Solving Triangular Intuitionistic Fuzzy Matrix Game by Applying the Accuracy Function Method
In this paper, the matrix game based on triangular intuitionistic fuzzy payoff is put forward. Then, we get a conclusion that the equilibrium solution of this game model is equivalent to the solution of a pair of the primal−dual single objective intuitionistic fuzzy linear optimization pro...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/10/1258 |
| _version_ | 1811304152005869568 |
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| author | Yumei Xing Dong Qiu |
| author_facet | Yumei Xing Dong Qiu |
| author_sort | Yumei Xing |
| collection | DOAJ |
| description | In this paper, the matrix game based on triangular intuitionistic fuzzy payoff is put forward. Then, we get a conclusion that the equilibrium solution of this game model is equivalent to the solution of a pair of the primal−dual single objective intuitionistic fuzzy linear optimization problems <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mi>F</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mi>F</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Furthermore, by applying the accuracy function, which is linear, we transform the primal−dual single objective intuitionistic fuzzy linear optimization problems <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mi>F</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mi>F</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> into the primal−dual discrete linear optimization problems <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> The above primal−dual pair <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>−<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is symmetric in the sense the dual of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Thus the primal−dual discrete linear optimization problems <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> are called the symmetric primal−dual discrete linear optimization problems. Finally, the technique is illustrated by an example. |
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| format | Article |
| id | doaj.art-11cf954adcec494f9ec281fd5fd55554 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-13T08:00:59Z |
| publishDate | 2019-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-11cf954adcec494f9ec281fd5fd555542022-12-22T02:55:16ZengMDPI AGSymmetry2073-89942019-10-011110125810.3390/sym11101258sym11101258Solving Triangular Intuitionistic Fuzzy Matrix Game by Applying the Accuracy Function MethodYumei Xing0Dong Qiu1Finance Department, Tianshui Normal University, Tianshui 741001, ChinaCollege of Science, Chongqing University of Post and Telecommunication, Chongqing 400065, ChinaIn this paper, the matrix game based on triangular intuitionistic fuzzy payoff is put forward. Then, we get a conclusion that the equilibrium solution of this game model is equivalent to the solution of a pair of the primal−dual single objective intuitionistic fuzzy linear optimization problems <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mi>F</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mi>F</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Furthermore, by applying the accuracy function, which is linear, we transform the primal−dual single objective intuitionistic fuzzy linear optimization problems <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mi>F</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>I</mi> <mi>F</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> into the primal−dual discrete linear optimization problems <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> The above primal−dual pair <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>−<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is symmetric in the sense the dual of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Thus the primal−dual discrete linear optimization problems <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>P</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>G</mi> <mi>L</mi> <mi>O</mi> <mi>D</mi> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> are called the symmetric primal−dual discrete linear optimization problems. Finally, the technique is illustrated by an example.https://www.mdpi.com/2073-8994/11/10/1258intuitionistic fuzzy matrix gameequilibrium solutionintuitionistic fuzzy linear optimization problemaccuracy function |
| spellingShingle | Yumei Xing Dong Qiu Solving Triangular Intuitionistic Fuzzy Matrix Game by Applying the Accuracy Function Method Symmetry intuitionistic fuzzy matrix game equilibrium solution intuitionistic fuzzy linear optimization problem accuracy function |
| title | Solving Triangular Intuitionistic Fuzzy Matrix Game by Applying the Accuracy Function Method |
| title_full | Solving Triangular Intuitionistic Fuzzy Matrix Game by Applying the Accuracy Function Method |
| title_fullStr | Solving Triangular Intuitionistic Fuzzy Matrix Game by Applying the Accuracy Function Method |
| title_full_unstemmed | Solving Triangular Intuitionistic Fuzzy Matrix Game by Applying the Accuracy Function Method |
| title_short | Solving Triangular Intuitionistic Fuzzy Matrix Game by Applying the Accuracy Function Method |
| title_sort | solving triangular intuitionistic fuzzy matrix game by applying the accuracy function method |
| topic | intuitionistic fuzzy matrix game equilibrium solution intuitionistic fuzzy linear optimization problem accuracy function |
| url | https://www.mdpi.com/2073-8994/11/10/1258 |
| work_keys_str_mv | AT yumeixing solvingtriangularintuitionisticfuzzymatrixgamebyapplyingtheaccuracyfunctionmethod AT dongqiu solvingtriangularintuitionisticfuzzymatrixgamebyapplyingtheaccuracyfunctionmethod |