analysis and Applications, vol. 22, no. 3, pp. 618–630, 1968. [25] P. Korhonen and J. Karaivanova, “An algorithm for projecting a reference direction onto the nondominated set of given points,” IEEE Trans. on Systems, Man and Cybernetics–Part A: Systems and Humans, vol. 29, no. 5, pp. 429–435, 1999. [26] S. Zionts and J. Wallenius, “An interactive programming method for solving the multiple criteria problem,” Management Science, vol. 22, pp. 656–663, 1976. [27] A. Mas-Colell, M. D. Whinston, and J. R. Green, Microeconomic Theory. New York: Oxford University Press, 1995. [28] A. P. Wierzbicki, “The use of reference objectives in multiobjective optimization,” in Multiple Criteria Decision Making Theory and Applications, G. Fandel and T. Gal, Eds. Berlin: Springer-Verlag, 1980, pp. 468–486. [29] K. Deb and R. B. Agrawal, “Simulated binary crossover for continuous search space,” Complex Systems, vol. 9, no. 2, pp. 115–148, 1995. [30] K. V. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization. Berlin: Springer-Verlag, 2005. [31] R. H. Byrd, J. Nocedal, and R. A. Waltz, KNITRO: An integrated package for nonlinear optimization. Springer-Verlag, 2006, pp. 35– 59. [32] E. Zitzler, K. Deb, and L. Thiele, “Comparison of multiobjective evolutionary algorithms: Empirical results,” Evolutionary Computation Journal, vol. 8, no. 2, pp. 125–148, 2000. [33] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, “Scalable test problems for evolutionary multi-objective optimization,” in Evolutionary Multiobjective Optimization, A. Abraham, L. Jain, and R. Goldberg, Eds. London: Springer-Verlag, 2005, pp. 105–145. [34] D. Saxena and K. Deb, “Trading on infeasibility by exploiting constraint’s criticality through multi-objectivization: A system design perspective,” in Proceedings of the Congress on Evolutionary Computation (CEC-2007), in press. [35] D. W. Corne, J. D. Knowles, and M. Oates, “The Pareto envelope-based selection algorithm for multiobjective optimization,” in Proceedings of the Sixth International Conference on Parallel Problem Solving from Nature VI (PPSN-VI), 2000, pp. 839–848. 46
2 Progressively interactive evolutionary multi-objective optimization method using generalized polynomial value functions A. Sinha, K. Deb, P. Korhonen, and J. Wallenius In Proceedings of the 2010 IEEE Congress on Evolutionary Computa- tion (CEC-2010), pages 1-8. IEEE Press, 2010. 47
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Department of Business Technology P
Page 3 and 4: Aalto University publication series
Page 5 and 6: Progressively Interactive Evolution
Page 7: Acknowledgements The dissertation h
Page 10 and 11: II List of Papers 27 1 An interacti
Page 12 and 13: 1 Introduction Many real-world appl
Page 14 and 15: • f(x (1) ) ≥ f(x (2) ) fi(x (1
Page 16 and 17: for a maximization problem. Mathema
Page 18 and 19: • x (1) ∼ x (2) ⇔ x (1) and x
Page 20 and 21: Start Initialise Population Assign
Page 22 and 23: ithm can approximate the Pareto-opt
Page 24 and 25: Minimize/Maximize F(x) = (f1(x), f2
Page 26 and 27: the search is chosen. A scalarizing
Page 28 and 29: of the papers is provided in this s
Page 30: problems. The study discusses some
Page 33 and 34: [9] K. Deb and A. Sinha. An efficie
Page 35 and 36: [31] L. Thiele, K. Miettinen, P. Ko
Page 37 and 38: 1 An interactive evolutionary multi
Page 39 and 40: DM and an MCDM-based EMO algorithm
Page 41 and 42: In Step 2, points in the best non-d
Page 43 and 44: esulting constraints then become ki
Page 45 and 46: mechanisms) and their emphasis of n
Page 47 and 48: f2 10 8 6 4 2 P1 P2 Most preferred
Page 49 and 50: DM calls. As mentioned earlier, the
Page 51 and 52: A linear value function similar to
Page 53: on developed value function. For ex
Page 57 and 58: DM one or more pairs of alternative
Page 59 and 60: f 2 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 P
Page 61 and 62: TABLE III DISTANCE OF OBTAINED SOLU
Page 63: The PI-EMO-VF algorithm has been te
Page 66 and 67: An Interactive Evolutionary Multi-O
Page 68 and 69: direction has been done by Phelps a
Page 70 and 71: 0 ∀ i ∈ {1, . . . , M}, then th
Page 72 and 73: f2 0000000000000 1111111111111 0000
Page 74 and 75: Table 1. Final solutions obtained b
Page 76 and 77: Table 4. Distance of obtained solut
Page 78 and 79: DM Calls and Func. Evals. (in thous
Page 80 and 81: 13. P. Korhonen and J. Laakso, “A
Page 82 and 83: An EfficientandAccurateSolution Met
Page 84 and 85: eight differentproblems areshown. A
Page 86 and 87: 3.2 AlgorithmicDevelopments Onesimp
Page 88 and 89: more studies are performed, the alg
Page 90 and 91: 4.4 TestProblemTP4 The next problem
Page 92 and 93: F2 Step 3: Map (U1,U2) to (f1*,f2*)
Page 94 and 95: F2 1.4 1.2 1 0.8 0.6 0.4 0.2 0 A y1
Page 96 and 97: • Theupperlevelproblemhas multi-m
Page 98 and 99: are K + L + 1real-valuedvariablesin
Page 100 and 101: thereafterinaself-adaptivemannerbyd
Page 102 and 103: ulation of Nl(x (1) u ) lower level
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NSGA-II is able to bring the member
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F2 0 −0.2 −0.4 −0.6 −0.8
Page 108 and 109:
LL Function Evals. UL Function Eval
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Page 112 and 113:
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Table 1: Total function evaluations
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Difference in HV, DH(T) 350000 3000
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Table 4: Comparison of function eva
Page 120 and 121:
Deb, K. and Sinha, A. (2009a). Cons
Page 122 and 123:
Sun, D., Benekohal, R. F., and Wall
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Bilevel Multi-Objective Optimizatio
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The constraint functions g(x) and h
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value function which are required t
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to get close to the most preferred
Page 132 and 133:
F2 0.4 0.2 0 −0.2 Lower level P
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F2 2 1.5 1 0.5 Most Preferred Point
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provement in terms of function eval
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9HSTFMG*aeafcd+ ISBN: 978-952-60-40
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