Gray Method for Multiple Attribute Decision Making with Incomplete Weight Information under the Pythagorean Fuzzy Setting

1 Introduction

To address the issue of difficulties of acquiring sufficient and accurate data for real decision making due to the imprecision and ambiguity of socio-economics, Zadeh introduced the concept of the fuzzy set [39], and it has been used in a wide range of scientific fields. Vahdani et al. [32] proposed an integrated model based on a compromised solution method to solve fuzzy belief multi-objective large-scale non-linear programming problem with block angular structure. The authors developed a new method to transfer each belief decision-making problem into some fuzzy problems. Mojtahedi et al. [24] developed a systematic decision process for identifying and analyzing risks concurrently by applying multiple attribute group decision making (MAGDM) in a fuzzy environment. Because selecting a transport project to invest is an important task, Mohagheghi et al. [22] developed a sustainable transport investment selection method under an interval-valued fuzzy sets setting to address uncertainty. To address the importance of criteria, the authors proposed relative preference relation. Vahdani et al. [31] developed a compromise model to solve the multi-objective large-scale linear programming problems with block angular structure involving fuzzy parameters. After the appearance of a fuzzy set, a series of extensions both in theoretical and practical areas have been presented. Among these extensions, intuitionistic fuzzy set (IFS) [1], initiated by Atanassov, seems to be the most reasonable and acceptable. IFS is a powerful tool to deal with the imprecision and ambiguity of things more comprehensively. Since the establishment of IFS, it has been successfully applied in many areas of decision-making problems [2; 3; 17; 29; 33; 34]. Zhang et al. [44] developed a gray relational projection method for multi-attribute decision making (MADM) based on intuitionistic trapezoidal fuzzy number. Liu and Qin [18] developed the linguistic intuitionistic fuzzy Maclaurin symmetric mean operator, weighted linguistic intuitionistic fuzzy Maclaurin symmetric mean operator, linguistic intuitionistic fuzzy dual Maclaurin symmetric mean operator, and weighted linguistic intuitionistic fuzzy dual Maclaurin symmetric mean operator. Ebrahimnejad et al. [8] developed a multi-criteria decision-making (MCDM) model under an interval-valued intuitionistic fuzzy environment to select the best outsourcing provider. Foroozesh et al. [9] proposed the VIKOR (vlsekriterijumska optimizacija i kompromisno resenje, in Serbian) method under a hesitant fuzzy set environment by utilizing the weights of decision makers and the distance measure along with a new index for the ranking process of compromise solution procedure. To decrease the loss for industrial selection problems, Gitinavard et al. [12] developed a soft computing approach based on interval-valued hesitant fuzzy (IVHF) complex proportional assessments and IVHF compromise solution methods with last aggregation. Mousavi et al. [25] purposed a modified compromise ranking method (VIKOR), known as sorting the possible alternatives and determining the compromise solution under IVHF sets for solving group decision-making problems in manufacturing systems. However, in some practical decision-making process, the sum of the membership degree and the non-membership degree to which an alternative satisfying a criterion provided by a decision maker may be >1, but their square sum is ≤1. To overcome this situation, Yager [35; 36] initiated the concept of Pythagorean fuzzy set (PFS) as a generalization of IFS and characterized by a membership degree and a non-membership degree, which satisfies the condition that their square sum is ≤1. Zhang and Xu [43] established an extension of TOPSIS to MADM with PFS information. The error to the proof-of-distance measure in Zhang and Xu [43] has been pointed out by Yang et al. [38]. For MADM problems under a Pythagorean fuzzy environment, Yager and Abbasov [37] developed a series of aggregation operators. Peng and Yang [26] discussed their relationship among these aggregation operators and established the superiority and inferiority ranking MAGDM method. Using Einstein operation, Garg [10] generalized Pythagorean fuzzy information aggregation. Gou et al. [13] studied several Pythagorean fuzzy functions and investigated their fundamental properties, such as continuity, derivability, and differentiability, in detail. Zhang [42] put forward a hierarchical qualitative flexible (QUALIFLEX) multi-criteria approach with the closeness index-based ranking methods for multi-criteria Pythagorean fuzzy decision analysis. Zeng et al. [41] explored a hybrid method for Pythagorean fuzzy MCDM. Ren et al. [28] extended an acronym in Portuguese for Interactive Multi-criteria Decision-Making approach to solve the MCDM problems with Pythagorean fuzzy information. Ma and Xu [21] proposed symmetric Pythagorean fuzzy weighted geometric/averaging operators, and investigated the relationships among these operators and those existing ones. Peng and Yang [27] presented a new Pythagorean fuzzy Choquet integral-based multi-attribute border approximation area comparison method for MCGDM. Garg [11] developed the confidence Pythagorean fuzzy weighted averaging operator, confidence Pythagorean fuzzy weighted ordered averaging operator, confidence Pythagorean fuzzy weighted geometric operator, and confidence Pythagorean fuzzy weighted ordered geometric operator, along with their desired properties. Zeng [40] proposed the Pythagorean fuzzy probabilistic ordered weighted average operator and applied it to the MAGDM problem. Liu et al. [20] developed the Pythagorean uncertain linguistic Bonferroni mean operator, as well as the Pythagorean uncertain linguistic partitioned Bonferroni mean operator and its weighted form to solve the MADM problem. Lui et al. [20] proposed the Pythagorean fuzzy uncertain linguistic prioritized weighted averaging aggregation operator and the Pythagorean fuzzy uncertain linguistic prioritized weighted geometric aggregation operators. Mohagheghi et al. [23] developed a last aggregation group decision-making process for weighting and evaluating PFS. The authors developed a method in computing the weight of decision makers. The gray relational analysis (GRA) method was first proposed by Deng [5; 6], and was successfully applied to solve MCDM problems. Khan et al. [15] developed the interval-valued Pythagorean fuzzy TOPSIS based on the Choquet integral. Khan and Abdullah [14] proposed the interval-valued Pythagorean fuzzy GRA method for MADM problems.

In dealing with MADM, there may be a situation where the decision maker may provide the degree of membership and non-membership of a particular attribute in such a way that their sum is >1; then, it is better to use Pythagorean fuzzy information. PFSs in comparison with IFSs provide more flexibility and power in expressing uncertainty. This enhancement in flexibility and power to express membership degree, non-membership degree, and hesitancy is caused by improving the space in which those degrees can be expressed. Sometimes, the attribute values take the form of Pythagorean fuzzy information and the information about attribute weights is incompletely known because of time stress, lack of data or knowledge, and the expert’s limited knowledge about the problem domain. The traditional GRA method [5; 6] and intuitionistic fuzzy GRA method [33; 34] will fail in dealing with the above Pythagorean fuzzy MADM problems with incomplete weight information. How to derive the attribute weights from both the given Pythagorean fuzzy information and incompletely known attribute weight information based on the basic ideal of the traditional GRA method is an interesting and important research topic. Therefore, it is essential to concentrate on this problem. The purpose of this paper is to extend the concept of GRA to develop a methodology for solving MADM problems under a Pythagorean fuzzy environment in which the information about attribute weights is incompletely known and the attribute values take the form of Pythagorean fuzzy numbers (PFNs). To do this, the remainder of the paper is structured as follows.

In Section 2, we briefly review some basic definitions and results about Choquet integral and PFSs. In Section 3, we introduce the GRA method for Pythagorean fuzzy MADM problems with incomplete weight information. In Section 4, we illustrate our proposed algorithmic method with an example. The conclusion is in Section 5.

2 Preliminaries

In this section, fuzzy measure, the Choquet integral, and the definition of PFSs are reviewed. Some operations and comparison laws of PFSs, which will be utilized in the latter analysis, are also presented.

3 GRA Method for MADM with Incomplete Weight Information in the Pythagorean Fuzzy Setting

Let X={X1,X2,…,Xm} be a discrete set of alternatives and A={A1,A2,…,An} be the set of attributes, w=(w1,w2,…,wn) is the weighting vector of the attributes Aj(j=1,2,…n), where wj∈[0,1], ∑j=1nwj=1. Suppose that the decision makers provide the attribute weight information may be presented in the following forms [16], for i ≠ j:

1. A weak ranking: {wi≥wj};

2. A strict ranking: {wi−wj≥γi(>0)};

3. A ranking with multiples: {wi≥γiwj}, 0≤γi≤1;

4. An interval form: {λi≤wi≤λi+δi}, 0≤γi≤γi+δi≤1;

5. A ranking of differences: {wi−wj≥wk−wl}, for j≠k≠l.

For convenience, we denote by Δ the set of the known information about attribute weights provided by the experts.

Let Rk=[pij(k)]m×n be a Pythagorean fuzzy decision matrix, provided by decision maker dk(k=1,2,…,l), as the following form:

where pij(k)=(aij(k),bij(k)) is a PFN representing the performance rating of the alternative xj∈X with respect to the attribute Ai∈A provided by the decision makers d_k.

To extend the GRA method in the process of group decision making, we first need to fuse all individual decision matrices into a collective matrix by using the PFCIG operator.

Step 1: As for every alternative Xi(i=1,2,…,m), each expert e_k (k=1,2,…,r) is invited to express their individual evaluation or preference according to each attribute Aj(j=1,2,…,n) by a PFN pij(k)=(aij(k),bij(k))(i=1,2,…,m;j=1,2,…,n;k=1,2,…,r) expressed by the experts e_k. Then, we can obtain a decision-making matrix Rk=[pij(k)]m×n as follows:

Step 2: Confirm the fuzzy density μi=μ(Ai) of each expert. According to Eq. (4), parameter λ₁ of expert can be determined.

Step 3: By Definition 5, pijk is reordered such that pij(k)≥pij(k−1). Utilize the Pythagorean fuzzy Choquet integral average operator

to aggregate all the Pythagorean fuzzy decision matrices Rk=[pij(k)]m×n(k=1,2,…,r) into a collective Pythagorean fuzzy decision matrix R=[pij(k)]m×n, where pij=(aij,bij) (i=1,2,…,m;j=1,2,…,n),A(k)={e(k),…,e(r)},A(r−1)=ϕ, and μ(A(k)) can be calculated by Eq. (3).

Step 4: Let H₁ be a collection of benefit criteria (i.e. the larger c_j, the greater preference) and H₂ be a collection of cost criteria (i.e. the smaller c_j, the greater preference). The Pythagorean fuzzy positive-ideal solution (PFPIS), denoted p+=(p1+,p2+,…,pn+), and the Pythagorean fuzzy negative-ideal solution (PFNIS), denoted p−=(p1−,p2−,…,pn−), are defined as follows:

and

where p+=(ai+,bi+) and p−=(ai−,bi−)(i=1,2,…,n).

Step 5: According to Pythagorean fuzzy distance, calculate the distance between the alternative X_i and the PFPIS p⁺ and the distance between the alternative X_i and the PFNIS p⁻, respectively:

and construct a Pythagorean fuzzy positive-ideal separation matrix D⁺ and Pythagorean fuzzy negative-ideal separation matrix D⁻ as follows:

and

Procedure I

Step 6: Calculate the gray relational coefficient of each alternative from positive-ideal solution (PIS) and negative-ideal solution (NIS) using the following equations, respectively. The gray relational coefficient of each alternative from PIS is given as

i=1,2,…,m;j=1,2,…,n.

Similarly, the gray relational coefficient of each alternative from NIS is given as

i=1,2,…,m;j=1,2,…,n, where the identification coefficient is ρ = 0.5.

Step 7: Calculate the degree of gray relational coefficient of each alternative from the PIS and NIS using the following equations, respectively:

The basic principle of the GRA method is that the chosen alternative should have the largest degree of gray relation from the PIS and the smallest degree of gray relation from the NIS. Obviously, for the weight vector given, the smaller ξi− and the larger ξi+, the better alternative A_i is. However, the information about attribute weights is incompletely known. Thus, in order to get ξi− and ξi+, we must first calculate the weight information. Thus, we can establish the following multiple objective optimization models to calculate the weight information:

As each alternative is non-inferior, there exists no preference relation on all the alternatives. Then, we may aggregate the above multiple objective optimization models with equal weights into the following single objective optimization model:

By solving the model (M2), we get the optimal solution w=(w1,w2,…,wn), which can be used as the weight vector of attributes. Then, we can get ζi+(i=1,…,m) and ζi−(i=1,…,m) by Eqs. (27) and (28), respectively.

Step 8: Calculate the relative relational degree of each alternative from PIS using the following equation:

Step 9: Rank all the alternatives Ai(i=1,2,…,m) and select the best one(s) in accordance with ζi(i=1,2,…,m). If any alternative has the highest ζ_i value, then it is the most important alternative.

Procedure II

Step 6: Calculate the gray relational coefficient of each alternative from PIS and NIS using the following equation. The gray relational coefficient of each alternative from PIS is given as

i=1,2,…,m;j=1,2,…,n.

Step 7: Case a: If the information about attribute weights is incompletely known, in order to get the ξi+, we must first calculate the weight information. The gray relational coefficient between PIS and itself is (1,1,…,1), so the comprehensive gray relational coefficient deviation sum is

Therefore, we can establish the following multiple objective optimization models to calculate the weight information:

As each alternative is non-inferior, there exists no preference relation on all the alternatives. Then, we may aggregate the above multiple objective optimization models with equal weights into the following single objective optimization model:

By solving the model (M4), we get the optimal solution w=(w1,w2,…,wn), which can be used as the weight vector of attributes. Then, we can get ζi+ (i=1,2,…,m) by Eq. (27).

Case b: If the information about attribute weights is completely unknown, we can establish another multiple objective programming model as follows:

Similarly, we may aggregate the above multiple objective optimization models with equal weights into the following single objective optimization model:

To solve this model, we construct the Lagrange function:

where θ is the Lagrange multiplier. The partial derivatives of L with respect to w_j and θ are computed as follows:

Putting Eq. (40) in Eq. (41), we get

Combining Eqs. (40) and (42), we get

Then, we can get ζi+(i=1,…,m) by Eq. (27).

Step 8: Rank all the alternatives Xi(i=1,2,…,m) and select the best one(s) in accordance with ζi+(i=1,…,m). If any alternative has the highest ζi+ value, then it is the most important alternative.

4 Illustrative Examples

In this section, we shall present a numerical example with Pythagorean fuzzy information in order to illustrate the method developed in this paper.

The three experts’ own opinions regarding the results obtained with each emerging technology enterprise are shown in Tables 1–3.

Assume that the information about attribute weights, given by experts, is partly known: Δ={0.2≤w1≤0.25,0.15≤w2≤0.2,0.28≤w3≤0.32,0.35≤w4≤0.4} wj≥0, j=1,2,3,4, ∑j=14wj=1. Then, we utilize the developed approach to get the most desirable alternative(s).

Step 2: We first determine the fuzzy density of each decision maker and its λ parameter. Suppose that μ(A1)=0.30, μ(A2)=0.40, μ(A3)=0.50. Then, λ of expert can be determined as λ=−0.45. By Eq. (6), we have μ(A1,A2)=0.65, μ(A1,A3)=0.73, μ(A2,A3)=0.81, and μ(A1,A2,A3)=1.

Step 3: According to Definition 5, pij(k) is reordered such that pij(k)≥pij(k−1). Then, utilize the PFCIG operator

to aggregate all the Pythagorean fuzzy decision matrices R(k)=[pij(k)]m×n into a collective Pythagorean fuzzy decision matrix R=[pij]m×n, as below.

Step 4: Utilizing Eqs. (20) and (21), we get the PIS and NIS, respectively, as

Step 5: Utilize Eqs. (23) and (24) to get the positive-ideal separation matrix and negative-ideal separation matrix, respectively, as follows:

Step 6: Utilizing Eqs. (25) and (26), we get the gray relational coefficient matrices in which each alternative is calculated from PIS and NIS as follows:

Step 7: Utilize the model (M2) to establish the following single-objective programming model: {minξ(w)=−0.6554w1+0.4124w2−0.0364w3−0.5455w4.

Solving this model, we get the weight vector of attributes:

Then, we can get the degree of gray relational coefficient of each alternative from PIS and NIS:

Step 8: Utilizing Eq. (29), we get the relative relational degree of each alternative from PIS as follows:

Step 9: According to the relative relational degree, the ranking order of the five alternatives is X4>X3>X5>X1>X2, and thus the most desirable alternative is X₄.

In the following example, we will use Procedure II to get the gray relation coefficient matrix and optimal weight of the attributes.

Assume that the four decision makers di(i=1,2,3,4) provide his/her preference information on candidates with regard to attributes by using PFNs, as shown in Tables 4–7.