1. Introduction

Decision-making information provided by decision makers is often imprecise and uncertain because of a lack of data, time pressure, or the decision maker’s limited attention and information-processing capabilities. Accordingly, research pertaining to multiple criteria decision analysis (MCDA) problems has often been performed within a fuzzy uncertain environment^1,4,11,18.

The concept of Pythagorean fuzzy (PF) sets, which was introduced by Yager²³ and Yager and Abbasov²⁴, is a useful and valuable extension of Atanassov’s intuitionistic fuzzy sets², in which the sum of the degree of membership and the degree of non-membership is less than or equal to one. Intuitionistic fuzzy sets are the most widely used type of nonstandard fuzzy modelsbecause of their great ability to handle imprecise and ambiguous information^3,4,18. In a similar manner of the intuitionistic fuzzy model, PF sets are also characterized by degrees of membership and non-membership. However, their sum is not required to be less than one. Instead, the square sum of degrees of membership and non-membership is less than or equal to one^8,12,17,25. Because of the relaxed constraint conditions, PF sets are capable of managing more-complex uncertainty in real-world decision situations than intuitionistic fuzzy sets^6,7,25. From this perspective, PF sets are becoming increasingly popular in the MCDA field by the day^5–7,15.

Because PF sets possess superior ability to reflect the ambiguous nature of subjective judgments and model highly uncertain information in realistic applications, numerous valuable methods have been developed to solve MCDA problems within the PF environment. For example, Garg¹⁰ presented some series of geometric-aggregated operators within the PF environment and employed Einstein t-norm and t-conorm for MCDA. By combining PF sets with hesitant fuzzy sets, Liang and Xu¹² extended the technique for order preference by similarity to ideal solution (TOPSIS) to the hesitant PF environment. Mohagheghi et al.¹³ introduced a last aggregation evaluating approach based on a group decision-making procedure and PF sets to enhance decision flexibility. Rahman et al.¹⁶ developed useful PF weighted geometric aggregation operators to solve a group decision-making problem concerning plant location selection. Wei¹⁹ proposed some PF interaction aggregation operators to solve MCDA problems in the PF context. Xu et al.²¹ proposed PF induced generalized ordered weighted averaging (OWA) operators to manage group decision-making problems.

Based on useful concepts of displaced and fixed remoteness indices, Chen⁷ developed a novel VlseKriterijumska Optimizacija I Kompromisno Resenje (i.e., multicriteria optimization and compromise solution) (VIKOR) method for MCDA involving Pythagorean fuzzy information. Wei and Lu²⁰ utilized power aggregation operators to develop some useful PF power aggregation operators for addressing MCDA problems. Xue et al.²² proposed the linear programming technique for multidimensional analysis of preference (LINMAP) based on the entropy theory and PF sets, and they applied the PF LINMAP to solve multiple attribute group decision-making problems.

Considering the powerfulness of the PF theory in tackling imprecise and ambiguous information, the purpose of this paper is to propose an assignment-based MCDA method using correlation-based precedence indices for addressing MCDA problems under complex uncertainty based on PF sets. Concretely speaking, this paper first presents a valuable concept of correlation-based precedence indices that can fully take into account the amount of information associated with PF sets and can effectively discriminate among evaluative ratings under complex uncertainty. In the PF context, this paper subsequently discusses some valuable properties of the developed correlation-based precedence index in detail. By use of correlation-based precedence indices, this paper constructs some useful concepts of a discordance indicator, a weighted discordance indicator, a comprehensive discordance indicator, and a comprehensive discordance index within the PF environment. Based on these valuable concepts and a permutation matrix, this paper establishes an assignment model for the purpose of addressing MCDA problems involving PF information. Furthermore, an effective algorithm is provided to help decision makers manage vagueness and uncertainty and conduct multiple criteria evaluation and selection in the PF context. Finally, a practical example is provided to illustrate the application of the proposed methodology and demonstrate its practicality and effectiveness.

The remainder of this paper is organized as follows. Section 2 briefly introduces some basic concepts related to PF sets that are used throughout this article. Section 3 develops a new concept of correlation-based precedence indices and investigates some interesting properties. Section 4 formulates an MCDA problem under complex uncertainty based on PF information. Section 5 proposes a novel assignment-based MCDA method for acquiring a comprehensive ranking of competing alternatives. Section 6 applies the developed methodology to a practical problem regarding a financing problem of working capital policies in order to demonstrate its feasibility and applicability. Section 7 investigates a comparison of the effects between the Pythagorean fuzzy and intuitionistic fuzzy approaches through a sensitivity analysis in the context of intuitionistic fuzziness. Finally, Section 8 presents the conclusions.

2. Basic Concept of PF Sets

This section first briefly reviews some basic concepts related to PF sets that are used throughout this article.

Fig. 1

Comparison of spaces for a PF value and an intuitionistic fuzzy value.

3. Correlation-Based Precedence Index

Inspired by the idea of correlation coefficients for PF sets⁹, this section introduces a novel concept of correlation-based precedence indices within the PF environment and investigates their useful and desirable properties.

It is worthwhile to mention that the core concept of the proposed methodology is an extended definition of correlation coefficients in the PF context. The concept of correlation coefficients is one of the most used indices in statistics and engineering sciences. With the aid of a measure of interdependency, correlation can reflect the joint relationship between two variables and indicate how well two variables move together in a linear fashion. However, the existing correlation coefficients based on the probability theory cannot be employed within the PF environment. In particular, massive fuzzy uncertainty exists in a complicated decision-making system, which leads to the difficulty in exact estimations of the probability of events. Accordingly, the correlation coefficients yielded by the probabilistic approach have limitations to deal with built-in uncertainties contained in the PF information.

Considering the usefulness of correlation coefficients in many real-world decision-making problems, this paper proposes a novel correlation coefficient to measure the relationship between two PF values. Different from the classical definition, pairs of membership, non-membership, and indeterminacy degrees within the PF values will be considered during formulation of PF correlation coefficients. Based on the extended correlation coefficient, a useful concept of correlation-based precedence indices is then constructed in this section. These new concepts can facilitate to establish an easy-to-use assignment-based model to manage MCDA problems with PF information. As can be expected, the results based on the proposed correlation coefficients and correlation-based precedence indices can provide useful and influential information to decision makers and hence the corresponding MCDA approach is adequate to account for highly complicated uncertainties within the PF environment.

4. MCDA Problem Under PF Uncertainty

The discussed problems in this paper mainly are a type of decision-making problems which have limited numbers of candidate alternatives evaluated on multiple criteria. Additionally, in view that there are many real-world situations where due to insufficiency in information availability, PF sets are appropriate to deal with such problems under complex uncertainty.

Consider an MCDA problem in which Z = {z₁, z₂,· · ·, z_m} is a set of m (m ≥ 2) candidate alternatives and C = {c₁, c₂, · · ·,c_n} is a finite set of n (n ≥ 2) criteria. For numerous practical MCDA problems, the decision maker’s evaluations related to grades of importance of criteria and evaluative ratings of alternatives with respect to each criterion are often expressed by linguistic terms comprising vagueness and uncertainty. These uncertain, vague and hesitant judgments provided by the decision maker can be represented more comprehensively by using the PF sets. For example, Chen⁷ suggested a PF linguistic rating system to transform five- and seven-point linguistic rating scales into suitable PF values.

By use of a linguistic rating system, the evaluative rating of an alternative z_i ∈ Z with respect to a criterion c_j ∈ C can be expressed as a PF value p_ij. = (µ_ij, ν_ij), such that µ_ij∈ [0,1], ν_ij ∈[0,1], and 0 ≤ (µ_ij)² + (ν_ij)² ≤ 1. In particular, µ_ij and ν_ij represent the degrees to which z_i performs well and poorly, respectively, in terms of c_j. The indeterminacy degree that corresponds to each p_ij. is computed as πij=1−(μij)2−(νij)2 . The MCDA problem within the PF environment can be concisely expressed in the following PF decision matrix:

In the PF context, let a PF value w_j = (ω_j, ϖ_j). denote the importance weight of criterion c_j ∈ C, such that ω_j ∈ [0,1], ϖ_j ∈ [0,1], and 0 ≤ (ω_j)²+ (ϖ_j)² ≤ 1. The indeterminacy degree τ_i, that corresponds to each w_j is determined as 1−(ωj)2−(ϖj)2 .

Having formulating the MCDA problem involving PF information, the next part of this research is to postulate and define a new assignment model by utilizing the developed concept of correlation-based precedence indices.

5. A Novel Assignment-Based MCDA Method

Based on the correlation-based precedence index, this section attempts to establish a novel assignment-based MCDA method to determine a comprehensive ranking of candidate alternatives that is in the closest agreement with the criterion-wise precedence ranks among alternatives.

As demonstrated in the previous theorems, the correlation-based precedence index within the PF environment possesses certain useful and interesting properties. Consider a PF evaluative rating p_ij in the PF decision matrix p. The larger the I(p_ij) value is, the better the performance on the evaluation result it is, and the greater the preference is for p_ij. In contrast, the smaller the I(p_ij) value is, the worse the performance on the evaluation result it is, and the lesser the preference is for p_ij. Additionally, the same I(p_ij) value represents indifferent preference. Consequently, the criterion-wise precedence ranks among competing alternatives can be precisely determined in descending order of the I(p_ij) values.

To show the usefulness of the permutation matrix Γ consider the following simple example. Assume that there are three candidate alternatives z₁, z₂, and z₃. For convenience, let Z=(z₁, z₂, z₃) in this reference order. The ordering (z₃, z₁, z₂) (i.e., the ranking z₃ ≻ z₁ ≻ z₂) could be obtained by multiplying Z by Γ, where

This gives that:

Therefore, by varying the position of the zeroes and ones in the permutation matrix Γ, all of the possible rank orders can be generated correspondingly.

As far as the objective function in the developed assignment model is concerned, one can see that it combines the criterion-wise precedence rankings linearly by means of comparisons of the correlation-based precedence indices. Nevertheless, the overall compensation hypothesis among criteria is captured by the choice of the sum of comprehensive discordance indices which is to be minimized. More specifically, the decision maker would like to determine an optimal comprehensive ranking that effects a best compromise among all criterion-wise precedence rankings. Namely, the objective function should assign an alternative to that rank order which has the smallest comprehensive discordance index.

Reconsider the reference order Z=(z₁, z₂, z₃) as an example. Suppose that the following comprehensive discordance indices are obtained: D(d11)=0.92 , D(d12)=0.83 , D(d13)=0.77 , D(d21)=0.85 , D(d22)=0.76 , D(d23)=0.94 , D(d31)=0.75 , D(d32)=0.88 , and D(d33)=0.86 . If one arbitrarily chooses the ranking z₃ ≻ z₁ ≻ z₂, then the binary variables in Γ are as follows: γ12=γ23=γ31=1 and γ11=γ13=γ21=γ22=γ32=γ33=0 . Accordingly, the sum of the comprehensive discordance indices corresponding to the ordering (z₃, z₁, z₂) is 2.52(=∑i=13∑k=13(γik⋅D(dik))=D(d12)+D(d23)+D(d31)=0.83+0.94+0.75) . Generally, the decision maker would think of examining all criterion-wise precedence rankings and choosing the one which yields the smallest value of ∑i=13∑k=13(γik⋅D(dik)) . Because the ordering (z₃, z₂, z₁) yields the smallest value 2.28(=D(d13)+D(d22)+D(d31)=0.77+0.76+0.75) , the optimal comprehensive ranking in this example is z₃ ≻ z₂ ≻ z₁.

The smaller the degree of discordance indicated by D(dik) is, the greater the concordance will be from assigning z_i, to the k-th comprehensive rank. It is clear to see that the decision maker prefers that overall ranking where ∑i=1m∑k=1m(γik⋅D(dik)) is the smallest because it represents that ranking which effects the best compromise among all criterion-wise precedence rankings. Such an exhaustive search can be formulated using a simple linear programming model. Simply note that γik is unknown to be determined by the model. Consequently, the constraints on the problem are the constraints for γik . Remember that ∑k=1mγik=1 and ∑i=1mγik=1 . For solution purposes, the following assignment model can be established:

Another way is to arbitrarily choose each of the m! possible rank orders to scrutinize which one yields the smallest value of ∑i=1m∑k=1m(γik⋅D(dik)) . However, a linear programming procedure affects a solution easier and quicker. More precisely, solving the linear programming problem given by Eq. (23) yields the optimal permutation matrix Γ^ as would be obtained by complete enumeration but much more efficiently.

It can be verified that the Γ^ yields the smallest value of the total sum of the comprehensive discordance indices as required. Let the reference order Z = (z₁, z₂, · · ·, z_m) for notational convenience. Based on Eq. (24), the decision maker can find the optimal order by applying the permutation matrix Γ^ to Z. Namely, the optimal comprehensive ranks of the m alternatives can be determined by multiplying Z by the optimal permutation matrix Γ^ , as follows:

By using correlation-based precedence indices, the proposed assignment-based MCDA method can be summarized as follows:

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  Step 1: Formulate an MCDA problem with the set of candidate alternatives Z = {z₁, z₂,· · ·,z_m} and the set of criteria C = {c₁, c₂,· · ·, c_n}.

• •

  Step 2: Establish the PF evaluative rating p_ij of each alternative z_i ∈ Z with respect to criterion c_j ∈ C and the PF importance weight w_i of each c_j Form a PF decision matrix p = [p_ij]_m×n.

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  Step 3: Apply Eq. (14) to calculate the correlation-based precedence index I(p_ij) for each p_ij in the PF decision matrix p.

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  Step 4: Form the rank evaluation matrix r based on the descending order of the I(p_ij) values with respect to each c_j. A mean rank should be assigned in case of the occurrence of ties.

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  Step 5: Employ Eq. (20) to compute the comprehensive discordance indicator dik for z_i, to become the comprehensive rank k

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  Step 6: Use Eq. (21) to determine the comprehensive discordance index D(dik) of each dik .

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  Step 7: Construct a permutation matrix Γ=[γik]m×m and an assignment model using Eqs. (22) and (23), respectively.

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  Step 8: Solve for the optimal permutation matrix Γ^ . Obtain the optimal comprehensive ranks of the m alternatives using Eq. (24).

6. Practical Application

As an application of the proposed methodology, this section investigates a financing decision problem on aggressive/conservative policies of working capital management for a medical institution to demonstrate the feasibility and effectiveness of the developed approach in practice.

Working capital whose main components are accounts payable, accounts receivable, and inventory concerns managing the day-to-day short-term operations of a firm. Working capital management has been playing a pivotal role for corporate finance due to its significant influence on firms’ liquidity and profitability. Financing policy refers to the financing models implemented by firms to satisfy working capital requirements. The financing policy of working capital management can be generally determined as aggressive and conservative. Because of the particularity of the medical and health care system, financing decision on aggressive or conservative working capital policies becomes a highly complicated and ambiguous MCDA problem.

The medical institution investigated in this real-world case study is Chang Gung Memorial Hospital (CGMH). The main branch of CGMH is located in Linkou. Linkou CGMH is not only a multi-specialty hospital but also the largest medical center in Taiwan. Working capital management is important to Linkou CGMH because it can grant this medical center financial flexibility and reduce the dependence on external sources of finance. This case study comprises five plans related to working capital financing policy: aggressive dominant (z₁) (i.e., 100% aggressive plan-oriented), aggressive-leaning (z₂) (i.e., 75% aggressive and 25% conservative), balanced aggressive and conservative (z₃) (i.e., 50% aggressive and 50% conservative), conservative-leaning (z₄) (i.e., 25% aggressive and 75% conservative), and conservative dominant (z₅) (i.e., 100% conservative plan-oriented). To carefully evaluate the five types of alternative plans, managers consider the following six evaluation criteria: cash reserves (c₁), maturity hedging (c₂), interest rate fluctuation (c₃), financial leverage (c₄), return on assets (c₅), and financing cost (c₆).

The proposed assignment-based MCDA method using correlation-based precedence indices was employed to help Linkou CGMH to select the most appropriate financing policy of working capital management. In Step 1, this MCDA problem is defined by five financing policies and six criteria for evaluating the alternatives. The set of alternatives is denoted by Z = {z₁, z₂,· · ·, z₅}, and the set of criteria is denoted by C = {c_l, c₂, · · ·, c₆}.

In Step 2, based on managers’ knowledge and expertise in Linkou CGMH, the PF decision matrix p (= [p_ij]_5×6) was constructed as follows:

The grades of criterion importance were expressed using a seven-point scale, and the obtained linguistic evaluation terms were as follows: low for c₁, fair for c₂, high for c₃, very low for c₄, very high for c₅, and extremely high for c₆. Applying Chen’s seven-point rating system⁷, the PF importance weights of the six criteria were acquired as follows: w₁=(0.35, 0.65), w₂=(0.55, 0.45), w₃=(0.65, 0.35), w₄=(0.25, 0.75), w₅=(0.75, 0.25), and w₆=(0.85, 0.15).

In Step 3, the computed results of the correlation-based precedence index I(p_ij) for each p_ij in p are presented in the top part of Table 1. Consider I(p₁₅) as an example. By using Eq. (14), one can obtain: