Progressively Interactive Evolutionary Multi-Objective Optimization ...

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the search is chosen. A scalarizing function is formulated based on the information and a search is performed with a fixed number of function evaluations (say nf ). This leads to a progress from point P1 to P2. At this instant, another decision maker call is made and more questions are asked from the decision maker. Based on the answers provided, the search direction is modified to D2, and another scalarizing function is formulated based on point P2 and new direction D2. With nf number of function evaluations, further progress is made from P2 to P3. Another decision maker call is executed and the process is repeated until no further progress is possible. In the figure it is shown that a satisfactory point is found in four decision maker calls. The point finally achieved by the algorithm is very close to the most preferred point. If the step size (P1 to P2, P2 to P3, P3 to P4 and P4 to P5 are the steps) of the algorithm is reduced, an even higher accuracy could be obtained, but with a higher number of decision maker calls. This procedure has potential to get close to the most preferred point which is, otherwise, difficult in other approaches. 1.6 Motivation f2 P1 Decision Making Instances D3 P2 D2 P3 D1 D4 P5 P4 Final Solution Most Preferred Point f1 Figure 1.8: <strong>Progressively</strong> <strong>Interactive</strong> Approach. As already pointed out, the target of the EMO algorithms has been to find a set of well-converged and well diversified Pareto-optimal solutions. Once the optimization process is started there is no interaction involved with the decision maker until a set of representative Pareto solutions are found. 17
However, it is never guaranteed that a set of representative Pareto-optimal solutions can be obtained. There can be difficulties involved, for example, the algorithm is unable to converge to the optima, or the entire Paretofront is not represented by the set of solutions. Though EMO procedures have shown their efficacy in solving multi-objective problems, they are not equipped to handle a high number of objectives. The challenges posed by high objective problems make the evolutionary multi-objective algorithms suffer in convergence as well as maintaining diversity. Moreover, the decision making task also becomes demanding when the non-dominated front cannot be represented geometrically. The difficulties necessitate coming up with procedures which can effectively handle the challenges offered by high objective optimization problems. As there is just a single point which is most preferred to a decision maker, and finding the entire Pareto-optimal front has its own difficulties, there is motivation to aim for the most preferred solution by judiciously using search and decision making. The manual and the computational resources available can be effectively mobilized if the single point of interest is perpetuated as the target right from the start of the optimization process. It also alleviates the problems associated with generating the entire Pareto-optimal set. Therefore, it would be advisable to begin with the exploration along with inputs from a decision maker and advance towards the region or point of interest. Moreover, the conjugation is expected to find the most preferred solution with less computational expense and a high accuracy for difficult problems. 1.7 Summary of Research Papers The dissertation consists of five papers which concern multi-objective optimization in general, and specifically deal with progressively interactive methods and bilevel optimization. The first paper is about a progressively interactive methodology which uses an implicitly defined value function and the second paper is an extension of the work. The third paper proposes a different progressively interactive methodology for the decision maker to interact with the algorithm and provide preferences. The fourth paper focusses on the less explored area of bilevel multi-objective optimization, and takes the domain a step forward by developing a generic evolutionary algorithm to handle the problem and also proposing test problems to evaluate the procedure. The fifth and final paper develops the previous paper by incorporating decision making in the suggested algorithm for bilevel multi-objective optimization. A short summary for each 18
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 22 and 23: ithm can approximate the Pareto-opt
Page 24 and 25: Minimize/Maximize F(x) = (f1(x), f2
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
Page 72 and 73: f2 0000000000000 1111111111111 0000
Page 74 and 75: 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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