A Computational Method for Fuzzy Arithmetic Operations on Bipolar Fuzzy Numbers and Its Application

Subject Areas : Statistics

Fazlollah Abbasi ¹ , sahleh shakeri ²

1 - Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran
2 - Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University,Amol, Iran

Received: 2019-10-26 Accepted : 2020-01-14 Published : 2021-09-23

Keywords: اعداد فازی دو قطبی, معادله خطی فازی دو قطبی, حساب فازی دو قطبی, اعمال انتقال-میانگین,

Abstract :

A bipolar fuzzy set is a powerful tool for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to the fuzzy model. We define certain notions, including a bipolar fuzzy number and bipolar fuzzy arithmetic. In this paper we propose the new fuzzy arithmetic operations based on on bipolar fuzzy numbers from [1,2]. We define 2- dipole fuzzy numbers and cutting them. The properties of these propose operations and their fundamental qualities are discussed in detail. Several illustrative examples were given to show the accomplishment and ability of the proposed method. At the end it is shown that the solution of the proposed method in comparison to other methods of solving fuzzy equations are more realistic, that is, they have smaller support. Further, we analyze our new approach to find the solutions of a bipolar fuzzy linear equations. In this paper, in addition to familiarity with bipolar fuzzy number operations based on transmission- average and presenting practical solutions for calculations in specific cases, this problem is identified.

References:

[1] Abbasi, F., Tofigh Allahviranloo, and Saeid Abbasbandy. "A new attitude coupled with fuzzy thinking to fuzzy rings and fields." Journal of Intelligent & Fuzzy Systems 29.2 (2015): 851-861.

 

[2] Abbasi, F., S. Abbasbandy, and J. J. Nieto. "A new and efficient method for elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers." Journal of Fuzzy Set Valued Analysis 2 (2016): 156-173.

 

[3] Alghamdi, M. A., Noura Omair Alshehri, and Muhammad Akram. "Multi-criteria decision-making methods in bipolar fuzzy environment." International Journal of Fuzzy Systems 20.6 (2018): 2057-2064.

 

[4] Akram, Muhammad, and Maham Arshad. "A novel trapezoidal bipolar fuzzy TOPSIS method for group decision-making." Group Decision and Negotiation 28.3 (2019): 565-584.

 

[5] Akram, Muhammad, Ghulam Muhammad, and Tofigh Allahviranloo. "Bipolar fuzzy linear system of equations." Computational and Applied Mathematics 38.2 (2019): 69.

 

[6] Allahviranloo, Tofigh, Irina Perfilieva, and F. Abbasi. "A new attitude coupled with fuzzy thinking for solving fuzzy equations." Soft Computing 22.9 (2018): 3077-3095.

 

[7] Biacino, Loredana, and Ada Lettieri. "Equations with fuzzy numbers." Information Sciences 47.1 (1989): 63-76.

 

[8] Buckley, James J. "Solving fuzzy equations." Fuzzy sets and systems 50.1 (1992): 1-14.

 

[9] Buckley, J. J., and Yunxia Qu. "Solving linear and quadratic fuzzy

equations." Fuzzy sets and systems 38.1 (1990): 43-59.

 

[10] Buckley, James J., Esfandiar Eslami, and Yoichi Hayashi. "Solving fuzzy equations using neural nets." Fuzzy Sets and Systems 86.3 (1997): 271-278.

 

[11] Chen, Juanjuan, et al. "m-Polar Fuzzy Sets: An Extension of Bipolar Fuzzy Sets." The Scientific World Journal (2014).

 

[12] Dubois D, Prade H, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.

 

[13] Dubois D, Prade H, Fuzzy numbers: an overview In: Bezdek, J. C., ed., Analysis of Fuzzy Information Vol.1: Mathematics and Logic. CRC Press, Boca, Raton, FL, 1987, 3-29

 

[14] Lee, K. M. "Bipolar-valued fuzzy sets and their basic operations: In: Proceedings of the International Conference." (2000): 307-317.

 

[15] Moore, Ramon E. Methods and applications of interval analysis. Society for Industrial and Applied Mathematics, 1979.

 

[16] Mizumoto, M., and K. Tanaka. "The four operations of arithmetic on fuzzy numbers." Syst. Comput. Controls 7.5 (1976): 73-81.

 

[17] Sanchez, Elie. "Solution of fuzzy equations with extended operations." Fuzzy sets and Systems 12.3 (1984): 237-248.

 

[18] Zadeh, Lotfi A. "Fuzzy sets." Information and control 8.3 (1965): 338-353.

 

[19] Zhang, Wen-Ran. "Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis." NAFIPS/IFIS/NASA'94. Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intellige. IEEE, 1994.

 

[20] Zhang, W-R. "(Yin)(Yang) bipolar fuzzy sets." 1998 IEEE International Conference on Fuzzy Systems Proceedings. IEEE World Congress on Computational Intelligence (Cat. No. 98CH36228). Vol. 1. IEEE, 1998.