Elsevier

Applied Soft Computing

Volume 61, December 2017, Pages 111-121
Applied Soft Computing

A new approach for multiple attribute group decision making with interval-valued intuitionistic fuzzy information

https://doi.org/10.1016/j.asoc.2017.07.008Get rights and content

Highlights

  • An effective algorithm for the MAGDM with IVIFN is proposed.

  • The distinctive feature of the proposed method is its excellent ability to handle IVIFSs with a value of zero.

  • The proposed method provides a new method to improve the quality of the collective matrix.

Abstract

This paper proposes a novel method for multiple attribute group decision making (MAGDM) with interval-valued intuitionistic fuzzy information. The interval-valued intuitionistic fuzzy numbers of each expert preference matrix are first mapped into two dimensions. Thus, the values of each membership degree and non-membership degree are considered as points in the two-dimensional representation. Moreover, the distance between the points represents the variance among the different experts preferences. The preference points of the same character are considered as a point set. We employ the plant growth simulation algorithm (PGSA) to calculate the optimal rally points of every point set, the sum of whose Euclidean distances to other given points is minimal, and these optimal rally points reflect the preferences of the entire expert group. These points are used to establish an expert preference aggregation matrix. Suitable points from the matrix are chosen to constitute an ideal point matrix, a projection method is employed to calculate the sum of its Euclidean distance to the expert preference aggregation matrix, and the score of each alternative is evaluated. Finally, the overall ranking of alternatives is obtained. In addition, this study develops a process to evaluate the pros and cons of different aggregation methods. Two typical examples are presented to illustrate the feasibility and effectiveness of the proposed approach.

Introduction

Multi-attribute group decision making (MAGDM) is a classical decision-making construct applied in various areas such as emergency management, economics, society, etc. [1], [2], [3], [4], [5]. Herrera [6] classified the decision making process into three parts: translate experts preferences, aggregate experts opinions (i.e., establish the collective matrix) and select the best alternative (the aggregation of expert opinion is the core step among these three steps). The experts preference was described using distinct numbers in the early multi-criteria decision making problem. Yager [7] proposed the ordered weighted averaging operators (OWA) to integrate the experts preference, which provided a broader idea for future research on the multi-attribute decision making problem. However, with the increasing complexity of decision making, it is difficult to make the right decision by analyzing the opinion of a single expert. In addition, on account of their limited knowledge related to the problem research area and lack of adequate information as well as time, the experts cannot express their opinions with distinct numbers. Therefore, in recent years, many researchers have begun to develop various new operators to solve the MAGDM problem with attribute values that define interval numbers or linguistic variables. Yu [8] developed the uncertain linguistic power weighted geometric (ULPWG) operator and the uncertain linguistic power ordered weighted geometric (ULPOWG) operator. In his research, he also utilized the proposed operators to develop approaches to solve the uncertain linguistic MAGDM problems. Merigo [9] presented an overview of fuzzy research with bibliometric indicators, which provided a general overview identifying some of the most influential research in this area. However, the novel indicators have certain limitations due to the differences between researchers, the specificity of research area, and the difficulty faced in quantification of key information. Yu [10] developed an effective interval-valued multiplicative intuitionistic fuzzy preference relation, which analyzed the basic operations for interval-valued multiplicative intuitionistic preference information and its aggregation techniques. Further, the aggregation operators for interval-valued multiplicative intuitionistic fuzzy sets (IMIFs) were also developed in this study. Hashemi [11] developed an enhanced version of the ELECTRE method, called ELECTRE III, for multicriteria group decision making under the interval-valued intuitionistic fuzzy environment. And it had an advantage in handling a data set with a high degree of uncertainty. Sahin [12] proposed a novel function of interval-valued intuitionistic fuzzy numbers (IVIFNs) and established a new process by using the proposed function to rank the IVIFNs. However, the proposed method still used the aggregation operators developed by Xu in 2007 to establish the collective matrix, which may not deliver a precise outcome. In addition, certain straightforward methods were proposed to calculate experts weights. Yue [13], [14], [15] proposed several excellent methods for determining weights of experts in MAGDM based on an extended technique for order preference by similarity to an ideal solution (TOPSIS). Liu and LI [16] developed an approach to determine the integrated weights of experts based on interval-valued preference matrices. They used the generalized Fermat point to determine the weights of experts, which delivered good simulation results.

The purpose of this study is to develop a novel method for the accurate aggregation of interval-valued intuitionistic fuzzy matrices (IVIFMs), so that the MAGDM problems with IVIFN can be solved effectively. An iterative algorithm, called plant growth simulation algorithm (PGSA), is used to replace the traditional operators for determining the aggregation of experts preferences. The two-dimensional model of experts preference is established, which places expert preference points into their respective points set. Using the PGSA to calculate the optimal rally points of every point set, the collective matrix is established with the optimal rally point set. Then, the projection method is used to calculate the sum of Euclidean distances between the expert preference aggregation matrix and ideal point matrix. Finally, the best alternative is chosen by obtaining the ranking of alternatives after evaluating the score of each alternative.

The remainder of this paper is organized as follows: Section 2 briefly describes the research problem statement and contributions to this paper. Section 3 introduces the preliminaries, including the interval-valued intuitionistic fuzzy set, the concept of optimal rally point and the principle of the PGSA. Section 4 details the decision making process of the proposed method and focuses on the core steps of the expert preference aggregation. In Section 5, two practical examples are presented to illustrate the efficiency of the proposed method. Section 6 compares the solution results yielded by other relevant methods with those of the proposed method. Section 7 presents the conclusions and prospects for further research.

Section snippets

Research problem statement and contributions

This research investigates the MAGDM problem based on interval-valued intuitionistic fuzzy sets (IVIFS). In the recent past, owing to the fact that the IVIFN can express expert preference more accurately than a crisp number, interval-valued number, and linguistic value, several researchers have attempted to develop various operators for aggregating IVIFS.

Interval-valued intuitionistic fuzzy set (IVIFS)

IVIFSA˜, a concept introduced by Atanassov and Gargov [25], can be described as follows:

Let a set X be fixed:A˜={x,μA˜(x),νA˜(x)|xX}whereμA˜=[μA˜L(x),μA˜U(x)][0,1]νA˜=[νA˜L(x),νA˜U(x)][0,1]μA˜(x) and νA˜(x) are intervals, μA˜L(x)= inf μA˜(x), μA˜U(x)= sup μA˜(x), νA˜L(x)= inf νA˜(x), νA˜U(x)= sup νA˜(x)

andμA˜U(x)+νA˜U(x)1,xX

Additionally,πA˜(x):=[πA˜L(x),πA˜U(x)],xXwhereπA˜L(x)=1μA˜U(x)νA˜U(x)πA˜U(x)=1μA˜L(x)νA˜L(x)μA˜(x) and νA˜(x) represent the membership degree and the

Aggregation idea

In MAGDM, the preference information of the tth (1 < t < k) expert can be expressed using the following matrix:D(t)=[dij(t)]m×n,(1im,1jn,1tk).

In this study, expert preference is expressed by interval-valued fuzzy numbers. Eq. (21) can be expressed as follows:D(t)=[dij(t)]m×n=[μij(t),νij(t)]=[(μij(t)L),(μij(t)U),(νij(t)L),(νij(t)U)],(1im,1jn,1tk).

We establish two coordinates to aggregate membership degree and non-membership degree information separately. As can be seen from Fig. 5(a),

Example 1

In this section, the performance of the proposed method is evaluated through a case study, which was first presented by Herrera and subsequently modified by Xu [31].

  • Let us assume that someone intends to buy a car and consults a set of experts. An expert ek(k = 1, 2, 3, 4) uses an IVIFN d˜ijk(i,j=1,2,3,4,5) to describe the characteristics Cj(j = 1, 2, 3, 4, 5) of each supplier Ai(i = 1, 2, 3, 4, 5), The weighted vector of the four experts is ζ = (0.3, 0.2, 0.3, 0.2)T, the weighted vector of the five

Discussion

The above two examples are solved by the IIFHA operator. This section will discuss the different results obtained by the proposed method and the IIFHA operator and illustrate the advantages of the proposed method.

Definition 8

[1]

Let α˜i=([ai,bi],[ci,di])(i=1,2,,n) be IVIFNs, α˜˙σ(j) be the jth largest element of weighted IVIFNs α˜˙i(i=1,2,,n), where α˜˙i=nωiα˜i(i=1,2,,n) and ω = (ω1, ω2, …, ωn)T be the weighting vector of α˜˙i(i=1,2,,n). Let wj be the weight of α˜j, wj[0,1], j=1n=1, and n be the balance

Conclusions and suggestions for future research

In this paper, a novel method based on the PGSA is proposed for MAGDM problems with IVIFS. The distinctive feature of the proposed method is its ability to handle IVIFSs with a value of zero. Unlike traditional operators, our method translates the IVIFNs of expert preference matrices as preference points instead of handling them directly. The differences between the experts preferences are expressed as distances between expert preference points, thereby ensuring that the preference aggregation

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