Optimization of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mtext>LR</a:mtext> </a:math>-Type Fully Bipolar Fuzzy Linear Programming Problems
Muhammad Athar Mehmood, Muhammad Akram, Majed Alharbi, Shahida Bashir
Abstract
In this study, we present a technique to solve <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"> <a:mtext>LR</a:mtext> </a:math> -type fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints. We define <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"> <c:mtext>LR</c:mtext> </c:math> -type bipolar fuzzy numbers and their arithmetic operations. We discuss multiplication of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M4"> <e:mtext>LR</e:mtext> </e:math> -type bipolar fuzzy numbers. Furthermore, we develop a method to solve <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M5"> <g:mtext>LR</g:mtext> </g:math> -type FBFLPPs with equality constraints involving <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M6"> <i:mtext>LR</i:mtext> </i:math> -type bipolar fuzzy numbers as parameters and variables. Moreover, we define ranking for <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M7"> <k:mtext>LR</k:mtext> </k:math> -type bipolar fuzzy numbers which transform the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M8"> <m:mtext>LR</m:mtext> </m:math> -type FBFLPP into a crisp linear programming problem. Finally, we consider numerical examples to illustrate the proposed method.