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.

3. An Adjustable Approach to Fuzzy Soft Set-Based Decision-Making

Generally, the existing approaches to fuzzy soft set based-decision-making are mainly based on different kinds of level soft sets. However, it is hard for decision-makers to select a suitable level soft set. In dealing with decision-making problems, it is obvious that the smaller the distance between the alternative and the decision-maker’s ideal object, the better the alternative is. So in this section, we present an adjustable approach to fuzzy soft set-based decision-making problems using the distance of fuzzy soft sets in [32]. This approach is effective and reasonable under uncertain conditions. It not only allows us to avoid the problem of selecting the suitable level soft set but also helps reduce the complexity of computations in the process of decision-making.

Definition 3.1 (see [32]). Suppose and be two fuzzy soft sets over , the distance between and can be defined as Next, to cope with weighted fuzzy soft set-based decision-making problems, we introduce a new distance between the two fuzzy soft sets as follows.

Definition 3.2. Suppose and be two fuzzy soft sets over , the distance between and can be defined as where and is the weight of the th parameter .
It is easy to prove that (6) and (7) satisfy the following properties:

In the following, we develop an algorithm to deal with decision-making problems based on fuzzy soft sets.

Algorithm 1 (decision-making based on fuzzy soft sets). Input: A fuzzy soft set over as given in Table 2, where is a set of parameters denoted by and is an initial universe set denoted by .
Output: The order relation of all the alternatives.

Step 1. Construct an ideal fuzzy soft set with a single object as given in Table 3.

Step 2. Obtain a fuzzy soft set with respect to a single alternative as given in Table 4.

Step 3. Calculate the distance between fuzzy soft sets and using (6). If is a weighted fuzzy soft set and is a weighting vector of parameters, we will use (7) to calculate the distance, where .
It is clear that the smaller the is, the closer the approaches to the ideal fuzzy soft set . Thus, is better.

Step 4. Rank all alternatives according to or .
In order to better understand the above idea, let us consider the following example.

Example 3.1 (see [14]). Let be the fuzzy soft set given in Table 5.

Then, we can use the proposed decision-making method to get the ranking of the alternatives. (1)Construct an ideal fuzzy soft set with a single alternative as given in Table 6.(2)Based on the above fuzzy soft set , we get the fuzzy soft set with respect to a single alternative as given in Tables 7–9.(3)Utilize (6) to get the distance : (4)Rank all alternatives according to the distance .

Then, the order relation among all the alternatives is , and the best alternative is the alternative .

For comparison with different decision-making methods, we give the ranking orders of the proposed method and other methods [12, 14, 15] as shown in Table 10. Therefore, from Table 10 we can see that the three ranking orders of them are the same, and the alternative is the best choice in the proposed method and the other two methods [14, 15]. But Basu et al.’s method [12] is another ranking order, and the alternative is the best choice.

From above, we can see that the proposed method is reliable and reasonable. Moreover, the complexity of the proposed method of decision-making in this paper is lower than that of Tang and Li et al.’s methods.

4. An Approach to Fuzzy Soft Set-Based Group Decision-Making

As we all know, experts’ weights play an important role in integrating individual fuzzy soft sets into a collective one for group decision-making problems. In [23], the authors comprehensively considered the similarity and proximity degrees and employed a control parameter to construct the combined weight. In [24], considering different knowledge, experiences, and preferences of diverse decision-makers, the authors seek a weight vector such that the deviations between the individual preferences and the group opinion are minimized. In [25], to derive decision-makers’ weights, the authors constructed an optimization model by maximizing the group consensus. In [33], the authors proposed a method based on maximizing consensus for determining experts’ weight for interval-valued intuitionistic fuzzy group decision-making problems. Inspired by these works, we introduce a new approach to determine experts’ weight for group decision-making based on fuzzy soft sets.

In the practical group decision-making process, the experts usually come from different research fields, and each expert has his unique characteristics with regard to knowledge, skills, and practical experience. Thus, they are familiar with some of the attributes, but not others. In other words, there usually exists the fuzziness of the information provided for decision-makers by the experts and the divergence among the individual experts’ opinions. Therefore, we consider the group decision-making problem from the perspective of the group and from the perspective of the individual. That is to say, we consider not only the consistency between the individual expert and the group but also how useful he can provide information for decision-makers as the individual expert.

4.1. Method Based on Knowledge Measure of Fuzzy Soft Set for Determining Experts’ Weights

In the practical multiexpert group decision-making process, the weights of experts usually play an important role in determining the final decision results and should be taken fully into account. Generally, the existing methods for determining the experts’ weights are mainly based on the relation between each expert and the other experts or the ideal expert. Few methods are proposed to obtain the individual expert’s weight by considering the fuzziness of the information provided for decision-makers by the expert. So in this subsection, we propose a knowledge measure to measure the degree of fuzziness of fuzzy soft set. Then, we develop a method for obtaining appropriate experts’ weights by using the proposed knowledge measure. In other words, the weights of experts can be obtained from the perspective of the individual.

Definition 4.1.1. Let be a fuzzy soft set over , the knowledge measure is defined as It is easy to prove that the presented knowledge measure satisfies the following properties: (1), , or .(2).(3) is less fuzzy than , that is, , , or .(4) is the fuzziest, that is, , .These properties show that the smaller the knowledge measure is, the fuzzier the available information becomes. During multiexpert group decision-making process, if the knowledge measure of a fuzzy soft set given by an expert is larger, he can provide decision-makers with more useful information, then the expert plays a relatively more important role in the group decision-making process. Therefore, the expert should be assigned a bigger weight. Otherwise, such an expert will be judged unimportant by most decision-makers. In other words, such an expert should be assigned a smaller weight.
Suppose there are experts and they give evaluation values in the form of fuzzy soft sets, respectively, namely, . We can get the weight of the expert as follows:

4.2. Method Based on Divergence Degree for Determining the Experts’ Weights

In order to solve group decision-making problems, flexible and agile -similarity relation is introduced in fuzzy soft sets. Similar to the ideas in [24, 25, 33], to measure the deviations between the individual preferences and the groups’ opinion, the concept of divergence degree based on the -similarity relation is first proposed to determine the experts’ weights.

Definition 4.2.1. Let be a fuzzy soft set over , and , a -similarity relation over fuzzy soft set can be defined as

From this definition, it can be observed that if a pair of objects from belongs to , then they are perceived as similar. It is easy to test that the -similarity relation is a tolerance relation (a relation that is reflexive and symmetric, but not necessarily transitive).

Let be a fuzzy soft set over ; for ; then, the -similarity class of object with respect to is defined as

In dealing with multiexpert group decision-making problems, the -similarity relation under the expert can be expressed as

Then, the -similarity classes of object with respect to under the expert can be expressed as

Thus, the family set of the -similarity class under the expert for the alternative set with respect to the parameter set can be obtained as follows:

In the following, we introduce the concept of divergence degree between the experts and for all alternatives with respect to the parameter set , which is defined as

Equation (17) expresses the divergence degree between two experts for all alternatives with respect to parameter set . It is easy to verify that . The closer the is to 0, the poorer the divergence. That is to say, the smaller the is, the more similar the experts and are. Further, it is easily seen that the above definition of divergence degree is effective and reasonable because it avoids the use of distance or similarity functions to measure the divergence in group decision-making and thus reduces the effect of the application of some different distance or similarity functions for measuring divergence in group decision-making.

Then, according to (17), we can get the divergence degree of expert and all the other experts as follows:

Based on the above analyses, we can get a simple and exact formula for determining the weight of the expert as follows:

In practice, (11) and (19) can be integrated in accordance to attitudinal characteristics of decision-makers as the following (20) for the determination of experts’ weights in group decision-making based on fuzzy soft sets.

By integrating and , the ultimate weight of expert can be obtained: where is the parameter that reflects attitudinal characteristics of decision-makers, .

Finally, we could develop an algorithm to deal with multiexpert group decision-making problems with unknown experts’ weights.

Algorithm 2 (group decision-making based on fuzzy soft sets). Input: The fuzzy soft sets over a finite initial universe and a finite parameter set .
Output: The order relation of all the alternatives.

Step 1. Calculate the knowledge measures of all individual fuzzy soft set of expert , and then get experts’ weighting vector , through (10) and (11).

Step 2. Set the value of , and calculate the divergence degree between each expert and the other experts, and then ensure the experts’ weighting vector using (17), (18), and (19).

Step 3. Set the value of , and let be the ultimate weighting vector of experts, which can be formed through (20).

Step 4. Calculate the integrated fuzzy soft set according to (5).

Step 5. Apply Algorithm 1 to the integrated fuzzy soft set , then get the optimal alternatives.

5. Comparison Analysis

In this section, we consider an illustrative example and comparison analyses to demonstrate the practicability, feasibility, and effectiveness of the proposed method.

5.1. Illustrative Example

In this subsection, to demonstrate the applicability of the proposed approach for effectively solving the group decision-making problem based on fuzzy soft sets, we present an example modified from [33] as follows.

Example 5.1. Suppose a company is considering 4 short-listed candidates for a job position. An interview session is held where three experts are to evaluate each candidate over four criteria, namely, good attitude , pleasant personality , good command in English , and competent communication skills . is the set of parameters. Here, we assume that the weighting vector of the parameters is determined as . And the experts’ weights is unknown. is a fuzzy soft set given by expert (Tables 11–13).

Then, we utilize the developed approach to get the ranking of the alternatives. (1)Calculate the knowledge measure of fuzzy soft set . According to Definition 4.1.1, we can get the knowledge measure as the following: (2)Determine the weights of experts. By utilizing (11), the weighting vector of experts can be obtained as the following: (3)Set , and obtain the family set of the -similarity classes under the expert for the alternative set with respect to the parameter set . Through (14), (15), and (16), we can get the family set of -similarity class as the following: (4)Calculate the divergence degree between the two experts. According to (17), the divergence degree between the two experts can be calculated based on the -similarity class: (5)Obtain the weights of experts. By utilizing the (18) and (19), the weighting vector of experts can be obtained as the following: (6)Set , and determine the ultimate weight vector of experts. According to (20), the ultimate weight vector of experts can be obtained as the following: (7)Calculate the integrated fuzzy soft set. Through the (5), we can aggregate the overall individual fuzzy soft set to get the integrated fuzzy soft set as given in Table 14.(8)Rank the alternatives. By utilizing the Algorithm 1, the ranking order of all the alternatives can be obtained as the following:  Thus, the optimal alternative is .

5.2. Sensitivity Analysis on the Parameters

In the above example, the computation results are obtained by given the parameters a priori in (14) and (20). Normally, the different values of parameters can lead to different weights of experts, then different ranking order of alternatives. To inspect the influence of different parameters on the experts’ weights, it is necessary to do sensitivity analysis of the parameters.

From Example 5.1, we can see that if or , the divergence degree between any two experts is 0, which is meaningless. So it is significant for decision-makers to know the range of the parameter . Let and