Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 875460, 8 pages

http://dx.doi.org/10.1155/2015/875460

## The Knowledge of Expert Opinion in Intuitionistic Fuzzy Linear Programming Problem

^{1}PG & Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli, Tamil Nadu 620 020, India^{2}Department of Mathematics, Faculty of Science, Technology and Human Development, Universiti Tun Hussein Onn Malaysia, 86400 Johor, Malaysia^{3}Department of Mathematics, Sri Krishna Arts and Science College (Autonomous), Coimbatore, Tamil Nadu 641 008, India

Received 26 March 2015; Revised 30 June 2015; Accepted 2 July 2015

Academic Editor: Yan-Jun Liu

Copyright © 2015 A. Nagoorgani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In real life, information available for certain situations is vague and such uncertainty is unavoidable. One possible solution is to consider the knowledge of experts on the parameters involved as intuitionistic fuzzy data. We examine a linear programming problem in which all the coefficients are intuitionistic in nature. An approach is presented to solve an intuitionistic fuzzy linear programming problem. In this proposed approach, a procedure for allocating limited resources effectively among competing demands is developed. An example is given to highlight the illustrated study.

#### 1. Introduction

A linear programming problem is a mathematical program in which the objective function is linear and the constraints consist of linear equalities and linear inequalities. The first and most fruitful industrial applications of linear programming can be found in the petroleum industry, including oil extraction, refining, blending, and distribution. The computational task is then to devise an algorithm for these systems to choose the best schedule of actions from among the possible alternatives. Some people learn to make such selections via intuitive processes. In some cases, making the right selection is mainly a problem of organizing and interpreting facts. Linear programming has proved to be extremely useful for solving certain types of industrial problems for it provides a precise way of using statements of limitations such as “not more than” and “not less than” in mathematical computations. When applying OR methods to industrial problems, for instance, the problems to be modelled and solved are normally quite clear cut, well described, and crisp. They can generally be modelled and solved by using classical linear programming methods. If uncertainty occurs, it can be properly modelled using fuzzy theory.

The linear programming problems in which at least one coefficient is a fuzzy number when one or more coefficients of linear programming problems have uncertain values are known as fuzzy linear programming problems (FLPP). It is regarded to treat uncertainty of optimization problems, such as fuzzy data envelopment analysis and fuzzy network optimization [1, 2]. The fuzzy linear programming problems in which all the parameters as well as the variables are represented by fuzzy numbers are known as fully fuzzy linear programming problems. The main advantage of fuzzy linear programming problems compared to the crisp problem formulation is the fact that the decision maker is not forced into a precise formulation. Over the past decades, solving fuzzy linear programming has become one of the fundamental research subjects in the field of fuzzy sets and systems. Fuzzy linear programming applications in real world situations are numerous and diverse.

Though fuzzy optimization formulations are more flexible, one of the poorly studied problems in this field is the definition of membership degrees. Fuzzy set theory has been widely developed and various modifications have been done. Out of several higher order fuzzy sets, intuitionistic fuzzy sets (IFS) have been found to be highly useful in dealing with vagueness. Here, the degrees of satisfaction and rejection are considered so that the sum of both the values is always less than or equal to one. The concept of IFS was viewed as an alternative approach for imprecise data. Therefore, considering nonmembership function as the complement of membership function developed Intuitionistic Fuzzy Optimization (IFO) problems. The main advantage of IFO problems is that they are given the richest apparatus for the formulation of optimization problems and the solution of IFO problems satisfies the objective with a higher degree of determinacy compared to the fuzzy and crisp cases. In order to avoid unrealistic modelling, the use of intuitionistic fuzzy linear programming problem (IFLPP) can be recommended. Their application implies that the problems will be solved in an interactive way. In this paper, we consider a problem in which all coefficients and variables are intuitionistic fuzzy triangular numbers in nature. In this manner, we want to propose a new matrix-analysis method to improve the efficiency of solving large-scale IFLPP, which will reduce the number of steps in the classical simplex method. The basic idea of this method is to arrange IFLPP data in matrix form and solve for various determinants to obtain the optimum solution of IFLPP. To illustrate the proposed method, numerical example is solved and the obtained result is discussed.

The paper is organized as follows. We present the works related to finding the optimal solution of an intuitionistic fuzzy linear programming problem (IFLPP) in Section 2. Section 3 provides preliminary background on intuitionistic fuzzy sets (IFS), intuitionistic fuzzy numbers (IFN), and IFLPP. The procedure for the proposed method is described in Section 4. An illustrative example is explained briefly in Section 5. Finally, conclusions are presented in Section 6.

#### 2. Related Works

The research towards uncertain systems has attracted a lot of attention [3] especially the adaptive control of linear and nonlinear systems with completely unknown functions. The fuzzy logic systems (FLS), the neural networks (NN), and the fuzzy-neural networks (FNN) are very effective tools for controlling uncertainty systems [4]. As an application, the FLS, NN, and FNN have been widely used in the area of system modeling [5], fuzzy control [6], and fuzzy optimization problems [7]. Fuzzy control, which directly uses fuzzy rules, is the most important application in fuzzy theory. In a practical situation, sometimes it is quite difficult to obtain an optimal solution for fuzzy optimization problems; the use of fuzzy controls helps to find better solutions by the decision maker in order to terminate mathematical programming algorithms [8]. The intuitionistic fuzzy set theory is an extension of the fuzzy set theory by Atanassov [9] and intuitionistic fuzzy linear programming problem (IFLPP) is a special type of fuzzy linear programming problem (FLPP). There are lots of articles in this area which cannot be reviewed completely and only a few of them are reviewed here. Interval valued intuitionistic fuzzy sets were first introduced by Atanassov and Gargov [10]; since then there has been many types of intuitionistic fuzzy numbers (IFN) addressed such as interval valued intuitionistic fuzzy numbers (IVIFN), triangular intuitionistic fuzzy numbers (TIFNs), and trapezoidal intuitionistic fuzzy numbers. Mahapatra and Roy [11] discussed briefly intuitionistic fuzzy numbers and their arithmetic operations. The arithmetic operations and logic operations of triangular intuitionistic fuzzy numbers have been addressed by Wang et al. [12]. Ranking of intuitionistic fuzzy numbers plays a vital role in practical problems and so Li [13] developed a new ranking method based on the concept of a ratio of the index of the ambiguity index. The article by Wu and Chiclana [14] describes new score and accuracy functions for interval valued intuitionistic fuzzy numbers. A ranking procedure for triangular intuitionistic fuzzy numbers was developed by Wan and Dong [15] and its applications to multiattribute decision making was also given. Evaluation and ranking of fuzzy quantities were dealt with by Anzilli et al. [16]. The concept of the FLP was first proposed by Tanaka et al. [17], which were based on the concept of decision analysis in fuzzy environment by Bellman and Zadeh [18]. Zimmermann [19, 20] introduced fuzzy sets in operations research and presented a fuzzy approach to multiobjective linear programming problems. A new concept of the optimization problem under uncertainty was proposed and treated in [21]. On the other hand, Zhu and Xu [22] developed a fuzzy linear programming method to deal with group decision-making problems. The optimal solution for several degrees of feasibility of fuzzy linear and nonlinear programming problems was given by Mohtashami [23]. A real life multiobjective linear programming problem was taken into an intuitionistic fuzzy environment and solved by Nishad and Singh [24]. Moreover, Ye [25] proposed a linear programming model to solve interval valued intuitionistic multicriteria decision-making problems. Li [26] used interval valued intuitionistic fuzzy sets to capture fuzziness in linear programming.

Motivated by these articles, we proposed a study on the solutions of intuitionistic fuzzy linear programming problem (IFLPP). The classical simplex method requires much iteration to solve IFLPP. To overcome this limitation, a new matrix-analysis method is proposed in this paper. The IFLPP is represented in matrix format and various matrix operations are performed to obtain the optimum solution.

#### 3. Preliminaries

In this section, the basic notations and definitions are presented. We start by defining an intuitionistic fuzzy set.

##### 3.1. Intuitionistic Fuzzy Set (IFS)

Given a fixed set , an intuitionistic fuzzy set is defined as which assigns to each element a membership degree and a nonmembership degree under the condition , for all .

##### 3.2. Intuitionistic Fuzzy Number (IFN)

An intuitionistic fuzzy number is(i)an intuitionistic fuzzy subset of the real line;(ii)normal; that is, there is some such that , ;(iii)convex for the membership function , that is; , for every , ;(iv)concave for the nonmembership function , that is; , for every , .

##### 3.3. Triangular Intuitionistic Fuzzy Number (TIFN)

A triangular intuitionistic fuzzy number is an intuitionistic fuzzy set in with the following membership function and nonmembership function (Figure 1):where and , or , . This TIFN is denoted by