Progressively Interactive Evolutionary Multi-Objective Optimization ...

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LL Function Evals. UL Function Evals. 380000 340000 300000 260000 100 22000 20000 18000 16000 100 300 300 500 Population Size 500 Population Size Figure 16: Average lower and upper level function evaluations with different populationsizesforproblemTP2indicates Nu = 300isthebestchoice. 21runs areperformed ineachcase. tives. This multi-objective bilevel problem was solved in the original study (Zhang et al., 2007) by converting two objectives into a single objective by the weighted-sum approach. The reportedsolution is marked on the figure with a star. It is interesting to note thatthissolution is oneoftheextremesolutions ofour obtainedfront. For aproblem having a linear Pareto-optimal front, the solution of a weighted-sum optimization approach is usually one of the extreme points. To this account, this result signifies the accuracyand efficiencyofthe H-BLEMOprocedure. F2 −1550 −1600 −1650 −1700 −1750 −1800 −1850 −1900 −600 −580 −560 −540 −520 −500 −480 −460 F1 Figure 17: Final archive solutions for problem TP4. The point marked with a staris takenfromZhang etal. (2007). F2 −1550 −1600 −1650 −1700 −1750 −1800 −1850 800 800 −1900 −600 −580 −560 −540 −520 −500 −480 −460 F1 Figure 18: Attainment surfaces(0%, 50% and100%)forproblemTP4from21runs. Figure 18 shows three attainment surface plots which are close to each other, like they arein the previous two problems. The hypervolumes for the obtained attainment surfaces are 0.5239, 0.5255 and 0.5260, with a maximum difference of 0.4% only, indi- 100
cating the robustness of our procedure. 6.4 ProblemDS1 Thisproblemhas10upperand10lowerlevelvariables. Thus,anoverallpopulationof size Nu = 400 is used here. For this problem, we first consider τ = 1. Figure 19 shows the obtained archive solutions for a typical run. It is worth mentioning here that this problemwaspossibletobesolveduptoonlysixvariables(intotal)byourearlierfixed BLEMOapproach(Deb and Sinha, 2009a). But here with our hybrid and self-adaptive approach, we are able to solve 20-variable version of the problem. Later, we present results with40variablesas well for this problem. F2 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 F1 0.8 1 1.2 Figure 19: Final archive solutions for problemDS1. F2 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 F1 0.8 1 1.2 Figure 20: Attainment surfaces(0%, 50% and100%)forproblemDS1from21runs. The attainment surface plots in Figure 20 further show the robustness of the proposed algorithm. The hypervolumes for the obtained attainment surfaces are 0.7812, 0.7984and0.7992,withamaximumdifferenceofabout2%. Theparametricstudywith Nu in Figure 21 shows that Nu = 400 is the best choice in terms of achieving a similar performance on the hypervolume measure using the smallest number of function evaluations,thereby supporting our setting Nu = 20n. Sincethis problemismore difficultcomparedtothepreviousproblems (TP1,TP2, andTP4),we investigatethe effectofself-adaptivechanges inlower levelNSGA-IIparameters (Nl and tl) with the generation counter. In the left side plot of Figure 22, we show the average value of these two parameters for every upper level generation (T) from a typical simulation run. It is interesting to note that, starting with an average of 20 members in each lower level subpopulation, the number reduces with generation counter, meaning that smaller population sizes are needed for later lower level simulations. The average subpopulation size reduces to its lower permissible value of four in 32 generations. Occasional increase in average Nl indicates that an upper level variable vector may have been found near a previously undiscovered region of the Pareto-optimal front. The hybrid algorithm increases its lower level population to explorethe regionbetterinsuchoccasions. Thevariationofnumberoflowerlevelgenerations(tl)beforeterminationalsofollows a similar trend, except that at the end only 3-9 generations are required to fulfill 101
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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
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
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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
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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 110 and 111: LL Function Evals. UL Function Eval
Page 112 and 113: 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 122 and 123: Sun, D., Benekohal, R. F., and Wall
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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