A Pythagorean fuzzy ANP-QFD-Grey relational analysis approach to prioritize design requirements of sustainable supply chain

Abstract

A sustainable supply chain (SSC) is vital for company’s sustainability success, so it is imperative to identify and prioritize SSC’s design requirements (DRs) for better SSC planning. For customer-centric markets, the customer requirements (CRs) need to be integrated into SSC’s DRs. This paper thus proposes a customer-centric approach based on Analytic Network Process (ANP), Quality Function Deployment (QFD), Grey Relational Analysis (GRA), and Pythagorean Fuzzy Set (PFS) to rank SSC’s DRs, considering CRs and information ambiguity. The PFS is combined with ANP, QFD, and GRA to better handle uncertainty in the SSC. The Pythagorean fuzzy ANP is applied to analyze the correlations among the sustainable CRs and determine the corresponding weights. The sustainable CRs are transformed into the DRs using the Pythagorean fuzzy QFD. The relationships among the resulting DRs are analyzed through Pythagorean fuzzy GRA to prioritize DRs. The approach is validated through a case study. The results obtained in this paper shows that the proposed method is efficient to prioritize DRs of SSC with the consideration of sustainable CRs under uncertain environment. The novelties of proposed method are that it not only offers a customer-oriented SSC planning method through the integration of ANP, QFD and GRA, but also can reflect the uncertain information with a broader membership representation space via PFSs. Based on the proposed method, the decision-maker can conduct comprehensive analysis to prioritize DRs and design appropriate SSC to fulfill CRs under uncertain environment.

Keywords

Sustainable supply chain design requirements analytic network process pythagorean fuzzy set quality function deployment grey relational analysis

1 Introduction

As environmental and ethical awareness grow, supply chain (SC) advocates are demanding higher business performance on environmental concerns, social responsibility, and economic benefits [1, 2]. Consequently, enterprises are now pro-actively integrating sustainability factors into their SC and taking steps to support management in integrating sustainability [3 –5]. In this context, sustainable supply chain (SSC) design has become a critical strategy for enterprises to improve their performance on the economic, environmental and social aspects for overall business success [6, 7]. Thus, it is vital for enterprises to identify the critical design requirements (DRs) for the effective SSC design, which is the key driver of this study.

Since customer requirements (CRs) are the source and purpose of SSC development in a customer-oriented market, the CRs play an essential role in the successful design of SSC. The important goal of a successful SSC design is to fulfill the CRs and pursue higher sustainable performance. Thus, the first step of SSC development is to capture CRs and integrate them into DRs of SSC for an effective SSC design solution. As well known to all, QFD is a customer-driven method, which is effective to identify the relationship between CRs and DRs, and determine the priority of DRs considering CRs. Although there are many studies that utilize QFD to integrate CRs into SSC, QFD has difficulty to understand and analyze CRs’ hierarchical relationships and complex interrelations in a systematic and comprehensive manner due to the inherent deficiencies [8]. In addition, the understanding of DRs, especially the priority analysis of DRs is an important decision basis for SSC development [9]. The traditional QFD prioritizes DRs based on the relationships between CRs and DRs evaluated by experts, which ignores the errors caused by small sample size and cannot illustrate the intercorrelations among DRs, and may have a negative impact on the proper prioritization of DRs. Finally, the traditional QFD is insufficient in dealing with uncertainty and fuzziness of evaluation information involved in the SSC development, which may influence the priority analysis of DRs and subsequently result in an inappropriate decision support in SSC development.

To bridge the research gaps mentioned above, this study proposes an integrative methodology to prioritize DRs in customer-oriented SSC development under fuzzy environment, which is achieved by a hybrid analytical approach combining of analytic network process (ANP), quality function deployment (QFD), grey relational analysis (GRA) and Pythagorean fuzzy set (PFS). In the proposed method, PFS is applied to express the objective evaluation information through the SSC planning, including evaluation of correlations among CRs, relationship between CRs and DRs. At first, the Pythagorean fuzzy ANP (PF-ANP) is developed to prioritize sustainable CRs of SSC by analyzing the hierarchical relationships and correlations of sustainable CRs. Next, PFS is incorporated into QFD to transform sustainable CRs into SSC’s DRs using Pythagorean fuzzy numbers (PFNs), so as to effectively handle the uncertainty and fuzziness of experts’ evaluation through describing the membership and non-membership. Then, based on the integration of the developed PF-ANP and Pythagorean fuzzy QFD (PF-QFD), and the Pythagorean fuzzy GRA (PF-GRA) is adopted to analyze the interrelations among DRs considering the sustainable CRs and prioritize DRs. Finally, a case is studied to validate the proposed method and a comparative analysis is conducted.

The novelties of proposed method are described as following. Firstly, the integration of ANP, QFD and GRA offers a customer-oriented SSC planning method through incorporating CRs into DRs of SSC. Secondly, expressing evaluation information with PFSs, the proposed method can effectively handle the uncertain information with a broader membership representation space of PFSs. Thirdly, based on the analysis results obtained from PF-ANP and PF-QFD, the PF-GRA can prioritize DRs and express the interrelation of DRs through subjective analysis, which can compensate for subjective errors resulted from small sample size. Thus, with the proposed method, the decision-maker can conduct comprehensive analysis to prioritize DRs and design appropriate SSC to fulfill CRs under uncertain environment.

The structure of this paper is set as follows. Section 2 offers a review of the research status of SSC development, and integrated QFD. In Section 3, the concepts related to PFS are introduced. In Section 4, the integrative method based on ANP, QFD, GRA, and PFS is introduced in detail. In Section 5, a case study is used to illustrate the feasibility of this method. Subsequently, Section 6 discusses the effectiveness of the method and Section 7 concludes the study.

2 Literature review

In a customer-oriented market, successful SSC development should begin with understanding sustainable CRs and integrating them into SSC’s DRs. As well known to all, QFD is a customer-driven method, which transforms CRs into relevant attributes effectively and enables designers accurately to develop the products or services [10 –13]. Thus, QFD is viewed as an effective method to incorporate CRs into strategic management and operations management [14 –16], and it has been applied it in various fields, such as product development [17 –21], service design [22]. In addition, QFD has been successfully applied to transform CRs into SSC’s DRs [6, 22]. Thus, QFD has great potential in integrating sustainable CRs into DRs to support customer-oriented SSC development. However, some limitations exist in QFD, which should be handled to support SSC design.

First, there are complex hierarchical relationships and interrelationships in CRs, but QFD cannot analyze CRs’ complex relationships systematically [8], which may have impact on the appropriate importance analysis of SSC’s DRs. While, ANP enables managers to deeply understand the hierarchical relationships and interrelationships in CRs, and many studies combine ANP with QFD to analyze CRs’ interrelations [20]. Aiming to select supplier based on CRs, Asadabadi Mehdi Rajabi [23] applied ANP in QFD to describe the internal dependencies of CRs, product requirements, and suppliers’ qualifications. As for the customer-oriented SSC design, ANP has been combined with QFD to prioritize CRs considering the interrelations and transform CRs into SSC’s DRs [22, 24]. The existing studies provide valuable insights into the integration of ANP and QFD to tackle the deficiency of QFD in CRs analysis.

Second, the traditional QFD determines the priority of DRs based on the relationships between CRs and DRs evaluated by experts, which may have subjective errors caused by the small sample size. Moreover, the relationships between CRs and DRs may imply the interrelations among DRs, but the traditional QFD cannot extract and utilize this information. While, GRA proposed by Julong Deng [25] can provide precise interrelationship information through the mathematical analysis of a relatively small amount of data. In this respect, GRA has been extended into multi-criteria decision making (MCDM) problems to provide more objective and accurate decision support. Avinash Samvedi et al. [26] have integrated GRA with analytical hierarchy process (AHP) to rank the alternative machine tools. Aiming at selecting the optimal aviation fuel, the fuzzy ANP and fuzzy GRA are developed to rank the alternative aviation fuels. Xu Wang et al. [9] and Wenyan Song et al. [27] have employed GRA to prioritize technical attributes in QFD. These researches indicate that GRA has advantages to analyze SSC’s DRs based on QFD.

In addition, as a typical MCDM method, various information uncertainty is involved in QFD. While, the traditional QFD mentioned in above researches are insufficient to deal with information uncertainty, which may result in a negative impact on the accuracy of DRs’ ranking results in SSC design. Recognizing the imprecise information and ambiguity issues, fuzzy set (FS) is employed into QFD to express decision maker’s evaluation [28, 29]. As FS expresses experts’ preference using only membership degree, it neglects the non-membership and hesitance degrees involved in the subjective evaluation. So, to overcome the drawbacks of FS, the intuitionistic fuzzy set (IFS) developed by Atanassov Krassimir T. [30] is extended to QFD, which can effectively describe experts’ uncertainty with membership, non-membership and hesitance degrees. However, the sum of membership degree and non-membership degree of IFS should be less than or equal to 1, and it fails to deal with the situation when the sum is greater than 1. Aiming to address the shortcomings existing in IFS, PFS proposed by Yager Ronald R. [31] is considered as an effective method, whose sum of membership degree and non-membership degree can exceed 1, but the sum of squares may not. Thus, PFS enables a larger domain to express the uncertainty, and is more powerful on the issues of uncertainty. Motivated by this, a great attention is paid toward applying PFS in MCDM issues to handle the information uncertainty. Ahme Çalık [32] has integrated AHP and the technique for order preference by similarity to ideal solution (TOPSIS) methods under the Pythagorean fuzzy environment to select the best green supplier. The Pythagorean fuzzy TOPSIS method based on similarity measure has been developed for sustainable recycling partner selection [33]. To handle the Pythagorean fuzzy information effectively in the group decision-making process, some new operational laws and operators has been established [34 –37]. For example, Garg Harish [36] has developed some new operational laws and the corresponding weighted aggregation operators for Pythagorean fuzzy information to neutrally treat the membership and non-membership degrees, and applied them in multiple attribute group decision-making problems. It is observed form the above studies that the fuzzy MCDM problems can be successfully solved under PFS environment. Thus, PFS is an appropriate method to compensate for the shortcomings of QFD when dealing with information uncertainty in SSC’s DRs analysis.

Therefore, aiming to solve the shortcomings mentioned above, a novel customer-oriented method integrating ANP, QFD and GRA under Pythagorean fuzzy environment is presented to prioritize DRs of SSC.

3 Preliminaries of PFS

The PFS is an extension of the IFS [37]. Compared to the classical FS, the sum of membership and non-membership degrees can exceed 1, but the sum of their squares cannot exceed 1 [38]. Therefore, PFS can have a larger domain to express the uncertainty, and be more powerful on the issues of uncertainty. In this section, some basic definitions related to PFS have been presented, and more descriptions in details can be seen in the corresponding literatures.

Definition 1. [35 , 39] A PFS P on a non-empty set X ={ x₁, x₂, ⋯ , x_n } is defined as: $P = {〈 x, P (u_{p} (x), v_{p} (x)) 〉 | x \in X}$

Where u_p (x) ∈ [0, 1] and v_p (x) ∈ [0, 1] denote the degrees of belonging and non-belonging on the objective x ∈ X respectively, and 0 ⩽ (u_p (x)) ² + (v_p (x)) ² ⩽ 1. The degree of hesitancy is expressed as $π_{p} (x) = \sqrt{1 - {(u_{p} (x))}^{2} - {(v_{p} (x))}^{2}}$ , x ∈ X.

Definition 2. [33 , 39] Let β₁ = P (u_{β
₁}, v_{β
₁}), β₂ = P (u_{β
₂}, v_{β
₂}) and β = P (u_β, v_β) be PFNs. The operations for these PFNs are given by:

β₁ ∪ β₂ = P (max {u_β
₁, u_β
₂} , min {v_β
₁, v_β
₂});

β₁ ∩ β₂ = P (min {u_β
₁, u_β
₂} , max {v_β
₁, v_β
₂});

$β_{1} \oplus β_{2} = P (\sqrt{u_{β_{1}}^{2} + u_{β_{2}}^{2} - u_{β_{1}}^{2} u_{β_{2}}^{2}}, v_{β_{1}} v_{β_{2}})$ ;

$β_{1} \otimes β_{2} = P (u_{β_{1}} u_{β_{2}}, \sqrt{v_{β_{1}}^{2} + v_{β_{2}}^{2} - v_{β_{1}}^{2} v_{β_{2}}^{2}})$ ;

$β_{1} ⊖ β_{2} = P (\sqrt{\frac{u_{β_{1}}^{2} - u_{β_{2}}^{2}}{1 - u_{β_{2}}^{2}}}, \frac{v_{β_{1}}}{v_{β_{2}}}), u_{β_{1}} ⩾ u_{β_{2}}, v_{β_{1}} ⩽ min {v_{β_{2}}, \frac{v_{β_{2}} π_{β_{1}}}{π_{β_{2}}}}$ ;

$β_{1} ø β_{2} = P (\frac{u_{β_{1}}}{u_{β_{2}}}, \sqrt{\frac{v_{β_{1}}^{2} - v_{β_{2}}^{2}}{1 - v_{β_{2}}^{2}}}), u_{β_{1}} ⩽ min {u_{β_{2}}, \frac{u_{β_{2}} π_{β_{1}}}{π_{β_{2}}}}, v_{β_{1}} ⩾ v_{β_{2}}$ ;

$λ β = P (\sqrt{1 - {(1 - u_{β}^{2})}^{λ}}, {(v_{β})}^{λ}), λ > 0$ ;

$β^{λ} = P ({(u_{β})}^{λ}, \sqrt{1 - {(1 - v_{β}^{2})}^{λ}})$ , λ > 0;

Definition 3. [34, 40] Let β = P (u_β, v_β) be a PFN. The scoring function of β can be defined as follows: $S (β) = u_{β}^{2} - v_{β}^{2}, S (β) \in [- 1, 1]$ (1)

Definition 4. [40] Let β_j = P (u_{β
_j}, v_{β
_j}) (j = 1, 2) be two PFNs. The distance between β₁ and β₂ can be defined as follows:

$\begin{matrix} d (β_{1}, β_{2}) = \frac{1}{2} \\ (| u_{β_{1}}^{2} - u_{β_{2}}^{2} | + | v_{β_{1}}^{2} - v_{β_{2}}^{2} | + | π_{β_{1}}^{2} - π_{β_{2}}^{2} |) \end{matrix}$ (2)

Definition 5. [35, 41] Let β_j = P (u_{β
_j}, v_{β
_j}) (j = 1, 2, ⋯ , n) be PFNs, the Pythagorean fuzzy weighted averaging operator (PFWAO) are defined as below: $PFWAO = P (\sqrt{1 - \prod_{j = 1}^{n} {(1 - {(u_{β_{j}})}^{2})}^{w_{j}}}, \prod_{j = 1}^{n} {(v_{β_{j}})}^{w_{j}})$ (3) where w_j indicates the importance degree of β_j, satisfying w_j ⩾ 0 and $\sum_{j = 1}^{n} w_{j} = 1$ .

4 Methodology

This paper develops a customer-driven method to determine the priorities of SSC’s DRs under the Pythagorean fuzzy environment. In the section, the approach of combining ANP-QFD-GRA with PFS is employed to analyze SSC’s DRs. There is an assumption is that the experts involved in the evaluation process are of the same importance. First, the PF-ANP is employed to analyze the interrelated hierarchical relationships of sustainable CRs and to determine the priorities of sustainable CRs under fuzzy environment. Then, the analysis results of sustainable CRs are integrated into DRs through the PF-QFD for developing SSC more effectively, and PF-GRA is employed to determine the priority of DRs. Figure 1 shows the process. In the proposed method, the inputs of the process are including the Pythagorean fuzzy evaluation of sustainable CRs’ correlation, Pythagorean fuzzy evaluation of relationships between sustainable CRs and DRs. The priority of sustainable CRs and DRs are the outputs of the method.

Fig. 1

The process of proposed method.

4.1 Identification of the sustainable CRs

Identifying CRs is critical for enterprises to efficiently design the supply chain (SC) to satisfy CRs for greater customer satisfaction. CRs considered in this paper are sustainable requirements. As the main objectives of sustainability development for SC involve economic, environmental, and social sustainability, CRs are mainly divided into three categories, including economic requirements (CR₁), environmental requirements (CR₂) and social requirements (CR₃). CR_i denotes CRs in the ith category, i = 1, 2, 3. Then, the major CRs in each category are identified by employing questionnaires to customers, literature surveys, or expert interviews. The identified CRs in the ith category can be denoted by CR_ij, j = 1, 2, ⋯ m_i and m_i is the number of CRs in the ith category. The hierarchy diagram of sustainable CRs for SSC is described as follows (Fig. 2).

Fig. 2

Hierarchical structure of sustainable CRs for SSC.

4.2 Analysis of the sustainable CRs’ weights

As the limitation of SSC’s development resources, the companies cannot fulfill all the sustainable CRs at one time. Thus, it’s important to conduct priority analysis of the identified sustainable CRs. In this section, PF-ANP is employed to understand and analyze sustainable CRs, which can effectively model the interdependency among sustainable CRs, and address the imprecision and uncertainty involved in the experts’ subjective evaluations using PFNs.

4.2.1 The Pythagorean fuzzy pairwise comparison matrix between sustainable CRs

In this part, experts are invited to use PFNs to conduct pairwise comparisons to evaluate the inner dependency of sustainable CRs in the ith category with respect to CR_gl (g = 1, 2, 3 ; j = 1, 2, ⋯ , m_g), and the corresponding Pythagorean fuzzy pairwise comparison matrix ${\tilde{B}}_{gl}$ is shown in Table 1.

Table 1
The Pythagorean fuzzy pairwise comparison matrix of sustainable CRs in the i_th category with respect to CR_gl

CR _gl CR _i1 ⋯ CR _ik ⋯ CR _{im
_i} Normalized feature vector

CR _i1 ${\tilde{α}}_{11}^{gl}$ ⋯ ${\tilde{α}}_{1 k}^{gl}$ ⋯ ${\tilde{α}}_{1 m_{i}}^{gl}$ ${\tilde{w}}_{i 1}^{gl}$

⋮ ⋮ ⋱ ⋮ ⋯ ⋮ ⋮

CR _ij ${\tilde{α}}_{j 1}^{gl}$ ⋯ ${\tilde{α}}_{jk}^{gl}$ ⋯ ${\tilde{α}}_{{jm}_{i}}^{gl}$ ${\tilde{w}}_{ij}^{gl}$

⋮ ⋮ ⋯ ⋮ ⋯ ⋮ ⋮

CR _{im
_i} ${\tilde{α}}_{m_{i} 1}^{gl}$ ⋯ ${\tilde{α}}_{m_{i} k}^{gl}$ ⋯ ${\tilde{α}}_{m_{i} m_{i}}^{gl}$ ${\tilde{w}}_{{im}_{i}}^{gl}$

CR _gl	CR _i1	⋯	CR _ik	⋯	CR _{im _i}	Normalized feature vector
CR _i1	${\tilde{α}}_{11}^{gl}$	⋯	${\tilde{α}}_{1 k}^{gl}$	⋯	${\tilde{α}}_{1 m_{i}}^{gl}$	${\tilde{w}}_{i 1}^{gl}$
⋮	⋮	⋱	⋮	⋯	⋮	⋮
CR _ij	${\tilde{α}}_{j 1}^{gl}$	⋯	${\tilde{α}}_{jk}^{gl}$	⋯	${\tilde{α}}_{{jm}_{i}}^{gl}$	${\tilde{w}}_{ij}^{gl}$
⋮	⋮	⋯	⋮	⋯	⋮	⋮
CR _{im _i}	${\tilde{α}}_{m_{i} 1}^{gl}$	⋯	${\tilde{α}}_{m_{i} k}^{gl}$	⋯	${\tilde{α}}_{m_{i} m_{i}}^{gl}$	${\tilde{w}}_{{im}_{i}}^{gl}$

In Table 1, the PFN ${\tilde{α}}_{jk}^{gl} = (u_{jk}^{gl}, v_{jk}^{gl})$ denotes the relative importance of CR_ij compared to CR_ik with respect to CR_gl, $u_{jk}^{gl}$ and $v_{jk}^{gl}$ represent membership and non-membership degrees respectively. If j = k, $u_{jk}^{gl} = v_{jk}^{gl} = 0.5$ , it means that CR_ij and CR_ik are of the same importance. Based on the eigenvalue method proposed by Selim Zaim et al. [42] the normalized feature vector of the Pythagorean fuzzy pairwise comparison matrix ${\tilde{B}}_{gl}$ can be obtained, and is shown in Table 1, where the PFN ${\tilde{w}}_{ij}^{gl}$ denotes the relative weight of CR_ij under the criterion of CR_gl.

4.2.2 The Pythagorean fuzzy super matrix

Following the steps described in section 4.2.1, each CR in the gth category is taken as the criterion in turn, and the Pythagorean fuzzy pairwise comparison matrixes for the inner dependency evaluation of sustainable CRs in the ith category with respect to each criterion are constructed. Then the normalized feature vector of each Pythagoras fuzzy matrix is integrated as the Pythagorean fuzzy matrix ${\tilde{W}}_{ig}$ , which expresses the relative importance of each sustainable CR in the ith category with respect to each sustainable CR in the gth category. ${\tilde{W}}_{ig} = [\begin{matrix} \begin{matrix} {\tilde{w}}_{i 1}^{g 1} & {\tilde{w}}_{i 1}^{g 2} \\ {\tilde{w}}_{i 2}^{g 1} & {\tilde{w}}_{i 2}^{g 2} \end{matrix} & \begin{matrix} \dots & {\tilde{w}}_{i 1}^{{gm}_{g}} \\ \dots & {\tilde{w}}_{i 2}^{{gm}_{g}} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ \\ {\tilde{w}}_{{im}_{i}}^{g 1} & {\tilde{w}}_{{im}_{i}}^{g 2} \end{matrix} & \begin{matrix} \dots & ⋮ \\ \dots & {\tilde{w}}_{{im}_{i}}^{{gm}_{g}} \end{matrix} \end{matrix}]$

Following the steps mentioned above, the Pythagorean fuzzy matrixes for sustainable CRs in each category with respect to sustainable CRs in each category can be determined and they can be integrated as Pythagorean fuzzy super matrix $\tilde{W}$ . $\tilde{W} = [\begin{matrix} {\tilde{W}}_{11} & {\tilde{W}}_{12} & {\tilde{W}}_{13} \\ {\tilde{W}}_{21} & {\tilde{W}}_{22} & {\tilde{W}}_{23} \\ {\tilde{W}}_{31} & {\tilde{W}}_{32} & {\tilde{W}}_{33} \end{matrix}]$

4.2.3 The Pythagorean fuzzy weight matrix

According to the steps described in section 4.2.1, taking each CR_g as the criterion, the experts are invited to conduct the pairwise comparisons among CR₁, CR₂ and CR₃ according to its’ correlation degree with CR_g using the PFNs, and the Pythagorean fuzzy weight matrix $\tilde{A}$ of CR_i is obtained as follows: $\tilde{A} = [\begin{matrix} {\tilde{a}}_{11} & {\tilde{a}}_{12} & {\tilde{a}}_{13} \\ {\tilde{a}}_{21} & {\tilde{a}}_{22} & {\tilde{a}}_{23} \\ {\tilde{a}}_{31} & {\tilde{a}}_{32} & {\tilde{a}}_{33} \end{matrix}]$

The PFN ${\tilde{a}}_{ig} = (u_{ig}, v_{ig})$ denotes the relative weight of CR_i under the criterion CR_g, where i, g = 1, 2, 3.

4.2.4 The Pythagorean fuzzy weighted super matrix

Based on the Pythagorean fuzzy weight matrix $\tilde{A}$ and the Pythagorean fuzzy super matrix $\tilde{W}$ , the Pythagorean fuzzy weighted super matrix $\tilde{M}$ can be obtained through Equation (4). $\begin{matrix} \tilde{M} = \tilde{A} \tilde{W} = [\begin{matrix} {\tilde{a}}_{11} {\tilde{W}}_{11} & {\tilde{a}}_{12} {\tilde{W}}_{12} & {\tilde{a}}_{13} {\tilde{W}}_{13} \\ {\tilde{a}}_{21} {\tilde{W}}_{21} & {\tilde{a}}_{22} {\tilde{W}}_{22} & {\tilde{a}}_{23} {\tilde{W}}_{23} \\ {\tilde{a}}_{31} {\tilde{W}}_{31} & {\tilde{a}}_{32} {\tilde{W}}_{32} & {\tilde{a}}_{33} {\tilde{W}}_{33} \end{matrix}] \\ = [\begin{matrix} {\tilde{b}}_{11}^{11} & {\tilde{b}}_{11}^{12} & \dots & {\tilde{b}}_{11}^{3 m_{3}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ {\tilde{b}}_{ij}^{11} & {\tilde{b}}_{ij}^{12} & \dots & {\tilde{b}}_{ij}^{3 m_{3}} \\ ⋮ & ⋮ & \dots & ⋮ \\ {\tilde{b}}_{3 m_{3}}^{11} & {\tilde{b}}_{3 m_{3}}^{12} & \dots & {\tilde{b}}_{3 m_{3}}^{3 m_{3}} \end{matrix}] \end{matrix}$ (4)

Where, the PFN ${\tilde{b}}_{ij}^{gl} = (u_{ij}^{gl}, v_{ij}^{gl})$ , and i, g = 1, 2, 3 ; j = 1, 2, ⋯ , m_i; l = 1, 2, ⋯ , m_g.

4.2.5 Determine the importance weights of sustainable CRs

Based on the Pythagorean fuzzy weighted super matrix $\tilde{M}$ , CR_ij’s Pythagorean fuzzy importance weight ${\tilde{w}}_{ij}$ is obtained through the PFWAO and is shown in Equation (5). Then, the score of ${\tilde{w}}_{ij}$ denoted as S_ij is calculated by Equation (1), and indicates the priority of CR_ij. $\begin{array}{l} {\tilde{w}}_{i j} = P F W A O ({\tilde{b}}_{i j}^{11}, \dots {\tilde{b}}_{i j}^{1 m_{1}}, {\tilde{b}}_{i j}^{21}, \dots {\tilde{b}}_{i j}^{2 m_{2}}, {\tilde{b}}_{i j}^{31}, \dots, {\tilde{b}}_{i j}^{3 m_{3}}) \\ = (\sqrt{1 - {\prod_{g = 1, 2, 3}}_{l = 1, \dots, m_{g}}^{m_{1} + m_{2} + m_{3}} {(1 - {(u_{g l})}^{2})}^{1 / (m_{1} + m_{2} + m_{3})}}, {\prod_{g = 1, 2, 3}}_{l = 1, \dots, m_{g}}^{m_{1} + m_{2} + m_{3}} {(v_{g l})}^{1 / (m_{1} + m_{2} + m_{3})}) \\ = (u_{i j}, v_{i j}) \end{array}$ (5)

4.3 Identify DRs of sustainable supply chain

In SSC, sustainable CRs are satisfied with related DRs. To identify the main DRs, interviews with the experts in SSC design and literature study are conducted. The identified DRs are represented as DR_h (h = 1, 2, ⋯ , k), k is the number of DRs. Based on the identified major DRs, the PF-QFD and PF-GRA are integrated to analyze DRs of SSC.

4.4 Construct the Pythagorean fuzzy HoQ

To incorporate sustainable CRs into SSC design, HoQ of QFD is employed to express the relationships among sustainable CRs and DRs, which is evaluated by experts using PFNs. The established Pythagorean fuzzy HoQ (PF-HoQ) can be seen in Table 2, and ${\tilde{β}}_{ij}^{h} (i = 1, 2, 3; j = 1, 2, \dots, m_{i}; h = 1, 2, \dots, k)$ represents the relationship intensity between CR_ij and DR_h.

Table 2
The Pythagorean fuzzy HoQ

DR ₁ DR ₂ ⋯ DR _h ⋯ DR _k

CR ₁₁ ${\tilde{β}}_{11}^{1}$ ${\tilde{β}}_{11}^{2}$ ⋯ ${\tilde{β}}_{11}^{h}$ ⋯ ${\tilde{β}}_{11}^{k}$

CR ₁₂ ${\tilde{β}}_{12}^{1}$ ${\tilde{β}}_{12}^{2}$ ⋯ ${\tilde{β}}_{12}^{h}$ ⋯ ${\tilde{β}}_{12}^{k}$

⋮ ⋮ ⋮ ⋯ ⋮ ⋯ ⋮

CR _1m₁ ${\tilde{β}}_{1 m_{1}}^{1}$ ${\tilde{β}}_{1 m_{1}}^{2}$ ⋯ ${\tilde{β}}_{1 m_{1}}^{h}$ ⋯ ${\tilde{β}}_{1 m_{1}}^{k}$

⋮ ⋮ ⋮ ⋱ ⋮ ⋱ ⋮

CR _ij ${\tilde{β}}_{ij}^{1}$ ${\tilde{β}}_{ij}^{2}$ ⋯ ${\tilde{β}}_{ij}^{h}$ ⋯ ${\tilde{β}}_{ij}^{k}$

⋮ ⋮ ⋮ ⋱ ⋮ ⋱ ⋮

CR _3m₃ ${\tilde{β}}_{3 m_{3}}^{1}$ ${\tilde{β}}_{3 m_{3}}^{2}$ ⋯ ${\tilde{β}}_{3 m_{3}}^{h}$ ⋯ ${\tilde{β}}_{3 m_{3}}^{k}$

	DR ₁	DR ₂	⋯	DR _h	⋯	DR _k
CR ₁₁	${\tilde{β}}_{11}^{1}$	${\tilde{β}}_{11}^{2}$	⋯	${\tilde{β}}_{11}^{h}$	⋯	${\tilde{β}}_{11}^{k}$
CR ₁₂	${\tilde{β}}_{12}^{1}$	${\tilde{β}}_{12}^{2}$	⋯	${\tilde{β}}_{12}^{h}$	⋯	${\tilde{β}}_{12}^{k}$
⋮	⋮	⋮	⋯	⋮	⋯	⋮
CR _1m₁	${\tilde{β}}_{1 m_{1}}^{1}$	${\tilde{β}}_{1 m_{1}}^{2}$	⋯	${\tilde{β}}_{1 m_{1}}^{h}$	⋯	${\tilde{β}}_{1 m_{1}}^{k}$
⋮	⋮	⋮	⋱	⋮	⋱	⋮
CR _ij	${\tilde{β}}_{ij}^{1}$	${\tilde{β}}_{ij}^{2}$	⋯	${\tilde{β}}_{ij}^{h}$	⋯	${\tilde{β}}_{ij}^{k}$
⋮	⋮	⋮	⋱	⋮	⋱	⋮
CR _3m₃	${\tilde{β}}_{3 m_{3}}^{1}$	${\tilde{β}}_{3 m_{3}}^{2}$	⋯	${\tilde{β}}_{3 m_{3}}^{h}$	⋯	${\tilde{β}}_{3 m_{3}}^{k}$

Then, the sustainable CRs’ Pythagorean fuzzy importance weights derived in Section 4.2.5 are extended into PF-HoQ, and the weighted Pythagorean fuzzy relationship matrix between sustainable CRs and DRs, denoted as G , can be obtained with Equations (6) and (7). ${\tilde{γ}}_{ij}^{h} = {\tilde{w}}_{ij} \times {\tilde{β}}_{ij}^{h} = (u_{ij}^{h}, v_{ij}^{h})$ (6) $G = {[{\tilde{γ}}_{ij}^{h}]}_{(m_{1} + m_{2} + m_{3}) \times k} = {[(u_{ij}^{h}, v_{ij}^{h})]}_{(m_{1} + m_{2} + m_{3}) \times k}$ (7)

4.5 Determine DRs’ weights based on Pythagorean fuzzy GRA

In the weighted Pythagorean fuzzy relationship matrix G , the PFS $G_{h} = ({\tilde{γ}}_{11}^{h}, \dots, {\tilde{γ}}_{ij}^{h}, \dots, {\tilde{γ}}_{3 m_{3}}^{h})^{T}$ representing the correlation among CRs and DR_h, can be selected as the alternative sequence to describe DR_h considering CRs. Assume that $G_{ref} = ({\tilde{γ}}_{11}^{0}, \dots, {\tilde{γ}}_{ij}^{0}, \dots, {\tilde{γ}}_{3 m_{3}}^{0})^{T}$ is the ideal reference sequence, ${\tilde{γ}}_{ij}^{0}$ is obtained by Equation (8). ${\tilde{γ}}_{ij}^{0} = \max ({\tilde{γ}}_{ij}^{1}, \dots, {\tilde{γ}}_{ij}^{h}, \dots, {\tilde{γ}}_{ij}^{k})$ (8)

Then, the grey relational degree of the ideal reference sequence and alternative sequence can be determined with Equations (9) and (10).

$\begin{matrix} ξ ({\tilde{γ}}_{ij}^{0}, {\tilde{γ}}_{ij}^{h}) = \\ \frac{min_{h} min_{ij} d ({\tilde{γ}}_{ij}^{0}, {\tilde{γ}}_{ij}^{h}) + ρ min_{h} max_{ij} d ({\tilde{γ}}_{ij}^{0}, {\tilde{γ}}_{ij}^{h})}{d ({\tilde{γ}}_{ij}^{0}, {\tilde{γ}}_{ij}^{h}) + ρ max_{h} max_{ij} d ({\tilde{γ}}_{ij}^{0}, {\tilde{γ}}_{ij}^{h})}, \\ ρ \in (0, 1) \end{matrix}$ (9) $R (G_{ref}, G_{h}) = \frac{1}{m_{1} + m_{2} + m_{3}} \sum_{i = 1}^{3} \sum_{j = 1}^{m_{i}} ξ ({\tilde{γ}}_{ij}^{0}, {\tilde{γ}}_{ij}^{h})$ (10)

Where, $d ({\tilde{γ}}_{ij}^{0}, {\tilde{γ}}_{ij}^{h})$ denotes distance between ${\tilde{γ}}_{ij}^{0}$ and ${\tilde{γ}}_{ij}^{h}$ , $ξ ({\tilde{γ}}_{ij}^{0}, {\tilde{γ}}_{ij}^{h})$ is the corresponding grey relational coefficients. ρ is the distinguishing coefficient, and the smaller ρ, the higher distinguishing ability. In general, ρ is taken as 0.5 [43]. R ( G _ref, G _h) represents the grey relational degree between G _ref and G _h, and reflects the closeness degree between DR_h and the ideal reference sequence. Thus, if DR_h possesses higher closeness with the ideal reference sequence, it means DR_h has greater impact on CRs, and higher importance weight should be assigned to DR_h. Then, the weight of the DR_h can be calculated as follows. $w_{h}^{DR} = \frac{R (G_{ref}, G_{h})}{\sum_{h = 1}^{k} R (G_{ref}, G_{h})}$ (11)

5 Case study

In order to illustrate the practical application of the proposed method, Company M, a household appliance manufacturer in China, is introduced to conduct the case study. Due to the increasing customer requirements on sustainability, SSC development with a strong focus on customer requirements has become an important strategy for Company M to improve the enterprise’s performance on the economic, environmental and social aspects. While, there is a lack of methods and framework to guide the managers to analyze sustainable CRs, incorporate sustainable CRs into DRs of SSC, and rank DRs of SSC, which may cause negative impact on SSC design and management. In this context, to provide reasonable decision support for SSC development, a systematic approach to prioritize SSC’s DRs with the consideration of CRs is required by Company M.

5.1 Identification of the Sustainable CRs

According to customers’ questionnaires, literature surveys and expert interviews, 12 sustainable CRs of SSC under the three sustainable aspects of economy, environment and society are identified, and the hierarchical structure is shown in Fig. 3.

Fig. 3

Hierarchical structure of sustainable CRs.

5.2 Sustainable CRs’ weights analysis based on Pythagorean fuzzy ANP

5.2.1 The Pythagorean pairwise comparison matrix between sustainable CRs

After identifying sustainable CRs, taking CR₁₁ as criterion, experts are then invited to use PFNs to evaluate the correlation among CRs under CR₁ (including CR₁₂, CR₁₃ and CR₁₄) with respect to the criteria through pairwise comparison, and the Pythagorean fuzzy pairwise comparison matrix of CRs under CR₁ with respect to CR₁₁ is shown in Table 3. Next, the corresponding normalized feature vector is obtained based on the eigenvalue method, and is listed in Table 3.

Table 3
The Pythagorean fuzzy pairwise comparison matrix of CRs under CR₁ with respect to CR₁₁

CR11 CR12 CR13 CR14 Weight

CR12 (0.5,0.5) (0.7,0.3) (0.7,0.3) (0.854,0.495)

CR13 (0.2,0.8) (0.5,0.5) (0.6,0.4) (0.423,0.879)

CR14 (0.3,0.7) (0.4,0.6) (0.5,0.5) (0.432,0.868)

CR11	CR12	CR13	CR14	Weight
CR12	(0.5,0.5)	(0.7,0.3)	(0.7,0.3)	(0.854,0.495)
CR13	(0.2,0.8)	(0.5,0.5)	(0.6,0.4)	(0.423,0.879)
CR14	(0.3,0.7)	(0.4,0.6)	(0.5,0.5)	(0.432,0.868)

Following the same steps, the Pythagorean fuzzy pairwise comparison matrix and the relative weight for CRs under CR₂ with respect to CR₁₁ can be obtained, and are shown in Table 4. Table 5 expresses the related information for CRs under CR₃ with respect to CR₁₁.

Table 4

The Pythagorean fuzzy pairwise comparison matrix of CRs under CR₂ with respect to CR₁₁

CR11	CR21	CR22	CR23	CR24	Weight
CR21	(0.5,0.5)	(0.4,0.6)	(0.1,0.9)	(0.6,0.4)	(0.196,0.965)
CR22	(0.6,0.4)	(0.5,0.5)	(0.3,0.7)	(0.6,0.4)	(0.381,0.862)
CR23	(0.9,0.1)	(0.7,0.3)	(0.5,0.5)	(0.7,0.3)	(0.840,0.534)
CR24	(0.4,0.6)	(0.4,0.6)	(0.7,0.3)	(0.5,0.5)	(0.423,0.856)

Table 5

The Pythagorean fuzzy pairwise comparison matrix of CRs under CR₃ with respect toCR₁₁

CR11	CR31	CR32	CR33	CR34	Weight
CR31	(0.5,0.5)	(0.4,0.6)	(0.4,0.6)	(0.3,0.7)	(0.311,0.919)
CR32	(0.6,0.4)	(0.5,0.5)	(0.3,0.7)	(0.4,0.6)	(0.381,0.890)
CR33	(0.6,0.4)	(0.7,0.3)	(0.5,0.5)	(0.7,0.3)	(0.770,0.625)
CR34	(0.7,0.3)	(0.6,0.4)	(0.3,0.7)	(0.5,0.5)	(0.504,0.835)

5.2.2 The Pythagorean fuzzy super matrix

According to the steps described in Section 5.2.1, each sustainable CR is taken as the criterion in turn, and the Pythagorean fuzzy pairwise comparison matrix among sustainable CRs in each group with respect to the criterion are obtained. Then, the normalized eigenvectors of each matrix are integrated into the Pythagorean fuzzy super matrix, and the results are listed in Table 6.

Table 6
The Pythagorean fuzzy HoQ

CR11 CR24 CR31 CR34

CR11 / … (0.423,0.856) (0.311,0.919) … (0.504,0.835)

CR12 (0.884,0.460) … (0.240,0.946) (0.365,0.892) … (0.851,0.519)

CR13 (0.852,0.509) … (0.520,0.809) (0.266,0.946) … (0.852,0.519)

CR14 (0.726,0.684) … (0.706,0.702) (0.439,0.853) … (0.821,0.569)

CR21 (0.705,0.690) … (0.827,0.559) (0.663,0.696) … (0.752,0.656)

CR22 (0.708,0.687) … (0.768,0.617) (0.742,0.640) … (0.687,0.722)

CR23 (0.520,0.821) … (0.864,0.480) (0.377,0.892) … (0.622,0.762)

CR24 (0.569,0.791) … / (0.813,0.580) … (0.443,0.848)

CR31 (0.502,0.840) … (0.469,0.848) / … (0.627,0.745)

CR32 (0.453,0.854) … (0.367,0.896) (0.804,0.582) … (0.614,0.746)

CR33 (0.316,0.916) … (0.579,0.791) (0.828,0.543) … (0.290,0.932)

CR34 (0.573,0.796) … (0.266,0.936) (0.728,0.657) … /

	CR11		CR24	CR31		CR34
CR11	/	…	(0.423,0.856)	(0.311,0.919)	…	(0.504,0.835)
CR12	(0.884,0.460)	…	(0.240,0.946)	(0.365,0.892)	…	(0.851,0.519)
CR13	(0.852,0.509)	…	(0.520,0.809)	(0.266,0.946)	…	(0.852,0.519)
CR14	(0.726,0.684)	…	(0.706,0.702)	(0.439,0.853)	…	(0.821,0.569)
CR21	(0.705,0.690)	…	(0.827,0.559)	(0.663,0.696)	…	(0.752,0.656)
CR22	(0.708,0.687)	…	(0.768,0.617)	(0.742,0.640)	…	(0.687,0.722)
CR23	(0.520,0.821)	…	(0.864,0.480)	(0.377,0.892)	…	(0.622,0.762)
CR24	(0.569,0.791)	…	/	(0.813,0.580)	…	(0.443,0.848)
CR31	(0.502,0.840)	…	(0.469,0.848)	/	…	(0.627,0.745)
CR32	(0.453,0.854)	…	(0.367,0.896)	(0.804,0.582)	…	(0.614,0.746)
CR33	(0.316,0.916)	…	(0.579,0.791)	(0.828,0.543)	…	(0.290,0.932)
CR34	(0.573,0.796)	…	(0.266,0.936)	(0.728,0.657)	…	/

5.2.3 The Pythagorean fuzzy weighted super matrix

Following the steps described in Section 5.2.1, taking CR₁, CR₂ and CR₃ as the criterion in turn, experts are invited to construct the pairwise comparison matrix between them with respect to each criterion, and the relative weights for CR₁, CR₂, CR₃ under each criterion are obtained, as shown in Table 7.

Table 7
The Pythagorean fuzzy weight matrix for CR₁, CR₂ and CR₃

CR1 CR2 CR3

CR1 (0.933,0.355) (0.516,0.825) (0.528,0.839)

CR2 (0.411,0.873) (0.788,0.610) (0.249,0.951)

CR3 (0.190,0.966) (0.486,0.834) (0.863,0.487)

	CR1	CR2	CR3
CR1	(0.933,0.355)	(0.516,0.825)	(0.528,0.839)
CR2	(0.411,0.873)	(0.788,0.610)	(0.249,0.951)
CR3	(0.190,0.966)	(0.486,0.834)	(0.863,0.487)

According to Equation (4), the Pythagorean fuzzy super matrix is weighted, and then the Pythagorean fuzzy weighted super matrix is obtained, as shown in Table 8.

Table 8

The Pythagorean fuzzy weighted super matrix

	CR11	…	CR24	CR31	…	CR34
CR11	/	…	(0.174,0.968)	(0.059,0.995)	…	(0.096,0.990)
CR12	(0.825,0.557)	…	(0.099,0.987)	(0.069,0.993)	…	(0.162,0.975)
CR13	(0.795,0.594)	…	(0.214,0.958)	(0.051,0.996)	…	(0.162,0.975)
CR14	(0.497,0.732)	…	(0.290,0.938)	(0.084,0.991)	…	(0.156,0.977)
CR21	(0.363,0.913)	…	(0.651,0.754)	(0.322,0.918)	…	(0.366,0.909)
CR22	(0.365,0.912)	…	(0.605,0.782)	(0.361,0.905)	…	(0.334,0.924)
CR23	(0.268,0.946)	…	(0.680,0.719)	(0.183,0.969)	…	(0.302,0.934)
CR24	(0.294,0.938)	…	/	(0.395,0.894)	…	(0.215,0.956)
CR31	(0.265,0.956)	…	(0.117,0.986)	/	…	(0.541,0.813)
CR32	(0.240,0.959)	…	(0.091,0.990)	(0.694,0.704)	…	(0.530,0.813)
CR33	(0.167,0.976)	…	(0.144,0.982)	(0.715,0.680)	…	(0.250,0.949)
CR34	(0.303,0.944)	…	(0.066,0.994)	(0.628,0.753)	…	/

5.2.4 Determine the Pythagorean fuzzy weights of sustainable CRs

According to Equation (5), the Pythagorean fuzzy weighted super matrix is aggregated to achieve the Pythagorean fuzzy weight of sustainable CRs. Then, the score of weight degree for sustainable CRs is obtained by Equation (1), and the sustainable CRs are prioritized by the obtained scores, as shown in Table 9. In Table 9, CR₁₃ (Quality improvement) and CR₂₃ (Energy efficiency) are the two most important sustainable CRs, CR₃₃ (Strengthened relationships) is of the least importance.

Table 9
The Pythagorean fuzzy weights and priority of sustainable CRs

Weight Score Ranking Weight Score Ranking

CR11 (0.855,0.170) 0.703 4 CR23 (0.888,0.057) 0.785 2

CR12 (0.693,0.343) 0.363 8 CR24 (0.886,0.294) 0.689 5

CR13 (0.962,0.003) 0.926 1 CR31 (0.926,0.311) 0.761 3

CR14 (0.668,0.474) 0.222 10 CR32 (0.837,0.442) 0.505 6

CR21 (0.762,0.480) 0.350 9 CR33 (0.515,0.751) -0.298 12

CR22 (0.649,0.563) 0.104 11 CR34 (0.800,0.408) 0.474 7

	Weight	Score	Ranking	Weight	Score	Ranking
CR11	(0.855,0.170)	0.703	4	CR23	(0.888,0.057)	0.785	2
CR12	(0.693,0.343)	0.363	8	CR24	(0.886,0.294)	0.689	5
CR13	(0.962,0.003)	0.926	1	CR31	(0.926,0.311)	0.761	3
CR14	(0.668,0.474)	0.222	10	CR32	(0.837,0.442)	0.505	6
CR21	(0.762,0.480)	0.350	9	CR33	(0.515,0.751)	-0.298	12
CR22	(0.649,0.563)	0.104	11	CR34	(0.800,0.408)	0.474	7

5.3 Identify DRs of sustainable supply chain

According to the expert consultation and literature research, the critical DRs of SSC for Company M are identified. Table 10 expresses the detail information of the identified SSC DRs.

Table 10
The critical DRs of SSC

Notations Design requirements Notations Design requirements

DR1 Total cost management DR11 Renewable energy consumption

DR2 SC optimize DR12 Sustainable product design

DR3 Inventory management DR13 Socially responsible investing

DR4 Forecast management DR14 Efficient processing and storage

DR5 Life cycle management DR15 Green transport

DR6 Supplier scorecard DR16 Cooperation with partners

DR7 Flexible and clean technology DR17 Employee practice

DR8 Delivery time and capacity DR18 Outsourcing

DR9 The use of effective system and tool DR19 Stakeholders’ rights

DR10 Environmental management system DR20 Monitoring and maintenance

Notations	Design requirements	Notations	Design requirements
DR1	Total cost management	DR11	Renewable energy consumption
DR2	SC optimize	DR12	Sustainable product design
DR3	Inventory management	DR13	Socially responsible investing
DR4	Forecast management	DR14	Efficient processing and storage
DR5	Life cycle management	DR15	Green transport
DR6	Supplier scorecard	DR16	Cooperation with partners
DR7	Flexible and clean technology	DR17	Employee practice
DR8	Delivery time and capacity	DR18	Outsourcing
DR9	The use of effective system and tool	DR19	Stakeholders’ rights
DR10	Environmental management system	DR20	Monitoring and maintenance

5.4 Establish the Pythagorean fuzzy HoQ for sustainable CRs and DRs

The experts are invited to evaluate the correlation strength between the identified sustainable CRs and DRs with PFNs, so as to construct the PF-HoQ of SSC, which is shown in Table 11.

Table 11
The Pythagorean fuzzy HoQ for sustainable CRs and DRs

CRs’ weight DR1 DR2 … DR11 … DR19 DR20

CR11 (0.855,0.170) (0.8,0.3) (0.4,0.6) … (0.8,0.4) … (0.8,0.5) (0.3,0.7)

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

CR14 (0.668,0.474) (0.1,0.8) (0.1,0.9) … (0.2,0.7) … (0.4,0.5) (0.4,0.6)

CR21 (0.762,0.480) (0.6,0.5) (0.5,0.6) … (0.9,0.2) … (0.8,0.3) (0.7,0.2)

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

CR24 (0.886,0.294) (0.6,0.4) (0.8,0.3) … (0.8,0.1) … (0.7,0.2) (0.8,0.2)

CR31 (0.926,0.311) (0.3,0.5) (0.6,0.4) … (0.6,0.3) … (0.7,0.3) (0.8,0.3)

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

CR34 (0.800,0.408) (0.3,0.9) (0.2,0.7) … (0.6,0.3) … (0.7,0.3) (0.8,0.2)

	CRs’ weight	DR1	DR2	…	DR11	…	DR19	DR20
CR11	(0.855,0.170)	(0.8,0.3)	(0.4,0.6)	…	(0.8,0.4)	…	(0.8,0.5)	(0.3,0.7)
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
CR14	(0.668,0.474)	(0.1,0.8)	(0.1,0.9)	…	(0.2,0.7)	…	(0.4,0.5)	(0.4,0.6)
CR21	(0.762,0.480)	(0.6,0.5)	(0.5,0.6)	…	(0.9,0.2)	…	(0.8,0.3)	(0.7,0.2)
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
CR24	(0.886,0.294)	(0.6,0.4)	(0.8,0.3)	…	(0.8,0.1)	…	(0.7,0.2)	(0.8,0.2)
CR31	(0.926,0.311)	(0.3,0.5)	(0.6,0.4)	…	(0.6,0.3)	…	(0.7,0.3)	(0.8,0.3)
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
CR34	(0.800,0.408)	(0.3,0.9)	(0.2,0.7)	…	(0.6,0.3)	…	(0.7,0.3)	(0.8,0.2)

5.5 Determine the weights of DRs

Based on the Pythagorean fuzzy relation matrix between sustainable CRs and DRs in Table 11, the weighted Pythagorean fuzzy relation matrix can be obtained by employing Equations (6)-(7). Then, Equation (8) is applied to determine the ideal reference sequence of DRs. According to Equations (9)-(10), the grey relational degree for each DR is calculated. Finally, the importance weights of DRs based on the grey relational degree are obtained through Equation (11), and the results are shown in Table 12. The DRs of SSC can be prioritized as follows: DR₁₀ ≻ DR₂ ≻ DR₁₂ ≻ DR₁₁ ≻ DR₂₀ ≻ DR₁₃ ≻ DR₁₈ ≻ DR₉ ≻ DR₁₅ ≻ DR₇ ≻ DR₁₄ ≻ DR₁₉ ≻ DR₅ ≻ DR₁ ≻ DR₄ ≻ DR₃ ≻ DR₈ ≻ DR₁₆ ≻ DR₆ ≻ DR₁₇. In Table 12, the DR₁₀ (Environmental management system), DR₂ (SC optimize), DR₁₂ (Sustainable product design) are the top three DRs considering the importance weight and the DR₁₇ (Employee practice) is of the least importance. In the SSC planning, the managers should allocate more resource to improve the performance of DR₁₀, DR₂ and DR₁₂.

Table 12
The weights and ranking of SSC’ DRs

Grey relational degree Weight Ranking Grey relational degree Weight Ranking

DR1 0.6684 0.0484 14 DR11 0.7423 0.0537 4

DR2 0.7440 0.0539 2 DR12 0.7438 0.0538 3

DR3 0.6446 0.0467 16 DR13 0.7254 0.0525 6

DR4 0.6557 0.0475 15 DR14 0.6828 0.0494 11

DR5 0.6787 0.0491 13 DR15 0.7025 0.0508 9

DR6 0.6339 0.0459 19 DR16 0.6398 0.0463 18

DR7 0.6959 0.0504 10 DR17 0.6233 0.0451 20

DR8 0.6438 0.0466 17 DR18 0.7134 0.0517 7

DR9 0.7031 0.0509 8 DR19 0.6806 0.0493 12

DR10 0.7510 0.0544 1 DR20 0.7362 0.0533 5

	Grey relational degree	Weight	Ranking	Grey relational degree	Weight	Ranking
DR1	0.6684	0.0484	14	DR11	0.7423	0.0537	4
DR2	0.7440	0.0539	2	DR12	0.7438	0.0538	3
DR3	0.6446	0.0467	16	DR13	0.7254	0.0525	6
DR4	0.6557	0.0475	15	DR14	0.6828	0.0494	11
DR5	0.6787	0.0491	13	DR15	0.7025	0.0508	9
DR6	0.6339	0.0459	19	DR16	0.6398	0.0463	18
DR7	0.6959	0.0504	10	DR17	0.6233	0.0451	20
DR8	0.6438	0.0466	17	DR18	0.7134	0.0517	7
DR9	0.7031	0.0509	8	DR19	0.6806	0.0493	12
DR10	0.7510	0.0544	1	DR20	0.7362	0.0533	5

6 Discussion

Based on the case study, the developed methodology named as PF-ANP-QFD is compared with the approaches of ANP-QFD [24], QFD [44] and pairwise comparison method [45] to verify its advantages, and the results are shown in Table 13 and Fig. 4.

Table 13
The comparisons among different methods

PF-ANP-QFD ANP-QFD [24] QFD [44] Pairwise comparison method [45]

Weight Ranking Weight Ranking Weight Ranking Weight Ranking

DR1 0.0484 14 0.1258 1 0.1080 1 0.0593 2

DR2 0.0539 2 0.0526 9 0.0516 5 0.0719 1

DR3 0.0467 16 0.0565 8 0.0657 3 0.0494 11

DR4 0.0475 15 0.0201 17 0.0188 10 0.0494 11

DR5 0.0492 13 0.0260 15 0.0188 10 0.0549 6

DR6 0.0459 19 0.0291 14 0.0329 8 0.0390 15

DR7 0.0504 10 0.0895 2 0.1080 1 0.0379 16

DR8 0.0466 17 0.0074 20 0.0047 12 0.0467 12

DR9 0.0510 8 0.0699 4 0.0657 3 0.0571 4

DR10 0.0544 1 0.0494 12 0.0657 3 0.0560 5

DR11 0.0537 4 0.0797 3 0.1033 2 0.0538 7

DR12 0.0538 3 0.0611 6 0.0516 5 0.0521 9

DR13 0.0525 6 0.0608 7 0.0610 4 0.0401 14

DR14 0.0494 11 0.0523 10 0.0423 7 0.0532 8

DR15 0.0509 9 0.0197 18 0.0235 9 0.0401 14

DR16 0.0463 18 0.0230 16 0.0235 9 0.0587 3

DR17 0.0451 20 0.0136 19 0.0141 11 0.0390 15

DR18 0.0517 7 0.0631 5 0.0469 6 0.0510 10

DR19 0.0493 12 0.0490 13 0.0469 6 0.0467 12

DR20 0.0533 5 0.0515 11 0.0469 6 0.0439 13

	PF-ANP-QFD	ANP-QFD [24]	QFD [44]	Pairwise comparison method [45]
DR1	0.0484	14	0.1258	1	0.1080	1	0.0593	2
DR2	0.0539	2	0.0526	9	0.0516	5	0.0719	1
DR3	0.0467	16	0.0565	8	0.0657	3	0.0494	11
DR4	0.0475	15	0.0201	17	0.0188	10	0.0494	11
DR5	0.0492	13	0.0260	15	0.0188	10	0.0549	6
DR6	0.0459	19	0.0291	14	0.0329	8	0.0390	15
DR7	0.0504	10	0.0895	2	0.1080	1	0.0379	16
DR8	0.0466	17	0.0074	20	0.0047	12	0.0467	12
DR9	0.0510	8	0.0699	4	0.0657	3	0.0571	4
DR10	0.0544	1	0.0494	12	0.0657	3	0.0560	5
DR11	0.0537	4	0.0797	3	0.1033	2	0.0538	7
DR12	0.0538	3	0.0611	6	0.0516	5	0.0521	9
DR13	0.0525	6	0.0608	7	0.0610	4	0.0401	14
DR14	0.0494	11	0.0523	10	0.0423	7	0.0532	8
DR15	0.0509	9	0.0197	18	0.0235	9	0.0401	14
DR16	0.0463	18	0.0230	16	0.0235	9	0.0587	3
DR17	0.0451	20	0.0136	19	0.0141	11	0.0390	15
DR18	0.0517	7	0.0631	5	0.0469	6	0.0510	10
DR19	0.0493	12	0.0490	13	0.0469	6	0.0467	12
DR20	0.0533	5	0.0515	11	0.0469	6	0.0439	13

Fig. 4

The weights of DRs determined by different methods.

As shown in Table 13, the priorities of DRs determined by the proposed method are different from those obtained through ANP-QFD, QFD and pairwise comparison method. This is because that different manners are applied to deal with sustainable CRs, analyze SSC’s DRs and tackle the uncertain information. The qualitative comparison between the proposed PF-ANP-QFD and the existing methods is summarized in Table 14. It is obvious that the proposed method can provide comprehensive analysis on sustainable CRs and DRs.

Table 14

Main differences between PF-ANP-QFD and the listed methods

	PF-ANP-QFD	ANP-QFD [24]	QFD [44]	Pairwise comparison [45]
Consideration of the hierarchical relationships and correlations of sustainable CRs	Yes	Yes	No	No
Integrating sustainable CRs into DRs	Yes	Yes	Yes	No
Handling the small sample size problem	Yes	No	No	No
Expressing the intercorrelations among DRs	Yes	No	No	Yes
Conducting objective analysis on DRs	Yes	No	No	No
Manipulation of uncertainty	Yes	No	No	No

The first comparison is conducted with the results obtained from the AHP-QFD method. According to Table 13 and Fig. 4, the top three DRs obtained through PF-ANP-QFD are DR₁₀ (Environmental management system), DR₂ (SC optimize), DR₁₂ (Sustainable product design). However, in AHP-QFD, the top three DRs are DR₁ (Total cost management), DR₇ (Flexible and clean technology) and DR₁₁ (Renewable energy consumption). The main reasons for the differences are discussed as follows. Firstly, although ANP-QFD incorporates sustainable CRs into DRs’ analysis and analyzes the intercorrelations with sustainable CRs, it ignores the uncertainty and impreciseness in the experts’ judgment, which is now handled through PFS in the proposed method. Also, PFN applied in this paper enables a larger domain to express the uncertainty, and is more powerful on the issues of uncertainty. Secondly, the ANP-QFD has a limitation on analyzing the intercorrelations of SSC’s DRs and dealing with errors caused by small sample size in HoQ, while PF-GRA adopted in the proposed method allows for a mathematical analysis to determine the objective priorities of DRs, thus not only addresses the errors caused by small sample size, but also reflects the correlations of SSC’s DRs.

The second comparison is conducted with the results obtained from QFD method. As shown in Table 13 and Fig. 4, the DR’s ranking results of PF-ANP-QFD and QFD are different. For example, the most important DRs obtained by PF-ANP-QFD is DR₁₀ (Environmental management system), but it is ranked the 3rd in QFD method. This is because that DR₁₀ is considered to be influenced by other DRs in PF-ANP-QFD, and the influence is analyzed through PF-GRA. On the contrary, DR₁ is considered as an independent requirement in QFD. Moreover, QFD ignores the uncertainty and fuzziness of experts’ judgment in evaluation, while it can be handled through PFS in the proposed method. The most important is that, although QFD enables integration of sustainable CRs into SSC development, it neglects the complex hierarchical relationships and complex interrelations of sustainable CRs, which has great impact on the priorities of sustainable CRs and DRs. This is now addressed by the PF-ANP in the proposed method.

The last comparative method is the pairwise comparison method. As can be seen from Table 13, except for DR₁₉ (stakeholders’ rights), the priorities of other DRs determined by PF-ANP-QFD are different from those obtained with pairwise comparison method. The main reason for the differences between PF-ANP-QFD and pairwise comparison method is that pairwise comparison method has inner deficiency to incorporate sustainable CRs into DRs. While, sustainable CRs have a significant impact in SSC’s DRs management in customer-oriented market. In addition, pairwise comparison method has other limitations mentioned in QFD and ANP-QFD.

Thus, through the above comparison with ANP-QFD, QFD and pairwise comparison method, the integrative method can provide comprehensive insights on DRs for designing more effective SSC to satisfy sustainable CRs proactively through analyzing sustainable CRs, integrating sustainable CRs into SSC’s DRs directly, and prioritizing DRs objectively in an uncertain environment.

7 Conclusions

With the increasing CRs on sustainability and the uncertainty involving in SSC, a hybrid analytical approach integrating ANP, QFD and GRA with PFS is proposed to prioritize SSC’ DRs aiming to provide robust decision support for SSC development to proactively fulfill CRs under an uncertain environment. The PFS based method provides a larger domain to express the uncertainty, and is more powerful to deal with the uncertainty in SSC. The sustainable CRs of SSC are first prioritized using PF-ANP by considering the interrelations among them. Then, the sustainable CRs are integrated into SSC’s DRs through PF-QFD in order to achieve customer-oriented SSC development. Based on the analysis results obtained from PF-ANP and PF-QFD, the PF-GRA is developed to prioritize DRs through interrelation analysis of DRs considering the sustainable CRs, which can compensate for subjective errors resulted from small sample size. The case study and comparative analysis indicate the practical significance of the proposed approach, and it is efficient to provide decision support for customer-oriented SSC design. The proposed method is hoped to rich the literature on the improvement of QFD in SSC development.

In conclusion, the main contributions of this research are described as follows. Firstly, this study proposes a novel approach to provide decision support for SSC development under the Pythagorean fuzzy environment, which can accommodate more uncertainties involved in expert’s subjective and vague judgments. Secondly, the hierarchical relationships and correlations of CRs are explored and integrated into SSC’s DRs, which can realize customer-oriented SSC design. Thirdly, the weight of each DR is successfully determined by exploring complex interrelationships among DRs and dealing with the errors caused by small sample size, which can provide decision support for SSC development.

However, some limitations exist and need to be studied in future. First, the competitive analysis of SSC’s DRs is an important factor for SSC design, which should be taken into consideration in the future study. Next, a decision support system based on the proposed method should be developed to assist experts to conduct related activities with friendly interface and software tool. Further, the linguistic interval-valued Pythagorean fuzzy set proposed by Garg Harish [46] can be extended to the proposed method to offer even greater flexibility in analysis.

Footnotes

Acknowledgments

The work is supported by the National Natural Science Foundation of China (Grant No. 51705436), the National Science and Technology Major Project (Grant No. 2017-I-0011-0012), and Sichuan Science and Technology Program (Grant No. 2021JDRC0174). The authors would like to thank the anonymous reviewers for the valuable comments and suggestions.

Declaration of conflicting interests

The Authors declare that there is no conflict of interest.

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A Pythagorean fuzzy ANP-QFD-Grey relational analysis approach to prioritize design requirements of sustainable supply chain

Abstract

Keywords

1 Introduction

2 Literature review

3 Preliminaries of PFS

4.2.1 The Pythagorean fuzzy pairwise comparison matrix between sustainable CRs

4.2.3 The Pythagorean fuzzy weight matrix

4.2.4 The Pythagorean fuzzy weighted super matrix

4.4 Construct the Pythagorean fuzzy HoQ

5.1 Identification of the Sustainable CRs

5.2.1 The Pythagorean pairwise comparison matrix between sustainable CRs

Table 3 The Pythagorean fuzzy pairwise comparison matrix of CRs under CR1 with respect to CR11 CR11 CR12 CR13 CR14 Weight CR12 (0.5,0.5) (0.7,0.3) (0.7,0.3) (0.854,0.495) CR13 (0.2,0.8) (0.5,0.5) (0.6,0.4) (0.423,0.879) CR14 (0.3,0.7) (0.4,0.6) (0.5,0.5) (0.432,0.868)

Table 7 The Pythagorean fuzzy weight matrix for CR1, CR2 and CR3 CR1 CR2 CR3 CR1 (0.933,0.355) (0.516,0.825) (0.528,0.839) CR2 (0.411,0.873) (0.788,0.610) (0.249,0.951) CR3 (0.190,0.966) (0.486,0.834) (0.863,0.487)

Footnotes

Acknowledgments

Declaration of conflicting interests

References

Table 3
The Pythagorean fuzzy pairwise comparison matrix of CRs under CR₁ with respect to CR₁₁

CR11 CR12 CR13 CR14 Weight

CR12 (0.5,0.5) (0.7,0.3) (0.7,0.3) (0.854,0.495)

CR13 (0.2,0.8) (0.5,0.5) (0.6,0.4) (0.423,0.879)

CR14 (0.3,0.7) (0.4,0.6) (0.5,0.5) (0.432,0.868)

Table 7
The Pythagorean fuzzy weight matrix for CR₁, CR₂ and CR₃

CR1 CR2 CR3

CR1 (0.933,0.355) (0.516,0.825) (0.528,0.839)

CR2 (0.411,0.873) (0.788,0.610) (0.249,0.951)

CR3 (0.190,0.966) (0.486,0.834) (0.863,0.487)