Progressively Interactive Evolutionary Multi-Objective Optimization ...

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• x (1) ∼ x (2) ⇔ x (1) and x (2) are equally preferable, • x (1) x (2) ⇔ x (1) and x (2) are incomparable, where the preference relation, ≻, is asymmetric, the indifference relation, ∼, is reflexive and symmetric and the incomparability relation, , is irreflexive and symmetric. A weak preference relation can be established as =≻ ∪ ∼ such that, • x (1) x (2) ⇔ x (1) is either preferred over x (2) or they are equally preferred. As already mentioned, preference can easily be established for pairs where one solution dominates the other. However, for pairs which are non-dominated with respect to each other, a decision is required to establish a preference. The following are the inferences for preference choice which can be drawn from dominance: • If x (1) strongly dominates x (2) ⇒ x (1) ≻ x (2) , • If x (1) weakly dominates x (2) ⇒ x (1) x (2) . The binary relations ≻, ∼ and , are also explained in Figure 1.2 for a two objective case. Multiple points have been shown in the objective space and when comparisons are made between solutions in pairs then one of the binary relations will hold. For example, points 1 and 2 are close to each other; therefore, a decision maker may be indifferent between the two points. Points 3 and 4 lie on the extremes and are far away from each other; therefore, a decision maker may find such points incomparable. When points 2 and 5 are considered, a decision maker is not required as 2 dominates point 5; it can be directly inferred that a rational decision maker will prefer 2 over 5. It is common to emulate a decision maker with an non-decreasing value function, V (f(x)) = V (f1(x), . . . , fM(x)), which is scalar in nature and assigns a value or a measure of satisfaction to each of the solution points. For two solutions, x (1) and x (2) , • If x (1) ≻ x (2) ⇔ V (f(x (1) )) > V (f(x (2) )), • If x (1) ∼ x (2) ⇔ V (f(x (1) )) = V (f(x (2) )). 9
1.4 Evolutionary Multi-objective Optimization (EMO) Algorithms An evolutionary algorithm is a generic population based optimization algorithm which uses a mechanism inspired by biological evolution, i.e., selection, mutation, crossover and replacement. The common underlying idea behind an evolutionary technique is that, for a given population of individuals, the environmental pressure causes natural selection which leads to a rise in fitness of the population. A comprehensive discussion of the principles of an evolutionary algorithm can by found in [16, 24, 12, 1, 25]. In contrast to classical algorithms which iterate from one solution point to the other until termination, an evolutionary algorithm works with a population of solution points. Each iteration of an evolutionary algorithm results in an update of the previous population by eliminating inferior solution points and including the superior ones. In the terminology of evolutionary algorithms an iteration is commonly referred to as a generation and a solution point as an individual. A pseudo code for a generic evolutionary algorithm is provided next: Step 1: Create a random initial population Step 2: Evaluate the individuals in the population and assign fitness Step 3: Repeat the generations until termination Sub-step 1: Select the most fit individuals (parents) from the population for reproduction Sub-step 2: Produce new individuals (offsprings) through Crossover and Mutation operators Sub-step 3: Evaluate the new individuals and assign fitness Sub-step 4: Replace low fitness members with high fitness members in the population Step 4: Output Along with the pseudo code presented above, a flowchart for a general evolutionary algorithm has also been presented in Figure 1.4. A pool of individuals is generated by randomly creating points in the search space which is called the population. Each member in the population is evaluated and assigned a fitness. For instance, while solving a single objective 10
Page 1 and 2: 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 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
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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
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Page 47 and 48: f2 10 8 6 4 2 P1 P2 Most preferred
Page 49 and 50: DM calls. As mentioned earlier, the
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Page 61 and 62: TABLE III DISTANCE OF OBTAINED SOLU
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direction has been done by Phelps a
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0 ∀ i ∈ {1, . . . , M}, then th
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f2 0000000000000 1111111111111 0000
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Table 1. Final solutions obtained b
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Table 4. Distance of obtained solut
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DM Calls and Func. Evals. (in thous
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13. P. Korhonen and J. Laakso, “A
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An EfficientandAccurateSolution Met
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eight differentproblems areshown. A
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3.2 AlgorithmicDevelopments Onesimp
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more studies are performed, the alg
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4.4 TestProblemTP4 The next problem
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F2 Step 3: Map (U1,U2) to (f1*,f2*)
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F2 1.4 1.2 1 0.8 0.6 0.4 0.2 0 A y1
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• Theupperlevelproblemhas multi-m
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are K + L + 1real-valuedvariablesin
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thereafterinaself-adaptivemannerbyd
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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
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LL Function Evals. UL Function Eval
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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
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Deb, K. and Sinha, A. (2009a). Cons
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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
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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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