Advances in Fuzzy Systems

Volume 2016 (2016), Article ID 6475403, 12 pages

http://dx.doi.org/10.1155/2016/6475403

## An Efficient Ranking Technique for Intuitionistic Fuzzy Numbers with Its Application in Chance Constrained Bilevel Programming

^{1}Department of Mathematics, University of Kalyani, Kalyani 741235, India^{2}Department of Mathematics, Government College of Engineering and Textile Technology, Serampore 712201, India

Received 22 November 2015; Revised 24 March 2016; Accepted 3 April 2016

Academic Editor: Kemal Kilic

Copyright © 2016 Animesh Biswas and Arnab Kumar De. 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

The aim of this paper is to develop a new ranking technique for intuitionistic fuzzy numbers using the method of defuzzification based on probability density function of the corresponding membership function, as well as the complement of nonmembership function. Using the proposed ranking technique a methodology for solving linear bilevel fuzzy stochastic programming problem involving normal intuitionistic fuzzy numbers is developed. In the solution process each objective is solved independently to set the individual goal value of the objectives of the decision makers and thereby constructing fuzzy membership goal of the objectives of each decision maker. Finally, a fuzzy goal programming approach is considered to achieve the highest membership degree to the extent possible of each of the membership goals of the decision makers in the decision making context. Illustrative numerical examples are provided to demonstrate the applicability of the proposed methodology and the achieved results are compared with existing techniques.

#### 1. Introduction

The concept of bilevel programming problem (BLPP) was first introduced by Candler and Townsley [1]. The BLPP is considered as a class of optimization problems where two decision makers (DMs) locating at two different hierarchical levels independently control a set of decision variables paying serious attention to the benefit of the others in a highly conflicting decision making situation. The upper level DM is termed as leader and the lower level DM as follower. There are several applications of BLPP in many real life problems such as agriculture, biofuel production, economic systems, finance, engineering, banking, management sciences, and transportation problem. Several methods were proposed to solve BLPPs by different researchers [2, 3] in the past. But these traditional approaches are unable to provide a satisfactory solution if the parameter values involved with a BLPP inevitably contain some uncertain data or linguistic information. Stochastic programming (SP) and fuzzy programming (FP) are two powerful techniques to handle such type of problems. Using probability theory, Dantzig [4] introduced SP. The SP was developed in various directions like chance constrained programming (CCP), recourse programming, multiobjective SP, and so forth. Charnes and Cooper [5] developed the concept of CCP.

Again, from the viewpoint of uncertainty or fuzziness involved in human’s judgments, Zimmermann [6] first applied fuzzy set theory [7] in decision making problems with several conflicting objectives. The concept of membership functions in BLPPs was introduced by Lai and Hwang [8]. Lai’s solution concept was then extended by Shih et al. [9] and a supervised search procedure with the use of max-min operator of Bellman and Zadeh [10] was proposed. The basic concept of this procedure is that the follower optimizes his/her objective function, taking into consideration leader’s goal. Recently, Lodwick and Kacprizyk [11] developed a methodology for solving decision making problems under fuzziness. The main difficulty of FP approach is that the objectives of the DMs are conflicting. So there is possibility of rejecting the solution again and again by the DMs and the solution process is continued by redefining the membership functions repeatedly until a satisfactory solution is obtained. This makes the solution process a very lengthy and tedious one. To remove these difficulties fuzzy goal programming (FGP) [12–14] is used as an efficient tool for making decision in an imprecisely defined multiobjective decision making (MODM) arena. Baky [15] developed a FGP technique for solving multiobjective multilevel programming problem.

Also it is observed that the fuzzy sets (FSs) are not always capable of dealing with lack of knowledge with respect to degrees of membership. Realizing the fact Atanassov [16–18] introduced the concept of intuitionistic FSs (IFSs) by implementing a nonmembership degree which can handle the drawback of FSs and express more abundant and flexible information than the FSs. In recent years, there is a growing interest in the study of decision making problems with intuitionistic fuzzy numbers (IFNs) [19–21] and with interval-valued intuitionistic fuzzy information [22–25].

The ranking of IFNs [26, 27] plays an important role in dealing with IFNs as ranking of fuzzy numbers (FNs). Grzegoraewski [28] suggested some methods for measuring distances between IFNs and interval-valued FNs, based on Hausdorff metric. The methodology for solving intuitionistic fuzzy linear programming problems with triangular IFNs (TIFNs) was developed by Dubey and Mehra [29] by converting the model into crisp linear programming problem. Li [30] developed a ratio ranking method for the TIFNs. Nehi [31] put forward a new ordering method of IFNs in which two characteristic values for IFNs are defined by the integral of the inverse fuzzy membership and nonmembership functions multiplied by the grade with powered parameters. Recently, numerous ranking methods for IFNs have been proposed in literature to rank IFNs [32–34]. Although many defuzzification methods have already been proposed so far, no method gives a right effective defuzzification output. Most of the existing defuzzification methods tried to make the estimation of IFNs in an objective way. This paper proposes a method using the concept probability density function of IFNs and Mellin’s transform [35, 36] to find the ranking of normal IFNs. This ranking method removes the ambiguous outcomes and distinguishes the alternatives clearly.

In fuzzy BLPP [37] it is sometimes realized that the concept of membership function does not provide satisfactory solutions in a highly conflicting decision making situation. In this context IFNs can be used to capture both the membership and nonmembership degrees of uncertainties of both the DMs. Also there are some real world situations, where randomness and fuzziness occur simultaneously. The decision making problem having such type of ambiguous information is known as fuzzy stochastic programming problem. Many researchers [38–41] derived different methods to solve such type of decision making problems. But FGP approach for solving fuzzy stochastic linear BLPP with IFNs is yet to appear in the literature.

In the present study FGP process is adopted for solving bilevel intuitionistic FP problems where the parameters are expressed in terms of normal IFNs. A ranking technique for normal IFNs is first proposed and then using the proposed technique the model is converted into a deterministic problem. The individual optimal value of each objective is found in isolation to construct the fuzzy membership goals of each of the objectives. Finally, FGP model is developed for the achievement of highest degree of each of the defined membership goals to the extent possible by minimizing group regrets in the decision making context. To explore the potentiality of the proposed approach, two illustrative examples are considered and solved and the achieved solutions are compared with the predefined technique developed by Dubey and Mehra [29].

#### 2. Preliminaries

In this section some basic concepts on FNs, triangular FNs (TFNs), IFNs, and triangular IFNs (TIFNs) are discussed.

##### 2.1. FN [42]

A fuzzy set defined on the set of real numbers, , is said to be an FN if its membership function satisfies the following characteristics:(i) is continuous.(ii) for all .(iii) is strictly increasing on and strictly decreasing on .(iv) for all , where .

##### 2.2. TFN [43]

An FN is said to be TFN if its membership function is given by

The TFN can be expressed in the form of Figure 1.