Progressively Interactive Evolutionary Multi-Objective Optimization ...

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5 Bilevel multi-objective optimization problem solving using progressively interactive EMO A. Sinha In Proceedings of the Sixth International Conference on Evolutionary Multi-criterion Optimization (EMO-2011), In Press, 2011. 115
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Department of Business Technology P
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Aalto University publication series
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Progressively Interactive Evolution
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Acknowledgements The dissertation h
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II List of Papers 27 1 An interacti
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1 Introduction Many real-world appl
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• f(x (1) ) ≥ f(x (2) ) fi(x (1
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for a maximization problem. Mathema
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• x (1) ∼ x (2) ⇔ x (1) and x
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Start Initialise Population Assign
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ithm can approximate the Pareto-opt
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Minimize/Maximize F(x) = (f1(x), f2
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the search is chosen. A scalarizing
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of the papers is provided in this s
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problems. The study discusses some
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[9] K. Deb and A. Sinha. An efficie
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[31] L. Thiele, K. Miettinen, P. Ko
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1 An interactive evolutionary multi
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DM and an MCDM-based EMO algorithm
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In Step 2, points in the best non-d
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esulting constraints then become ki
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mechanisms) and their emphasis of n
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f2 10 8 6 4 2 P1 P2 Most preferred
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DM calls. As mentioned earlier, the
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A linear value function similar to
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on developed value function. For ex
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2 Progressively interactive evoluti
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DM one or more pairs of alternative
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f 2 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 P
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TABLE III DISTANCE OF OBTAINED SOLU
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The PI-EMO-VF algorithm has been te
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An Interactive Evolutionary Multi-O
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direction has been done by Phelps a
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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
Page 104 and 105: NSGA-II is able to bring the member
Page 106 and 107: F2 0 −0.2 −0.4 −0.6 −0.8
Page 108 and 109: LL Function Evals. UL Function Eval
Page 114 and 115: Table 1: Total function evaluations
Page 116 and 117: Difference in HV, DH(T) 350000 3000
Page 118 and 119: Table 4: Comparison of function eva
Page 120 and 121: Deb, K. and Sinha, A. (2009a). Cons
Page 124 and 125: Bilevel Multi-Objective Optimizatio
Page 126 and 127: The constraint functions g(x) and h
Page 128 and 129: value function which are required t
Page 130 and 131: to get close to the most preferred
Page 132 and 133: F2 0.4 0.2 0 −0.2 Lower level P
Page 134 and 135: F2 2 1.5 1 0.5 Most Preferred Point
Page 136 and 137: provement in terms of function eval
Page 138: 9HSTFMG*aeafcd+ ISBN: 978-952-60-40
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