Application of Genetic Algorithm in Multi-objective Optimization

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2. The solution is strictly better than in at least one objective, or mathematically, ) ⊲ ) for at least one ∈1,2,3,….,M The dom inance relation does not have reflexive property as no solution dominates itself. It also does not exhibit symmetric characteristic as the dominance of one solution x over another solution y does not mean the dominance of y over x. If x dominates y and y dominates z (a third solution), then x dominates z which shows the transitive property of the dominance relation. Pareto-Optimal Set (Non-dominated set): A set is said to be a non-dominated set or Pareto-optimal set if it is not dominated by any other solution that belongs to the solution set. The Pareto-optimal set is the best optimal solution for all objective functions and cannot be improved with respect to one objective by worsening another one. Mathematically, if P is a set of solutions, the non-dominated set of solutions P * comprises those solutions which are not dominated by any member of the set P. The non-dominated set of solutions can be generated by comparing all possible pairs of a given solution set and determining which solutions dominate which, and which are not dominated by each other. The Pareto-optimal set, P * can be written as: ∗ ∈ | ∃ ∈ ≼ Pareto-optimal sets are called global when the set of solutions, P, is the entire search space. If for every member x in a set P, there exist no solutions y in the neighborhood of x, then‖‖ , dominating any member of the set, then P establishes a locally Pareto-optimal set. Pareto-front: The Pareto-front contains the values of objective functions for all solutions in the Pareto-optimal set in the objective space. If the Pareto-front is ∗ for a given MOOP having objective function F(x), then mathematically: 26
∗ ≔ | ∈ ∗ Methods for solving the non-dominated set: In each iteration of a MOOP, the non-dominated set is needed to be determined. Thus, a computationally efficient approach is required to perform the determination step. Three different approaches were mentioned which are (i) Naïve and slow, (ii) Continuously updated and (iii) Kung et al.’s efficient method [1]. All of these methods use the concept of domination to determine the non-dominant set with respect to different objective functions. But the most efficient [17] and least computationally complex method is the third one. 2.2.3 Approaches to solve Multi-objective Optimization: Extensive studies have been conducted in multi-objective optimization algorithms. But most of the research work has avoided the complexity by transforming the problem into single- objective optimization with the use of some user-defined parameters. Deb [1] classifies the approaches towards solving multi-objective optimization in two groups. Ideal multi-objective optimization, where a set of solutions in the form of a trade-off curve is obtained, and the desired solution is selected based on some higher level information of the problem. Preference based multi-objective optimization, where using the higher level information a preference vector transforms the multi-objective problem to a single-objective optimization. The optimal solution is obtained by solving the single-objective problem. The ideal approach is less subjective than the preference-based approach. It requires analysis of nontechnical, qualitative and experimental information to find the preference vector. In the absence of higher level information in an optimization problem within the ideal approach, none of the Paretooptimal solutions is preferred over others. Therefore, in the ideal approach, the main objective is to 27
Page 1 and 2: Grand Valley State University Schol
Page 3 and 4: Dedication To my parents, Engr. Md.
Page 5 and 6: Abstract In this thesis, an indeter
Page 7 and 8: Table of Content Dedication……
Page 9 and 10: 2.3.2. Genetic Algorithm with Multi
Page 11 and 12: 4.2.1. Continuous Method………
Page 13 and 14: List of Tables Table 3.1: GA parame
Page 15 and 16: GAs have been used in numerous opti
Page 17 and 18: 2. Literature Review: 2.1 Optimizat
Page 19 and 20: other evolutionary operators named
Page 21 and 22: 2.1.3 Differences between Evolution
Page 23 and 24: espectively, and are the constr
Page 25: equiring a better solution than any
Page 29 and 30: to find a set of non-dominated solu
Page 31 and 32: There are some other approaches whi
Page 33 and 34: individual may be selected. And the
Page 35 and 36: 2.3.1.5. Mutation: Mutation is a ra
Page 37 and 38: Multi-objective genetic algorithms
Page 39 and 40: In 1940, Courant [62] first had the
Page 41 and 42: violated constraints. If a problem
Page 43 and 44: of test problems with a conclusion
Page 45 and 46: (Though in figure 3.1, due to 2D dr
Page 47 and 48: GA operators, such as population in
Page 49 and 50: 1 1 1 , where, , ,
Page 51 and 52: 3.2.1.3.1. GA Operator: In the “C
Page 53 and 54: In reality, objects, similar to the
Page 55 and 56: So, if there are n supports, the to
Page 57 and 58: 4. Results & Discussion: 4.1. Uncon
Page 59 and 60: Heuristically, it was assumed that
Page 61 and 62: The nodal points were generated in
Page 63 and 64: and , which were the head and the
Page 65 and 66: These two approaches were also comp
Page 67 and 68: 0.8 MINIMUM VALUE OF OBJECTIVE FUNC
Page 69 and 70: (a) (b) (c) Figure 4.7: Indetermina
Page 71 and 72: As one of the objectives was to max
Page 73 and 74: this thesis work, one of the popula
Page 75 and 76: Paired t-test ( 5%) was conducted t
Page 77 and 78:
5. Conclusion: In this work, a stan
Page 79 and 80:
Appendices A. Table 4.7 Results of
Page 81 and 82:
C. Paired t-test analysis of Discre
Page 83 and 84:
References [1] Deb, Kalyanmoy. Mult
Page 85 and 86:
[20] Rawlins, Gregory J. E. Foundat
Page 87 and 88:
[39] Deb, Kalyanmoy, et al. "A fast
Page 89 and 90:
[59] Takahama, Tetsuyuki, and Setsu
Page 91:
[87] Hwang, C-L., and Abu Syed Md M
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Application of Genetic Algorithm in Multi-objective Optimization

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