Certain methods to solve bipolar fuzzy linear system of equations
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Certain methods to solve bipolar fuzzy linear system of equations Muhammad Akram1 · Muhammad Ali1 · Tofigh Allahviranloo2 Received: 17 December 2019 / Revised: 29 June 2020 / Accepted: 7 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this article, we present two different analytical methods based on embedding technique and bipolar fuzzy center to solve bipolar fuzzy linear system (BFLS) of equations. In the first method, to solve BFLS of equations, we replace BFLS of equations by a pair of positive(∗) and negative(•) two n × n crisp linear systems. We provide the necessary and sufficient conditions for the solution of BFLS of equations. In the second method, we use the graphical technique and apply bipolar fuzzy center to draw a graph at some specific end points to solve the BFLS of equations. Further, we develop a technique to solve the fully bipolar fuzzy linear system of equations. We present solutions of some numerical examples to show the effectiveness of the proposed techniques. Keywords Bipolar fuzzy linear system · Bipolar fuzzy number · Fully bipolar fuzzy linear system · Interval model Mathematics Subject Classification 15A06
1 Introduction The notion of fuzzy set was introduced by Zadeh (1965, 1975) to deal with uncertain information. Dubois and Prade (1978) discussed the basic arithmetic operations of fuzzy numbers. Zhang (1998, 1994) introduced the concept of bipolar fuzzy set in (1994) which is an extension of fuzzy set. A bipolar fuzzy set (BFS) is a powerful mathematical tool for expressing
Communicated by Leonardo Tomazeli Duarte.
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Tofigh Allahviranloo [email protected] Muhammad Akram [email protected] Muhammad Ali [email protected]
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Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan
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Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey 0123456789().: V,-vol
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fuzziness and uncertainty. The simultaneous linear system of equations play an important role in different kinds of fields including mathematics, physics, statistics, networking and attribute decision-making. In many places, a system of fuzzy linear equation in bipolar form is used for linear optimization of system when system is represented in bipolar form. Firstly, the fuzzy linear system (FLS) of equations was examined by Friedman et al. (1998) and they used the embedding method to solve fuzzy system. They used the procedure in which n × n FLS of equations whose coefficient matrix is a crisp number matrix and right-side column vector is fuzzy number vector which is replaced by 2n × 2n crisp linear systems. Abbasbandy and Alavi (2005) used another method to solve n × n FLS of equations whose coefficient matrix is a crisp number matrix and right-side column vector is fuzzy number vector. They used a procedure in which n × n FLS of equations is replaced by two n × n crisp linear systems. Allahviranloo (2019) proposed different methods to solve uncerta
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