# Decision Making Approach to Fuzzy Linear Programming (FLP) Problems with Post Optimal Analysis

DOI: 10.4018/IJORIS.2015100105
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## Abstract

This paper finds solutions to the fuzzy linear program where some parameters are fuzzy numbers. In practice, there are many problems in which all decision parameters are fuzzy numbers, and such problems are usually solved by either probabilistic programming or multi objective programming methods. Unfortunately all these methods have shortcomings. In this paper, using the concept of comparison of fuzzy numbers, the author introduces a very effective method for solving these problems. This paper extends linear programming based problem in fuzzy environment. With the problem assumptions, the optimal solution can still be theoretically solved using the simplex based method. To handle the fuzzy decision variables can be initially generated and then solved and improved sequentially using the fuzzy decision approach by introducing robust ranking technique. The model is illustrated with an application and a post optimal analysis approach is obtained. The proposed procedure was programmed with MATLAB (R2009a) version software, the four dimensional slice diagram is represented to the application. Finally, numerical example is presented to illustrate the effectiveness of the theoretical results, and to gain additional managerial insights.
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## 1. Introduction

Over the last few years, more and more manufacturers had applied the optimization technique most frequently in linear programming to solve real-world problems and there it is important to introduce new tools in the approach that allow the model to fit into the real world as much as possible. Any linear programming model representing real-world situations involves a lot of parameters whose values are assigned by experts’ opinion, and in the conventional approach, they are required to fix an exact value to the aforementioned parameters. However, both experts and the decision maker frequently do not precisely know the value of those parameters. If exact values are suggested these are only statistical inference from past data and their stability is doubtful, so the parameters of the problem are usually defined by the decision maker in an uncertain space. Therefore, it is useful to consider the knowledge of experts’ opinion about the parameters as fuzzy data. Fuzzy data helps the decision maker to take the decision in open ended space; since the market is volatile it is very difficult to take the optimum decision of the decision parameters. In the mean time fuzzy related data helps for obtaining the optimal solution and then the post optimal solution gives the managerial implications for the given problem.

Two significant questions may be found in these kinds of problems: how to handle the relationship between the fuzzy parameters, and how to find the optimal values for the fuzzy multi-objective function. The answer is related to the problem of ranking fuzzy numbers.

In fuzzy decision making problems, the concept of optimizing the decision was introduced by (Bellman & Zadeh, 1970). (Zimmerman, 1978) presented a fuzzy approach to multi-objective linear programming problems in his classical paper. (Lai & Hwang, 1992) considered the situations where all parameters are in fuzzy number. (Lai & Huang, 1992) assume that the parameters have a triangular possibility distribution. (Gani, Duraisamy & Veeramani, 2009) introduce fuzzy linear programming problem based on L-R fuzzy number. (Jimenez, Arenas, Bilbao & Rodriguez, 2005) propose a method for solving linear programming problems where all coefficients are, in general, fuzzy numbers and using linear ranking technique. (Bazaraa, Jarvis & Sherali, 1990) and (Nasseri, Ardil, Yazdani & Zaefarjan, 2005) define linear programming problems with fuzzy numbers and simplex method is used for finding the optimal solution of the fuzzy problem. (Rangarajan & Solairaju, 2010) compute improved fuzzy optimal Hungarian assignment problems with fuzzy numbers by applying robust ranking techniques to transform the fuzzy assignment problem to a crisp one. (Pattnaik, 2012) presented a fuzzy approach to several linear and nonlinear inventory models. (Swarup, Gupta & Mohan, 2006) explain the method to obtain sensitivity analysis or post optimality analysis of the different parameters in the linear programming problems. (Ebrahimnejad, Nasseri & Mansourzadeh 2011) introduce an efficient approach to overcome this shortcoming. The bounded fuzzy primal simplex algorithm starts with a primal feasible basis and moves towards attaining primal optimality while maintaining primal feasibility throughout. This algorithm will be useful for sensitivity analysis using primal simplex tableaus. (Hanafizadeh, Ghaemi & Tavana 2011) study the sensitivity analysis for a class of linear programming (LP) problems with a functional relation among the objective function parameters or those of the right-hand side (RHS). The classical methods and standard sensitivity analysis software packages fail to function when a functional relation among the LP parameters prevail. In order to overcome this deficiency, the authors derive a series of sensitivity analysis formulae and devise corresponding algorithms for different groups of homogenous LP parameters. (Nasseri & Ebrahimnejad, 2011) developed the duality results in fuzzy environment and presented a dual simplex algorithm for solving linear programming problems with trapezoidal fuzzy variables. Here, the authors show that this presented dual simplex algorithm directly using the primal simplex tableau algorithm tenders the capability for sensitivity (or post optimality) analysis using primal simplex tableaus.

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