Elsevier

Applied Soft Computing

Volume 31, June 2015, Pages 317-325
Applied Soft Computing

A novel fuzzy soft set approach in decision making based on grey relational analysis and Dempster–Shafer theory of evidence

https://doi.org/10.1016/j.asoc.2015.03.015Get rights and content

Highlights

  • We introduce a novel fuzzy soft set approach in decision making based on grey relational analysis and D–S theory of evidence (see Section 4).

  • We give an application for medical diagnosis (see Section 6).

  • We propose an illustrative example (see Example 5.1).

Abstract

This paper proposes a novel fuzzy soft set approach in decision making based on grey relational analysis and Dempster–Shafer theory of evidence. First, the uncertain degrees of various parameters are determined via grey relational analysis, which is applied to calculate the grey mean relational degree. Second, suitable mass functions of different independent alternatives with different parameters are given according to the uncertain degree. Third, to aggregate the alternatives into a collective alternative, Dempster's rule of evidence combination is applied. Finally, the alternatives are ranked and the best alternatives are obtained. The effectiveness and feasibility of this approach are demonstrated by comparing with the mean potentiality approach because the measure of performance of this approach is the same as the mean potentiality approach's, the belief measure of the whole uncertainty falls from 0.4723 to 0.0782 (resp. 0.3821 to 0.0069) in the example of Section 5 (resp. Section 6).

Introduction

To solve complicated problems in economics, engineering, environmental science and social science, classical mathematical methods are not always successful because of various types of uncertainties present in these problems. There are several theories: probability theory, fuzzy set theory [30], rough set theory [22] and interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. To overcome these kinds of difficulties, Molodtsov [19] proposed soft set theory for modeling uncertainty.

Recently works on soft set theory are progressing rapidly. Maji et al. [20] defined fuzzy soft sets by combining soft sets with fuzzy sets, in other words, a degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. The study of hybrid models combining soft sets or fuzzy soft sets with other mathematical structures and new operations are emerging as an active research topic of soft set theory [10], [29]. Aktas et al. [1] initiated soft groups. Jun applied soft sets to BCK/BCI-algebras [9]. Jiang et al. [10] extended soft sets with description logics. Li et al. [18] investigated relationships among soft sets, soft rough sets and topologies.

At the same time, there have been some progress concerning applications of soft set theory, especially the usage of soft sets in decision making. Using soft set theory to describe or set objects with traditional mathematics tools is very different. We can describe approximately the original objects in soft set theory. There is no limiting condition when objects are described. Researchers can choose parameters and their forms according needs. The fact that setting parameters is non-binding greatly simplifies decision-making process and then we can still do effective decisions under the circumstance of the absence of partial information.

Maji et al. [21] first applied soft sets to solve decision making problems by means of rough set theory. Chen et al. [3] defined the parameterization reduction of soft set and discussed its application of decision making problem. Cagˇman et al. [4] constructed an uni-int decision making method which selects a set of optimum elements from the alternatives by using uni-int decision functions. Roy et al. [23] discussed score value as the evaluation basis to finding an optimal choice object in fuzzy soft sets. Kong et al. [13] argued that the Roy's method was incorrect in general and they proposed a revised algorithm. Feng et al. [7] applied level soft sets to discuss fuzzy soft sets based decision making. Jiang et al. [11] generalized the adjustable approach to fuzzy soft sets based decision making and presented an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets. Based on Feng’ works, Basu et al. [2] further investigated the previous methods to fuzzy soft sets in decision making and introduced the mean potentiality approach, which was showed more efficient and more accurate than the previous methods.

The existing approaches have significant contributions to solve fuzzy soft sets in decision making. However, these approaches are mainly based on the level soft set, and the decision makers select any level soft set with much subjectivity and uncertainty [2]. Moreover, there exists no unique or uniform criterion for the selection, the same decision problem may induce many different results by using different evaluation criteria. As a result, it is difficult to judge that which result is adequate, and which method or level soft sets should be chosen for selecting the optimal choice object. The key to this problem is how to reduce subjectivity and uncertainty when we choose making decisions method. Then it is necessary to pay attention to this issue.

Grey relational analysis initiated by Deng [6] is utilized for generalizing estimates under small samples and uncertain conditions, and it can be regarded an effective method to solve decision making problems [12], [26], [33]. Dempster–Shafer theory of evidence is a new important reasoning method under uncertainty, which has an advantage to deal with subjective judgments and to synthesize the uncertainty of knowledge [32].

Compared to probability theory, Dempster–Shafer theory of evidence [5], [24] can capture more information to support decision-making by identifying the uncertain and unknown evidence. It provides a mechanism to derive solutions from various vague evidences without knowing much prior information and has been successfully applied into many fields such as intelligent medical diagnosis [8], knowledge reduction [27], fault diagnosis [28], multi-class classification [17], supplier selection [25], etc. Moreover, applying both theories enables the ultimate decision makers to take advantage of both methods’ merits and make evaluation experts to deal with uncertainty and risk confidently [16], [25]. The hybrid model has been proved to have its usefulness and versatility in successfully solving a variety of problems in the information sciences, such as data mining, knowledge discovery, and decision making.

Therefore, it is very meaningful to explore an approach to fuzzy soft set in decision making by combining Dempster–Shafer theory of evidence with grey relational analysis.

The purpose of this paper is to give a novel fuzzy soft set approach in decision making based on grey relational analysis and Dempster–Shafer theory of evidence. The novelty aspects or advantages of this approach is avoiding the selection of appropriate level soft sets and distribution of parameters’ weight, reducing significantly the uncertainty of decision-making and the fuzziness of people's subjective understanding, improving the reliability of decision making and increasing the level of decision-making.

Section snippets

Preliminaries

Throughout this paper, U denotes an initial universe, E denotes the set of all possible parameters, IU denotes the family of all fuzzy sets in U. We only consider the case where U and E are both nonempty finite sets.

In this section, we briefly recall some basic concepts about fuzzy soft sets, the measure of performance of methods and Dempster–Shafer theory of evidence.

Mean potentiality approach

Like most of decision making problems, fuzzy soft sets based on decision making involve the evaluation of all decision alternatives. Recently, applications of fuzzy soft set based on decision making have attracted more and more attentions. The works of Roy et al. [13], [23], [7] are fundamental and significant. Later Kong et al. [14] applied grey relational analysis to solve fuzzy soft sets in decision making. Generally, there does not exist any unique or uniform criterion for the evaluation of

A novel fuzzy soft set approach in decision making

The existing approaches to fuzzy soft sets in decision making are mainly based on the level soft set to obtain useful information such as choice values and score values. However, it is very difficult for decision makers to select a suitable level soft set. Inspired by the work of Wu et al. [16], [25], we introduce a new approach to fuzzy soft sets in decision making based on grey relational analysis and Dempster–Shafer theory of evidence. It not only allows us to avoid the problem of selecting

An illustrative example

In this section, we give the following example to illustrate our approach.

Example 5.1

Using our approach, we reconsider the fuzzy soft set (F, A) given in Example 3.1.

Now, we suppose that the three mutually exclusive and exhaustive alternatives construct a frame of discernment, denoted by Θ = {x1, x2, x3}. We consider the set of parameters A = {e1, e2, e3, e4, e5} as a set of evidences.

  • (1)

    Construct a fuzzy soft decision matrix induced by (F, A) as follows:D=(xij)3×5=0.850.710.380.320.750.560.820.760.640.430.840.51

An application for medical diagnosis

A major task of medical science is to diagnose diseases. Generally a patient suffering from a disease may have multiple symptoms and the information available to physician about his patient is inherently uncertain. Again it is also observed that there are certain symptoms which may be common to more than one diseases leading to diagnostic dilemma. Doctors always detect clinical manifestations by the comparison with predefined classes to find the most similar disease. Only one comprehensive

Conclusions

In this paper, we have introduced a novel fuzzy soft set approach in decision making based on grey relational analysis and Dempster–Shafer theory of evidence and given an application for medical diagnosis. Detailed flow chart and schematic block diagram of the proposed approach is as follows:

Since the measure of performance of this approach is the same as the mean potentiality approach's, the belief measure of the whole uncertainty falls from 0.4723 to 0.0782 (resp. 0.3821 to 0.0069) in the

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This work is supported by the National Social Science Foundation of China (No. 12BJL087).

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