1 Introduction

The green concept which is one of the important paradigms in supply chain management may be considered as an organizational philosophy. The concept of green supply chain management (GSCM) has attracted more attention due to environmental regulations and consumer pressures on sustainability (Govindan et al. 2015). GSCM is a form of management style that integrates structure of environmental thinking into all supply chain operations such as product design, material selection, purchasing and production process across enables companies to gain more profits and improve their environmental performance by reducing the effects of environmental risks (Mishra et al. 2019).

The GSCM has to start at the beginning of the supply chain, namely procurement of raw materials, and continue at every stage, including recycling or disposal of the product. It is not sufficient to focus on only greenness at the inbound supply chain operations for environmental goals and solutions, and companies should attain the environmental burdens of outbound operations among partners or stakeholders to raise the performance of their suppliers (Banaeian et al. 2018). Therefore, suppliers play a vital role in providing environmental improvements for companies (Mathiyazhagan et al. 2018). Consequently, companies have paid attention to the green supplier selection (GSS) problem while establishing GSCM.

A new and smart (digital) supply chain is created by accessing more information and technology in modern supply chains than ever before. In the current digitalization period, companies are looking for new ways to design supply chain applications and are increasingly dependent on the use of “smart technologies” such as smart supply chain, big data analysis, cloud systems and the Internet of Things (IoT). With the strategic initiative called “Industry 4.0,” the introduction of smart technologies into production processes has led companies to seek more innovative ways to achieve greater value for themselves and their stakeholders (Wang et al. 2015). Thus, adequate supplier selection and evaluation policies should be reorganized for new business models that come with Industry 4.0 or adapted to new models on existing policies.

Multi-criteria decision-making (MCDM) approaches can be a suitable tool to deal with the GSS problem and compare suppliers. Although there are various studies regarding GSS in the literature under crisp and fuzzy environments, further research is needed with different criteria, expertise and linguistic variables that take into account from the Industry 4.0 window.

The aim of this study is to develop a new hybrid group decision-making approach with the AHP and TOPSIS methods based on Pythagorean fuzzy sets (PFSs) for GSS from the Industry 4.0 window. In order to decide the most suitable supplier, an integrated three-stage MCDM approach is proposed by PFSs. In the first stage, a panel is constituted to gather the opinions of all experts. Then, GSS criteria are narrowed according to the literature survey and experts’ opinions from the Industry 4.0 perspective. PFSs are extension of intuitionistic fuzzy sets that gives more freedom to express experts’ judgments on the uncertainty and vagueness in decision-making problems. In the second stage, the identified criteria weights are obtained with Pythagorean fuzzy AHP (PFAHP), and in the third stage, potential suppliers are ranked with Pythagorean fuzzy TOPSIS (PFTOPSIS). A case study from the agricultural tools and machinery industry is implemented, and finally a sensitivity analysis is implemented for the validity of the results in terms of the solution of the weights of experts in the TOPSIS method.

2 Literature review

In recent years, many of the studies in the field of GSS have performed with MCDM methods based on standalone and integrated use of fuzzy and conventional sets. In this section, firstly the results of the literature review related to GSS are presented, and then the results of Industry 4.0 components including the green paradigm are provided. Since the 1960s, supplier selection and evaluation have been the focus of many studies not only with traditional criteria but also considering environmental criteria that have revealed GSS. Table 1 provides the details of these studies that are utilized for the solution and evaluation of the GSS problem. Table 2 shows the summary of the studies in which basic components of Industry 4.0 and green paradigm are used together.

Table 1 A brief summary about GSS studies
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Table 2 Summary of basic components of Industry 4.0 with green paradigm
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As it can be seen from the studies summarized below, the studies that contain the basic components of Industry 4.0 and GSS have been lacking. Moreover, traditional and green criteria are generally taken into consideration in the evaluation of suppliers in GSS. Although supplier selection has investigated with the components of Industry 4.0; almost no attention has been paid to GSS considering the components both PFAHP and PFTOPSIS methods and the components of Industry 4.0 have not been evaluated with environmental concerns. According to Tables 1 and 2, there are few case studies on PFSs; studies generally focus on conventional and type-2 fuzzy sets. Despite GSS has been investigated in the literature in various sectors such as automobile, battery and textile—to the best knowledge of the authors—a new GSS approach for the agricultural tools and machinery industry needs extra research.

3 Proposed green supplier selection approach under Pythagorean fuzzy environment

In this study, a group decision-making approach is proposed for GSS approach by using hybrid AHP with TOPSIS methods under PFSs. In the first stage, GSS criteria and basic components of Industry 4.0 with green paradigm are identified from the literature review. In the second stage, using the opinion of decision-makers, the PFAHP method is applied to specify the weight of all criteria. At the last stage, the PFTOPSIS method is utilized to rank each alternative. Figure 1 shows the flow chart of the proposed GSS approach.

Fig. 1
figure 1

Flow chart of the proposed GSS approach

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3.1 Preliminaries

In this section, basic concepts and definitions of PFSs are presented. PFSs are firstly proposed by Yager (2014) as the generalization to the intuitionistic fuzzy sets. Therefore, PFSs are more powerful and flexible for solving problems involving uncertainty (Ilbahar et al. 2018; Gul 2018; Gul and Ak 2018; Karasan et al. 2018).

Fuzzy set theory has been applied in a wide range of areas by different research works. Additionally, fuzzy set theory has been utilized in different parts of mathematics such as convergence of series for fuzzy real numbers (Tripathy and Das 2012), paranormed sequence spaces of fuzzy numbers (Tripathy and Debnath 2013), p-absolutely summable sequence of interval numbers (Dutta and Tripathy 2016), p-bounded variation of fuzzy real number sequences (Tripathy and Das 2019) and neutrosophic indeterminacy function (Das et al. 2020).

In PFSs, unlike the intuitionistic fuzzy sets, the sum of membership and non-membership degrees may exceed 1, but the sum of squares may not (Zhang and Xu 2014; Zeng et al. 2016; Karasan et al. 2018). This situation is given below in Definition 1.

Definition 1

Let a set \( X \) be a universe of discourse. A PFS \( P \) is an object having the form (Zhang and Xu 2014):

$$ P = \left\{ {\left\langle {x,P\left( {\mu_{P} \left( x \right),v\left( x \right)} \right)} \right\rangle {\mid }x \in X} \right\} $$
(1)

where \( \mu_{P} \left( x \right):X \mapsto \left[ {0,1} \right] \) defines the degree of membership and \( v_{P} \left( x \right):X \mapsto \left[ {0,1} \right] \) defines the degree of non-membership of the element \( x \in X \) to \( P \), respectively, and, for every \( x \in X \), it holds:

$$ 0 \le \mu_{P} \left( x \right)^{2} + v_{P} \left( x \right)^{2} \le 1 $$
(2)

For any PFS \( P \) and \( x \in X \), \( \pi_{P} \left( x \right) = \sqrt {1 - \mu_{P}^{2} \left( x \right) - v_{P}^{2} \left( x \right)} \) is called the degree of indeterminacy of \( x \) to \( P \).

Definition 2

Let \( \beta_{1} = P\left( {\mu_{{\beta_{1} }} ,v_{{\beta_{1} }} } \right) \) and \( \beta_{2} = P\left( {\mu_{{\beta_{2} }} ,v_{{\beta_{2} }} } \right) \) be two Pythagorean fuzzy numbers, and \( \lambda > 0 \), then the operations on these two Pythagorean fuzzy numbers are defined as follows (Zhang and Xu 2014; Zeng et al. 2016):

$$ \beta_{1} \oplus \beta_{2} = P\left( {\sqrt {\mu_{{\beta_{1} }}^{2} + \mu_{{\beta_{2} }}^{2} - \mu_{{\beta_{1} }}^{2} \mu_{{\beta_{2} }}^{2} } ,v_{{\beta_{1} }} v_{{\beta_{2} }} } \right) $$
(3)
$$ \beta_{1} \otimes \beta_{2} = P\left( {\mu_{{\beta_{1} }} \mu_{{\beta_{2} }} ,\sqrt {v_{{\beta_{1} }}^{2} + v_{{\beta_{2} }}^{2} - v_{{\beta_{1} }}^{2} v_{{\beta_{2} }}^{2} } } \right) $$
(4)
$$ \lambda \beta_{1} = P\left( {\sqrt {1 - \left( {1 - \mu_{{\beta_{1} }}^{2} } \right)^{\lambda } } ,\left( {v_{{\beta_{1} }} } \right)^{\lambda } } \right),\quad \lambda > 0 $$
(5)
$$ \beta_{1}^{\lambda } = P\left( {\left( {\mu_{{\beta_{1} }} } \right)^{\lambda } ,\sqrt {1 - \left( {1 - v_{{\beta_{1} }}^{2} } \right)^{\lambda } } } \right),\quad \lambda > 0 $$
(6)

Definition 3

Let \( \beta_{1} = P\left( {\mu_{{\beta_{1} }} ,v_{{\beta_{1} }} } \right) \) and \( \beta_{2} = P\left( {\mu_{{\beta_{2} }} ,v_{{\beta_{2} }} } \right) \) be two Pythagorean fuzzy numbers, a nature quasi-ordering on the Pythagorean fuzzy numbers is defined as follows (Zhang and Xu 2014):

$$ \beta_{1} \ge \beta_{2} \;{\text{if}}\;{\text{and}}\;{\text{only}}\;{\text{if}}\;\mu_{{\beta_{1} }} \ge \mu_{{\beta_{2} }} \;{\text{and}}\;v_{{\beta_{1} }} \le v_{{\beta_{2} }} $$

To compare the magnitude of two Pythagorean fuzzy numbers, a score function is developed by (Zhang and Xu 2014) as follows:

$$ s\left( {\beta_{1} } \right) = \left( {\mu_{{\beta_{1} }} } \right)^{2} - \left( {v_{{\beta_{1} }} } \right)^{2} . $$
(7)

Definition 4

Based on the score functions proposed above, the following laws are defined to compare two Pythagorean fuzzy numbers (Zhang and Xu 2014):

  1. (i)

    If \( s\left( {\beta_{1} } \right) < s\left( {\beta_{2} } \right) \), then \( \beta_{1} < \beta_{2} , \)

  2. (ii)

    If \( s\left( {\beta_{1} } \right) > s\left( {\beta_{2} } \right) \), then \( \beta_{1} > \beta_{2} , \)

  3. (iii)

    If \( s\left( {\beta_{1} } \right) = s\left( {\beta_{2} } \right) \), then \( \beta_{1} \sim\beta_{2} . \)

Definition 5

After the judgments of the decision-makers are converted into the interval-valued Pythagorean fuzzy numbers (IVPFNs), these IVPFNs are aggregated using the interval-valued Pythagorean fuzzy weighted geometric (IVPFWG) operator presented in Eq. (8): IVPFWG: \( \tilde{P}^{n} \to \tilde{P} \), where

$$ {\text{IVPFWG}}\left( {\tilde{p}_{1} ,\tilde{p}_{2} , \ldots ,\tilde{p}_{n} } \right) = \left( {\left[ {\mathop \prod \limits_{j = 1}^{n} \left( {\mu_{{\alpha_{j} }}^{L} } \right)^{{w_{j} }} ,\mathop \prod \limits_{j = 1}^{n} \left( {\mu_{{\alpha_{j} }}^{U} } \right)^{{w_{j} }} } \right],\left[ {\mathop \prod \limits_{j = 1}^{n} \left( {v_{{\alpha_{j} }}^{L} } \right)^{{w_{j} }} ,\mathop \prod \limits_{j = 1}^{n} \left( {v_{{\alpha_{j} }}^{U} } \right)^{{w_{j} }} } \right]} \right) $$
(8)

where \( n \) is the number of decision-makers, and \( w_{j} = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right)^{\text{T}} \) be the weight vector of \( \tilde{p}_{j} \) with \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \) (Peng and Yang 2016).

Definition 6

The individual assessments of decision-makers are combined using the Pythagorean fuzzy weighted averaging (PFWA) operator as follows:

$$ \tilde{r}_{ij} = {\text{PFWA}}\left( {\tilde{r}_{ij}^{1} ,\tilde{r}_{ij}^{2} , \ldots ,\tilde{r}_{ij}^{k} } \right) = = \left( {\sqrt {1 - \mathop \prod \limits_{k = 1}^{l} \left( {1 - \left( {u_{ij}^{k} } \right)^{2} } \right)^{{w_{k} }} } ,\mathop \prod \limits_{k = 1}^{l} \left[ {v_{ij}^{k} } \right]^{{w_{k} }} } \right), $$
(9)

where \( \tilde{r}_{ij}^{k} = \left( {u_{ij}^{k} ,v_{ij}^{k} } \right) \) is the Pythagorean fuzzy value provided by decision-maker \( k \), on the assessment of \( A_{i} \) in relation to \( C_{j} \) (\( i = 1,2, \ldots ,m \), \( = 1, 2, \ldots , n \)).

3.2 PFAHP

The steps of PFAHP are presented as follows.

  • Step 1: The pairwise comparison matrix \( A = \left( {a_{ik} } \right)_{m \times m} \) is constructed based on the linguistic evaluation of experts. The linguistic terms that are given (Ilbahar et al. 2018) are presented in Table 3.

    Table 3 Linguistic terms for importance weights of criteria
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  • Step 2: The difference matrices \( D = \left( {d_{ik} } \right)_{m \times m} \) between the lower and upper values of the membership and non-membership functions are calculated using Eqs. (10) and (11):

    $$ d_{{ik_{L} }} = \mu_{{ik_{L} }}^{2} - v_{{ik_{U} }}^{2} , $$
    (10)
    $$ d_{{ik_{U} }} = \mu_{{ik_{U} }}^{2} - v_{{ik_{L} }}^{2} . $$
    (11)

  • Step 3: Interval multiplicative matrix \( S = \left( {s_{ik} } \right)_{m \times m} \) is computed using Eqs. (12) and (13):

    $$ s_{{ik_{L} }} = \sqrt {1000^{{d_{{ik_{L} }} }} } , $$
    (12)
    $$ s_{{ik_{U} }} = \sqrt {1000^{{d_{{ik_{U} }} }} } . $$
    (13)

  • Step 4: The determinacy value \( \tau = \left( {\tau_{ik} } \right)_{m \times m} \) is calculated using Eq. (14):

    $$ \tau_{ik} = 1 - \left( {\mu_{{ik_{U} }}^{2} - \mu_{{ik_{L} }}^{2} } \right) - \left( {v_{{ik_{U} }}^{2} - v_{{ik_{L} }}^{2} } \right). $$
    (14)

  • Step 5: The determinacy degrees are multiplied with \( S = \left( {s_{ik} } \right)_{m \times m} \) matrix for obtaining the matrix of weights, \( T = \left( {t_{ik} } \right)_{m \times m} \) before normalization using Eq. (15).

    $$ t_{ik} = \left( {\frac{{s_{{ik_{L} }} + s_{{ik_{U} }} }}{2}} \right)\tau_{ik} . $$
    (15)

  • Step 6: The priority weights \( w_{i} \) of criteria are normalized by using Eq. (16):

    $$ w_{i} = \frac{{\mathop \sum \nolimits_{k = 1}^{m} t_{ik} }}{{\mathop \sum \nolimits_{i = 1}^{m} \mathop \sum \nolimits_{k = 1}^{m} t_{ik} }}. $$
    (16)

3.3 PFTOPSIS

The five steps of the procedure of PFTOPSIS approach under PFSs environment are presented as follows:

  • Step 1: Pythagorean fuzzy number-based decision matrix \( R = \left( {C_{j} \left( {x_{i} } \right)} \right)_{m \times n} \) is constructed. Here, \( C_{j} \left( {j = 1,2, \ldots ,n} \right) \) and \( x_{i} \left( {i = 1,2, \ldots ,m} \right) \) refer to the values of criteria and alternatives. The matrix form is as follows:

    $$ R = \left( {C_{j} \left( {x_{i} } \right)} \right)_{m \times n} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {P\left( {u_{11} ,v_{11} } \right)} & {P\left( {u_{12} ,v_{12} } \right)} \\ \end{array} } & \ldots & {P\left( {u_{1n} ,v_{1n} } \right)} \\ {\begin{array}{*{20}c} {P\left( {u_{21} ,v_{21} } \right)} & {P\left( {u_{22} ,v_{22} } \right)} \\ \end{array} } & \ldots & {P\left( {u_{2n} ,v_{2n} } \right)} \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} \vdots \\ {P\left( {u_{m1} ,v_{m1} } \right)} \\ \end{array} } & {\begin{array}{*{20}c} \vdots \\ {P\left( {u_{m2} ,v_{m2} } \right)} \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} \vdots \\ \ldots \\ \end{array} } & {\begin{array}{*{20}c} \vdots \\ {P\left( {u_{mn} ,v_{mn} } \right)} \\ \end{array} } \\ \end{array} } \right). $$

  • Step 2: Pythagorean fuzzy positive ideal solution (PIS) and negative ideal solutions (NIS) are determined using Eqs. (17, 18) as follows:

    $$ x^{ + } = \left\{ {C_{j} , \mathop {\hbox{max} }\limits_{i} s\left( {C_{j} \left( {x_{i} } \right)} \right){\mid }j = 1,2, \ldots ,n} \right\} $$
    $$ = \left\{ {\langle C_{1} ,P\left( {u_{1}^{ + } ,v_{1}^{ + } } \right)\rangle ,\langle C_{2} ,P\left( {u_{2}^{ + } ,v_{2}^{ + } } \right)\rangle , \ldots ,\langle C_{n} ,P\left( {u_{n}^{ + } ,v_{n}^{ + } } \right)\rangle } \right\} , $$
    (17)
    $$ x^{ - } = \left\{ {C_{j} , \mathop {\hbox{min} }\limits_{i} s\left( {C_{j} \left( {x_{i} } \right)} \right){\mid }j = 1,2, \ldots ,n} \right\} $$
    $$ = \left\{ {\langle C_{1} ,P\left( {u_{1}^{ - } ,v_{1}^{ - } } \right)\rangle ,\langle C_{2} ,P\left( {u_{2}^{ - } ,v_{2}^{ - } } \right)\rangle , \ldots ,\langle C_{n} ,P\left( {u_{n}^{ - } ,v_{n}^{ - } } \right)\rangle } \right\} . $$
    (18)

  • Step 3: In the third step, distances from Pythagorean fuzzy PIS and NIS are determined using Eqs. (19, 20) as follows:

    $$ \begin{aligned} D\left( {x_{i} ,x^{ + } } \right) & = \mathop \sum \limits_{j = 1}^{n} w_{j} d\left( {C_{j} \left( {x_{i} } \right),C_{j} \left( {x^{ + } } \right)} \right) \\ & = \frac{1}{2}\mathop \sum \limits_{j = 1}^{n} w_{j} \left( {\left| {\left( {\mu_{ij} } \right)^{2} - \left( {\mu_{j}^{ + } } \right)^{2} } \right| + \left| {\left( {v_{ij} } \right)^{2} - \left( {v_{j}^{ + } } \right)^{2} } \right| + \left| {\left( {\pi_{ij} } \right)^{2} - \left( {\pi_{j}^{ + } } \right)^{2} } \right|} \right), \\ \end{aligned} $$
    (19)
    $$ \begin{aligned} D\left( {x_{i} ,x^{ - } } \right) & = \mathop \sum \limits_{j = 1}^{n} w_{j} d\left( {C_{j} \left( {x_{i} } \right),C_{j} \left( {x^{ - } } \right)} \right) \\ & = \frac{1}{2}\mathop \sum \limits_{j = 1}^{n} w_{j} \left( {\left| {\left( {\mu_{ij} } \right)^{2} - \left( {\mu_{j}^{ - } } \right)^{2} } \right| + \left| {\left( {v_{ij} } \right)^{2} - \left( {v_{j}^{ - } } \right)^{2} } \right| + \left| {\left( {\pi_{ij} } \right)^{2} - \left( {\pi_{j}^{ - } } \right)^{2} } \right|} \right). \\ \end{aligned} $$
    (20)

    for Eqs. (17, 18) \( i = 1,2, \ldots ,n \). In general, the smaller \( D\left( {x_{i} ,x^{ + } } \right) \) the better the alternative \( x_{i} \) and the bigger \( D\left( {x_{i} ,x^{ - } } \right) \) the better the alternative \( x_{i} \) and let \( D_{\hbox{min} } \left( {x_{i} ,x^{ + } } \right) = \mathop {\hbox{min} }\limits_{1 \le i \le m} D\left( {x_{i} ,x^{ + } } \right) \) and \( D_{\hbox{max} } \left( {x_{i} ,x^{ - } } \right) = \mathop {\hbox{max} }\limits_{1 \le i \le m} D\left( {x_{i} ,x^{ + } } \right) \).

  • Step 4: Fourthly, the revised closeness \( \xi \left( {x_{i} } \right) \) of the alternative \( x_{i} \) is computed using Eq. (21) as follows:

    $$ \xi \left( {x_{i} } \right) = \frac{{D\left( {x_{i} ,x^{ - } } \right)}}{{D_{\hbox{max} } \left( {x_{i} ,x^{ - } } \right)}} - \frac{{D\left( {x_{i} ,x^{ + } } \right) }}{{D_{\hbox{min} } \left( {x_{i} ,x^{ + } } \right)}}. $$
    (21)

  • Step 5: Finally, the best-ranking order of the alternatives is determined. The alternative with the highest revised coefficient value is the best alternative.

4 An empirical case study

Production systems have been passed to new levels in parallel with the advancement in information technologies. In response to global competition, it has caused organizations to change their core competencies, improve the current environment, and develop new business models for themselves and their stakeholders. In this section, the application of the proposed GSS approach is carried out by the managers of an agricultural tool manufacturer (expressed as XYZ) located in Turkey. The purpose of the managers is to determine the performance of the suppliers by exploring the ranking of importance of the GSS criteria from the Industry 4.0 window. The company that generally produces lawnmowers continues development in the sector by increasing its product range without compromising the understanding of quality and continuous development, and desires to apply its environmental policy in the entire supply chain, including cooperation actions with all suppliers considering Industry 4.0 practices. In view of this situation, GSS has been identified as a necessary decision-making activity for the XYZ company.

4.1 Problem description

XYZ company wants to choose the best green supplier considering the Industry 4.0 components. A panel consisting of four decision-makers was designed to evaluate suppliers. The data collection process was carried out by face to face interview. Four decision-makers denoted by {DM1, DM2, DM3, DM4} from different departments of the company have been invited to provide feedback on the proposed approach. It has been decided that five suppliers specified by {S1, S2, S3, S4, S5} can procure the required part. In order to select green suppliers and create a decision-making tool in the evaluation, at first the relevant criteria must be determined correctly. In Sect. 2, the ten most common environmental criteria are identified. The selection of criteria and sub-criteria in the evaluation process was limited based on the opinions of the decision-makers and the literature review and given in Table 4.

Table 4 Evaluation criteria and their descriptions for assessment of suppliers
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4.2 Application of PFAHP to determine of criteria weights

The PFAHP method is used to obtain the weights of the criteria. All calculations are carried out by considering all criteria are benefit ones. The decision-makers expressed their views about the criteria by using linguistic terms that are determined according to PFSs as presented in Table 3. Then, the linguistic terms are converted to the corresponding interval-valued Pythagorean fuzzy numbers (Table 5). Then, the converted interval-valued Pythagorean numbers are aggregated according to Eq. (8) and shown in Table 6. Finally, the importance level of each criterion is determined according to the opinions of the four decision-makers. Similar operations are carried out within the sub-criteria, and the local and global weights of the main and sub-criteria are computed and illustrated in Table 7.

Table 5 Pairwise comparison of main with respect to experts’ judgments
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Table 6 Aggregated pairwise comparison matrix of main criteria
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Table 7 Local and global weights the criteria
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4.3 Implementation of PFTOPSIS to specify green suppliers

The PFTOPSIS method is carried out by using the weights obtained from PFAHP to select the best supplier. The linguistic variables for the assessment of potential suppliers are defined by the Pythagorean fuzzy numbers that can be seen in Table 8. The decision-makers evaluated the suppliers from the use of linguistic terms, and the terms are transformed to Pythagorean fuzzy numbers according to the scale given in Table 8. Since the evaluations of the decision-makers are different, the four individual evaluations have been aggregated according to Eq. (9), and the aggregated decision matrix is given in Table 9.

Table 8 Pythagorean fuzzy linguistic ratings and Pythagorean fuzzy numbers
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Table 9 Aggregated Pythagorean decision matrix
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After the Pythagorean fuzzy decision matrix is created in the PFTOPSIS method, PIS and NIS values are obtained, and the closeness coefficients are calculated and presented in Table 10. The priority order of the green supplier is \( S4 > S2 > S3 > S5 > S1 \): thus, S4 is the best green supplier.

Table 10 Closeness coefficients of suppliers
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4.4 Sensitivity analysis

Sensitivity analysis is performed to test the results of the criterion weights. If the existing ranking order is changed when the weight of the criteria change, it can be said that the results obtained are in nature robust; otherwise, they are sensitive. In this context, ten different experimental sets were generated, the weights of the criteria were changed and the ranking results are given in Table 11.

Table 11 Sensitivity analysis for different criterion weights on ranking results
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Based on the results presented in Table 11, the place of the best supplier does not change when the weight of the first and third criteria is dominant (the sum of the two criteria is at least 0.60) and the ranking results are robust. On the other hand, if the weights of the second, fourth and fifth criterion are increased, the ranking of best supplier is changed and the ranking results are sensitive.

5 Results and conclusion

Companies must give importance to environmental competencies due to important environmental issues such as global warming, epidemics and demands from governments and consumers. The emergence of new technologies has led to changes in the purchasing and production processes, and exposed modification in the decision-making processes of companies. The purpose of this study is to investigate how the practices of Industry 4.0 technologies are integrated into the GSS problem. Many methods are presented to identify the best green supplier. In recent years, PFSs have emerged as an effective tool to depict the ambiguity of MCDM problems. PFSs are capable of expressing ambiguity and uncertainty in the opinions of decision-makers, and PFSs are more capable of addressing the uncertainty of real-life problems. For this reason, in this study, a new method for GSS has been proposed by considering Industry 4.0 applications under Pythagorean knowledge.

In the proposed approach, the evaluation criteria were determined by the experts of the case company, and the linguistic variables of Pythagorean fuzzy numbers were used in the evaluations of the experts. The weights of the evaluation criteria were determined by the interval-valued PFAHP method, and suppliers were evaluated by PFTOPSIS taking into account the distance and similarity between alternatives. Finally, a case study was executed to verify the feasibility of the proposed GSS approach.

Our case study results reveal that three different criteria, production, delivery and quality, are the most important factors from the Industry 4.0 window for GSS. These three criteria cover 88% of the total weight for GSS. Decision-makers want to increase their presence in the world market by offering quality products. Thus, these criteria take maximum weightage by decision-makers. Konya holds over 65% of the agricultural tools and machinery sector which is one of the important centers of Turkey in internal and external markets. According to the PFAHP results (Table 7), Quality 4.0, service level and IoT and CPS criteria obtained the most important sub-criterion in the assessment. In order to increase the efficiency of the lawnmowers production that keeps with 16% of the most produced agricultural tools and machinery, the production process must be carried out in high quality. Although techniques such as just-in-time, kaizen and six sigma are the main factors for quality (Sinha and Anand 2018), the importance of criteria change when we focus on industry 4.0 components. Implementing key components of Industry 4.0 leads to digitalization of the supplier selection process (Ghadimi et al. 2019). While companies have difficulty in developing products, optimizing production processes and using product usage data for their products, IoT technology enables to overcome these difficulties (Tao et al. 2016). Data collected from products, logistics operations and production processes help to improve products and services (Kamble et al. 2018). The fact that companies are ahead of the competition stems from the basic components of Industry 4.0. However, without the high and comprehensive quality management system in the use of new technologies, they will not be able to have the necessary qualifications for Industry 4.0 success.

As a result of this study, the dimensions that should be taken into consideration for the companies aiming to improve the supplier selection process in agricultural production enterprises are revealed through the Industry 4.0 window. Owing to the MCDM methods used in the study, it was aimed that decision-makers overcome uncertainty more easily. In future studies, the different criteria, which affect the purchasing process can be included, and the approach can be expanded and also can be applied for different sectors.