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

^{1}

^{2}

^{3}

^{4}

^{*}

## 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

- Chanas, S. The use of parametric programming in fuzzy linear programming. Fuzzy Sets Syst.
**1983**, 11, 229–241. [Google Scholar] [CrossRef] - Werner, B. An interactive fuzzy programming systems. Fuzzy Sets Syst.
**1987**, 23, 131–147. [Google Scholar] [CrossRef] - Werner, B. Interactive multiple object programming subject to flexible constraints. Eur. J. Oper. Res.
**1987**, 31, 342–349. [Google Scholar] [CrossRef] - Zimmerman, H.J. Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst.
**1978**, 1, 45–55. [Google Scholar] [CrossRef] - Zimmerman, H.J. Latent connectives in human decision making. Fuzzy Sets Syst.
**1980**, 4, 291–298. [Google Scholar] [CrossRef] - Zimmerman, H.J. Application of fuzzy set theory to mathematical programming. Inf. Sci.
**1985**, 36, 29–58. [Google Scholar] [CrossRef] - Bector, C.R.; Chandra, S. Fuzzy Mathematical Programming and Fuzzy Matrix Games; Springer: Berlin, Germany, 2005; Volume 169. [Google Scholar]
- Faizi, S.; Rashid, T.; Sałabun, W.; Zafar, S.; Wątróbski, J. Decision making with uncertainty using hesitant fuzzy sets. Int. J. Fuzzy Syst.
**2017**, 1–11. [Google Scholar] [CrossRef] - Sharifian, S.; Chakeri, A.; Sheikholeslam, F. Linguisitc representation of Nash equilibriums in fuzzy games. In Proceedings of the 2010 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), Toronto, ON, Canada, 12–14 July 2010; pp. 1–6. [Google Scholar]
- Faizi, S.; Sałabun, W.; Rashid, T.; Wątróbski, J.; Zafar, S. Group decision-making for hesitant fuzzy sets based on characteristic objects method. Symmetry
**2017**, 9, 136. [Google Scholar] [CrossRef] - Sun, L.; Liu, Y.; Zhang, B.; Shang, Y.; Yuan, H.; Ma, Z. An integrated decision-making model for transformer condition assessment using game theory and modified evidence combination extended by D numbers. Energies
**2016**, 9, 697. [Google Scholar] [CrossRef] - Chen, B.S.; Tseng, C.S.; Uang, H.J. Fuzzy differential games for nonlinear stochastic systems: Suboptimal approach. IEEE Trans. Fuzzy Syst.
**2002**, 10, 222–233. [Google Scholar] [CrossRef] - Chakeri, A.; Dariani, A.N.; Lucas, C. How can fuzzy logic determine game equilibriums better? In Proceedings of the 4th International IEEE Conference Intelligent Systems, Varna, Bulgaria, 6–8 September 2008; Volume 1, pp. 2–51. [Google Scholar]
- Chakeri, A.; Habibi, J.; Heshmat, Y. Fuzzy type-2 Nash equilibrium. In Proceedings of the Computational Intelligence for Modelling Control & Automation, Vienna, Austria, 10–12 December 2008; pp. 398–402. [Google Scholar]
- Chakeri, A.; Sadati, N.; Dumont, G.A. Nash equilibrium strategies in fuzzy games. In Game Theory Relaunched; InTech: Rijeka, Croatia, 2013. [Google Scholar]
- Chakeri, A.; Sadati, N.; Sharifian, S. Fuzzy Nash equilibrium in fuzzy games using ranking fuzzy numbers. In Proceedings of the 2010 IEEE International Conference on Fuzzy Systems (FUZZ), Barcelona, Spain, 18–23 July 2010; pp. 1–5. [Google Scholar]
- Chakeri, A.; Sheikholeslam, F. Fuzzy Nash equilibriums in crisp and fuzzy games. IEEE Trans. Fuzzy Syst.
**2013**, 21, 171–176. [Google Scholar] [CrossRef] - Garagic, D.; Cruz, J.B. An approach to fuzzy noncooperative nash games. J. Optim. Theory Appl.
**2003**, 118, 475–491. [Google Scholar] [CrossRef] - Qiu, D.; Xing, Y.; Chen, S. Solving multi-objective matrix games with fuzzy payoffs through the lower limit of the possibility degree. Symmetry
**2017**, 9, 130. [Google Scholar] [CrossRef] - Tan, C.; Jiang, Z.Z.; Chen, X.; Ip, W.H. A Banzhaf function for a fuzzy game. IEEE Trans. Fuzzy Syst.
**2014**, 22, 1489–1502. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Sałabun, W.; Piegat, A. Comparative analysis of MCDM methods for the assessment of mortality in patients with acute coronary syndrome. Artif. Intell. Rev.
**2016**, 1–15. [Google Scholar] [CrossRef] - Bellman, R.E.; Zadeh, L.A. Decision making in fuzzy environment. Manag. Sci.
**1970**, 17, B141–B146. [Google Scholar] [CrossRef] - Tanaka, H.; Okuda, T.; Asai, K. On fuzzy mathematical programming. J. Cybern.
**1974**, 3, 37–46. [Google Scholar] [CrossRef] - Campos, L. Fuzzy linear programming models to solve fuzzy matrix games. Fuzzy Sets Syst.
**1989**, 32, 275–289. [Google Scholar] [CrossRef] - Nishizaki, I.; Sakwaw, M. Fuzzy and Multi-Criteria Games for Conflict Resolution; Physica-Verlag: Berlin, Germany, 2001. [Google Scholar]
- Sakawa, M.; Nishizaki, I. Max-min solutions for fuzzy multiobjective matrix games. Fuzzy Sets Syst.
**1994**, 61, 265–275. [Google Scholar] [CrossRef] - Bector, C.R.; Chandra, S.; Vidyottama, V. Matrix gameswith fuzzy goals and fuzzy linear programming duality. Fuzzy Optim. Decis. Mak.
**2004**, 3, 255–269. [Google Scholar] [CrossRef] - Vijay, V.; Chandra, S.; Bector, C.R. Matrix games with fuzzy goals and fuzzy payoffs. Omega
**2005**, 33, 425–429. [Google Scholar] [CrossRef] - Li, D.F. Lexicographic method for matrix games with payoffs of triangular fuzzy numbers. Int. J. Uncertain. Fuzziness Knowl. Based Syst.
**2008**, 16, 371–389. [Google Scholar] [CrossRef] - Li, D.F. Linear programming approach to solve interval-valued matrix games. Omega
**2011**, 39, 655–666. [Google Scholar] [CrossRef] - Nayak, P.K.; Pal, M. Linear programming technique to solve two person matrix gams with interval pay-offs. Asia Pac. J. Oper. Res.
**2009**, 26, 285–305. [Google Scholar] [CrossRef] - Aggarwal, A.; Khan, I. Solving multi-criteria fuzzy matrix games via multi-criteria linear programming approach. Kybernetika
**2016**, 52, 153–168. [Google Scholar] - Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Atanassov, K. Intuitionistic Fuzzy Sets. Theory and Applications; Physica-Verlag: Heidelberg, Germany, 1999. [Google Scholar]
- Aggarwal, A.; Mehra, A.; Chandra, S. Application of linear programming with I-fuzzy sets to matrix games with I-fuzzy goals. Fuzzy Optim. Decis. Mak.
**2012**, 11, 465–480. [Google Scholar] [CrossRef] - Angelov, P.P. Optimization in an intuitionistic fuzzy environment. Fuzzy Sets Syst.
**1997**, 86, 299–306. [Google Scholar] [CrossRef] - Chakrabortty, S.; Pal, M.; Nayak, P.K. Intuitionistic fuzzy optimization technique for Pareto optimal solution of manufacturing inventory models with shortages. Eur. J. Oper. Res.
**2013**, 228, 381–387. [Google Scholar] [CrossRef] - Dubey, D.; Chandra, S.; Mehra, A. Fuzzy linear programming under interval uncertainty based on IFS representation. Fuzzy Sets Syst.
**2012**, 188, 68–87. [Google Scholar] [CrossRef] - Razim, J.; Jafarian, E.; Amin, S.H. An intuitionistic fuzzy goal programming approach for finding pareto-optimal solutions to multi-criteria programming problems. Exp. Syst. Appl.
**2016**, 65, 181–193. [Google Scholar] [CrossRef] - Fernandez, F.R.; Puerto, J. Vector linear programming in zero sum multicriteria matrix games. J. Optim. Theory Appl.
**1996**, 89, 115–127. [Google Scholar] [CrossRef] - Garg, H.; Rani, M.; Sharma, S.P.; Vishwakarma, Y. Intuitionistic fuzzy optimization technique for solving multi-criteria reliability optimization problems in interval environment. Exp. Syst. Appl.
**2014**, 41, 3157–3167. [Google Scholar] [CrossRef] - Steuer, R.E. Multi Criteria Optimization: Theory, Computation and Application; John Wiley: New York, NY, USA, 1986. [Google Scholar]
- Bubeck, S. Convex optimization: Algorithms and complexity. Found. Trends Mach. Learn.
**2015**, 8, 231–357. [Google Scholar] [CrossRef] - Cook, C.R. Zero-sum games with multiple goals. Nav. Res. Logist. Quart.
**1976**, 23, 615–622. [Google Scholar] [CrossRef]

# | 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 $ |

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

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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