Complexity

Volume 2018, Article ID 2501489, 12 pages

https://doi.org/10.1155/2018/2501489

## A Novel Approach to Fuzzy Soft Set-Based Group Decision-Making

^{1}School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041004 Shanxi, China^{2}Computer and Information Engineering Academy, Shanxi Technology and Business College, Taiyuan, 030006 Shanxi, China

Correspondence should be addressed to Qinrong Feng; moc.361@27rqgnef

Received 4 January 2018; Revised 28 April 2018; Accepted 24 May 2018; Published 12 July 2018

Academic Editor: Thierry Floquet

Copyright © 2018 Qinrong Feng and Xiao Guo. 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

There are many uncertain problems in practical life which need decision-making with soft sets and fuzzy soft sets. The purpose of this paper is to develop an approach to effectively solve the group decision-making problem based on fuzzy soft sets. Firstly, we present an adjustable approach to solve the decision-making problems based on fuzzy soft sets. Then, we introduce knowledge measure and divergence degree based on -similarity relation to determine the experts’ weights. Further, we develop an effective group decision-making approach with unknown experts’ weights. Finally, sensitivity analysis about the parameters and comparison analysis with other existing methods are given.

#### 1. Introduction

The mathematical modelling of vagueness and uncertainty has become an increasingly important issue in diverse research areas. In recent years, uncertain theories such as rough set theory [1], fuzzy set theory [2], and intuitionistic fuzzy set theory [3] and other mathematical tools have been widely applied in lots of social fields. But all these theories have their own difficulties as pointed out in [4]. To overcome these difficulties, Molodtsov [4] proposed the soft set theory for modeling uncertainty.

Recently, works on soft set theory are progressing rapidly. Many efforts have been devoted to further generalizations and extensions of Molodtsov’s soft sets. Maji et al. [5] defined fuzzy soft sets by combining soft sets with fuzzy sets. Yang et al. [6] initiated the notion of interval-valued fuzzy soft set by combining the interval-valued fuzzy sets and soft sets. Maji et al. [7, 8] introduced the concept of the intuitionistic fuzzy soft set which is a combination of the soft set and the intuitionistic fuzzy set. Xu et al. [9] defined a concept of vague soft set. Moreover, they also studied its basic properties and applications. By integrating the interval-valued intuitionistic fuzzy sets with soft sets, Jiang et al. [10] proposed a more general soft set model called interval-valued intuitionistic fuzzy soft sets.

Applications of fuzzy soft sets have made great progress, especially in decision-making. Feng et al. [11] applied level soft sets to discuss fuzzy soft set-based decision-making. Based on Feng et al.’s works, Basu et al. [12] further investigated the previous methods to fuzzy soft set-based decision-making and introduced the mean potentiality approach, which was showing more efficiency and more accuracy than the previous methods. Alcantud [13] presented two innovations that produced a novel approach to the problem of fuzzy soft set-based decision-making in the presence of multiobserver input parameter data sets. Tang [14] proposed a novel fuzzy soft set approach in decision-making based on grey relational analysis and Dempster-Shafer theory of evidence. Li et al. [15] introduced an approach to fuzzy soft set-based decision-making by combining grey relational analysis with Dempster-Shafer theory of evidence and given a practical application to medical diagnosis problems. Liu et al. [16] proposed a decision model based on fuzzy soft set and ideal solution. Alcantud et al. [17] put forward an algorithmic solution for the diagnosis of glaucoma through a hybrid model of fuzzy and soft set-based decision-making techniques.

Group decision-making is an important research topic in decision theory. In recent years, a lot of methods have been developed for solving group decision-making problems in existing literatures. Yue [18] presented a multiple-attribute group decision-making model based on aggregating crisp values into intuitionistic fuzzy numbers. Xu and Shen [19] proposed an outranking method aimed at solving multicriteria group decision-making problems under Atanassov’s interval-valued intuitionistic fuzzy environment. Wan and Dong [20] developed some power geometric operators of trapezoidal intuitionistic fuzzy numbers and applied them to multiattribute group decision-making with trapezoidal intuitionistic fuzzy numbers. Sun and Ma [21] studied the group decision-making problem with linguistic preference relations. Qin and Liu [22] investigated a new method to handle multiple-attribute group decision-making problems based on a combined ranking value under interval type-2 fuzzy environment. Wan et al. [23] developed a new method for solving multiple-attribute group decision-making (MAGDM) problems with Atanassov’s interval-valued intuitionistic fuzzy values (AIVIFVs) and incomplete attribute weight information. In [24], Wan et al. investigated the group decision-making (GDM) problems with interval-valued Atanassov intuitionistic fuzzy preference relations (IV-AIFPRs) and developed a novel method for solving such problems. In [25], Wan et al. investigated a group decision-making (GDM) method based on additive consistent interval-valued Atanassov intuitionistic fuzzy (IVAIF) preference relations (IVAIFPRs) and likelihood comparison algorithm.

In dealing with multiexpert group decision-making problems, experts have their own characteristics and structure of knowledge; normally, each expert should have different weights. So experts’ weights play an important role in group decision-making. In this paper, we suppose that the weights of experts are different and unknown. How to measure the experts’ weights? Up to now, some methods have been developed to do this. Yue and Jia [26] used an extended TOPSIS method and an optimistic coefficient to obtain the weights of decision-makers. Mao et al. [27] introduced a method for determining the weights of experts by using the distance between intuitionistic fuzzy soft sets. Wan et al. [28] constructed an intuitionistic fuzzy linear programming model to derive experts’ weights. Based on the generalized cross-entropy measure, Qi et al. [29] developed a method to determine unknown experts’ weights by considering divergence of decision matrices from positive or negative ideal decision matrix and similarity degree between individual decision matrices. Zhang and Xu [30] proposed the consensus index from the perspective of the ranking of decision information and constructed an optimal model based on the maximizing consensus in order to derive the experts’ weights.

In this paper, we present an adjustable approach to fuzzy soft set-based decision-making problems using the distance measure. Moreover, we introduce a new knowledge measure and -similarity relation over fuzzy soft sets. Based on the proposed knowledge measure and -similarity relation, we develop two methods for obtaining appropriate experts’ weights. Then, an effective group decision-making approach is constructed by integrating the aforepresented methods, from which we can find the optimal object with minor risk by tuning the value of parameters.

The rest of this paper is organized as follows. In Section 2, some basic notions of soft sets and fuzzy soft sets are reviewed. In Section 3, a new method based on the distance is proposed to solve the problems of decision-making. In Section 4, a knowledge measure and -similarity relation are proposed for obtaining the experts’ weights. Then, an approach integrating the above methods for group decision-making is developed. In Section 5, an example and a comparative analysis are given to illustrate effectiveness and practicality of presented methods. Finally, conclusions are stated in Section 6.

#### 2. Preliminaries

In this section, some basic notions of soft sets and fuzzy soft sets are reviewed, which will be required in the later sections. Let be an initial universe set and be a set of parameters.

*Definition 2.1 (see [4]). *A pair is called a soft set over , if is a mapping of into the set of all subsets of .

In other words, the soft set is a parameterized family of subsets of the set . Every set () from this family may be considered as the set of -elements of the soft set or as the set of -approximate elements of the soft set.

In [5], Maji et al. introduced the definition of fuzzy soft set by combining fuzzy set and soft set, which can be shown as follows.

*Definition 2.2 (see [5]). *Let be the universe and be the parameter set. denotes the set of all fuzzy subsets of ; a pair is called a fuzzy soft set over , where is a mapping from into .

*Definition 2.3 (see [5]). *Let be the universe and and be two fuzzy soft sets over ; is called a fuzzy soft subset of , denoted by ; if and , .

*Example 2.1. *Consider a fuzzy soft set , which describes the “attractiveness of houses” that Mr. X is considering for purchase. Suppose that there are four houses under consideration, namely, the universes and the parameter set , where stands for “beautiful,” “large,” “modern,” and “cheap,” respectively. Let

The tabular form of such fuzzy soft set is represented in Table 1.

*Definition 2.4 (see [31]). *Let be a fuzzy soft set over and be the parameter set, . Fuzzy soft matrix , where . That is,
From this definition, we can see that there is a one-to-one correspondence between the fuzzy soft set and fuzzy soft matrix, so we will use fuzzy soft set and fuzzy soft matrix without distinction in the following.

*Definition 2.5 (see [31]). *Let and be two fuzzy soft matrices over and . Then,

Theorem 2.1 (see [31]). *Let and be two fuzzy soft matrices over and , then we have
*

Theorem 2.2 (see [31]). *Let be fuzzy soft matrices over and be a given weight vector, where , , then
**From this theorem, we know how to integrate multiple fuzzy soft sets into one fuzzy soft set.*