# Intuitionistic-Fuzzy Goals in Zero-Sum Multi Criteria Matrix Games

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 3. Multi-Criteria Zero Sum Game

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Theorem**

**1.**

**Theorem**

**2.**

## 4. The Proposed Multi-Criteria Matrix Game Model with I-Fuzzy Goals

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 5. Illustrative Example

**Example**

**1.**

#### 5.1. Comparison with Existing Models

- From Table 5, the disadvantages of Nishizaki and Sakwaw’s model are apparent, and it does not give information about fuzzy goals and strategies regarding the individual criteria, whereas the inspection of Table 1, Table 2 will reveal the strategies optimized for all the three criteria: cost, time and productivity.
- In addition, there are some other lapses in the Nishizaki and Sakwaw’s model, like the slight error in the linear programming model for Player II explained in [28,33], and our model produces much better strategies with higher securities as $max({\alpha}_{1},$ ${\alpha}_{2},$ ${\alpha}_{3})\ge 0.88>\lambda =$ $0.33088$ and $max({\gamma}_{1},$ ${\gamma}_{2},$ ${\gamma}_{3})\ge 0.9>1-\sigma =0.4196.$
- For Player I, $max({\alpha}_{1},$ ${\alpha}_{2},$ ${\alpha}_{3})\ge 0.88,$ whereas $max({\alpha}_{1}^{**},$ ${\alpha}_{2}^{**},$ ${\alpha}_{3}^{**})<0.58$ for all the strategies in Aggarwal and Khan’s model.
- For Player II, $max({\gamma}_{1},$ ${\gamma}_{2},$ ${\gamma}_{3})\ge 0.9,$ but in Aggarwal and Khan’s model $max({\gamma}_{1}^{**},$ ${\gamma}_{2}^{**},$ ${\gamma}_{3}^{**})<0.55$ for all the strategies.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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# | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

${x}_{1}$ | 0.330458768 | 0.328177309 | 0.326990679 | 0.331013375 | 0.329643138 | 0.32678683 |

${x}_{2}$ | 0.003067144 | 0.005351084 | 0.006804512 | 0.002408675 | 0.003936089 | 0.007106936 |

${x}_{3}$ | 0.666474094 | 0.666471604 | 0.666204822 | 0.666577952 | 0.666420764 | 0.666106235 |

${\alpha}_{1}$ | 0.186152005 | 0.18253803 | 0.212667119 | 0.135107449 | 0.190664067 | 0.212773933 |

${\alpha}_{2}$ | 0.263940075 | 0.40147779 | 0.160051536 | 0.400629476 | 0.307761745 | 0.277666181 |

${\alpha}_{3}$ | 0.990850397 | 0.981218877 | 0.950636349 | 0.999575878 | 0.987843841 | 0.886780913 |

${\beta}_{1}$ | 0.000154645 | 0.03633369 | 0.0000135 | 0.0374 | 0.001713378 | 0.0000102 |

${\beta}_{2}$ | 0.000340823 | 0.0000183 | 0.000795207 | 0.0000582 | 0.000686912 | 0.000344612 |

${\beta}_{3}$ | 0.002762645 | .0000751 | 0.013691788 | 0.000154497 | 0.003703514 | 0.045478111 |

# | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

${y}_{1}$ | 0.208396956 | 0.147116772 | 0.077061037 | 0.122099413 | 0.137881486 | 0.317982986 |

${y}_{2}$ | 0.468619872 | 0.50322759 | 0.527975471 | 0.512031939 | 0.501345421 | 0.431959882 |

${y}_{3}$ | 0.322983166 | 0.349655633 | 0.394963479 | 0.365868639 | 0.360773083 | 0.250057132 |

${\gamma}_{1}$ | 0.972875407 | 0.906662826 | 0.140586213 | 0.400312721 | 0.310226077 | 0.998287523 |

${\gamma}_{2}$ | 0.736265443 | 0.708827346 | 0.96036924 | 0.956430645 | 0.974640435 | 0.547342286 |

${\gamma}_{3}$ | 0.647858021 | 0.673578085 | 0.70315437 | 0.684380799 | 0.677435326 | 0.603149025 |

${\xi}_{1}$ | 0.002326489 | 0.009271841 | 0.001231738 | 0.00174913 | 0.001736754 | 0.000188018 |

${\xi}_{2}$ | 0.014342864 | 0.011291209 | 0.001163753 | 0.00456181 | 0.000373398 | 0.443368563 |

${\xi}_{3}$ | 0.351815861 | 0.325604484 | 0.296726416 | 0.315289635 | 0.322547636 | 0.39676953 |

**Table 3.**Aggarwal and Khan [33] for Player I.

# | ${\mathit{x}}_{1}^{**}$ | ${\mathit{x}}_{2}^{**}$ | ${\mathit{x}}_{3}^{**}$ | ${\mathit{\alpha}}_{1}^{**}$ | ${\mathit{\alpha}}_{2}^{**}$ | ${\mathit{\alpha}}_{3}^{**}$ |
---|---|---|---|---|---|---|

1 | $0.875$ | $0.125$ | $0.0$ | $0.4531$ | $0.0375$ | $0.5769$ |

2 | $0.8098$ | $0.125$ | $0.0651$ | $0.4368$ | $0.0776$ | $0.5719$ |

3 | $0.7446$ | $0.125$ | $0.1303$ | $0.4205$ | $0.1157$ | $0.5668$ |

4 | $0.6794$ | $0.125$ | $0.1995$ | $0.4042$ | $0.1548$ | $0.5618$ |

5 | $0.6142$ | $0.125$ | $0.2607$ | $0.3879$ | $0.1939$ | $0.5568$ |

6 | $0.5491$ | $0.125$ | $0.3258$ | $0.3716$ | $0.2330$ | $0.5518$ |

**Table 4.**Aggarwal and Khan [33] for Player II.

# | ${\mathit{y}}_{1}^{**}$ | ${\mathit{y}}_{2}^{**}$ | ${\mathit{y}}_{3}^{**}$ | ${\mathit{\gamma}}_{1}^{**}$ | ${\mathit{\gamma}}_{2}^{**}$ | ${\mathit{\gamma}}_{3}^{**}$ |
---|---|---|---|---|---|---|

1 | $0.625$ | $0.0$ | $0.375$ | $0.5468$ | $0.2875$ | $0.1442$ |

2 | $0.6299$ | $0.0249$ | $0.3451$ | $0.5337$ | $0.3064$ | $0.1442$ |

3 | $0.6349$ | $0.0498$ | $0.3152$ | $0.5207$ | $0.3253$ | $0.1442$ |

4 | $0.6399$ | $0.0747$ | $0.2853$ | $0.5076$ | $0.3442$ | $0.1442$ |

5 | $0.6449$ | $0.0996$ | $0.2554$ | $0.4945$ | $0.3632$ | $0.1442$ |

6 | $0.6499$ | $0.1245$ | $0.2255$ | $0.4814$ | $0.3821$ | $0.1442$ |

**Table 5.**Nishizaki and Sakwaw [26].

Player I | Player II | ||||||

${x}_{1}^{*}$ | ${x}_{2}^{*}$ | ${x}_{3}^{*}$ | $\lambda $ | ${y}_{1}^{*}$ | ${y}_{2}^{*}$ | ${y}_{3}^{*}$ | $\sigma $ |

$0.3860$ | $0.1250$ | $0.48897$ | $0.33088$ | $0.25595$ | $0.3469$ | $0.3972$ | $0.5804$ |

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**MDPI and ACS Style**

Bashir, Z.; Wątróbski, J.; Rashid, T.; Sałabun, W.; Ali, J.
Intuitionistic-Fuzzy Goals in Zero-Sum Multi Criteria Matrix Games. *Symmetry* **2017**, *9*, 158.
https://doi.org/10.3390/sym9080158

**AMA Style**

Bashir Z, Wątróbski J, Rashid T, Sałabun W, Ali J.
Intuitionistic-Fuzzy Goals in Zero-Sum Multi Criteria Matrix Games. *Symmetry*. 2017; 9(8):158.
https://doi.org/10.3390/sym9080158

**Chicago/Turabian Style**

Bashir, Zia, Jarosław Wątróbski, Tabasam Rashid, Wojciech Sałabun, and Jawad Ali.
2017. "Intuitionistic-Fuzzy Goals in Zero-Sum Multi Criteria Matrix Games" *Symmetry* 9, no. 8: 158.
https://doi.org/10.3390/sym9080158