Progressively Interactive Evolutionary Multi-Objective Optimization ...

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ithm can approximate the Pareto-optimal front for a high objective problem with a huge set of points, the herculean task of choosing the best point from the set still remains. For two and three objectives where the solutions in the objective space could be represented geometrically, making decisions might be easy (though even such an instance could be, in reality, a difficult task for a decision maker). Imagine a multi-objective problem with more than three objectives for which an evolutionary multi-objective algorithm is able to produce the entire front. The front is approximated with high accuracy and high number of points. Since a graphical representation is not possible for the Pareto-points, how is a decision maker going to choose the most preferred point? There are of course decision aids available, but the limited accuracy with which the final choice could be made using these aids, questions the purpose of producing the entire front with a high accuracy. Binary comparisons can be a solution to choose the best point out of a set, but this can only be utilized if the points are very few in number. Therefore, offering the entire set of Pareto-points should not be considered as a complete solution to the problem. However, the difficulties related to decision making have been realized by EMO researchers only after copious research has already gone towards producing the entire Pareto-front for many objective problems. Minimize/Maximize F(x) = (f1(x), f2(x)) Computational Pareto−optimal Resources Solutions Most Preferred Solution Figure 1.5: A posteriori approach. 13 Decision Maker
1.5.2 A priori Approach In this approach, decision making is performed before the start of the algorithm, then the optimization algorithm is executed by incorporating the preference rules, and the most preferred solution is identified. Figure 1.6 shows the process followed to arrive at the most preferred solution. This approach has been common among MCDM practitioners, who realized the complexities involved in decision making for such problems. Their approach to the problem is to ask simple questions from the decision maker before starting the search process. The initial queries usually include the direction of search, aspiration levels for the objectives, or preference information for one or more given pairs. After eliciting such information from the decision maker, the multi-objective problem is usually converted into a single objective problem. One of the early approaches, that is, <strong>Multi</strong>- Attribute Utility Theory (MAUT) [21] used the initial information from the decision maker to construct a utility function which reduced the problem to a single objective optimization problem. Scalarizing functions (for example, [32]) are also commonly used by the researchers in this field to convert a multi-objective problem into a single objective problem. Other techniques which are used to elicit information from a decision maker can be found in the review [23] on multi-criteria decision support. Since information is elicited towards the beginning, the solution obtained after executing the algorithm is usually a satisfactory solution and may not be close to the most preferred solution. Moreover, the decision makers’ preferences might be different for solutions close to the Paretooptimal front and the initial inputs taken from them may not confirm it. Therefore, it will be difficult to get close to the actual solution which confirms to the requirements of the decision maker. The approach is also highly error prone as even slight deviations in providing preference information at the beginning may lead to entirely different solutions. To avoid the errors due to deviations, researchers in the EMO field used the approach in a slightly modified way. They produced multiple solutions in the region of interest to the decision maker [2, 11, 31, 17], instead of a single solution, therefore, giving choices to the decision maker at the end of the EMO search. However, researchers in the MCDM field recognized the enormous possibilities which could lead to erroneous results and therefore there exists a different school of thought which focusses on interactive approaches. 14
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 18 and 19: • x (1) ∼ x (2) ⇔ x (1) and x
Page 20 and 21: Start Initialise Population Assign
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 and 54: on developed value function. For ex
Page 55 and 56: 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
Page 70 and 71: 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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