Supplier evaluation and selection in a sustainable supply chain based on fuzzy-BWM,entropy method and grey relational TOPSIS

Abstract

Suppliers significantly affect the effectiveness of sustainable supply chain management. Hence, it is extremely important to evaluate and select suppliers scientifically and objectively. Based on the theory of triple bottom line (economic, social, and environmental dimension) and a balanced scorecard, a measureable supplier evaluation framework in a sustainable supply chain is first formulated. Second, to reduce the defects of the single weight method, the subjective and objective weights of evaluation indicators are determined by combining the fuzzy best-worst method (BWM) and the entropy method, and then the combination weights are obtained through linear weighting. Third, the grey relational technique for order performance by similarity to ideal solution (TOPSIS) method is further adopted to evaluate and rank the suppliers. Finally, a case study illustrates and demonstrates the availability of the proposed supplier evaluation index system and evaluation method. Subsequently, some suggestions are proposed according to the results.

Keywords

Sustainable supply chain management supplier evaluation the fuzzy BWM grey relational TOPSIS

1 Introduction

Owing to the fierce market competition, rigorous regulations, and increasing sustainable demand from stakeholders, sustainability has been integrated into organizational strategy, such as supply chain strategy.

However, the sustainable supply chain management (SSCM) involves several issues, typically from their suppliers [10, 16], as they provide raw materials, services and finished products as inputs to the supply chain to meet operation needs [7, 14]. For example, poor testing of material by suppliers may cause dangerous and harmful products flowing to consumers, with incalculable losses, poorer corporate reputation, and lowered revenue as outcomes [33].

Supplier behavior directly affects the performance of organization [7, 42]. With the closer relationship between organization and suppliers, it is imperative for organization to evaluate and select their suppliers with sustainable performance. For example, the Rana Plaza disaster in Bangladesh in 2013, which involved perilous working conditions at a garment supplier that lead to the death of more than 1100 people. It turned out that the supplier selection processes of the organization were inadequate [27]. Therefore, selection of sustainable suppliers is of paramount significance for organization in their pursuit of SSCM and sustainable performance.

As a result, the supplier evaluation framework and methods are particular significant. The early supplier evaluation criteria only involved economic sustainability, for instance, dominated by criteria such as price, quality, and delivery [6, 39]. With more attention paid to environmental issues, sustainable supplier evaluation has adapted environmental factor. The implement of Solar Roadways Technology proved it [2]. Like environmental sustainability, social sustainability of suppliers also affects the financial performance, market competitiveness, and anti-risk ability of purchasers [25]. Owing to the difficulty and challenge of social sustainability and because it is difficult to equate high social sustainability with high performance, the organization often hesitates to invest in and implement social sustainability. Thus, in this paper we aim to present a sustainable supplier evaluation indicator framework based on Triple Bottom Line (TBL). At the same time, since all current evaluation methods have certain deficiencies, and there is no optimal evaluation method in the field of supplier evaluation, on the basis of the proposed evaluation framework, we propose combination approach of fuzzy-BWM and grey relational TOPSIS to evaluate the sustainability of suppliers.

The rest of the paper is organized as follows. In Section 2, we reviewed the related literature. Section 3 presents a sustainable supplier evaluation indicator framework and proposes a methodology to evaluate the sustainability of suppliers. Section 4 provides a case application adopting the proposed framework to prove the methodology’s availability. Finally, Section 5 summarizes the conclusions and presents future research.

2 Literature review

The literature related to this study mainly includes the supplier evaluation framework and methods.

2.1 The supplier evaluation framework

The early supplier evaluation criteria only involved economic sustainability, for instance, dominated by criteria such as price, quality, and delivery [6, 39]. With more attention paid to environmental issues, sustainable supplier evaluation has adapted environmental factor. Noci [13] introduced environmental sustainability into the supplier evaluation framework. Kumar et al. [4] established a supplier evaluation framework including eight modules: cost, distance, carbon emissions, industry status and reputation, historical cooperation, and so on. These studies take into account environmental sustainability but not refer to social sustainability.

However, Ferri and Pedrini [24] proposed that both social and environmental sustainability of suppliers affect the financial performance, market competitiveness, and anti-risk ability of purchasers. Owing to the difficulty and challenge of social sustainability and because it is difficult to equate high social sustainability with high performance, the organization often hesitates to invest in and implement social sustainability. In recent years, there were some studies on social sustainability, but it mostly referred to qualitative indicators. For example, Nouri et al. [11] developed a framework for evaluating sustainable service supply chains based on a balanced scorecard, which includes four aspects—finance, service supply chain operation, stakeholders, and learning, growth, and innovation. Neri et al. [5] proposed a set of key performance indicators that are balanced on TBL based on the balanced scorecard and supply chain operations to measure the sustainability performance of industrial supply chains. As a result, our work constructs a framework including quantitative indicators in economic, social and environmental performance.

2.2 The supplier evaluation method

To better evaluate and select suppliers, it is important to adopt appropriate evaluation methods. There are many multi-criteria decision support tools to build and support supplier evaluation and selection [9], such as grey relational analysis (GRA), fuzzy set theory (FST), TOPSIS, analytic hierarchy process (AHP), fuzzy analytic hierarchy process (FAHP); and their improvement and their hybrids [23].

Many studies have adopted these classical methods [38]. However, classical methods have deficiencies or shortcomings when faced with fuzzy situation. Compared with precise system, fuzzy system is more common. For example, Fuzzy logic system is an intelligent system, which is easy to understand, simple to design and better than using the other type of controller [26]. In addition, researchers also introduce hybrid evaluation method to meet with different needs. A brief summary of fuzzy or hybrid methodologies by various researchers and practitioners in supplier selection towards green and sustainable practices is given in Table 1.

Table 1
Summary of the methodologies/techniques by various researchers and practitioners in supplier selection

Literature Methodologies/techniques used Issues addressed (domain) (supplier number)

[7] Interval grey numbers, best-worst method (BWM) and TODIM Socially sustainable suppliers selection (Iranian vehicle manufacturing company) (5)

[19] Rough set theory and rough cloud TOPSIS Sustainable supplier selection based on SSCM practices (sustainable photovoltaic modules supplier) (4)

[46] Rough-fuzzy DEMATEL and Rough-fuzzy TOPSIS Sustainable supplier selection (vehicle manufacturing company) (4)

[3] AHP and TOPSIS Green supplier selection (manufacturing company assembling measurement equipment) (5)

[25] triangular fuzzy approach, fuzzy BWM and α-cut analysis Sustainable supplier selection (vehicle manufacturing company) (3)

[18] Fuzzy BWM and fuzzy TOPSIS Green supplier evaluation and selection (electronics manufacturer) (6)

[43] ANP, preference ranking organization method Sustainable supplier selection (chemical manufacturer company) (69)

[28] AHP, entropy weight method, grey clustering and grey relational analysis Green supplier selection (vehicle manufacturing company) (10)

[12] Fuzzy-Delphi method, BWM and grey relational TOPSIS green supplier evaluation and optimal order allocation under uncertainty (Delta Industrial Group) (6)

Literature	Methodologies/techniques used	Issues addressed (domain) (supplier number)
[7]	Interval grey numbers, best-worst method (BWM) and TODIM	Socially sustainable suppliers selection (Iranian vehicle manufacturing company) (5)
[19]	Rough set theory and rough cloud TOPSIS	Sustainable supplier selection based on SSCM practices (sustainable photovoltaic modules supplier) (4)
[46]	Rough-fuzzy DEMATEL and Rough-fuzzy TOPSIS	Sustainable supplier selection (vehicle manufacturing company) (4)
[3]	AHP and TOPSIS	Green supplier selection (manufacturing company assembling measurement equipment) (5)
[25]	triangular fuzzy approach, fuzzy BWM and α-cut analysis	Sustainable supplier selection (vehicle manufacturing company) (3)
[18]	Fuzzy BWM and fuzzy TOPSIS	Green supplier evaluation and selection (electronics manufacturer) (6)
[43]	ANP, preference ranking organization method	Sustainable supplier selection (chemical manufacturer company) (69)
[28]	AHP, entropy weight method, grey clustering and grey relational analysis	Green supplier selection (vehicle manufacturing company) (10)
[12]	Fuzzy-Delphi method, BWM and grey relational TOPSIS	green supplier evaluation and optimal order allocation under uncertainty (Delta Industrial Group) (6)

As shown in Table 1, Fuzzy-BWM and grey correction TOPSIS are widely used in green or sustainable supplier evaluation and selection. Mohammed et al. [3] calculated the weight of criteria using the AHP method and evaluated the suppliers using the TOPSIS method. However, compared with the BWM, more pairwise compassion data are required in the AHP method. TOPSIS method cannot show the closeness between the decision and the ideal solution from the shape. Amiri et al. [25] presented a new model with a triangular fuzzy approach for sustainable supplier selection. The proposed model is based on the BWM and α-cut analysis, which is greatly influenced by subjective factors. As a result, our work adopted a combination approach of the fuzzy-BWM and entropy method. The approach takes advantage of both methods. In the fuzzy-BWM method, less pairwise compassion data are required, which can adapt to uncertain situation. The entropy method has objectivity. The grey relational TOPSIS show the closeness between the decision and the ideal solution form the shape and distance.

Therefore, the contributions of our work are summarized as follows: First, based on the summary of the previous sustainable supplier evaluation indicators, this study adopts the balanced scorecard, measurability, and consistency with public policies to screen the indicators to build a quantitative, sustainable (economic, social and environmental) supplier evaluation indicator framework. Second, considering the uncertainty and ambiguity of objective things, the ambiguity of human thinking and the cognitive biases of some evaluators, the group aggregation value of fuzzy preference combined with fuzzy-BWM is determined to eliminate the impact of ambiguity on evaluation results. Third, owing to the shortcomings of the subjective evaluation method, the fuzzy BWM is combined with the entropy method to determine the combined weight of the indicators, and the grey relational TOPSIS method is adopted to rank and select sustainable suppliers.

3 Research method

In order to evaluate the overall sustainability of suppliers, this paper formulates the supplier evaluation model. First, we determine the subjective weight of the evaluation indicators based on the fuzzy BWM. Second, we determine the objective weight of the evaluation indicators based on the entropy method. Third, to compensate for the shortcomings of the single weighting method, we obtain the combination weight by combining the subjective and objective weight. Finally, we evaluate and select suppliers based on the grey relational TOPSIS method. The procedure is shown in Fig. 1.

Fig. 1

The research procedure.

3.1 Supplier evaluation framework in a sustainable supply chain

In evaluation framework, the indicator selection directly impacts the accuracy and reliability of the evaluation results. For example, during damage evaluation in beam-like structures, the Local Frequencies Change Ratio (LFCR) indicator and damage indicator were selected to locate damaged elements more effectively [41]. Similarly, Zenzen [36] adopted new indicator combining LFCR and Transmissibility with mode shape together, which can be sensitive enough to evaluate and identify damage. In this sustainable supplier evaluation framework, we first selected sustainability indicators by summarizing the relevant literature on the sustainable supplier evaluation and selection. Based on the TBL principle, we divided the evaluation indicators into three categories: economic, environmental, and social bottom line. Second, we screened the indicators based on the balanced scorecard, measurability, and consistency with public policies. Among them, measurability is a necessary condition, and the number of conditions that the indicator meets must be greater than or equal to 2. Third, we formulated a more objective and easier-to-measure sustainable supplier evaluation framework that comprised measurable indicators, which is shown in Table 2, and the measurable criteria are shown inTable 3.

Table 2
Supplier evaluation framework in a sustainable supply chain

Bottom line Indicator Brief description Reference

Economic bottom line (F) F1-Price of material A competitive material’s price that suppliers provide Pandey et al. [31], Luthra et al. [42], Ahmadi et al. [15]

F2-Quality of material A significant quality level that suppliers provide Jain and Singh [29], Khan et al. [37], Jain and Khan [44], Rajesh and Ravi [35]

F3-Technological innovation ability Number of scientific research patents that suppliers own Jain and Singh [29], Luthra et al. [42]

F4-Flexibility of supply chain Supplier should be sufficiently flexible to handle market variations Nouri et al. [11], Khan et al. [37], Luthra et al. [42]

F5-Delivery time Time required to deliver unit materials Khan et al. [37], Luthra et al. [42], Pandey et al. [31]

F6-Investment to sustainable processes / products Investment to sustainable products, processes, technologies, etc. Giannakis et al. [27], Erol et al. [17]

F7-Financial capability Financial conditions and stability Khan et al. [37], Ahmadi et al. [15]

Social bottom line (S) S1-Social investment in community Contribution to community Giannakis et al. [27], Bai et al. [7], Khan et al. [37], Veleva and Ellenbecker [45]

S2-Customer satisfaction with products / services Rate of customer satisfaction regarding products or services Giannakis et al. [27], Nouri et al. [11], Erol et al. [17]

S3-Number of health and safety incidents Number of health and safety incidents in every field Giannakis et al. [27], Veleva and Ellenbecker [45]

S4-Average days of employee training Average days of employee training per employee in every field provided by an organisation Giannakis et al. [27], Veleva and Ellenbecker [45]

S5-Information disclosure Information on the materials being used and carbon emissions that suppliers provide to their clients and stakeholders Bai et al. [7], Khan et al. [37], Luthra et al. [42], Kuo et al. [34]

S6-Allegations of discrimination Prosecuted for discrimination on race, religion, age, gender, sexual orientation, and other factors Jain and Singh [29], Govindan et al. [23]

S7-Salary of employees Employees’ remuneration for labour Bai et al. [7], Luthra et al. [42], Amindoust et al. [1], Kuo et al. [34]

Environmental bottom line (E) E1-Greenhouse gas emissions Greenhouse gas emissions per unit of product Giannakis et al. [27], Jain and Singh [29], Khan et al. [37]

E2-Sustainable energy efficiency The requirement for sustainable energy has grown up significantly over the years[22]. The indicator is the ratio of sustainable energy to energy used Khan et al. [37], Pandey et al. [31], Hsu and Hu [8]

E3- Amount of waste generated Amount of waste generated per unit of product Giannakis et al. [27], Erol et al. [17], Veleva and Ellenbecker [45]

E4-Green R&D and Innovation Investment in green technology, process, time, and method Luthra et al. [42], Rajesh and Ravi [35]

E5-ISO14001 : 2015 certification Possess ISO 14001 : 2015 certification valid for three years Jain and Singh [29], Nouri et al. [11]

E6-Energy consumption Energy consumption per unit of product Giannakis et al. [27], Nouri et al. [11], Khan et al. [37], Hsu and Hu [8]

E7-Waste recycling Usage rate of recycled material Jain and Singh [29], Khan et al. [37], Pandey et al. [31], Amindoust et al. [1]

Bottom line	Indicator	Brief description	Reference
Economic bottom line (F)	F1-Price of material	A competitive material’s price that suppliers provide	Pandey et al. [31], Luthra et al. [42], Ahmadi et al. [15]
	F2-Quality of material	A significant quality level that suppliers provide	Jain and Singh [29], Khan et al. [37], Jain and Khan [44], Rajesh and Ravi [35]
	F3-Technological innovation ability	Number of scientific research patents that suppliers own	Jain and Singh [29], Luthra et al. [42]
	F4-Flexibility of supply chain	Supplier should be sufficiently flexible to handle market variations	Nouri et al. [11], Khan et al. [37], Luthra et al. [42]
	F5-Delivery time	Time required to deliver unit materials	Khan et al. [37], Luthra et al. [42], Pandey et al. [31]
	F6-Investment to sustainable processes / products	Investment to sustainable products, processes, technologies, etc.	Giannakis et al. [27], Erol et al. [17]
	F7-Financial capability	Financial conditions and stability	Khan et al. [37], Ahmadi et al. [15]
Social bottom line (S)	S1-Social investment in community	Contribution to community	Giannakis et al. [27], Bai et al. [7], Khan et al. [37], Veleva and Ellenbecker [45]
	S2-Customer satisfaction with products / services	Rate of customer satisfaction regarding products or services	Giannakis et al. [27], Nouri et al. [11], Erol et al. [17]
	S3-Number of health and safety incidents	Number of health and safety incidents in every field	Giannakis et al. [27], Veleva and Ellenbecker [45]
	S4-Average days of employee training	Average days of employee training per employee in every field provided by an organisation	Giannakis et al. [27], Veleva and Ellenbecker [45]
	S5-Information disclosure	Information on the materials being used and carbon emissions that suppliers provide to their clients and stakeholders	Bai et al. [7], Khan et al. [37], Luthra et al. [42], Kuo et al. [34]
	S6-Allegations of discrimination	Prosecuted for discrimination on race, religion, age, gender, sexual orientation, and other factors	Jain and Singh [29], Govindan et al. [23]
	S7-Salary of employees	Employees’ remuneration for labour	Bai et al. [7], Luthra et al. [42], Amindoust et al. [1], Kuo et al. [34]
Environmental bottom line (E)	E1-Greenhouse gas emissions	Greenhouse gas emissions per unit of product	Giannakis et al. [27], Jain and Singh [29], Khan et al. [37]
	E2-Sustainable energy efficiency	The requirement for sustainable energy has grown up significantly over the years[22]. The indicator is the ratio of sustainable energy to energy used	Khan et al. [37], Pandey et al. [31], Hsu and Hu [8]
	E3- Amount of waste generated	Amount of waste generated per unit of product	Giannakis et al. [27], Erol et al. [17], Veleva and Ellenbecker [45]
	E4-Green R&D and Innovation	Investment in green technology, process, time, and method	Luthra et al. [42], Rajesh and Ravi [35]
	E5-ISO14001 : 2015 certification	Possess ISO 14001 : 2015 certification valid for three years	Jain and Singh [29], Nouri et al. [11]
	E6-Energy consumption	Energy consumption per unit of product	Giannakis et al. [27], Nouri et al. [11], Khan et al. [37], Hsu and Hu [8]
	E7-Waste recycling	Usage rate of recycled material	Jain and Singh [29], Khan et al. [37], Pandey et al. [31], Amindoust et al. [1]

Table 3

Supplier evaluation criteria in a sustainable supply chain

Bottom line	Indicator	Measurement criteria
Economic bottom line (F)	F1-Price of material	Cost price per unit of material, unit: yuan
	F2-Quality of material	Yield rate per unit of material—number of qualified materials divided by that of purchased materials
	F3-Technological innovation ability	Number of newly added patents in the past three years, unit: item
	F4-Flexibility of supply chain	Maximum quantity delivered within a week, unit: ten thousand
	F5-Delivery time	Average delivery days required to deliver each unit of material, unit: day
	F6-Investment in sustainable processes/products	Ratio of investment in sustainable products, processes, and technologies to annual operating profit
	F7-Financial capability	Current ratio = total current assets / total current liabilities
Social bottom line (S)	S1-Social investment in community	Ratio of investment in the community to annual operating profit
	S2-Customer satisfaction with products/services	Customer satisfaction of production or service = 1- (number of customer complaints about products or service / number of total customers)
	S3-Number of health and safety incidents	Number of health and safety incidents in the past three years, unit: item
	S4-Average days of employee training	Average days of employee training per employee in every field provided by suppliers, unit: day
	S5-Information disclosure	Number of times that suppliers release information on the materials used and carbon emission during the manufacturing process to their clients and stakeholders in the past three years, which can be obtained from the suppliers’ website, unit: item
	S6-Allegations of discrimination	Number of cases prosecuted for discrimination in the past three years, unit: item
	S7- Salary of employees	Average hourly wage of front-line production employees, unit: yuan/hour
Environmental bottom line (E)	E1-Greenhouse gas emissions	Greenhouse gas emissions per ton of product, unit: ton
	E2-Sustainable energy efficiency	Ratio of sustainable energy to energy used
	E3-Amount of waste generated	Amount of waste produced per ton of product, unit: ton
	E4-Green R&D and Innovation	Ratio of investment in green technology, process, time, and method to annual operating profit
	E5-ISO14001 : 2015 certification	Whether one owns an ISO14001 : 2015 certification valid for three years; if one has it, it is 1; otherwise, it is 0
	E6-Energy consumption	Amount of energy consumed per ton of product, unit: GJ
	E7-Waste recycling	Ratio of the amount of scrap to the total amount of raw materials in the production of materials

3.2 Determination of subjective weights based on the fuzzy BWM

Compared with the typical AHP method, less pairwise comparison data are required in the BWM, and the results generated by the BWM are more robust and reliable [20, 21]. Considering the vagueness frequently represented in decision data due to incomplete information and the ambiguity arising from the qualitative judgment of decision-makers, the crisp values of criteria may be inadequate to model the real-life MCDM issues. In this context, the BWM is extended to the fuzzy environment. Guo and Zhao [40] established a fuzzy BWM model by introducing triangular fuzzy numbers (TFNs) and verified its comparison consistency. As fuzzy sets and TFNs are the basis of the fuzzy BWM, we first introduce their definitions.

Definition 1. The fuzzy set theory

A fuzzy set $\tilde{a}$ is a pair (U, m), where U is a set, and m : U → [0, 1] is the membership function, denoted by $μ_{\tilde{a}} (x)$ . By referring to $μ_{\tilde{a}} (x)$ , each element x in a universe of discourse X can be mapped to a real number in the interval [0,1].

Definition 2. The triangular fuzzy number

A fuzzy number $\tilde{a} = (l, m, n) \in F (R), R = [0, 1]$ , $\tilde{a}$ is called a TFN in R, and its membership function is $μ_{\tilde{a}} (x) = {\begin{matrix} 0, \begin{matrix} \end{matrix} x < l \\ \frac{x - l}{m - l}, l ⩽ x ⩽ m \\ \frac{u - x}{u - m}, m ⩽ x ⩽ u \\ 0, \begin{matrix} \end{matrix} x > u \end{matrix}$ (1) where l, m, and u, respectively, represent the lower, modal, and upper values of the support of $\tilde{a}$ , all of which are crisp numbers (- ∝ < l ⩽ m ⩽ u < + ∝). A TFN can be represented as a triplet (l, m, n).

Definition 3. Let $\tilde{a} = (l, m, n)$ , and the grade mean integration representation (GMIR) $R (\tilde{a})$ of TFN $\tilde{a}$ can be calculated by Equation (2). $R (\tilde{a}) = \frac{l + 4 m + u}{6}$ (2)

Suppose that there are indicators for a research object, and the fuzzy pairwise comparison on these indicators can be performed based on the linguistic variables (terms) of decision-makers, such as ‘Equally import-ant (EI)’, ‘Weakly important (WI)’, ‘Fairly Important (FI)’, ‘Very important (VI)’, and ‘Absolutely important (AI)’. Subsequently, the linguistic evaluations of decision-makers need to be transformed to fuzzy ratings (represented by TFNs) [30], which is (1,1,1), (2/3,1,3/2), (3/2,2,5/2), (5/2,3,7/2), (7/2,4,9/2), respectively.

In order to reduce the influence of personal subjective preferences on the results, we adopt group aggregation value to aggregate fuzzy group evaluation information. The smaller the difference D (x, y) and the greater the similarity S (x, y) between the group aggregation value and the original data, the better the results. Here $\begin{matrix} D (x, y) = \\ \sqrt{\frac{1}{3} [{(l_{x} - l_{y})}^{2} + {(m_{x} - m_{y})}^{2} + {(u_{x} - u_{y})}^{2}]} \end{matrix}$ (3) $\begin{matrix} S (x, y) = \\ \frac{l_{x} l_{y} + m_{x} m_{y} + u_{x} u_{y}}{\sqrt{{(l_{x})}^{2} + {(m_{x})}^{2} + {(u_{x})}^{2}} \sqrt{{(l_{y})}^{2} + {(m_{y})}^{2} + {(u_{y})}^{2}}} \end{matrix}$ (4) Fuzzy preferences are given by multiple experts. Using Equation (5) to determine the group aggregation value (l_y, m_y, u_y). $\begin{matrix} min z = \sum_{a = 1}^{b} n_{a} D (x_{a}, y) \\ s . t . {\begin{matrix} S (x_{a}, y) ⩾ α, a = 1, 2, \dots, b \\ l_{y} ⩽ m_{y} ⩽ u_{y} \end{matrix} \end{matrix}$ (5) where n_a is the relative importance of ath expert, we sets that all experts have the same degree of importance.αis threshold, which is set as0.98.

The steps to determine the subjective weights of the evaluation indicators by the fuzzy BWM are as follows.

Step 1: Determine a set of decision indicators c_j (j = 1, 2, ⋯ , n).

Step 2: Determine the best indicator (most important) c_B and the worst indicator (least important) c_w among the selected decision indicators.

Step 3: Determine the fuzzy preference of the best indicator over all the other indicators. Subsequently, the obtained fuzzy preferences are transformed to TFNs based on the transformation rules. The obtained fuzzy vector of Best-to-Others (BO) is ${\tilde{A}}_{B} = ({\tilde{a}}_{B 1}, {\tilde{a}}_{B 2}, \dots, {\tilde{a}}_{Bn})$ , where ${\bar{a}}_{Bj}$ is the best indicator c_B’s fuzzy preference over the indicator c_j, and ${\tilde{a}}_{BB} = (1, 1, 1)$ .

Step 4: Determine the fuzzy preference of all indicators over the worst indicator. Subsequently, the obtained fuzzy preferences are transformed to TFNs based on the transformation rules. The obtained fuzzy vector of Others-to Worst (OW) is ${\tilde{A}}_{W} = ({\tilde{a}}_{1 W}, {\tilde{a}}_{2 W}, \dots {\tilde{a}}_{nW})$ , where ${\tilde{a}}_{iW}$ is the indicator c_j fuzzy preference over the worst indicator c_B, and ${\tilde{a}}_{WW} = (1, 1, 1)$ .

Step 5: Determine the optimal weights $({\tilde{w}}_{1}^{*}, {\tilde{w}}_{2}^{*}, \dots, {\tilde{w}}_{n}^{*})$ . The optimal weights of the indicators satisfy Equation (6). By solving Equation (6), the optimal fuzzy weights $({\tilde{w}}_{1}^{*}, {\tilde{w}}_{2}^{*}, \dots, {\tilde{w}}_{n}^{*})$ can be obtained. $\begin{matrix} min ξ^{*} \\ s . t . {\begin{matrix} | \frac{(l_{B}^{w}, m_{B}^{w}, u_{B}^{w})}{(l_{j}^{w}, m_{j}^{w}, u_{j}^{w})} - (l_{Bj}, m_{Bj}, u_{Bj}) | ⩽ (k^{*}, k^{*}, k^{*}) \\ | \frac{(l_{j}^{w}, m_{j}^{w}, u_{j}^{w})}{(l_{W}^{w}, m_{W}^{w}, u_{W}^{w})} - (l_{jW}, m_{jW}, u_{jW}) | ⩽ (k^{*}, k^{*}, k^{*}) \\ \sum_{j = 1}^{n} R ({\tilde{w}}_{j}) = 1 \\ l_{j}^{w} ⩽ m_{j}^{w} ⩽ u_{j}^{w} \\ l_{j}^{w} ⩾ 0 \\ j = 1, 2, \dots, n \end{matrix} \end{matrix}$ (6) where ${\tilde{w}}_{B} = (l_{B}^{w}, m_{B}^{w}, u_{B}^{w})$ is the weight of indicator c_B, ${\tilde{w}}_{j} = (l_{j}^{w}, m_{j}^{w}, u_{j}^{w})$ is the weight of other indicator, ${\tilde{w}}_{W} = (l_{W}^{w}, m_{W}^{w}, u_{W}^{w})$ is the weight of indicator c_W, ${\tilde{a}}_{Bj} = (l_{Bj}, m_{Bj}, u_{Bj})$ , ${\tilde{a}}_{jW} = (l_{jW}, m_{jW}, u_{jW})$ , ξ^* = (k^*, k^*, k^*) , k^* ⩽ l^ξ.

3.3 Determination of objective weights based on the entropy method

The entropy method calculates the weights of different indicators based on the original objective data and does not involve the personal opinions of the decision-makers. Thus, we adopt the entropy method to determine the objective weight of the indicators. Supposed that there are m suppliers. We therefore need to evaluate these m suppliers through n indicators. The procedures are as follows.

Step 1: Based on the extreme value method, dimensionless process the original data. All the values of the indicators are converted into the interval of [0-1 ]. In addition, to make the data meaningful, the dimensionless data can be shifted to a minimum unit value to meet with the calculation requirements. The translation unit in this study is 0.0001. Cost indicators can be processed by Equation (7), and efficiency indicators can be processed by Equation (8). $\begin{matrix} y_{ij} = \frac{max (x_{ij}) - x_{ij}}{max (x_{ij}) - min (x_{ij})} \\ (i = 1, 2, \dots, m; j = 1, 2, \dots, n) \end{matrix}$ (7) $\begin{matrix} y_{ij} = \frac{x_{ij} - min (x_{ij})}{max (x_{ij}) - min (x_{ij})} \\ (i = 1, 2, \dots, m; j = 1, 2, \dots, n) \end{matrix}$ (8)

Where x_ij is the original value of the jth indicator under the ith supplier. Thus, we can obtain a standardised decision matrix Y = (y_ij) _m×n.

Step 2: Normalise the standardised decision matrix by Equation (9) [28], and obtain the normalised decision matrix Q = (q_ij) _m×n. $q_{ij} = \frac{y_{ij}}{\sum_{j = 1}^{n} y_{ij}} (j = 1, 2, \dots, n)$ (9) where q_ij is the value of the jth element under the ith indicator.

Step 3: Calculate the entropy value of the jth indicator using Equation (10). $E_{j} = - I \sum_{i = 1}^{m} q_{ij} \cdot ln q_{ij} (j = 1, 2, \dots, n)$ (10) where $I = \frac{1}{ln m}$ , I > 0, and 0 ⩽ E_j ⩽ 1. m is the number of suppliers.

Step 4. Calculate the objective weight of each indicator using Equation (11). $w_{j} = \frac{1 - E_{j}}{\sum_{j = 1}^{n} (1 - E_{j})} (j = 1, 2, \dots, n)$ (11)

3.4 Determination of the combination weights

The entropy method has the advantage of objectivity, which makes it more credible and accurate than the BWM method. However, the weight of each indicator varies over the number of samples and data, which makes matters worse—the weighting result may not match the actual situation. The BWM method compensates for this shortcoming. Therefore, we combine the subjective and objective weights by adopting both the BWM and entropy methods to avoid the shortcoming of single weighting, increase the rationality and accuracy of the evaluation process, and improve the reliability of the evaluation results.

The combination weight of each indicator can be calculated by $W = α w_{BWM} + β w_{entropy} (j = 1, 2, \dots, n)$ (12)

3.5 Ranking suppliers’ sustainability performance based on grey relational TOPSIS method

As the TOPSIS method cannot show the closeness between the decision and the ideal solution from the shape, we combine the grey relational with TOPSIS method to overcome the deficiencies of the TOPSIS method and improve the accuracy of decision-making results. By this method, the procedures to evaluate and rank suppliers are as follows.

Step 1: Construct a weighted decision matrix F = (f_ij) _m×n as follows: $F = W \times Y$ (13) where Y is the standardised decision matrix determined by Equations (7) and (8).

Step 2: Determine the positive-ideal solution F⁺ and the negative-ideal solution F^- based on Equations (14)and (15) , respectively. $\begin{matrix} F^{+} = max_{1 ⩽ i ⩽ m} f_{i} (j) \\ A = (f^{+} (1), \dots, f^{+} (n)) (j = 1, 2, \dots, n) \end{matrix}$ (14) $\begin{matrix} F^{-} = min_{1 ⩽ i ⩽ m} f_{i} (j) \\ = (f^{-} (1), \dots, f^{-} (n)) (j = 1, 2, \dots, n) \end{matrix}$ (15)

Step 3: Calculate the Euclidean distance T + /T^- from each decision option to F + /F^- by Equations (16)/(17). $T_{i}^{+} = \sqrt{\sum_{j = 1}^{n} {(f_{ij} - f_{j}^{+})}^{2}} (i = 1, 2, \dots, m)$ (16) $T_{i}^{-} = \sqrt{\sum_{j = 1}^{n} {(f_{ij} - f_{j}^{-})}^{2}} (i = 1, 2, \dots, m)$ (17)

Step 4: Calculate the grey relational degree R + /R^- of each decision option to F + /F^- by Equations (18)/(19). $R_{i}^{+} = \frac{1}{n} \sum_{j = 1}^{n} r_{ij}^{+} (i = 1, 2, \dots, m)$ (18) $R_{i}^{-} = \frac{1}{n} \sum_{j = 1}^{n} r_{ij}^{-} (i = 1, 2, \dots, m)$ (19) $\begin{matrix} r_{ij}^{+} = \frac{min_{i} min_{j} V_{i}^{-} (j) + ρ max_{i} max_{j} V_{i}^{+} (j)}{V_{i}^{+} (j) + ρ max_{i} max_{j} V_{i}^{+} (j)} \\ (0 < ρ < 1) \end{matrix}$ (20) $\begin{matrix} r_{ij}^{-} = \frac{min_{i} min_{j} V_{i}^{-} (j) + ρ max_{i} max_{j} V_{i}^{-} (j)}{V_{i}^{-} (j) + ρ max_{i} max_{j} V_{i}^{-} (j)} \\ (0 < ρ < 1) \end{matrix}$ (21) where $V_{i}^{+} (j) = | f_{j}^{+} - f_{ij} |$ , $V_{i}^{-} (j) = | f_{j}^{-} - f_{ij} |$ . And ρ ∈ [0, 1] is the resolution coefficient.

Step 5: Euclidean distance and grey relational degree are processed dimensionless through Equation (22). $U = \frac{G_{i}}{max G_{i}} (i = 1, 2, \dots, m)$ (22) where G_i represents $T_{i}^{+}$ , $T_{i}^{-}$ , $R_{i}^{+}$ , and $R_{i}^{-}$ , respectively.

Step 6: Integrate the dimensionless Euclidean distance $T_{i}^{+}$ , $T_{i}^{-}$ and grey relational degree $R_{i}^{+}$ , $R_{i}^{-}$ through Equations (23) and (24). $V_{i}^{+} = ξ_{1} T_{i}^{-} + ξ_{2} R_{i}^{+} (i = 1, 2, \dots, m)$ (23) $V_{i}^{-} = ξ_{1} T_{i}^{+} + ξ_{2} R_{i}^{-} (i = 1, 2, \dots, m)$ (24)

According to the principle of the TOPSIS method, the greater the values of $T_{i}^{-}$ and $R_{i}^{+}$ , the closer the decision option to the ideal solution.

In Equations (23) and (24), the values of ξ₁ and ξ₂ represent the preference of evaluators on location and shape, and ξ₁ + ξ₂ = 1.

Step 7: Calculate the relative closeness S_i through Equation (25) and rank the suppliers. $S_{i} = \frac{V_{i}^{+}}{V_{i}^{+} + V_{i}^{-}} (i = 1, 2, \dots, m)$ (25)

The greater the value of S_i, the closer the evaluation result to the ideal solution.

4 Case study

We now provide a case study to illustrate the availability of the evaluation indicator system and evaluation method proposed in this study.

4.1 Introduction of company H

Company H is a construction machinery manufacturer, selling excavators, bulldozers, etc. Purchase parts from suppliers, such as lead batteries, sensors, turbochargers, etc., assemble them and produce finished products in their own factories. To comply with the development trend of the sustainable supply chain and enhance its competitiveness in the market, Company H needs to choose the best sustainable supplier among four suppliers to purchase lead batteries. It needs to establish a sustainable strategic partnership with this supplier to achieve the coordination and unification of economic, social, and environmental benefits.

Supplier sustainability implementation levels are used to evaluate the suppliers [7]. First, an evaluation team is formed, which is consisted of two relevant academic experts, one financial manager, one supply chain manager, one reliability manager, one material manager, one purchaser, one product manager, and two senior quality engineers.

The original data of the evaluation indicators are quantitative data obtained through suppliers’ and Company H’s historical data. Original evaluation data to four suppliers are listed in Table 4.

Table 4
Original data of the four suppliers

Indicator Supplier A Supplier B Supplier C Supplier D

F1 398 385 406 401

F2 99.80% 99.75% 99.91% 99.82%

F3 65 62 71 69

F4 5.76 4.32 5.84 5.53

F5 7 6 8 8

F6 10.26% 10.68% 12.53% 11.58%

F7 135.2% 145.5% 112.6% 97.3%

S1 2.5% 2.6% 2% 2.8%

S2 99% 98% 99% 97%

S3 3 2 2 3

S4 14 15 18 15

S5 32 35 40 33

S6 0 0 0 1

S7 20.08 21.43 22.32 20.98

E1 1.69 1.72 1.54 1.61

E2 20.11% 19.36% 25.43% 22.89%

E3 2.48 2.65 2.36 2.49

E4 9.25% 8.36% 10.23% 8.02%

E5 0 0 1 1

E6 12.99 12.30 12.13 13.18

E7 12% 15% 18% 14%

Indicator	Supplier A	Supplier B	Supplier C	Supplier D
F1	398	385	406	401
F2	99.80%	99.75%	99.91%	99.82%
F3	65	62	71	69
F4	5.76	4.32	5.84	5.53
F5	7	6	8	8
F6	10.26%	10.68%	12.53%	11.58%
F7	135.2%	145.5%	112.6%	97.3%
S1	2.5%	2.6%	2%	2.8%
S2	99%	98%	99%	97%
S3	3	2	2	3
S4	14	15	18	15
S5	32	35	40	33
S6	0	0	0	1
S7	20.08	21.43	22.32	20.98
E1	1.69	1.72	1.54	1.61
E2	20.11%	19.36%	25.43%	22.89%
E3	2.48	2.65	2.36	2.49
E4	9.25%	8.36%	10.23%	8.02%
E5	0	0	1	1
E6	12.99	12.30	12.13	13.18
E7	12%	15%	18%	14%

4.2 Determination of the indicators’ subjective weight

Sustainability is the harmonization of economic, social, and environmental performance. All indicators in TBL are significant in evaluating sustainable suppliers. Since it is difficult to compare the indicators among the three categories, we only compare the importance of the indicators within the same category. As shown in Table 1, there are seven indicators in each category. We adopt the fuzzy BWM method to determine the subjective weight of the indicators in different categories.

The data collection process is as follows. First, introducing the specific meaning of each indicator to experts, and then surveying some questions from them. Taking economic bottom line indicators as an example, each expert in the evaluation team was asked to choose the most and least important indicator among all the seven indicators. Table 5 shows the best and worst indicators determined by Experts 1–10 in the economic bottom line.

Table 5
Best and worst indicators in the economic bottom line determined by experts

Experts Most important Least important

1 F4 F5

2 F1 F7

3 F6 F5

4 F4 F5

5 F6 F5

6 F3 F5

7 F1 F7

8 F4 F5

9 F6 F5

10 F4 F7

Experts	Most important	Least important
1	F4	F5
2	F1	F7
3	F6	F5
4	F4	F5
5	F6	F5
6	F3	F5
7	F1	F7
8	F4	F5
9	F6	F5
10	F4	F7

According to the majority principle, the indicator with the most votes is selected. As shown in Table 4, the most important economic performance indicator is F4-Flexibility of supply chain, and the least important indicator is F5-Delivery time.

Second, issue the questionnaire, introduce the content of each indicator to the experts before filling out the questionnaire, and explain the scoring rules of fuzzy BWM in detail to ensure the rationality and accuracy of the scoring results. Each expert was then asked to determine the best indicator’s fuzzy preference over all other indicators based on the determined most important and least important indicators. Further, each expert was asked to determine the fuzzy preference of all indicators over the least important indicator. To reduce the influence of the evaluators’ personal cognitive bias on the evaluation results, the group aggregation value of the evaluation results is taken—ten experts respectively determine the fuzzy preference of F5 versus F1 to obtain ten vectors. Subsequently, we determine the group aggregation value using Equation (5), which is the final fuzzy preference result. This processing method synthesizes the experience and opinions of various experts and reduces the influence of the cognitive bias of individual evaluators on the evaluation results.

Table7

The fuzzy BWM implement result

Sustainable supplier selection (1)
Economic bottom line (0.4235)		Social bottom line (0.3261)		Environmental bottom line(0.2504)
Unweighted	Weighted	Unweighted	Weighted	Unweighted	Weighted
F1 (0.1923)	F1 (0.0814)	S1 (0.2021)	S1 (0.0659)	E1 (0.0920)	E1 (0.0230)
F2 (0.0749)	F2 (0.0317)	S2 (0.1821)	S2 (0.0594)	E2 (0.2167)	E2 (0.0542)
F3 (0.1140)	F3 (0.0483)	S3 (0.2433)	S3 (0.0793)	E3 (0.0759)	E3 (0.0190)
F4 (0.2458)	F4 (0.1041)	S4 (0.0928)	S4 (0.0303)	E4 (0.2077)	E4 (0.0520)
F5 (0.0666)	F5 (0.0282)	S5 (0.1380)	S5 (0.0450)	E5 (0.1234)	E5 (0.0309)
F6 (0.1923)	F6 (0.0815)	S6 (0.0821)	S6 (0.0268)	E6 (0.0711)	E6 (0.0179)
F7 (0.1140)	F7 (0.0483)	S7 (0.0596)	S7 (0.0194)	E7 (0.2133)	E7 (0.0534)

In this case, there are 10 BO and 10 OW evaluation preferences for 10 experts in the economic performance dimension. We need to determine the group aggregation value. Taking Expert 1 as an example, the BO evaluation preference is presented in Table 6. For brevity, the remaining 19 evaluation fuzzy preference languages are not shown.

Table 6

Best-to-Others evaluation fuzzy preference language for Expert 1

Most important	F1	F2	F3	F4	F5	F6	F7
F4	FI	VI	WI	EI	AI	FI	WI

Then we need to determine the group aggregation value of the fuzzy preference language and transform it into a TFN.

Therefore, the evaluation fuzzy preference matrix of F4 to all the other indicators and all the other indicators over F5 are ${\tilde{A}}_{B} = [\begin{matrix} (\frac{2}{3}, 1, \frac{3}{2}), (\frac{5}{2}, 3, \frac{7}{2}), (\frac{3}{2}, 2, \frac{5}{2}), (1, 1, 1), \\ (\frac{7}{2}, 4, \frac{9}{2}), (\frac{2}{3}, 1, \frac{3}{2}), (\frac{3}{2}, 2, \frac{5}{2}) \end{matrix}] .$ ${\tilde{A}}_{W} = [\begin{matrix} (\frac{5}{2}, 3, \frac{7}{2}), (\frac{2}{3}, 1, \frac{3}{2}), (\frac{3}{2}, 2, \frac{5}{2}), (\frac{7}{2}, 4, \frac{9}{2}), \\ (1, 1, 1), (\frac{3}{2}, 2, \frac{5}{2}), (\frac{2}{3}, 1, \frac{3}{2}) \end{matrix}]$

According to Equation (6), the fuzzy preference vector and weight of each indicator can be obtained.

The fuzzy preference vector = [(1641,0.1857,0.2469), (0.0720,0.0730,0.0853), (0.0977,0.1108,0.1433), (0.2386,0.2408,0.2733), (0.0650,0.0650,0.0745), (0.1641,0.1857,0.2469), (0.0977,0.1108,0.1433)]; the weights of F1 to F7 in economical bottom line are

w_F1 ∼ w_F7=(0.1923, 0.0749,0.1140, 0.2458, 0.0666, 0.1923, 0.1140)

Similarly, the weights of F1 to F7 in environmental and social bottom line are

w_E1 ∼ w_E7=(0.0920, 0.2167, 0.0759, 0.2077, 0.1234, 0.0711, 0.2133)

w_S1 ∼ w_S7=(0.2021, 0.1821, 0.2433, 0.0928, 0.1380, 0.0821, 0.0596)

The weights of the three categories (economic, social and environmental performance) are

w_F,S,E=(0.4235, 0.3261, 0.2504)

Weighting the above matrix, the final weight of each indicator determined by the fuzzy BWM is

w_BWM=(0.0814, 0.0317, 0.0483, 0.1041, 0.0282, 0.0815, 0.0483, 0.0659, 0.0594, 0.0793, 0.0303, 0.0450, 0.0268, 0.0194, 0.0230, 0.0542, 0.0190, 0.0520, 0.0309, 0.0179, 0.0534).

Table 7 shows the implement of the fuzzy-BWM and the results of weighting.

4.3 Determination of the indicators’ objective weight

According to Equations (7) and (8), the original data is processed dimensionless to obtain a standardized decision matrix and are translated by 0.0001. Then, it is normalized to obtain a normalized decision matrix according to Equation (9). Some calculation results are shown in Table 8.

Table 8
Normalised decision matrix (partial)

Indicator Supplier A Supplier B Supplier C Supplier D

F1 0.2353 0.6176 0.0001 0.1471

F2 0.1786 0.0001 0.5714 0.2500

F3 0.1579 0.0000 0.4736 0.3684

Indicator	Supplier A	Supplier B	Supplier C	Supplier D
F1	0.2353	0.6176	0.0001	0.1471
F2	0.1786	0.0001	0.5714	0.2500
F3	0.1579	0.0000	0.4736	0.3684

The entropy and weight of each indicator are calculated through Equations (10) and (11) , respectively. The results are

E=(0.6641, 0.7030, 0.7313, 0.7895, 0.4601, 0.6671, 0.7266, 0.7790, 0.7613, 0.5007, 0.6263, 0.5949, 0.7927, 0.7436, 0.6588, 0.6281, 0.7648, 0.6466, 0.5007, 0.6706, 0.7181)

w_entropy =(0.0489, 0.0432, 0.0391, 0.0306, 0.0786, 0.0484, 0.0398, 0.0322, 0.0347, 0.0727, 0.0544, 0.0589, 0.0302, 0.0373, 0.0497, 0.0541, 0.0342, 0.0514, 0.0727, 0.0479, 0.0410)

4.4 Combination weight

With α=0.5, β=0.5, based on Equation (12), the combination weights are

W=(0.0652, 0.0375, 0.0437, 0.0674, 0.0534, 0.0649, 0.0440, 0.0491, 0.0470, 0.0760, 0.0423, 0.0519, 0.0285, 0.0284, 0.0364, 0.0542, 0.0266, 0.0517, 0.0518, 0.0329, 0.0472)

4.5 Evaluation results

According to Equation (13), the weighted standardized decision matrix is obtained as in Table 9.

Table 9
Weighted standardised decision matrix (partial)

Indicator Supplier A Supplier B Supplier C Supplier D

F1 0.0248 0.0652 0.0000 0.0155

F2 0.0117 0.0000 0.0375 0.0164

F3 0.0146 0.0000 0.0437 0.0340

Indicator	Supplier A	Supplier B	Supplier C	Supplier D
F1	0.0248	0.0652	0.0000	0.0155
F2	0.0117	0.0000	0.0375	0.0164
F3	0.0146	0.0000	0.0437	0.0340

Based on Equations (14)and (15), the positive-ideal solution F⁺ and negative-ideal solution F^- are easily obtained. Through Equation (16), the Euclidean distance $T_{i}^{+}$ from each decision plan to the positive-ideal solution F⁺ is

$T_{i}^{+}$ =(0.1704, 0.1539, 0.1020, 0.1628)

Similarly, $T_{i}^{-}$ =(0.1100, 0.1409, 0.2000, 0.1153)

According to Equations (18) and (19), the grey relational matrices ( $R_{i}^{+}$ and $R_{i}^{-}$ ) are

$R_{i}^{+}$ =(0.6024, 0.6605, 0.8942, 0.6065)

$R_{i}^{-}$ =(0.7621, 0.7235, 0.5519, 0.7488)

According to Equation (22), the dimensionless processing is performed on T⁺, T^-, R⁺, and R^- respectively, and the results are

T⁺=(1.0000, 0.9032, 0.5986 0.9554)

T^-=(0.5500, 0.7045, 1.0000, 0.5765)

R⁺=(0.6736, 0.7386, 1.0000, 0.6782)

R^-=(1.0000, 0.9494, 0.7242, 0.9826)

By Equations (23)–(25) and ξ₁ = 0.5, ξ₂ = 0.5, $V_{i}^{+}$ , $V_{i}^{-}$ , and S_i are separately

$V_{i}^{+}$ =(0.6118, 0.7216, 1.0000, 0.6274)

$V_{i}^{-}$ =(1.0000, 0.9263, 0.6614, 0.9690)

S_i=(0.3796, 0.4379, 0.6019, 0.3930)

The larger the value of S_i, the closer the results to the optimal ideal solution. Thus, we conduct supplier selection with order C > B>D>A. It is obvious that Supplier C is the most sustainable supplier, which is consistent with the actual situation. In addition, the selection order of the remaining three is Supplier B, Supplier D and Supplier A.

We now alter the values of the position preference parameter ξ₁ and shape preference parameter ξ₂ to investigate the results’ robustness in the grey relational TOPSIS method based on the combination weights. We select different values of ξ₁ and ξ₂ in increments of 0.1, where 0≤ξ₁≤1, 0≤ξ₂≤1 and ξ₁+ξ₂ = 1. The results are represented in Fig. 2.

From Fig. 2, regardless of how ξ₁ and ξ₂ vary, Supplier C is the best. The ranking is relatively stable.

Fig. 2

The results when ξ₁ and ξ₂ changes.

5 Discussion and conclusion

With the increasing emphasis on sustainability at home and abroad, SSCM has become a trending topic. The sustainable supplier evaluation and selection is a significant link in a SSC. This study formulated a scientific and reasonable supplier evaluation framework based on the theory of sustainable supply chain, TBL, and balanced scorecard. The fuzzy BWM and entropy methods are adopted to determine the subjective and objective weights of the evaluation indicators, respectively. The combination weights of the subjective and objective weights are integrated to decrease the shortcomings of a single evaluation method. Finally, a case study verifies the effectiveness and stability of the proposed evaluation index system and evaluation method.

Compared with the existing methods, such as FAHP-TOPSIS and BWM-TOPSIS, the proposed approach has more advantages. BWM-TOPSIS cannot adapt to the fuzzy situation, while FAHP-TOPSIS needs more pair-wise comparison. The proposed approach in this paper has more advantages, which determines the weights of evaluation indicators more objective, and also can demonstrate the closeness between the decision and the ideal solution from the shape and distance.

In this case, even though Supplier C is considered to be the best sustainable supplier from the result and is recommended for contracting, some sustainability indicators had low ratings for supplier C. For the implementation of this selection recommendation, Company H may require specific post-selection negotiations with Supplier C for possible improvements in these indicators with low ratings by setting the other suppliers as benchmarks. For example, Supplier B has the highest rated performance for the indicator ‘price of material’ (F1). As a result, Company H can consider to negotiate with Supplier C to improve this indicator. Given the possibilities of interactions and trade-offs, care must be taken not to compromise the overall performance of supplier C. Thus, a supplier development process may be put into place that may help improve supplier C in a balanced way. Besides, based on the evaluation team’ s opinion, the most important indicator in economic bottom line is ‘flexibility of supply chain’ (F4), in social bottom line is ‘number of health and safety incidents’ (S3) and in environmental bottom line is ‘clean energy efficiency’ (E2). Suppliers should focus on improving these performance indicators.

The results of this study provide the following suggestions for the future development of sustainable suppliers. First, sustainability is the harmonization of economic, social, and environmental performances. Suppliers should make overall consideration in future development. Second, sustainable suppliers should not only focus on the present but also consider the future. A sustainable organization should formulate and implement actively sustainable strategies that meet the relevant requirements. Finally, sustainable suppliers must actively cooperate with upstream and downstream enterprises to jointly promote the sustainability of the entire supply chain.

Every study has certain limitations, and this study is no exception. One limitation is that we do not consider the influence of different cognitive biases of experts on the evaluation results when we determine the subjective weights by the fuzzy-BWM method. Another limitation is that the interrelationship among the indicators is not considered for the complexity. There is a certain synergetic relationship among them. Therefore, it is suggested to study the interrelationship in future research.

Funding

This work was supported by the National Natural Science Foundation of China with Grant Nos. 71971064 and 71801045, the Social Science Foundation of Fujian Province with Grant No. FJ2022B071, and the National Natural Science Foundation of Fujian Province with Grant No. 2020J01460.

Footnotes

Acknowledgments

The authors are very thankful for constructive comments and suggestions from anonymous reviewers and editors.

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Supplier evaluation and selection in a sustainable supply chain based on fuzzy-BWM,entropy method and grey relational TOPSIS

Abstract

Keywords

1 Introduction

2 Literature review

2.1 The supplier evaluation framework

2.2 The supplier evaluation method

4.1 Introduction of company H

Table 5 Best and worst indicators in the economic bottom line determined by experts Experts Most important Least important 1 F4 F5 2 F1 F7 3 F6 F5 4 F4 F5 5 F6 F5 6 F3 F5 7 F1 F7 8 F4 F5 9 F6 F5 10 F4 F7

Table 8 Normalised decision matrix (partial) Indicator Supplier A Supplier B Supplier C Supplier D F1 0.2353 0.6176 0.0001 0.1471 F2 0.1786 0.0001 0.5714 0.2500 F3 0.1579 0.0000 0.4736 0.3684

4.5 Evaluation results

Table 9 Weighted standardised decision matrix (partial) Indicator Supplier A Supplier B Supplier C Supplier D F1 0.0248 0.0652 0.0000 0.0155 F2 0.0117 0.0000 0.0375 0.0164 F3 0.0146 0.0000 0.0437 0.0340

Funding

Footnotes

Acknowledgments

References

Table 5
Best and worst indicators in the economic bottom line determined by experts

Experts Most important Least important

1 F4 F5

2 F1 F7

3 F6 F5

4 F4 F5

5 F6 F5

6 F3 F5

7 F1 F7

8 F4 F5

9 F6 F5

10 F4 F7

Table 8
Normalised decision matrix (partial)

Indicator Supplier A Supplier B Supplier C Supplier D

F1 0.2353 0.6176 0.0001 0.1471

F2 0.1786 0.0001 0.5714 0.2500

F3 0.1579 0.0000 0.4736 0.3684

Table 9
Weighted standardised decision matrix (partial)

Indicator Supplier A Supplier B Supplier C Supplier D

F1 0.0248 0.0652 0.0000 0.0155

F2 0.0117 0.0000 0.0375 0.0164

F3 0.0146 0.0000 0.0437 0.0340