Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary Algorithms
The fusion of evolutionary algorithms and the solution concepts of cooperative game theory is proposed in this paper to solve the fuzzy optimization problems. The original fuzzy optimization problem is transformed into a scalar optimization problem by assigning some suitable coefficients. The assign...
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| Format: | Article |
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MDPI AG
2023-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/24/4871 |
| _version_ | 1797380170125934592 |
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| author | Hsien-Chung Wu |
| author_facet | Hsien-Chung Wu |
| author_sort | Hsien-Chung Wu |
| collection | DOAJ |
| description | The fusion of evolutionary algorithms and the solution concepts of cooperative game theory is proposed in this paper to solve the fuzzy optimization problems. The original fuzzy optimization problem is transformed into a scalar optimization problem by assigning some suitable coefficients. The assignment of those coefficients is frequently determined by the decision-makers via their subjectivity, which may cause some biases. In order to avoid these subjective biases, a cooperative game is formulated by considering the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-level functions of the fuzzy objective function. Using the Shapley values of this formulated cooperative game, the suitable coefficients can be reasonably set up. Under these settings, the transformed scalar optimization problem is solved to obtain the nondominated solution, which will depend on the coefficients. In other words, we shall obtain a bunch of nondominated solutions depending on the coefficients. Finally, the evolutionary algorithms are invoked to find the best nondominated solution by evolving the coefficients. |
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| format | Article |
| id | doaj.art-07b29549396e4bf5bd4a75d78f819cc4 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T20:33:30Z |
| publishDate | 2023-12-01 |
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| spelling | doaj.art-07b29549396e4bf5bd4a75d78f819cc42023-12-22T14:23:09ZengMDPI AGMathematics2227-73902023-12-011124487110.3390/math11244871Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary AlgorithmsHsien-Chung Wu0Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, TaiwanThe fusion of evolutionary algorithms and the solution concepts of cooperative game theory is proposed in this paper to solve the fuzzy optimization problems. The original fuzzy optimization problem is transformed into a scalar optimization problem by assigning some suitable coefficients. The assignment of those coefficients is frequently determined by the decision-makers via their subjectivity, which may cause some biases. In order to avoid these subjective biases, a cooperative game is formulated by considering the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-level functions of the fuzzy objective function. Using the Shapley values of this formulated cooperative game, the suitable coefficients can be reasonably set up. Under these settings, the transformed scalar optimization problem is solved to obtain the nondominated solution, which will depend on the coefficients. In other words, we shall obtain a bunch of nondominated solutions depending on the coefficients. Finally, the evolutionary algorithms are invoked to find the best nondominated solution by evolving the coefficients.https://www.mdpi.com/2227-7390/11/24/4871cooperative gamesevolutionary algorithmsfuzzy optimizationscalar optimizationShapley value |
| spellingShingle | Hsien-Chung Wu Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary Algorithms Mathematics cooperative games evolutionary algorithms fuzzy optimization scalar optimization Shapley value |
| title | Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary Algorithms |
| title_full | Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary Algorithms |
| title_fullStr | Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary Algorithms |
| title_full_unstemmed | Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary Algorithms |
| title_short | Solving Fuzzy Optimization Problems Using Shapley Values and Evolutionary Algorithms |
| title_sort | solving fuzzy optimization problems using shapley values and evolutionary algorithms |
| topic | cooperative games evolutionary algorithms fuzzy optimization scalar optimization Shapley value |
| url | https://www.mdpi.com/2227-7390/11/24/4871 |
| work_keys_str_mv | AT hsienchungwu solvingfuzzyoptimizationproblemsusingshapleyvaluesandevolutionaryalgorithms |