Generalized Fuzzy Soft Power Bonferroni Mean Operators and Their Application in Decision Making

Since the decision-making problem exists in every field of life, it has always been paid close attention by the majority of scholars. As the social environment becomes increasingly complex, more and more factors are involved in the decision-making problems, which leads to the uncertainty, hesitation and fuzziness of decision makers (DMs) when they give evaluation opinions. Therefore, fuzzy set theory, which is used to fit people’s fuzzy opinions, has become one of the most commonly used tools to solve decision-making problems [1,2,3]. At the same time, as a more general form of fuzzy sets, soft sets have also been paid close attention by many scholars, and have been applied to uncertain decision-making problems in various fields [4,5,6,7,8]. The concepts of fuzzy soft sets (FSS) [9] and generalized fuzzy soft sets (GFSS) [10] are also proposed. In recent years, an increasing number of scholars have used generalized fuzzy soft sets to express people’s fuzzy views in order to solve decision-making problems [5,6,7,8,11,12]. Considering the complexity of practical problems, DMs with different backgrounds, and different levels of professional knowledge and experience, are often required to participate in the decision-making process, and different DMs often give different decision-making opinions. Therefore, in order to obtain a comprehensive opinion that is accepted by everyone, it is necessary to aggregate different opinions or carry out corresponding operation. At present, scholars have studied various forms of operator (such as the power average (PA) [13] operator, the Bonferroni mean (BM) [14] operator and the power Bonferroni mean (PBM) [15] operator) to carry out corresponding operations on different opinions in order to obtain as accurate a decision scheme as possible. These operators effectively integrate the information of individual DMs into the overall information, and better consider the possible correlation between different attribute variables.The existing literature on GFSS integration methods is mostly proposed under the condition that the attributes are independent of each other. In fact, there may be different degrees of correlation between different attributes. The PA operator and BM operator can solve these problems. Among them, the PA operator can determine the attribute weights according to the support relationship between the attributes to reduce the influence of the biased decision-maker’s abnormal preference value on the decision results, and the BM operator can fully consider the correlation between the attributes. However, there is no research on integrating GFSS using PBM operators. This paper proposes the generalized fuzzy soft power Bonferroni mean (GFSPBM) operator and the generalized fuzzy soft weighted power Bonferroni mean (GFSWPBM) operator by combining the advantages of the PA operator and BM operator. The new operator proposed in this paper can enrich the integration method of GFSS, expand the application field of PBM operator, and provide a new method for multi-attribute decision-making problems. The decision-making method proposed in this paper can be applied to fields such as supplier selection evaluation, product program selection evaluation, and recommendation of talent introduction in the human resources department.

In 1965, Zadeh [16] proposed the fuzzy set theory. This theory regards the object to be investigated and the fuzzy concept reflecting the object as a certain fuzzy set, establishes an appropriate membership function for this, and analyzes the fuzzy object through the related operations and transformations of the fuzzy set. In 1999, Molodtsov [4] introduced soft set theory, which is a mathematical tool for solving uncertain problems which can be widely used in economics, engineering, physics and other fields. Maji et al. further studied the theory of soft sets. They defined the operation of soft sets [17], applied the theory of soft sets to the solution of mathematical decision-making problems [18], and combined fuzzy sets with soft sets to introduce FSS [9]. In order to understand the influence of decision makers’ cognition on the effectiveness of information provided by them, Majumdar [10] further proposed GFSS on the basis of fuzzy soft sets. In recent years, the theoretical research and application exploration of generalized fuzzy soft sets have attracted the attention of many scholars. Among them, Chen et al. [5] applied the Bonferroni mean operator to GFSS and proposed the generalized fuzzy soft set Bonferroni mean (GFSSBM) operator, which solved the problem of group decision-making under limited cognition by decision-makers. Dey and Pal [11] introduced the concept of generalized multi-fuzzy soft sets and applied it to decision-making problems. Agarwal et al. [12] extended the intuitionistic soft set (IFSS) to the intuitionistic fuzzy set (IFS) and defined the generalized intuitionistic fuzzy soft set (GIFSS), which can provide a given standard evaluation and the host’s evaluation of the data. Xu et al. [6,7] combined the extreme learning machine and GFSS to establish an ensemble credit scoring model. Li et al. [8] combined GFSS with hesitant fuzzy sets and proposed a generalized hesitant fuzzy soft set (GHFSS). Due to the dynamic development of multi-criteria assessment methods, fuzzy criteria are being increasingly considered by scholars. Bazzocchi et al. [19] proposed a method to prioritize space debris by using multi-criteria decision-making (MCDM) methods and fuzzy logic. Dong et al. [20] propose a new fuzzy best–worst method (BWM) based on triangular fuzzy numbers for MCDM; this method is very useful in solving multi-attribute decision-making problems in a fuzzy environment. Thakur et al. [21] depict an MCDM issue and offer the means of the VIKOR approach inside the pythagorean fuzzy system. The information fusion operator of fuzzy numbers is a useful tool to integrate all input-independent variables into the composite total value. The existing literature on the integration methods and applications of GFSS are mostly proposed when the attributes are independent of each other. In many actual decision-making problems, the degree of correlation between different attributes may be different, such as complementarity, redundancy, and preference relationships. The BM operator, proposed by Bonferroni [14] in 1950, is a mean-type ensemble operator, which can effectively find the interrelationship between the variables we entered and aggregate multiple input variables into one variable, is a bounded ensemble operator. In recent years, the BM operator has been widely used in different multi-attribute decision-making problems. For example, Wei et al. [22] studied uncertain linguistic BM operators. The generalized BM operator was proposed by Yager [23]. Intuitionistic fuzzy BM operator is defined by Xu and Yager [24]. Liu and Zhang [8] defined four kinds of intuitionistic uncertain linguistic arithmetic Bonferroni mean (IULABM) operators. In addition, the PA [13] operator is also an integrated operator that can capture the correlation between existing data and cognitive information. It considers the support relationship between the input data to calculate the weight of the attribute, which can effectively reduce anomalies. The impact of data on decision-making results makes the processing of decision-making information more objective and fair and, therefore, has received widespread attention. For example, Liu et al. [25] proposed the generalized neutrosophic number weighted power average (GNNWPA) operator to solve the multi-attribute decision-making (MAGDM) problem. A PA integration operator is applied to an intuitionistic fuzzy number (IFN) environment by Xu [26]. To comprehensively utilize the advantages of the BM operator and the PA operator, He et al. [15] combined the PA operator with the BM operator and proposed the PBM operator. Now, PBM operators are used in various fuzzy environments, such as hesitant fuzzy sets [15], intuitionistic fuzzy sets [27,28], interval-value intuitionistic fuzzy sets [29], and linguistic intuitionistic fuzzy sets [30]. However, to the best of our knowledge, there is no research on how to use the PBM operator to integrate GFSS. Therefore, to enrich the GFSS integration method and expand the application field of the PBM operator, this paper will study the GFSS integration method based on the PBM operator and propose two new GFSS integration operators, namely, the GFSPBM Operator and GFSWPBM operator. In this paper, we discuss and study the excellent properties of operators carefully. On this basis, the idea of the bidirectional projection method is introduced, and a new GFSS multi-attribute decision-making method is given. The combination of different operators and different environments will form new environment operators. The environmental operator has the advantages of its component operator. We classify the environmental operator according to the difference in the component operator as follows

In decision-making problems, when we need to eliminate the influence of abnormal data on the decision-making results, we can use operators of the PA operator class. When we need to consider the correlation between attributes, we can use operators of the BM operator class. When we need to eliminate the influence of abnormal data on decision results, but also consider the correlation between attributes, we can use operators of the PBM operator class. When we need to consider the uncertainty of fuzzy evaluation information, and we also want to reduce the influence of abnormal preference values on decision results, and consider the correlation between attributes. At this time, the GFSPBM operator proposed in this paper can be used. Our arrangement for the rest of this paper is as follows. To make our discussion easier, we first introduce the definitions of GFSS, PA operator, BM operator and PBM operator in Section 2. In Section 3, we proposed the GFSPBM operator and the GFSWPBM operator and carefully analyze and discuss the excellent properties of these operators. In Section 4, we introduce the idea of a bidirectional projection method and provide a multi-attribute decision-making method based on the GFSWPBM operator. In Section 5, we provide a practical application example of software selection, which shows the feasibility of our decision-making method. We compare and analyze different operators to verify the applicability of our proposed method, and analyze the sensitivity of the decision-making process. In the end, the conclusion is given in Section 6.

To correct the shortcomings of similarity measures defined in the existing literature, Chen [5] defined a new GFSS similarity in 2020, as shown in Definition 4.3.

It is the vector formed by alternative $A_i$ and alternative $A_j$.

The corresponding vector modulus lengths are

Then we defined it is the cosine of the angle between $A^{-}{A}^{+}$ and $A^{-}{A}_i$.

Note 4. Ref. [33] The bidirectional projection has the following properties

These are, respectively, called the projection of the vector formed by the negative ideal alternative and the alternative on the vector formed by the positive ideal alternative and the negative ideal alternative, and the vector formed by the positive ideal alternative and the negative ideal alternative on the alternative and the projection on the vector formed by the positive ideal alternative.

Note 5. Ref. [33] The bigger $Pr{j}_{A^{-}{A}^{+}}\left(A^{-}{A}_i\right)$ is, the closer alternative $A_i$ is to the ideal alternative $A^{+}$. The smaller $Pr{j}_{A^{-}{A}^{+}}\left(A^{-}{A}_i\right)$ is, the farther away alternative $A_i$ is from the ideal alternative $A^{+}$. The bigger $Pr{j}_{A_i{A}^{+}}\left(A^{-}{A}^{+}\right)$ is, the closer alternative $A_i$ is to negative ideal alternative $A^{-}$. The smaller $Pr{j}_{A_i{A}^{+}}\left(A^{-}{A}^{+}\right)$ is, the farther away alternative $A_i$ is from negative ideal alternative $A^{-}$.

The above formula actually refers to the closeness formula of TOPSIS and other methods. Obviously, the larger the $C\left(A_i\right)$, the better the alternative $A_i$. The opposite is true. The above decision-making methods are defined when the importance of every attribute is the same. In the practical decision-making process, the importance of different attributes may be different, so they have different weights. Next, define the weighted bidirectional projection method of GFSS.

Refer to Equation (23) to define the closeness formula considering the attribute weight:

In this section, we will introduce a multi-attribute decision-making method based on the GFSS environment that considers the cognition of the decision maker. The algorithm steps are as follows.

Step 1: Suppose experts use generalized fuzzy soft sets to express their opinions, which contain decision information about attribute alternatives. Therefore, DM $l(l=1,2,\cdots ,L)$ expresses the judgment for attribute $e_j(j=1,2,\cdots ,n)$, which can be indicated as ${\tilde{F}}^l\left(e_j\right)$.

Step 2: Calculate the similarity measure between DMs, get the weight of DMs, and use Equation (13) to find the similarity coefficient $S\left({\tilde{F}}_{\gamma }^l,{\tilde{F}}_{\gamma }^k\right)$ between DMs. In this way, a consensus matrix of preferences of all DMs is obtained.

Next, we define the weight coefficient ${\omega }_l$ of DM l as follows

Step 3: Utilize the GFSWPBM operator, to aggregate all of the individual GFSSs ${\tilde{F}}_{\gamma }^l\left(e_j\right)(l=1,2,\cdots ,L)$ into a comprehensive GFSS ${\tilde{F}}_{\gamma }\left(e_j\right)$, then derive the comprehensive overall assessed value for attribute $e_j(j=1,2,\cdots ,n)$ of alternative $A_i(i=1,2,\cdots ,m)$.

Step 4: Determine the positive ideal alternative $A^{+}$ and negative ideal alternative $A^{-}$ in the decision matrix GFSS ${\tilde{F}}_{\gamma }\left(e_j\right)$. Take the adjustment factor ${\gamma }_j$ as the attribute weight vector, $w=\left(w_1,w_2,\cdots ,w_n\right)$, according to the Equation (30) to find $C{\left(A_i\right)}_w(i=1,2,\cdots ,m)$ and rank the alternatives.

We take the group decision-making problem proposed by Wang and Lee and Zhang [34,35] as an example. To improve work efficiency, the administrators of the university computer center need to consider a choice of computer software. There are four alternatives $U=\left\{A_1,A_2,A_3,A_4\right\}$ remaining on the alternative list. The expert evaluation team includes three DMs $D=\left\{d_1,d_2,d_3\right\}$. The DMs evaluate the four alternatives using a group of attributes: hardware/software is cheap $\left(e_1\right)$, benefits the organization $\left(e_2\right)$, easy migration from current system $\left(e_3\right)$, and Outsourcing software developers have high reliability $\left(e_4\right)$. Next, we use the decision-making method proposed in this paper to evaluate these four alternatives.

Step 1: Three evaluation experts DMs evaluated four alternatives $A_i(i=1,2,3,4)$ by using the information provided by the generalized fuzzy soft set. Table 1 shows the results of alternative $A_i$ evaluated by each DM under criteria $e_j(j=1,2,3,4)$.

Step 2: We calculate the similarity between DM and use it to calculate the weight of each DM.

By Equation (13), We calculate the similarity measure between DM1 and DM2.

In the same way, we can calculate the similarity measure between DM1 and DM3 and the similarity measure between DM2 and DM3: $S_{13}=0.7426$, $S_{23}=0.7697$.

Using Equation (33), we get the weights of the DMs. The calculation process of ${\omega }_1=0.3295$ is as follows

We use the same calculation method to get the values of ${\omega }_2$ and ${\omega }_3$: ${\omega }_2=0.3332$, ${\omega }_3=0.3373$.

Step 3: GFSWPBM operators combine the evaluation information of each DM to calculate the comprehensive decision-making GFSS. To facilitate the calculation, we choose the parameter $p=1,q=1$, as shown in Table 2.

Step 4: Calculate the score $C{\left(A_i\right)}_w(i=1,2,3,4)$ by Equation (30). The calculation process of $C{\left(A_1\right)}_w$ is as follows: