Elsevier

Information Sciences

Volume 381, March 2017, Pages 341-351
Information Sciences

Multiattribute decision making based on interval-valued intuitionistic fuzzy values and linear programming methodology

https://doi.org/10.1016/j.ins.2016.11.010Get rights and content

Abstract

In recent years, some multiattribute decision making (MADM) methods have been presented based on interval-valued intuitionistic fuzzy sets (IVIFSs). In this paper, we propose a new MADM method based on interval-valued intuitionistic fuzzy values (IVIFVs) and the linear programming methodology, where the weights of attributes and the evaluating values of attributes of the alternatives given by the decision maker are represented by IVIFVs. The linear programming methodology is used to obtain the optimal weights of the attributes. The proposed method has the advantage that it can overcome the drawbacks of the existing MADM methods for MADM in interval-valued intuitionistic fuzzy environments. The proposed method provides us with a very useful way for MADM in interval-valued intuitionistic fuzzy environments.

Introduction

In [3], Attanassov proposed the concepts of intuitionistic fuzzy sets (IFSs), where the degree of an element belonging to an IFS is represented by an intuitionistic fuzzy value (IFV). That is, the membership degree and the non-membership degree of an element belonging to an IFS are represented by crisp values in [0, 1], respectively. In [4], Atanassov and Gargov proposed the concepts of interval-valued intuitionistic fuzzy sets (IVIFSs), which are the extension of IFSs, where the degree of an element belonging to an IVIFS is represented by an interval-valued intuitionistic fuzzy value (IVIFV). That is, the membership degree and the non-membership degree of an element belonging to an IVIFS are represented by interval values in [0, 1], respectively. In recent years, some multiattribute decision making (MADM) methods based on IVIFSs have been presented [5], [6], [7], [8], [9], [10], [11], [12], [15], [17], [28], [31], [32], [33], [37], [38]. In [5], Chen presented an interval-valued intuitionistic fuzzy qualitative flexible (QUALIFLEX) multiple criteria decision making method with a likelihood-based comparison approach for multiple criteria decision analysis, where likelihood of fuzzy preference relations are used to compare interval-valued intuitionistic fuzzy numbers. In [6], Chen presented a prioritized aggregation operator-based method to handling multiple criteria decision making problems in which there exists a prioritization relationship over evaluative criteria. In [7], Chen presented a method for multiple criteria decision analysis using a likelihood-based outranking method based on interval-valued intuitionistic fuzzy sets. In [8], Chen presented an interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions for multiple criteria decision making analysis. In [9], Chen and Chiou presented a MADM method based on interval-valued intuitionistic fuzzy sets, particle swarm optimization (PSO) techniques and the evidential reasoning methodology. In [10], Chen et al. presented a MADM method based on their proposed interval-valued intuitionistic fuzzy weighted average operator and their proposed fuzzy ranking method for IVIFVs. In [11], Chen and Tsai presented a MADM method based on their proposed interval-valued intuitionistic fuzzy geometric averaging operators. In [12], Chen et al. presented a multicriteria fuzzy decision making method based on IVIFSs, where IVIFVs are used to represent the evaluating values of attributes of the alternatives given by the decision maker. In [15], Garg presented a generalized improved score function of IVIFSs by using the idea of weighted average of the degree of hesitation between their membership functions for multiple criteria decision making. In [17], Li presented a nonlinear-programming method based on the technique for order preference by similarity to ideal solution (TOPSIS) to solve MADM problems, where the ratings of alternatives on attributes and the weights of attributes are expressed by IVIFSs. In [28], Sahin presented a fuzzy multicriteria decision making method based on the improved accuracy function for IVIFSs by taking into account the hesitancy degree of IVIFSs. In [31], Tsao and Chen presented a projection-based compromising method for multiple criteria decision analysis with interval-valued intuitionistic fuzzy information, where the concept of projections considers the distance and the included angle between evaluative ratings of alternative actions with respect to a criterion. In [32], Wan and Li presented a fuzzy mathematical programming method for solving heterogeneous MADM problems with interval-valued intuitionistic fuzzy truth degrees based on the linear programming technique for multidimensional analysis of preference. In [33], Wang et al. presented a MADM method with interval-valued intuitionistic fuzzy assessments and incomplete attribute weight information, where individual assessments are represented by interval-valued intuitionistic fuzzy numbers (IVIFNs). In [37], Ze-Sui presented methods for aggregating interval-valued intuitionistic fuzzy information for MADM, where some operational laws of IVIFNs are defined, some aggregation operators (including the interval-valued intuitionistic fuzzy weighted arithmetic aggregation operator and the interval-valued intuitionistic fuzzy weighted geometric aggregation operator) are proposed and the score function and the accuracy function of IVIFNs are defined. In [38], Zhitao and Yingjun presented a MADM method in the frame of IVIFSs based on their proposed accuracy function to solve MADM problems in which both ratings of alternatives with respect to attributes and weights of attributes are expressed by IVIFSs.

However, in [9], Chen and Chiou pointed out that Li's method [17] and Zhitao and Yingjun's method [38] for MADM based on IVIFSs have the drawback of “the division by zero problem”, such that they cannot get the preference order of alternatives in some situations. However, we also find that Chen and Chiou's method [9] for MADM based on IVIFSs also has the drawback that it gets unreasonable preference orders of alternatives in some situations. Therefore, the motivation of this paper is to develop a new MADM method to overcome the drawbacks of Chen and Chiou's method [9], Li's method [17] and Zhitao and Yingjun's method [38] for MADM in interval-valued intuitionistic fuzzy environments.

In this paper, we propose a new MADM method based on IVIFVs [4] and the linear programming methodology [27], where the weights of attributes and the evaluating values of attributes of the alternatives given by the decision maker are represented by IVIFVs. The linear programming methodology is used to obtain the optimal weights of the attributes. The proposed method can overcome the drawbacks of Chen and Chiou's method [9], Li's method [17] and Zhitao and Yingjun's method [38] for MADM in interval-valued intuitionistic fuzzy environments. The proposed method provides us with a very useful way for dealing with MADM problems in interval-valued intuitionistic fuzzy environments.

The rest of this paper is organized as follows. In Section 2, we briefly review the concept of IVIFSs and IVIFVs from [4], briefly review the concepts of the linear programming methodology from [27], briefly review the score function of IVIFVs from [37], and briefly review the interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator of IVIFVs from [37]. In Section 3, we propose a new MADM method based on IVIFVs and the linear programming methodology. In Section 4, we make a comparison of the experimental results of the proposed method with the ones of the existing methods. The conclusions are discussed in Section 5.

Section snippets

Preliminaries

In this section, we briefly review the concept of IVIFSs and IVIFVs from [4], briefly review the concepts of the linear programming methodology from [27], briefly review the score function of IVIFVs from [37], and briefly review the interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator of IVIFVs from [37].

In [4], Atanassov and Gargov proposed the theory of IVIFSs. Let P={xi,μP(xi),υP(xi)|xiX} be an IVIFS in the universe of discourse X = {x1, x2, …, xn}, where μP and υP are

A new MADM method based on IVIFVs and the linear programming methodology

In this section, we propose a new MADM method based on IVIFVs and the linear programming methodology. Assume that x1, x2, …, and xm are alternatives, assume that A1, A2, …, and An are attributes, and assume that the weight of attribute Aj given by the decision maker is represented by an interval-valued intuitionistic fuzzy weight w˜j, where w˜j=([hj,yj],[zj,gj]) is an IVIFV, 0 ≤ hjyj ≤ 1, 0 ≤ zjgj ≤ 1, 0 ≤ yj + gj ≤ 1 and 1 ≤ jn. Let M˜=(m˜ij)m×n be a decision matrix given by the

Illustrative examples

In the following, we use some examples to compare the experimental results of the proposed method with the ones of Chen and Chiou's method [9], Li's method [17] and Zhitao and Yingjun's method [38].

Example 4.1

[9], [17], [38]: Assume that there are four alternatives x1, x2, x3, x4, assume that there are three attributes A1, A2, A3, assume that the interval-valued intuitionistic fuzzy weights w˜1, w˜2 and w˜3 of the attributes A1 , A2 and A3 given by the decision maker represented by IVIFVs are ([0.10,

Conclusions

In this paper, we have proposed a new multiattribute decision making (MADM) method based on IVIFVs and the linear programming methodology. The weights of attributes and the evaluating values of attributes of the alternatives given by the decision maker are represented by IVIFVs. From the experimental results, we can see that the proposed method can overcome the drawbacks of Chen and Chiou's method [9], Li's method [17] and Zhitao and Yingjun's method [38] for MADM in interval-valued

Acknowledgments

This work is supported by the Ministry of Science and Technology, Republic of China, under Grant MOST 104-2221-E-011-084-MY3.

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