Green supplier selection with undesirable outputs DEA under Pythagorean fuzzy environment

Abstract

With the enhancement of the global environmental awareness, green supplier selection is playing an increasingly important role in the development of enterprises. Green supplier evaluation criteria include quantitative and qualitative criteria. Due to the complexity and ambiguity of actual decision making problems, and the subjectivity of decision makers (DMs), it is difficult to describe qualitative criteria with precise data. In this paper, Pythagorean fuzzy numbers (PFNs) are used to describe the qualitative criteria, then a data envelopment analysis (DEA) model with undesirable outputs under Pythagorean fuzzy environment is developed. Finally, a numerical example ahout green supplier selection is provided to demonstrate the usefulness of the proposed method.

Keywords

DEA Pythagorean fuzzy set undesirable outputs green supplier selection

1 Introduction

In the modern society with rapid economic development, large scale development of industrialization and excessive consumption of social resources have led to the increasingly serious shortage of resources and global environmental problems. Environmental problems are no longer only concerned by environmental experts, environmental awareness affects almost every part of our society, especially in the industry. Enterprises should pay attention to environmental problems and take measures to reduce the impact of their services and products on environment. With the globalization of the economy and the intensification of market competition, green supply chain management (GSCM) has become a highly valued problem in the production and management of modern enterprises. GSCM is committed to reduce environmental pollution from upstream to downstream throughout the process from acquiring, processing, packaging, storage, transport, and using the raw material to the end-of-life disposal. Harmful substances contained in raw materials supplied by suppliers may cause serious environmental impacts on the supply chain. As the source of GSCM, green supplier selection affects the overall performance of green supply chain, and its importance is self-evident.

1.1 Green supplier selection

Some techniques have been used for green supplier selection in the available literature. The most commonly used techniques are AHP, ANP, TOPSIS, DEA and so on.

Noci [1] first pointed out the environmental performance criteria of suppliers evaluation include quantitative criteria and qualitative criteria. He incorporated green power, green image, life cycle cost and environmental efficiency into the supplier selection system, and addressed a green supplier evaluation system based on AHP. Handfield et al. [2] applied the Delphi method to collect 500 environmental management experts from IBM, Ford, Mascotech, Cone Drive, Herman Miller and DLSC to establish an evaluation criteria system and used AHP to evaluate environmental criteria. Lee et al. [3] applied the Delphi method to distinguish traditional suppliers and green suppliers. Considering the ambiguity of the opinions of DMs, green supplier evaluation model was constructed with fuzzy AHP. Hsu and Hu [4] incorporated hazardous substances management into green supplier selection and proposed a supplier evaluation framework based on ANP, which takes the links between the criteria into account. Although the preference for qualitative criteria can be well described by DMs using AHP and ANP, the data statistics are large and the weights are difficult to be determined when there are too many criteria.

Kannan and Jabbour [5] used TOPSIS and selected green supplier for Brazil electronics company based on GSCM practice under fuzzy environment. Freeman and Chen [6] constructed a framework based on AHP-Entropy-TOPSIS for green supplier selection. Hamdan and Cheaitou [7] combined fuzzy TOPSIS, AHP and optimization model for green supplier selection and order allocation. However, TOPSIS requires prior weight and does not test the consistency of DMs. In addition, when the DMUs are added, the ideal point may change and reverse order occurs.

Dobos and Vörösmarty [8] studied a DEA-comprehensive index method to select green supplier. Banaeian et al. [9] selected the green supplier with the Delphi method and green DEA in the edible oil sector. Based on genetic programming algorithm and DEA, Fallahpour et al. [10] selected the best supplier for textile companies.

1.2 Undesirable outputs DEA

The data envelopment analysis (DEA) is a non-parametric method for evaluating the relative efficiency of a series of decision making units (DMUs) with multiple inputs and multiple outputs. The weight of DEA model is obtained by the evaluation data, which is different from the conventional evaluation method that requires the attribute weight to be given in advance, so the results of DEA are more objective.

DEA measures the relative efficiency of DMUs with multiple evaluation criteria that are divided into inputs and outputs. Once the efficient frontier is determined, inefficient DMUs can improve their performance to reach the efficient frontier by either increasing their current output levels or decreasing their current input levels. In the assumption of the conventional DEA model, producing more outputs with less inputs means higher performance. However, such assumptions are not always reasonable. According to the global environmental awareness, the bad results produced in production processes and social activities, such as air pollutants and hazardous waste, are increasingly considered to be dangerous and unpopular, and these bad outputs are called undesirable outputs. When there are undesirable outputs in the efficiency evaluation, it is more efficient to produce more desirable outputs and less undesirable outputs with less inputs. Therefore, it is necessary to incorporate undesirable outputs into the efficiency evaluation of green supplier.

In the field of supplier selection, Farzipoor Saen [11, 12] presented a DEA model to select supplier in the presence of undesirable outputs and imprecise data. Noorizadeh et al. [13] simulated undesirable outputs in the cross efficiency model to obtain a complete ordering of suppliers. Noorizadeh et al. [14] applied the super efficiency model to rank suppliers in the presence of undesirable and non-discretionary outputs. Undesirable output in the above literature is the Parts Per Million (PPM). Mahdiloo et al. [15] developed a multi-objective linear programming (MOLP)-DEA model to measure technology, environment and ecological efficiency, in which the undesirable output is the carbon dioxide (CO₂) emissions.

1.3 Pythagorean fuzzy set

Zadeh [16] introduced fuzzy set theory and described the uncertainty of the problem with membership degree. Atanassov [17] presented the concept of intuitionistic fuzzy set (IFS), which introduced non-membership degree and hesitation degree on the basis of fuzzy set. The sum of its membership degree and non-membership degree is equal to or less than 1. However, IFS is not applicable when the sum of membership degree and non-membership degree is greater than 1. Yager [18, 19] proposed Pythagorean fuzzy set (PFS). The square sum of its membership degree and non-membership degree is equal to or less than 1. Membership degree and non-membership degree can express the satisfaction and dissatisfaction of decision-makers to some extent, so IFSs and PFSs can express the opinions of DMs more comprehensively. Moreover, If one is an intuitionistic fuzzy number (IFN), then it must also be a PFN, but not all PFNs are the IFNs. This result can be shown in Fig. 1. So PFSs is more powerful to describe the ambiguity nature than IFSs. PFSs are widely used in decision making [20 –26].

Fig.1

Compares of the IFNs and PFNs.

1.4 Gaps in the literature and purposeof the paper

Green supplier selection can be regarded as a multi-criteria decision making (MCDM) problem that involves many evaluation criteria with both qualitative and quantitative indicators. Quantitative indicators are usually expressed with precise data, while qualitative indicators are often subject to the subjective influence of DMs, and it is more convenient to express with fuzzy language or fuzzy numbers. In the field of supplier selection, fuzzy numbers have been widely used in efficiency evaluation [27 –29]. However, only fuzzy environment is considered in these literature, and precise environment is not considered. In other words, far few papers have considered both precise environment and fuzzy environment up to now. One of the characteristics of DEA is that it can process both quantitative and qualitative indicators. So DEA combine fuzzy environment and precise environment not only retain the subjectivity of DMs but also guarantee the objectivity of decision results to a higher degree.

It must be noted that, Hajiagha et al. [30] extended a DEA model in which partial inputs and outputs are expressed in the form of IFNs. But the model only considers membership degree and non-membership degree, and does not consider hesitation degree. And there is no further distinction between effective DMUs.

In view of the gaps mentioned above and the advantages of PFSs, this paper constructs DEA model under Pythagorean fuzzy environment. Considering the inevitability of undesirable outputs in industrial development, this paper develops DEA model combines PFSs and undesirable outputs, which is used for green supplier selection. The model built in this paper can get the efficiency values of all DMUs, thus realizing the complete ranking of DMUs.

This paper proceeds as follows. Section 2 introduces the definition and related operations of PFSs. In section 3 the model is proposed. Numerical example is discussed in Sections 4. In Section5, concluding remarks are presented.

2 PFSs

Definition 2.1. [18, 19]. Let a set X be a universe of discourse. An PFS P is an object having the form: $P = {〈 x, P (μ_{P} (x), ν_{P} (x)) 〉 | x \in X}$ (2.1)

For simplicity, Zhang and Xu [20] denoted β = P (μ_β, ν_β) as a PFN, where μ_β ∈ [0, 1] and ν_β ∈ [0, 1] is the degree of membership and the degree of non-membership of the element x ∈ X to P, respectively, and for every x ∈ X, it holds that $0 \leq μ_{β}^{2} + ν_{β}^{2} \leq 1$ . $π_{β} = \sqrt{1 - μ_{β}^{2} - ν_{β}^{2}}$ is called the degree of indeterminacy of x to P.

Definition 2.2. [20]. Let β₁ = P (μ_{β
₁}, ν_{β
₁}), β₂ = P (μ_{β
₂}, ν_{β
₂}) and β = P (μ_β, ν_β) are three PFNs, then their operations are defined as follows: $\begin{matrix} 1 . β_{1} \oplus β_{2} = P (\sqrt{μ_{β_{1}}^{2} + μ_{β_{2}}^{2} - μ_{β_{1}}^{2} μ_{β_{2}}^{2},} ν_{β_{1}} ν_{β_{2}}); \\ 2 . β_{1} \otimes β_{2} = P (μ_{β_{1}} μ_{β_{2}}, \sqrt{ν_{β_{1}}^{2} + ν_{β_{2}}^{2} - ν_{β_{1}}^{2} ν_{β_{2}}^{2}}); \\ 3 . λ β = P (\sqrt{1 - {(1 - μ_{β}^{2})}^{λ},} {(ν_{β})}^{λ}), λ > 0; \\ 4 . β^{λ} = P ({(μ_{β})}^{λ}, \sqrt{1 - {(1 - ν_{β}^{2})}^{λ},}), λ > 0; \\ 5 . β^{c} = P (ν_{β}, μ_{β}) . \end{matrix}$ (2.2)

Definition 2.3. [22]. Let β₁ = P (μ_{β
₁}, ν_{β
₁}), β₂ = P (μ_{β
₂}, ν_{β
₂}) be two PFNs, a strict quasi-ordering on the PFNs is defined as follows:

β₁ ≥ β₂ if and only if μ_{β
₁} ≥ μ_{β
₂}, ν_{β
₁} ≤ ν_{β
₂} and π_{β
₁} ≤ π_{β
₂}

Definition 2.4. [22]. Let β_j = P (μ_{β
_j}, ν_{β
_j}) (j = 1, 2, ⋯ , n) be a collection of PFNs and ω_j = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of β_j (j = 1, 2, ⋯ , n), with $\sum_{j = 1}^{n} ω_{j} = 1$ , then a PFWA operator is a mapping PFWA: Pⁿ → P, where $\begin{matrix} PFWA (β_{1}, β_{2}, \dots, β_{n}) \\ = P (\sqrt{1 - \prod_{j = 1}^{n} {(1 - {(μ_{β_{j}})}^{2})}^{ω_{j}}}, \prod_{j = 1}^{n} {(ν_{β_{j}})}^{ω_{j}}) \end{matrix}$ (2.3)

3 Proposed model

This paper supposes that n DMUs are evaluated and each DMU_j (j = 1, 2, ⋯ , n) is assumed to use m different positive inputs to produce s different positive outputs, denoted by x_ij (i = 1, 2, ⋯ , m) and y_rj (r = 1, 2, ⋯ , s), respectively. The BCC model [31] is expressed as model (3.1).

$\begin{matrix} min θ \\ s . t . \sum_{j = 1}^{n} λ_{j} x_{ij} \leq θ x_{io}, i = 1, 2, \dots, m (1) \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro}, r = 1, 2, \dots, s (2) \\ \sum_{j = 1}^{n} λ_{j} = 1 (3) \\ λ_{j} \geq 0, j = 1, 2, \dots, n \end{matrix}$ (3.1)

The BCC model have multiple DMUs with efficiency value of 1. Seiford and Zhu [32] proposed super-efficiency BCC (SE-BCC) model by removing the evaluated DMU from the reference set. The SE-BCC model can distinguish effective DMUs to some extent, but Seiford and Zhu pointed out that the SE-BCC model may be infeasible under VRS conditions. Chen [33] showed that infeasibility occurs under input-oriented (output-oriented) model when any output surplus (input saving) exists. Chen and Liang [34] proposed the following non-radial SE-BCC model (3.2), which can deal with the infeasible problems.

$\begin{matrix} min τ + M * \sum_{r = 1}^{s} β_{r} \\ s . t . \sum_{j = 1, \neq o}^{n} λ_{j} x_{ij} \leq (1 + τ) x_{io}, i = 1, 2, \dots, m (1) \end{matrix}$ $\begin{matrix} \sum_{j = 1, \neq o}^{n} λ_{j} y_{ij} \geq (1 - β_{r}) y_{ro}, r = 1, 2, \dots, s (2) \\ \sum_{j = 1, \neq o}^{n} λ_{j} = 1, λ_{j} \geq 0, j = 1, 2, \dots, n, j \neq o (3) \\ β_{r} \geq 0, r = 1, 2, \dots, s \end{matrix}$ (3.2)

$0 \leq β_{r}^{*} < 1$ , τ^* > -1 and M is a user-defined large positive number in model (3.2). The SE-BCC model is feasible if and only if all $β_{r}^{*} = 0$ , and the efficiency score of DMU_o obtained by model (3.2) is same as the efficiency score obtained by the SE-BCC model, which can be defined as 1 + τ^*. The SE-BCC model is infeasible if and only if some $β_{r}^{*} > 0$ , and the super-efficiency of DMU_o includes input super efficiency 1 + τ^* and output super efficiency $\frac{1}{| R |} \sum_{r \in R} \frac{1}{1 - β_{r}^{*}}$ , which can be defined as $1 + τ^{*} + \frac{1}{| R |} \sum_{r \in R} \frac{1}{1 - β_{r}^{*}}$ , R is the set of $β_{r}^{*} > 0$ .

Next, model (3.1) and model (3.2) under precise environment are extended to Pythagorean fuzzy environment. Consider the case where the inputs and outputs in model (3.1) are PFNs. According to the PFWA operator and multiplicative theory, (1) in model (3.1) can be expressed as: $\begin{matrix} 〈 \sqrt{1 - \prod_{j = 1}^{n} {(1 - μ_{x_{ij}}^{2})}^{λ_{j}}}, \prod_{j = 1}^{n} ν_{x_{ij}}^{λ_{j}}, \\ \sqrt{\prod_{j = 1}^{n} {(1 - μ_{x_{ij}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} ν_{x_{ij}}^{2 λ_{j}}} 〉 \\ \leq 〈 \sqrt{1 - {(1 - μ_{x_{io}}^{2})}^{θ}}, ν_{x_{io}}^{θ}, \\ \sqrt{{(1 - μ_{x_{io}}^{2})}^{θ} - ν_{x_{io}}^{2 θ}} 〉 \end{matrix}$ (3.3)

The data in (3.3) denotes the degree of membership, non-membership and hesitation of PFNs, respectively. According to Definition 2.3, (3.3) can be transformed into: $\begin{matrix} \sqrt{1 - \prod_{j = 1}^{n} {(1 - μ_{x_{ij}}^{2})}^{λ_{j}}} \leq \sqrt{1 - {(1 - μ_{x_{io}}^{2})}^{θ}} \\ \prod_{j = 1}^{n} ν_{x_{ij}}^{λ_{j}} \geq ν_{x_{io}}^{θ} \\ \sqrt{\prod_{j = 1}^{n} {(1 - μ_{x_{ij}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} ν_{x_{ij}}^{2 λ_{j}}} \geq \sqrt{{(1 - μ_{x_{io}}^{2})}^{θ} - ν_{x_{io}}^{2 θ}} \end{matrix}$ (3.4)

After some transformations, it can be obtained: $\begin{matrix} \sum_{j = 1}^{n} λ_{j} ln (1 - μ_{x_{ij}}^{2}) \geq θ n (1 - μ_{x_{io}}^{2}) \\ \sum_{j = 1}^{n} λ_{j} ln ν_{x_{ij}} \geq θ ln ν_{x_{io}} \\ \prod_{j = 1}^{n} {(1 - μ_{x_{ij}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} ν_{x_{ij}}^{2 λ_{j}} \geq {(1 - μ_{x_{io}}^{2})}^{θ} - ν_{x_{io}}^{2 θ} \end{matrix}$ (3.5)

Similarly, according to the PFWA operator, (2) in (3.1) can be expressed as: $\begin{matrix} 〈 \sqrt{1 - \prod_{j = 1}^{n} {(1 - μ_{y_{rj}}^{2})}^{λ_{j}}}, \prod_{j = 1}^{n} ν_{y_{rj}}^{λ_{j}}, \\ \sqrt{\prod_{j = 1}^{n} {(1 - μ_{y_{rj}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} ν_{y_{rj}}^{2 λ_{j}}} 〉 \\ \geq 〈 μ_{y_{ro}}, ν_{y_{ro}}, \sqrt{1 - μ_{y_{ro}}^{2} - ν_{y_{ro}}^{2}} 〉 \end{matrix}$ (3.6)

After some transformations, it can be obtained: $\begin{matrix} \sum_{j = 1}^{n} λ_{j} ln (1 - μ_{y_{rj}}^{2}) \leq ln (1 - μ_{y_{ro}}^{2}) \\ \sum_{j = 1}^{n} λ_{j} ln ν_{y_{rj}} \leq ln ν_{y_{ro}} \\ \prod_{j = 1}^{n} {(1 - μ_{y_{rj}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} ν_{y_{rj}}^{2 λ_{j}} \leq 1 - μ_{y_{ro}}^{2} - ν_{y_{ro}}^{2} \end{matrix}$ (3.7)

The BCC model is chosen because (3.3) and (3.6) require the sum of the weights to be 1 when using PFWA operator.

Theoretically, DMs hope that the greater desirable outputs and less undesirable outputs. Therefore, the desirable and undesirable outputs should be considered separately in DEA model. Seiford and Zhu [35] proposed a linear monotone decreasing transformation function ${\bar{y}}_{sj} = - y_{sj} + v > 0$ , which v is a proper translation vector that makes ${\bar{y}}_{sj} > 0$ .

DEA model is required to satisfy translation invariance when using monotonically decreasing transform function. Noorizadeh et al. [13] pointed out that the CCR model is not translation invariant, so it is impossible to process undesirable outputs through the monotone transformation. The input-oriented BCC model is invariant to outputs, so it is applicable, as shown in Fig. 2. The input-oriented BCC-efficiency of D is PR/PD. The ratio is invariant when the output value moves from Ox to O′x′. Therefore, the input-oriented BCC model can deal with undesirable outputs.

Fig.2

Translation invariant of the input-oriented BCC model.

This study supposes that n DMUs are evaluated and each DMU_j (j = 1, 2, ⋯ , n) is assumed to use m different positive inputs to produce p different positive outputs. The outputs that are corresponding to indices r = 1, 2, ⋯ , k are desirable outputs and the outputs that are corresponding to indices s = k + 1, k + 2, ⋯ , p are undesirable outputs. Then, the input-oriented BCC model dealing with undesirable outputs is expressed as model (3.8).

$\begin{matrix} min θ \\ s . t . \sum_{j = 1}^{n} λ_{j} x_{ij} \leq θ x_{io}, i = 1, 2, \dots, m (1) \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro}, r = 1, 2, \dots, k (2) \\ \sum_{j = 1}^{n} λ_{j} {\bar{y}}_{sj} \geq {\bar{y}}_{so}, s = k + 1, k + 2, \dots, p (3) \\ \sum_{j = 1}^{n} λ_{j} = 1 (4) \\ λ_{j} \geq 0, j = 1, 2, \dots, n \end{matrix}$ (3.8)

When the undesirable outputs are PFNs, the undesirable outputs can be transformed into the desirable outputs by complement theory of PFNs. which is (μ_{y
_sj}, ν_{y
_sj}) → (ν_{y
_sj}, μ_{y
_sj}). In principle, this method is consistent with the transformation between benefit attribute and cost attribute in the multi-attribute decision making problem. According to the PFWA operator, the transformed undesirable outputs should satisfy: $\begin{matrix} 〈 \sqrt{1 - \prod_{j = 1}^{n} {(1 - ν_{y_{sj}}^{2})}^{λ_{j}}}, \prod_{j = 1}^{n} μ_{y_{sj}}^{λ_{j}}, \\ \sqrt{\prod_{j = 1}^{n} {(1 - ν_{y_{sj}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} μ_{y_{sj}}^{2 λ_{j}}} 〉 \\ \geq 〈 ν_{y_{so}}, μ_{y_{so}}, \sqrt{1 - ν_{y_{so}}^{2} - μ_{y_{so}}^{2}} 〉 \end{matrix}$ (3.9)

After some transformations, it can be obtained: $\begin{matrix} \sum_{j = 1}^{n} λ_{j} ln μ_{y_{sj}} \leq ln μ_{y_{so}} \\ \sum_{j = 1}^{n} λ_{j} ln (1 - ν_{y_{sj}}^{2}) \leq ln (1 - ν_{y_{so}}^{2}) \\ \prod_{j = 1}^{n} {(1 - ν_{y_{sj}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} μ_{y_{sj}}^{2 λ_{j}} \leq 1 - ν_{y_{so}}^{2} - μ_{y_{so}}^{2} \end{matrix}$ (3.10)

By integrating (3.5), (3.7) and (3.10) under Pythagorean fuzzy environment and inputs-outputs under precise environment, the BCC model with undesirable outputs under Pythagorean fuzzy environment can be expressed as model (3.11). $\begin{matrix} min θ \\ s . t . \sum_{j = 1}^{n} λ_{j} ln (1 - μ_{x_{ij}}^{2}) \geq θ ln (1 - μ_{x_{io}}^{2}), i \in {PFN}_{x} (1) \\ \sum_{j = 1}^{n} λ_{j} ln ν_{x_{ij}} \geq θ ln ν_{x_{io}}, i \in {PFN}_{x} (2) \\ \prod_{j = 1}^{n} {(1 - μ_{x_{ij}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} ν_{x_{ij}}^{2 λ_{j}} \geq {(1 - μ_{x_{io}}^{2})}^{θ} - ν_{x_{io}}^{2 θ}, i \in {PFN}_{x} (3) \\ \sum_{j = 1}^{n} λ_{j} x_{ij} \leq θ x_{io}, i \in E_{x} (4) \\ \sum_{j = 1}^{n} λ_{j} ln (1 - μ_{y_{rj}}^{2}) \leq ln (1 - μ_{y_{ro}}^{2}), r \in {PFN}_{y_{r}} (5) \\ \sum_{j = 1}^{n} λ_{j} ln ν_{y_{rj}} \leq ln ν_{y_{ro}}, r \in {PFN}_{y_{r}} (6) \\ \prod_{j = 1}^{n} {(1 - μ_{y_{rj}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} ν_{y_{rj}}^{2 λ_{j}} \leq 1 - μ_{y_{ro}}^{2} - ν_{y_{ro}}^{2}, r \in {PFN}_{y_{r}} (7) \\ \sum_{j = 1}^{n} λ_{j} y_{rj} \geq y_{ro}, r \in E_{y_{r}} (8) \\ \sum_{j = 1}^{n} λ_{j} ln μ_{y_{sj}} \leq ln μ_{y_{so}}, s \in {PFN}_{y_{s}} (9) \\ \sum_{j = 1}^{n} λ_{j} ln (1 - ν_{y_{sj}}^{2}) \leq ln (1 - ν_{y_{so}}^{2}), s \in {PFN}_{y_{s}} (10) \\ \prod_{j = 1}^{n} {(1 - ν_{y_{sj}}^{2})}^{λ_{j}} - \prod_{j = 1}^{n} μ_{y_{sj}}^{2 λ_{j}} \leq 1 - ν_{y_{so}}^{2} - μ_{y_{so}}^{2}, s \in {PFN}_{y_{s}} (11) \\ \sum_{j = 1}^{n} λ_{j} {\bar{y}}_{sj} \geq {\bar{y}}_{so}, s \in E_{y_{s}} (12) \\ \sum_{j = 1}^{n} λ_{j} = 1, λ_{j} \geq 0, j = 1, 2, \dots, n (13) \\ i = 1, 2, \dots, m; r = 1, 2, \dots, k; s = k + 1, k + 2, \dots, p \end{matrix}$ (3.11)

In model (3.11), (1), (2) and (3) indicate the conditions that the degree of membership, non-membership and hesitation should be satisfied when the input is a PFN. (5), (6) and (7) represent the conditions that the degree of membership, non-membership and hesitation should be met when the desirable output is a PFN. When the undesirable output is a PFN, the degree of membership, non-membership and hesitation should satisfy (9), (10) and (11). (4), (8) and (12) are the conditions that the inputs, the desirable outputs and the undesirable outputs are precise data, respectively.

In model (3.11), (3), (7) and (11) are nonlinear programming, taking (3) as an example to show how to solve them. Logarithm of each item in (3), and convert (3) into (3.12). Formulas in (3.12) are linear programming, the solution of (7) and (11) can refer to (3) in model (3.11). Thus model (3.11) is a linear programming model in the calculation process, and the global solution is obtained. $\begin{matrix} A - B \geq C - D \\ ln A = \sum_{j = 1}^{n} λ_{j} (1 - μ_{x_{ij}}^{2}) \\ ln B = 2 \sum_{j = 1}^{n} λ_{j} ν_{x_{ij}} \\ ln C = θ ln (1 - u_{x_{io}}^{2}) \\ ln D = 2 θ ln ν_{x_{io}} \end{matrix}$ (3.12)

The SE-BCC model with undesirable outputs under Pythagorean fuzzy environment is to delete the DMU_o in the reference set in model (3.11), which is omitted. In order to solve the possible infeasible problems of the SE-BCC model, the non-radial model (3.2) can be extended to Pythagorean fuzzy environment based on the construction principle of model (3.11), which is expressed as model (3.13). $\begin{matrix} min τ + M * (\sum_{r = 1}^{k} β_{r} + \sum_{s = k + 1}^{p} β_{s}) \\ s . t . \sum_{j = 1, \neq o}^{n} λ_{j} ln (1 - μ_{x_{ij}}^{2}) \geq (1 + τ) ln (1 - μ_{x_{io}}^{2}), i \in {PFN}_{x} (1) \\ \sum_{j = 1, \neq o}^{n} λ_{j} ln ν_{x_{ij}} \geq (1 + τ) ln ν_{x_{io}}, i \in {PFN}_{x} (2) \\ \prod_{j = 1, \neq o}^{n} {(1 - μ_{x_{ij}}^{2})}^{λ_{j}} - \prod_{j = 1, \neq o}^{n} ν_{x_{ij}}^{2 λ_{j}} \geq {(1 - μ_{x_{io}}^{2})}^{(1 + τ)} \\ - ν_{x_{io}}^{2 (1 + τ)}, i \in {PFN}_{x} (3) \\ \sum_{j = 1, \neq o}^{n} λ_{j} x_{ij} \leq (1 + τ) x_{io}, i \in E_{x} (4) \\ \sum_{j = 1, \neq o}^{n} λ_{j} ln (1 - μ_{y_{rj}}^{2}) \leq (1 - β_{r}) ln (1 - μ_{y_{ro}}^{2}), r \in {PFN}_{y_{r}} (5) \\ \sum_{j = 1, \neq o}^{n} λ_{j} ln ν_{y_{rj}} \leq (1 - β_{r}) ln ν_{y_{ro}}, r \in {PFN}_{y_{r}} (6) \end{matrix}$

$\begin{matrix} \prod_{j = 1, \neq o}^{n} {(1 - μ_{y_{rj}}^{2})}^{λ_{j}} - \prod_{j = 1, \neq o}^{n} ν_{y_{rj}}^{2 λ_{j}} \leq {(1 - μ_{y_{ro}}^{2})}^{(1 - β_{r})} \\ - ν_{y_{ro}}^{2 (1 - β_{r})}, r \in {PFN}_{y_{r}} (7) \\ \sum_{j = 1, \neq o}^{n} λ_{j} y_{rj} \geq (1 - β_{r}) y_{ro}, r \in E_{y_{r}} (8) \\ \sum_{j = 1, \neq o}^{n} λ_{j} ln μ_{y_{sj}} \leq (1 - β_{s}) ln μ_{y_{so}}, s \in {PFN}_{y_{s}} (9) \\ \sum_{j = 1, \neq o}^{n} λ_{j} ln (1 - ν_{y_{sj}}^{2}) \leq (1 - β_{s}) ln (1 - ν_{y_{so}}^{2}), s \in {PFN}_{y_{s}} (10) \\ \prod_{j = 1, \neq o}^{n} {(1 - ν_{y_{sj}}^{2})}^{λ_{j}} - \prod_{j = 1, \neq o}^{n} μ_{y_{sj}}^{2 λ_{j}} \leq {(1 - ν_{y_{so}}^{2})}^{(1 - β_{s})} \\ - μ_{y_{so}}^{2 (1 - β_{s})}, s \in {PFN}_{y_{s}} (11) \\ \sum_{j = 1, \neq o}^{n} λ_{j} {\bar{y}}_{sj} \geq (1 - β_{s}) {\bar{y}}_{so}, s \in E_{y_{s}} (12) \\ \sum_{j = 1, \neq o}^{n} λ_{j} = 1, λ_{j} \geq 0, j = 1, 2, \dots, n, j \neq o (13) \\ i = 1, 2, \dots, m; r = 1, 2, \dots k; s = k + 1, k + 2, \dots, p \end{matrix}$ (3.13)

The solution of (3), (7) and (11) in model (3.13) can refer to (3) in model (3.11). $0 \leq β_{r}^{*} < 1$ , $0 \leq β_{s}^{*} < 1$ and τ^* > -1. The SE-BCC model under Pythagorean fuzzy environment is feasible if and only if all $β_{r}^{*} = 0$ and $β_{s}^{*} = 0$ , and the super efficiency score of DMU_o obtained by model (3.13) is same as the super efficiency score obtained by the SE-BCC model, which is 1 + τ^*. The SE-BCC model under Pythagorean fuzzy environment is infeasible if and only if some $β_{r}^{*} > 0$ or $β_{s}^{*} > 0$ , the super-efficiency score includes input super efficiency 1 + τ^* and output super efficiency $\frac{1}{| R | + | S |} (\sum_{r \in R} \frac{1}{1 - β_{r}^{*}} + \sum_{s \in S} \frac{1}{1 - β_{s}^{*}})$ , which can be defined as $1 + τ^{*} + \frac{1}{| R | + | S |} (\sum_{r \in R} \frac{1}{1 - β_{r}^{*}} + \sum_{s \in S} \frac{1}{1 - β_{s}^{*}})$ , R and S is the set of $β_{r}^{*} > 0$ and $β_{s}^{*} > 0$ , respectively.

4 Numerical example

To demonstrate the application of the proposed model in green supplier selection, the dataset is partially taken from Mahdiloo et al. [15]. This study selects the dataset of 20 green suppliers of Hyundai Steel in 2012. Two input criteria and three output criteria are used in the evaluation. The inputs include the number of employees (X1) and energy consumption (X2). The desirable outputs utilized in the study are sales (Y1) and green management system (Y2). The carbon dioxide (CO₂) emission (Y3) is considered as an undesirable output. Green management system is a qualitative criterion, DMs divides it into 1-5 types through the Likert scale, which means very poor, poor, medium, good, very good, then convert it into PFNs through linguistic variables, see Table 1. Table 2 shows the dataset for 20 green suppliers.

Table 1
Fuzzy language corresponding to PFNs

Digital Linguistic variables PFNs

1 very poor (0.1, 0.7)

2 poor (0.3, 0.65)

3 medium (0.5, 0.6)

4 good (0.7, 0.4)

5 very good (0.9, 0.2)

Digital	Linguistic variables	PFNs
1	very poor	(0.1, 0.7)
2	poor	(0.3, 0.65)
3	medium	(0.5, 0.6)
4	good	(0.7, 0.4)
5	very good	(0.9, 0.2)

Table 2

Dataset for 20 green suppliers.

DMUs	X1	X2	Y1	Y2	Y3
		(kW h/yr)	(1000 Korean Won)		(kg)
S1	1112	1267	119,477	1	43,562
S2	118	968	125,762	1	45,000
S3	458	1001	58,770	1	42,400
S4	416	1393	62,989	1	43,734
S5	413	1586	67,088	1	44,890
S6	430	1802	72,318	1	42,913
S7	426	1998	74,626	1	39,438
S8	452	1824	74,476	1	40,078
S9	503	1479	79,710	1	39,500
S10	498	1623	79,384	1	45,023
S11	192	1322	73,124	2	41,324
S12	171	831	62,529	4	45,000
S13	163	913	65,424	5	42,400
S14	161	893	71,027	5	43,734
S15	161	903	74,093	5	44,890
S16	162	778	72,830	4	42913
S17	159	710	71,940	4	39,438
S18	157	695	82,203	3	40,078
S19	151	637	55,681	3	39,500
S20	151	781	64839	3	38,570

The data in column 6 of Table 2 presents the undesirable output. The largest CO₂ emission is S10, which is 45023. Therefore, when the monotonic decreasing function is used to transform the undesirable output, it is assumed v = 50000. The efficiency values of each green supplier are solved by using the BCC model, the SE-BCC model and the non-radial SE-BCC model with undesirable outputs under Pythagorean fuzzy environment. The results are shown in Table 3.

Table 3

Efficiency value for 20 green supplier

DMUs	BCC model	SE-BCC model	Non-radial SE-BCC model			Comparative analysis
			Score	Solution details	Ranking	Score	Ranking
S1	1	Infeasible	1.9201	$β_{1}^{*} = 0.0482$	8	1.0158	8
S2	1	Infeasible	10.4763	$β_{1}^{*} = 0.0500$	1	10.4763	1
S3	0.6431	0.6431	0.6431		14	0.9097	15
S4	0.4688	0.4688	0.4688		15	0.8819	18
S5	0.4174	0.4174	0.4174		17	0.8623	20
S6	0.3737	0.3737	0.3737		18	0.9097	15
S7	0.3653	0.3653	0.3653		20	0.9935	11
S8	0.3718	0.3718	0.3718		19	0.9774	12
S9	1	Infeasible	1.5139	$β_{3}^{*} = 0.0344$	9	1.0092	10
S10	0.4244	0.4244	0.4244		16	0.8782	19
S11	0.7128	0.7128	0.7128		13	0.9500	14
S12	0.8974	0.8974	0.8974		12	0.9033	17
S13	1	Infeasible	2.2006	$β_{3}^{*} = 0.1755$	4	1.0315	6
S14	1	Infeasible	2.0301	$β_{1}^{*} = 0.0135$	7	1.0137	9
S15	1	Infeasible	2.0432	$β_{1}^{*} = 0.0414$	6	2.0432	2
S16	0.9570	0.9570	0.9570		11	0.9570	13
S17	1	Infeasible	2.2421	$β_{3}^{*} = 0.1083$	3	1.0629	5
S18	1	Infeasible	3.3926	$β_{3}^{*} = 0.0229$	2	1.0924	3
S19	1	1.1123	1.1123		10	1.0911	4
S20	1	Infeasible	2.1351	$β_{3}^{*} = 0.0759$	5	1.0249	7

When solving the relative efficiency of each green supplier with the above three models, the degree of membership, non-membership and hesitation of the PFNs are considered, which ensures the integrity of the fuzzy information.

Comparing the results of the three models, it is found that:

The BCC model with undesirable outputs under Pythagorean fuzzy environment can only divide the 20 green suppliers into efficient and inefficient types, as shown in column 2 of Table 3. There are 10 green suppliers that are relatively inefficient, with efficiency less than 1. The other 10 green suppliers are relatively efficient, with efficiency equal to 1. So the BCC model with undesirable outputs under Pythagorean fuzzy environment is impossible to distinguish efficient green suppliers.

The SE-BCC model with undesirable outputs under Pythagorean fuzzy environment is based on the BCC model by removing the evaluated green supplier. The relative efficiency of the 10 inefficient green suppliers are invariant. The relative efficiency of 10 efficient green suppliers change, and the super efficiency of S19 is 1.1123, while the other 9 efficient green suppliers are infeasible, see column 3 of Table 3.

The non-radial SE-BCC model with undesirable outputs under Pythagorean fuzzy environment can firstly determine whether the evaluated green supplier is feasible. Inefficient green suppliers are feasible, and their super efficiency scores are same as those obtained through the previous two models. The super efficiency scores of efficient green suppliers with feasible solutions are the same as those obtained by the second model. For efficient green suppliers with infeasible solutions, their super efficiency scores can be obtained by the corresponding formula, see column 4 of Table 4. S2 is selected as the best supplier.

Section 3 states that the reason why infeasible issues in the input-oriented model are output surplus. Let $β_{1}^{*}$ , $β_{2}^{*}$ and $β_{3}^{*}$ are correspond to the output surplus of Y1, Y2 and Y3, respectively. The outputs surplus obtained by the non-radial SE-BCC model are shown in column 5 of Table 3. S1, S2, S14 and S15 are infeasible because Y1 is surplus. S9, S13, S17, S18 and S20 are infeasible because Y3 is surplus. In column 4 of Table 3, the largest super efficiency is S2=10.4763, which is far from other green suppliers. The reasons for the analysis are as follows: Firstly, according to the input-output dataset in Table 2, the number of employees (X1) of S2 is the least, and sales (Y1) is the largest. Secondly, S2 has a larger proportion of sales surplus. Thirdly, the super efficiency model aims to determine the most beneficial efficiency for itself. So it is reasonable that the super efficiency value of S2 is larger than other green suppliers.

Another common way to deal with undesired output is to treat undesired outputs as inputs [36, 37]. Based on this method, the non-radial super efficiency value of all green suppliers are calculated, see column 7 of Table 3. The best green supplier selected by both methods are S2.

5 Conclusion

This paper considers environmental-ecological factors and incorporates undesired outputs into green supplier selection. Not only considering the ambiguity of decision-making problems and the subjectivity of DMs, but also considering the objectivity of decision-making results. The BCC model, the SE-BCC model and the non-radial SE-BCC model with undesirable outputs under Pythagorean fuzzy environment are constructed.

The results of numerical example show that the non-radial SE-BCC model can overcome the defects of the BCC model and the SE-BCC model, and can obtain the super-efficiency values of all DMUs, thus achieving the complete ranking of all DMUs.

In addition to green supplier selection, the model proposed in this paper can also be applied to the efficiency assessment of food safety, road safety, education sector, etc. It can also be expanded the fuzzy environment into hesitation fuzzy sets and neutrosophic sets.

Considering membership degree, nonmembership degree and hesitation degree at the same time can guarantee the complete of fuzzy information. However, converting an input or output criteria into three indicators is equivalent to increasing input or output criteria, thus reducing the discrimination degree. So it is the focus of next research.

Footnotes

Acknowledgments

This work was supported in part by the Fund for Shanxi “1331 Project” Key Innovative Research Team 2017, and in part by “The Discipline Group Construction Plan for Serving Industries Innovation”, Shanxi, China: The Discipline Group Program of Intelligent Logistics Management for Serving Industries Innovation 2018.

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