# Low-Carbon Supplier Selection Using Fuzzy AHP and Goal Programming Approach

Authors: Feipeng Huang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

It has become the leading world trend that all countries join hands to respond to climate change and promote green and low-carbon development. In this background, more and more enterprises seek to control carbon emissions from the source and reduce indirect carbon emissions, consequently paying increased importance to the carbon footprint of their suppliers. Most of the earlier studies concerning supplier selection mainly focus on traditional factors, such as quality, service, and lead time, but minimal importance to the carbon emission of supplies. This study fills this gap by incorporating carbon emission criteria into supplier selection and presenting a method of combining fuzzy analytic hierarchy process (AHP) and fuzzy goal programming (GP) to address the problem of supplier selection and order quota allocation. Firstly, we use fuzzy AHP to evaluate the relevance and importance of supplier selection criteria according to the experts’ opinions. Second, each objective is given weights according to the fuzzy AHP results, and then the fuzzy GP method is used for supplier selection and order quota allocation. Finally, we implement an example study with a data set from a realistic situation, and the results confirm the effectiveness of the proposed method in an uncertain environment.


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## Body

## 1. Introduction

Global warming caused by greenhouse gas emissions has attracted increasing attention in the past decade. Accordingly, consumers’ low-carbon awareness in choosing products is also gradually enhanced. According to Accenture research, with the gradual increase in consumers’ care for low-carbon products, more and more enterprises are considering producing low-carbon products to meet the needs of such low-carbon preference consumers. Enterprises also pay increasing attention to their carbon emissions. When studying the carbon emissions of enterprises, Shaw and other scholars [1] contend that we should focus on the direct carbon emissions of enterprises and consider the overall carbon emissions of the supply chain of enterprises from the perspective of the life cycle. According to the survey results of consulting company on the carbon emission of a typical enterprise’s supply chain, only 19% of the greenhouse gas emission comes from the direct operation activities of the enterprise. In comparison up to 81% of the greenhouse gas emission is the indirect emission generated by other members of the supply chain, such as the carbon emission of suppliers and the indirect emission of enterprises purchasing electricity. In this context, the selection of appropriate suppliers plays a vital role in reducing the indirect carbon emissions of enterprises and the overall carbon emissions of their supply chain.Although a large number of studies exist on the topic of supplier selection, the study on the green supplier selection in low-carbon supply chain is still insufficient [2–4]. Most of these studies mainly focus on economic supplier selection and they do not consider environmental and social attributes. In addition, low-carbon supply chain management has been considered as an integration and realization of an enterprise’s economic, environmental, and social objectives to improve its performance. When the enterprise selects its green supplier, it needs to pay special attention to proactively build low-carbon principles into its supply chain management, which is one of the most important factors for the success of low-carbon supply chain. However, many studies have ignored this point. Moreover, although extensive multicriteria decision-making (MCDM) approaches have been proposed for supplier selection, there are still some problems that need to be solved. The first is that they need to nondimensionalize the criteria, which will bring about a lot of calculating work, especially when there are many qualitative criteria. The second problem is how to determine the weight of each criterion. Different understanding of green supplier selection determines which approach should be used to compute the weight of each criterion.To fill the gaps in the existing studies of low-carbon supply chain management, this paper proposes determining the green attributes for the selection of supplier and then developing a model for assessing and ordering of low-carbon suppliers based on determined attributes. Firstly, the mission of low-carbon emission of products is accommodated in the process of selecting multiple suppliers for a particular enterprise. Specifically, in addition to product cost, product quality, and service quality, the carbon emission of products is added to the selection criteria for suppliers. In this way, we establish 4 main criteria and 11 subcriteria for green supplier selection. Then, a method based on the combination of fuzzy AHP and fuzzy goal programming (GP) is used to address the problems of supplier selection and order allocation, so as to provide decision support for enterprises to control their indirect carbon emission and achieve the mission of low-carbon supply chain management. The strength of the proposed method is that, despite the vagueness of experts’ opinions in the selection process, the model is easy to apply. Moreover, with the proposed method, enterprises can help their suppliers to improve sustainability for better management of low-carbon supply chain operations. A real case in the steel industry is also being studied to verify the applicability of the proposed criteria and methods for green supplier selection in low-carbon supply chain.The main contribution of this paper is developing a fuzzy MCDM approach for green supplier selection in low-carbon supply chain. The paper establishes the main criteria and subcriteria for green supplier selection after considering enterprise’ requirements in low-carbon supply chain management, which can help enterprise to identify the potential areas where green suppliers need to improve. The proposed method provides a mechanism of integrating the economic, social, and environmental criteria to fully reflect the requirements of low-carbon supply chain, which helps to avoid the potential risk of selecting the wrong suppliers. The proposed method has been successfully implemented in a case company to select its best green supplier and analyze its most appropriate alternative green supplier. This research’s results will improve the managers’ view on the nature of green supplier selection criteria. Besides, the proposed method can be widely used as a structural model for green supplier selection.The remainder of the paper is organized as follows: Section2 reviews the existing literature. Section 3 presents the methodology of supplier selection based on the fuzzy AHP-GP technique. Section 4 conducts an example analysis using a real case, while Section 5 concludes the study.


## 2. Literature Review

In this section, we classify the literature related to our work into two categories. The first includes the studies with a focus on the criteria for green supplier selection. The second category explores the literature on the method for green supplier selection. Here we review some recent representative works in the literature as follows.


### 2.1. Criteria for Green Supplier Selection

Many authors have stressed the importance of selecting suitable (qualitative and quantitative) criteria in the green supplier selection process. The traditional criteria for supplier selection have solely considered economic aspects for many years [5–7]. While in low-carbon supply chain management, enterprises must add the environmental, ecological, and social factors to the traditional supplier selection criteria such as quality, cost, delivery, and service to remain in the low-carbon supply chain [8].Lee et al. [9] proposed quality, technology capability, pollution control, green products, and green competencies for green supplier selection in the high-tech industry. Tseng [10] proposed 16 main green criteria, such as green technology capabilities, green purchasing capabilities, green design, and green certifications. Hsu et al. [11] established 13 criteria of carbon management with three dimensions (planning, implementation, and management). Gurel et al. [12] proposed a criteria list for green supplier selction for textile industry in a hierarchic structure which is useful to integrate multi criteria decision analysis. Awasthi et al. [13] proposed a criteria list for evaluating the environmental performance of suppliers. These criteria include the usage of environment-friendly technology, environment-friendly materials, green market share, partnership with green organizations, management commitment to green practices, adherence to environmental policies, involvement in green projects, staff training, lean process planning, design for the environment, environmental certification, and pollution control initiatives. Masoud et al. [14] integrated both classic and green key performance indicators for the aim of supplier selection and ranked these measures using expert’s opinions for electrical industries. Pang et al. [15] established 4 main criteria and 22 subcriteria for green supplier selection and propose a fuzz-grey multicriteria decision-making approach for green supplier selection in low-carbon supply chain.Most of the above literature focuses on economic supplier selection, while neglecting environmental and social aspects. It is inappropriate in the era of low-carbon economy, especially in low-carbon supply chain condition. The criteria determine what we should consider for green supplier selection, so they should adhere to scientific, dynamic, comprehensive, and oriented principles. That is to say, the criteria should include economic, environmental, and social aspects, which can meet the enterprise’s requirements for green supplier selection in low-carbon supply chain.


### 2.2. Methods for Green Supplier Selection

Extensive MCDM methods have been proposed for supplier selection, like the analytic hierarchy process (AHP), analytic network process (ANP), data envelopment analysis (DEA), fuzzy set theory, genetic algorithm (GA), mathematical programming, technique order preference by similarity to ideal solution (TOPSIS), and so forth and their hybrids [16–19].Amindoust et al. [20] proposed a new ranking method on the basis of fuzzy inference system (FIS) for the sustainable supplier selection problem. Darabi and Heydari [21] proposed an interval-valued hesitant fuzzy ranking method to rank the green suppliers candidates under conflicted criteria. Kuo et al. [22] developed a green supplier selection model which integrates artificial neural network (ANN), DEA, and ANP. Kannan et al. [3] presented an integrated approach of fuzzy multiattribute utility theory and multi-objective programming for rating and selecting the best green suppliers according to economic and environmental criteria and then allocating the optimum order quantities among them. Karsak and Dursun [23] proposed a fuzzy multicriteria group decision-making approach that makes use of quality function deployment and fusion of fuzzy information, and 2-tuple linguistic representation model was developed for supplier selection. Yamada et al. [24] proposed a low-carbon and economic supplier selection with an estimation method of the carbon emissions and conducted a material-based analysis for both the procurement costs minimization and carbon emissions reduction. Yazdani et al. [25] proposed an integrated approach for green supplier selection by considering various environmental performance requirements and criteria. Hamdan and Cheaitou [4] integrated three methods, fuzzy set theory, TOPSIS, and AHP, to solve a multiperiod green supplier selection and order allocation problem. Banaeian et al. [26] compared the application of three fuzzy MCDM methods, fuzzy set theory with TOPSIS, VIKOR, and GRA in an actual case study. Zavadskas et al. [27] utilize the fuzzy AHP method to choose the most suitable supplier for the purchase of materials necessary for the production of preinsulated pipes.The existing literature provides a valuable understanding of effectively selecting suppliers for enterprises [28]. However, when considering the supplier selection criteria, products’ carbon emissions are not given enough attention or even disregarded. Moreover, a minimal part of works considers minimizing the product’s carbon footprint as the objective in supplier selection. Due to the fuzziness of judgment, fuzzy set theory is always used in supplier selection, while fuzzy set theory needs to nondimensionalize the criteria, which will bring much calculating work. In particular, when there are many qualitative criteria, it is difficult to compare or quantify them. In this regard, this paper proposes the fuzzy AHP approach combined with fuzzy goal programming (GP) to address the problem of low-carbon supplier selection and order quota allocation.


## 2.1. Criteria for Green Supplier Selection

Many authors have stressed the importance of selecting suitable (qualitative and quantitative) criteria in the green supplier selection process. The traditional criteria for supplier selection have solely considered economic aspects for many years [5–7]. While in low-carbon supply chain management, enterprises must add the environmental, ecological, and social factors to the traditional supplier selection criteria such as quality, cost, delivery, and service to remain in the low-carbon supply chain [8].Lee et al. [9] proposed quality, technology capability, pollution control, green products, and green competencies for green supplier selection in the high-tech industry. Tseng [10] proposed 16 main green criteria, such as green technology capabilities, green purchasing capabilities, green design, and green certifications. Hsu et al. [11] established 13 criteria of carbon management with three dimensions (planning, implementation, and management). Gurel et al. [12] proposed a criteria list for green supplier selction for textile industry in a hierarchic structure which is useful to integrate multi criteria decision analysis. Awasthi et al. [13] proposed a criteria list for evaluating the environmental performance of suppliers. These criteria include the usage of environment-friendly technology, environment-friendly materials, green market share, partnership with green organizations, management commitment to green practices, adherence to environmental policies, involvement in green projects, staff training, lean process planning, design for the environment, environmental certification, and pollution control initiatives. Masoud et al. [14] integrated both classic and green key performance indicators for the aim of supplier selection and ranked these measures using expert’s opinions for electrical industries. Pang et al. [15] established 4 main criteria and 22 subcriteria for green supplier selection and propose a fuzz-grey multicriteria decision-making approach for green supplier selection in low-carbon supply chain.Most of the above literature focuses on economic supplier selection, while neglecting environmental and social aspects. It is inappropriate in the era of low-carbon economy, especially in low-carbon supply chain condition. The criteria determine what we should consider for green supplier selection, so they should adhere to scientific, dynamic, comprehensive, and oriented principles. That is to say, the criteria should include economic, environmental, and social aspects, which can meet the enterprise’s requirements for green supplier selection in low-carbon supply chain.


## 2.2. Methods for Green Supplier Selection

Extensive MCDM methods have been proposed for supplier selection, like the analytic hierarchy process (AHP), analytic network process (ANP), data envelopment analysis (DEA), fuzzy set theory, genetic algorithm (GA), mathematical programming, technique order preference by similarity to ideal solution (TOPSIS), and so forth and their hybrids [16–19].Amindoust et al. [20] proposed a new ranking method on the basis of fuzzy inference system (FIS) for the sustainable supplier selection problem. Darabi and Heydari [21] proposed an interval-valued hesitant fuzzy ranking method to rank the green suppliers candidates under conflicted criteria. Kuo et al. [22] developed a green supplier selection model which integrates artificial neural network (ANN), DEA, and ANP. Kannan et al. [3] presented an integrated approach of fuzzy multiattribute utility theory and multi-objective programming for rating and selecting the best green suppliers according to economic and environmental criteria and then allocating the optimum order quantities among them. Karsak and Dursun [23] proposed a fuzzy multicriteria group decision-making approach that makes use of quality function deployment and fusion of fuzzy information, and 2-tuple linguistic representation model was developed for supplier selection. Yamada et al. [24] proposed a low-carbon and economic supplier selection with an estimation method of the carbon emissions and conducted a material-based analysis for both the procurement costs minimization and carbon emissions reduction. Yazdani et al. [25] proposed an integrated approach for green supplier selection by considering various environmental performance requirements and criteria. Hamdan and Cheaitou [4] integrated three methods, fuzzy set theory, TOPSIS, and AHP, to solve a multiperiod green supplier selection and order allocation problem. Banaeian et al. [26] compared the application of three fuzzy MCDM methods, fuzzy set theory with TOPSIS, VIKOR, and GRA in an actual case study. Zavadskas et al. [27] utilize the fuzzy AHP method to choose the most suitable supplier for the purchase of materials necessary for the production of preinsulated pipes.The existing literature provides a valuable understanding of effectively selecting suppliers for enterprises [28]. However, when considering the supplier selection criteria, products’ carbon emissions are not given enough attention or even disregarded. Moreover, a minimal part of works considers minimizing the product’s carbon footprint as the objective in supplier selection. Due to the fuzziness of judgment, fuzzy set theory is always used in supplier selection, while fuzzy set theory needs to nondimensionalize the criteria, which will bring much calculating work. In particular, when there are many qualitative criteria, it is difficult to compare or quantify them. In this regard, this paper proposes the fuzzy AHP approach combined with fuzzy goal programming (GP) to address the problem of low-carbon supplier selection and order quota allocation.


## 3. Supplier Selection Model

We first use fuzzy AHP approach to calculate the weight of different selection criteria of suppliers (i.e., relative importance) based on the footprint of suppliers’ products. Then these values of weight are used as the coefficients of each objective function in fuzzy GP. Finally, the enterprise makes a decision on the order quota from each supplier based on the evaluation of the outcomes of the model. Figure1 illustrates the implementation steps of a fuzzy AHP-GP procedure.Figure 1
The AHP-GP procedure.We first use the fuzzy AHP method to calculate the coefficients of the objective function in the fuzzy GP, which includes the following four steps:Step 1: Determine the criteria for supplier selection and establish the corresponding indicators system.Step 2: Pairwise comparing the indicators selected by suppliers and determine the relative importance of indicators.Step 3: Calculate the weight of each selection criteriaStep 4: Derive the coefficients of each objective function (cost, quality, service, and carbon emission) in fuzzy GP.Secondly, we use the fuzzy GP approach to address the problems of supplier selection and order quota allocation, which includes the following four steps:Step 1: The indicator weights yield from the fuzzy AHP method are substituted into the objective function of fuzzy GP as the coefficients, and further establish the objective functions including the minimizing cost, maximizing quality, maximizing service quality, and minimizing product carbon emission.Step 2: Clarify the constraints for decision-makers to select suppliers, including the supplier’s supply capacity and the enterprise’s requirements for supplier products.Step 3: Solve the GP model.Step 4: The decision-maker selects low-carbon suppliers according to the outcomes of step 3, and determines the order quota of each supplier.


### 3.1. AHP Hierarchy of Low Carbon Supplier Selection

This study comprehensively concludes the related work of [29–31], and follows the principles of stability, comparability, flexible operation, and comprehensiveness to determine the supplier evaluation indicator system. In addition to the common selection criteria such as product cost, product quality, and service quality, we add the criteria of carbon emission of supplier products to establish the eventual AHP hierarchy of low-carbon supplier selection as illustrated in Figure 2.Figure 2
AHP hierarchy of low-carbon supplier selection.The overall objective of this AHP level is to select low-carbon suppliers; layer 2 includes four criteria: product cost (C1), product quality (C2), service quality (C3), and product carbon emission (C4); layer 3 specifically includes 11 different indicators, namely A1∼A11. The specific description of the indicators is present in Table 1.Table 1
Indicators selection and interpretation.
Selection criteriaIndicators selectionIndicator interpretationsProduct costProduct costIncludes production cost, maintenance cost and warranty costTransportation costSum of fixed transportation cost and variable transportation cost per unit productProduct qualityReject rateProportion of products returned due to quality problemsQuality assuranceThe supplier shall evaluate the product quality to ensure the qualityRemedies to quality problemsSupplier’s ability to solve product quality problemsService qualityLead timeTime interval between ordering and supplier deliverTimely deliveryThe supplier shall deliver the goods according to the specified timeTechnical supportSuppliers improve their technical level to meet the needs of enterprisesResponse efficiencySuppliers respond quickly to changes in the enterprise (product demand, ordering frequency, etc.)Product carbon emissionCarbon footprint verificationCarbon label of supplier’s productsTransport carbon emissionCarbon emissions generated during the transportation of products from suppliers to enterprises


### 3.2. Fuzzy GP

#### 3.2.1. Multi Objective Linear Programming

As discussed previously, the objectives of the model established in this study are as follows: (1) minimize procurement costs; (2) best product quality; (3) best service of supplier; (4) lowest product carbon emission. The model is further constructed after the setting of model assumptions, symbols, parameters, and decision variables.Model assumptions(i)
The enterprise only purchases the same product from each supplier(ii)
The discount brought by the purchase quota is not considered(iii)
There is no shortage of products provided by the supplierSymbolsi: Number of suppliersj: Target number of linear programming, j=1,2,…,JModel parametersD: The overall demand of the enterprise for products in a fixed planning cyclen: Amount of supplierspi: The price at which an enterprise purchases products from supplier iqi: Supplier i’s product qualitysi Supplier i’s service qualitygi: Carbon emission of supplier i’s productsUi: Supplier i’s supply capacityDecision variables:xi: Quota of products ordered by the enterprise from supplier i(1) Linear Programming Model. We establish the following multi-objective linear programming model by combining the aim of our study and the related work of [32] as follows:(1)MinimizeZ1=∑i=1npixi,(2)MinimizeZ2=∑i=1nqixi,(3)MinimizeZ3=∑i=1nsixi,(4)MinimizeZ4=∑i=1ngixi,(5)s.t.∑i=qnxi=D,(6)xi≤Ui,i=1,2,⋯,n,(7)xi≥0i,xiis an integer,i=1,2,⋯,n.Equations (1)–(4) are the objective functions with respect to minimize procurement costs, best product quality, best service of supplier, and lowest product carbon emission, respectively. Constraint condition (5) ensures that the products purchased by the enterprise meet its own needs; constraint condition (6) means that the products purchased by the enterprise cannot exceed its supply capacity, and constraint (7) means that the decision variable is not less than 0 and is an integer.


#### 3.2.2. Fuzzy Multi Objective Programming

In the process of selecting supplier selection, there are often some uncertainties in the relevant information of suppliers. When describing suppliers, the evaluation of their selection criteria may not be very accurate, such as “almost no quality problems” and “the supplier’s supply capacity is between 3000 and 3500.”To address the fuzziness of key information and make decisions in an uncertain environment, Bellman and Zadeh [33] proposed a fuzzy programming model, and Zimmermann [34] then used the model to deal with the problems of multi-objective programming. Follow these studies, the multi-objective linear programming in 3.2.1 can be transformed into the following form by considering product price, product quality, service quality, and product carbon emission as fuzzy information.(8)∑i=1npixi≺∼Z˜1,(9)∑i=1nqixi≻∼Z˜2,(10)∑i=1nsixi≻∼Z˜3,(11)∑i=1ngixi≻∼Z˜4,(12)s.t.∑i=qnxi=D,(13)xi≤Ui,i=1,2,⋯,n,14)xi≥0i,xiis an integer,i=1,2,⋯,n.In which∼ denotes a fuzzy environment. Z˜1, Z˜2, Z˜3, and Z˜4 are the enterprise’s expectations of product price, product quality, service quality, and product carbon emission. Sign “≺∼” means “basically less than or equal to,” while sign “≺∼≻∼” means “basically more than or equal to.” In order to find the membership function of fuzzy set, this study uses the reduced half trapezoid method to find the minimum value of the objective function, and uses the raised half trapezoid method to find the maximum value of the objective function. The membership functions of the minimization objective function Zm (m = 1, 4) and the maximization objective Zl (l = 2, 3) can be respectively expressed as:(15)μZmx=1,Zm≤Zm−,fμZm=Zm+−ZmxZm+−Zm−,Zm−≤Zmx≤Zm+m=1,40,Zm≥Zl−,,(16)μZlx=1,Zl≤Zl−,fμZl=Zmx−Zl−Zl+−Zl−,Zl−≤Zlx≤Zl+l=2,3,0,Zl≥Zl+.LetZj+ (k=1,2,3,4) and Zj− (k=1,2,3,4) denote the upper and lower bound of each objective function, respectively. According to [35], Zj+ is the maximum value of the objective function corresponding to the optimal solution of the following problem:(17)maxZjxs.t.∑i=1nxi=Dxi≤Ui,i=1,2,⋯,nxi≥0,xiis an integer,i=1,2,⋯,n.Zj− is the minimum value of the objective function corresponding to the optimal solution of the following problem:(18)minZjxs.t.∑i=1nxi=Dxi≤Ui,i=1,2,⋯,nxi≥0,xiis an integer,i=1,2,⋯,n.Refer to reference [36], we can transform the above problem into the deterministic form as follows:(19)maxλs.t.λ≤fμZjx,j=1,2,3,4∑i=1nxi=Dxi≤Ui,i=1,2,⋯,nxi≥0,i=1,2,⋯,n,λ∈0,1.In which different targets is allocated the same weight. In practice, different decision-makers might have different preferences for goals, so the above goals should not be treated equally. Consequently, the weighted maximum minimum method is presented to address this problem. Following this method, the different objectives in the model are endowed with different weights according to the results obtained in Section3.2.1. Then we use the weight maximum minimum model to find the optimal solution of the model.(20)maxλs.t.wjλ≤fμZjx,j=1,2,3,4,∑i=1nxi=D,∑i=14wj=1,xi≤Ui,i=1,2,…,n,xi≥0,i=1,2,…,n,λ∈0,1.


## 3.1. AHP Hierarchy of Low Carbon Supplier Selection

This study comprehensively concludes the related work of [29–31], and follows the principles of stability, comparability, flexible operation, and comprehensiveness to determine the supplier evaluation indicator system. In addition to the common selection criteria such as product cost, product quality, and service quality, we add the criteria of carbon emission of supplier products to establish the eventual AHP hierarchy of low-carbon supplier selection as illustrated in Figure 2.Figure 2
AHP hierarchy of low-carbon supplier selection.The overall objective of this AHP level is to select low-carbon suppliers; layer 2 includes four criteria: product cost (C1), product quality (C2), service quality (C3), and product carbon emission (C4); layer 3 specifically includes 11 different indicators, namely A1∼A11. The specific description of the indicators is present in Table 1.Table 1
Indicators selection and interpretation.
Selection criteriaIndicators selectionIndicator interpretationsProduct costProduct costIncludes production cost, maintenance cost and warranty costTransportation costSum of fixed transportation cost and variable transportation cost per unit productProduct qualityReject rateProportion of products returned due to quality problemsQuality assuranceThe supplier shall evaluate the product quality to ensure the qualityRemedies to quality problemsSupplier’s ability to solve product quality problemsService qualityLead timeTime interval between ordering and supplier deliverTimely deliveryThe supplier shall deliver the goods according to the specified timeTechnical supportSuppliers improve their technical level to meet the needs of enterprisesResponse efficiencySuppliers respond quickly to changes in the enterprise (product demand, ordering frequency, etc.)Product carbon emissionCarbon footprint verificationCarbon label of supplier’s productsTransport carbon emissionCarbon emissions generated during the transportation of products from suppliers to enterprises


## 3.2. Fuzzy GP

### 3.2.1. Multi Objective Linear Programming

As discussed previously, the objectives of the model established in this study are as follows: (1) minimize procurement costs; (2) best product quality; (3) best service of supplier; (4) lowest product carbon emission. The model is further constructed after the setting of model assumptions, symbols, parameters, and decision variables.Model assumptions(i)
The enterprise only purchases the same product from each supplier(ii)
The discount brought by the purchase quota is not considered(iii)
There is no shortage of products provided by the supplierSymbolsi: Number of suppliersj: Target number of linear programming, j=1,2,…,JModel parametersD: The overall demand of the enterprise for products in a fixed planning cyclen: Amount of supplierspi: The price at which an enterprise purchases products from supplier iqi: Supplier i’s product qualitysi Supplier i’s service qualitygi: Carbon emission of supplier i’s productsUi: Supplier i’s supply capacityDecision variables:xi: Quota of products ordered by the enterprise from supplier i(1) Linear Programming Model. We establish the following multi-objective linear programming model by combining the aim of our study and the related work of [32] as follows:(1)MinimizeZ1=∑i=1npixi,(2)MinimizeZ2=∑i=1nqixi,(3)MinimizeZ3=∑i=1nsixi,(4)MinimizeZ4=∑i=1ngixi,(5)s.t.∑i=qnxi=D,(6)xi≤Ui,i=1,2,⋯,n,(7)xi≥0i,xiis an integer,i=1,2,⋯,n.Equations (1)–(4) are the objective functions with respect to minimize procurement costs, best product quality, best service of supplier, and lowest product carbon emission, respectively. Constraint condition (5) ensures that the products purchased by the enterprise meet its own needs; constraint condition (6) means that the products purchased by the enterprise cannot exceed its supply capacity, and constraint (7) means that the decision variable is not less than 0 and is an integer.


### 3.2.2. Fuzzy Multi Objective Programming

In the process of selecting supplier selection, there are often some uncertainties in the relevant information of suppliers. When describing suppliers, the evaluation of their selection criteria may not be very accurate, such as “almost no quality problems” and “the supplier’s supply capacity is between 3000 and 3500.”To address the fuzziness of key information and make decisions in an uncertain environment, Bellman and Zadeh [33] proposed a fuzzy programming model, and Zimmermann [34] then used the model to deal with the problems of multi-objective programming. Follow these studies, the multi-objective linear programming in 3.2.1 can be transformed into the following form by considering product price, product quality, service quality, and product carbon emission as fuzzy information.(8)∑i=1npixi≺∼Z˜1,(9)∑i=1nqixi≻∼Z˜2,(10)∑i=1nsixi≻∼Z˜3,(11)∑i=1ngixi≻∼Z˜4,(12)s.t.∑i=qnxi=D,(13)xi≤Ui,i=1,2,⋯,n,14)xi≥0i,xiis an integer,i=1,2,⋯,n.In which∼ denotes a fuzzy environment. Z˜1, Z˜2, Z˜3, and Z˜4 are the enterprise’s expectations of product price, product quality, service quality, and product carbon emission. Sign “≺∼” means “basically less than or equal to,” while sign “≺∼≻∼” means “basically more than or equal to.” In order to find the membership function of fuzzy set, this study uses the reduced half trapezoid method to find the minimum value of the objective function, and uses the raised half trapezoid method to find the maximum value of the objective function. The membership functions of the minimization objective function Zm (m = 1, 4) and the maximization objective Zl (l = 2, 3) can be respectively expressed as:(15)μZmx=1,Zm≤Zm−,fμZm=Zm+−ZmxZm+−Zm−,Zm−≤Zmx≤Zm+m=1,40,Zm≥Zl−,,(16)μZlx=1,Zl≤Zl−,fμZl=Zmx−Zl−Zl+−Zl−,Zl−≤Zlx≤Zl+l=2,3,0,Zl≥Zl+.LetZj+ (k=1,2,3,4) and Zj− (k=1,2,3,4) denote the upper and lower bound of each objective function, respectively. According to [35], Zj+ is the maximum value of the objective function corresponding to the optimal solution of the following problem:(17)maxZjxs.t.∑i=1nxi=Dxi≤Ui,i=1,2,⋯,nxi≥0,xiis an integer,i=1,2,⋯,n.Zj− is the minimum value of the objective function corresponding to the optimal solution of the following problem:(18)minZjxs.t.∑i=1nxi=Dxi≤Ui,i=1,2,⋯,nxi≥0,xiis an integer,i=1,2,⋯,n.Refer to reference [36], we can transform the above problem into the deterministic form as follows:(19)maxλs.t.λ≤fμZjx,j=1,2,3,4∑i=1nxi=Dxi≤Ui,i=1,2,⋯,nxi≥0,i=1,2,⋯,n,λ∈0,1.In which different targets is allocated the same weight. In practice, different decision-makers might have different preferences for goals, so the above goals should not be treated equally. Consequently, the weighted maximum minimum method is presented to address this problem. Following this method, the different objectives in the model are endowed with different weights according to the results obtained in Section3.2.1. Then we use the weight maximum minimum model to find the optimal solution of the model.(20)maxλs.t.wjλ≤fμZjx,j=1,2,3,4,∑i=1nxi=D,∑i=14wj=1,xi≤Ui,i=1,2,…,n,xi≥0,i=1,2,…,n,λ∈0,1.


## 3.2.1. Multi Objective Linear Programming

As discussed previously, the objectives of the model established in this study are as follows: (1) minimize procurement costs; (2) best product quality; (3) best service of supplier; (4) lowest product carbon emission. The model is further constructed after the setting of model assumptions, symbols, parameters, and decision variables.Model assumptions(i)
The enterprise only purchases the same product from each supplier(ii)
The discount brought by the purchase quota is not considered(iii)
There is no shortage of products provided by the supplierSymbolsi: Number of suppliersj: Target number of linear programming, j=1,2,…,JModel parametersD: The overall demand of the enterprise for products in a fixed planning cyclen: Amount of supplierspi: The price at which an enterprise purchases products from supplier iqi: Supplier i’s product qualitysi Supplier i’s service qualitygi: Carbon emission of supplier i’s productsUi: Supplier i’s supply capacityDecision variables:xi: Quota of products ordered by the enterprise from supplier i(1) Linear Programming Model. We establish the following multi-objective linear programming model by combining the aim of our study and the related work of [32] as follows:(1)MinimizeZ1=∑i=1npixi,(2)MinimizeZ2=∑i=1nqixi,(3)MinimizeZ3=∑i=1nsixi,(4)MinimizeZ4=∑i=1ngixi,(5)s.t.∑i=qnxi=D,(6)xi≤Ui,i=1,2,⋯,n,(7)xi≥0i,xiis an integer,i=1,2,⋯,n.Equations (1)–(4) are the objective functions with respect to minimize procurement costs, best product quality, best service of supplier, and lowest product carbon emission, respectively. Constraint condition (5) ensures that the products purchased by the enterprise meet its own needs; constraint condition (6) means that the products purchased by the enterprise cannot exceed its supply capacity, and constraint (7) means that the decision variable is not less than 0 and is an integer.


## 3.2.2. Fuzzy Multi Objective Programming

In the process of selecting supplier selection, there are often some uncertainties in the relevant information of suppliers. When describing suppliers, the evaluation of their selection criteria may not be very accurate, such as “almost no quality problems” and “the supplier’s supply capacity is between 3000 and 3500.”To address the fuzziness of key information and make decisions in an uncertain environment, Bellman and Zadeh [33] proposed a fuzzy programming model, and Zimmermann [34] then used the model to deal with the problems of multi-objective programming. Follow these studies, the multi-objective linear programming in 3.2.1 can be transformed into the following form by considering product price, product quality, service quality, and product carbon emission as fuzzy information.(8)∑i=1npixi≺∼Z˜1,(9)∑i=1nqixi≻∼Z˜2,(10)∑i=1nsixi≻∼Z˜3,(11)∑i=1ngixi≻∼Z˜4,(12)s.t.∑i=qnxi=D,(13)xi≤Ui,i=1,2,⋯,n,14)xi≥0i,xiis an integer,i=1,2,⋯,n.In which∼ denotes a fuzzy environment. Z˜1, Z˜2, Z˜3, and Z˜4 are the enterprise’s expectations of product price, product quality, service quality, and product carbon emission. Sign “≺∼” means “basically less than or equal to,” while sign “≺∼≻∼” means “basically more than or equal to.” In order to find the membership function of fuzzy set, this study uses the reduced half trapezoid method to find the minimum value of the objective function, and uses the raised half trapezoid method to find the maximum value of the objective function. The membership functions of the minimization objective function Zm (m = 1, 4) and the maximization objective Zl (l = 2, 3) can be respectively expressed as:(15)μZmx=1,Zm≤Zm−,fμZm=Zm+−ZmxZm+−Zm−,Zm−≤Zmx≤Zm+m=1,40,Zm≥Zl−,,(16)μZlx=1,Zl≤Zl−,fμZl=Zmx−Zl−Zl+−Zl−,Zl−≤Zlx≤Zl+l=2,3,0,Zl≥Zl+.LetZj+ (k=1,2,3,4) and Zj− (k=1,2,3,4) denote the upper and lower bound of each objective function, respectively. According to [35], Zj+ is the maximum value of the objective function corresponding to the optimal solution of the following problem:(17)maxZjxs.t.∑i=1nxi=Dxi≤Ui,i=1,2,⋯,nxi≥0,xiis an integer,i=1,2,⋯,n.Zj− is the minimum value of the objective function corresponding to the optimal solution of the following problem:(18)minZjxs.t.∑i=1nxi=Dxi≤Ui,i=1,2,⋯,nxi≥0,xiis an integer,i=1,2,⋯,n.Refer to reference [36], we can transform the above problem into the deterministic form as follows:(19)maxλs.t.λ≤fμZjx,j=1,2,3,4∑i=1nxi=Dxi≤Ui,i=1,2,⋯,nxi≥0,i=1,2,⋯,n,λ∈0,1.In which different targets is allocated the same weight. In practice, different decision-makers might have different preferences for goals, so the above goals should not be treated equally. Consequently, the weighted maximum minimum method is presented to address this problem. Following this method, the different objectives in the model are endowed with different weights according to the results obtained in Section3.2.1. Then we use the weight maximum minimum model to find the optimal solution of the model.(20)maxλs.t.wjλ≤fμZjx,j=1,2,3,4,∑i=1nxi=D,∑i=14wj=1,xi≤Ui,i=1,2,…,n,xi≥0,i=1,2,…,n,λ∈0,1.


## 4. Example Analysis

### 4.1. Determination of the Weight of Selection Criteria

This paper takes a manufacturing enterprise as an example to illustrate the effectiveness of the method discussed above. In addition to the general concerns of supplier selection criteria when purchasing a product, such as the quality, cost, and service, the enterprise also takes into account the environmental friendliness of the product (i.e., the carbon emission information of the products), expecting to improve the environmental efficiency and economy simultaneous during the product purchase process. After establishing the supplier selection criteria (as shown in Figure2), we use the way of questionnaire to collect the evaluations of the relative importance of each criteria and index from purchasing managers, and then use the fuzzy AHP method to calculate the weight of the criteria and index. The fuzzy judgment matrix of experts is shown in Table 2.Table 2
Fuzzy judgment matrix of expert evaluation criteria.
Product cost (C1)Product quality (C2)Service quality (C3)Product carbon emission (C4)Product cost (C1)(1, 1, 1)(1/2, 1, 5/4)(1, 2, 5/2)(7/3, 3, 13/4)Product quality (C2)(4/5, 1, 2)(1, 1, 1)(1/2, 2, 9/4)(5/4, 2, 5/2)Service quality (C3)(2/5, 1/2, 1)(4/9, 1/2, 2)(1, 1, 1)(1, 2, 9/4)Product carbon emission (C4)(4/13, 1/3, 3/7)(2/5, 1/2, 4/5)(4/9, 1/2, 1)(1, 1, 1)According to equations (1)–(5), the relative importance of supplier selection criteria is calculated as follows:(21)∑i=1n∑j=1nMgij=1,1,1+0.5,1,1.25+⋯+1,1,1=13.37,19.33,25.23(22)∑i=1n∑j=1nMgij−1=0.0396,0.0517,0.0748(23)F1=∑i=1n∑j=1nMgij⊗∑i=1n∑j=1nMgij−1=4.83,7,8⊗0.0396,0.0517,0.0748=0.19,0.36,0.60(24)F2=3.55,6,7.75⊗0.0396,0.0517,0.0748=0.14,0.31,0.58(25)F3=2.84,4,6.25⊗0.0396,0.0517,0.0748=0.11,0.21,0.47(26)F4=2.15,2.33,3.23⊗0.0396,0.0517,0.0748=0.09,0.12,0.24(27)VF1≥F2=0.19−0.580.31−0.58−0.36−0.19=0.89,VF2≥F3,VF2≥F4=1(28)VF3≥F1=0.19−0.470.21−0.47−0.36−0.19=0.65(29)VF3≥F2=0.14−0.470.21−0.47−0.31−0.14=0.77,VF3≥F4=1(30)VF4≥F2=0.19−0.240.12−0.24−0.36−0.19=0.17(31)VF4≥F2=0.14−0.240.12−0.24−0.31−0.14=0.34(32)VF4≥F3=0.11−0.240.12−0.24−0.21−0.11=0.59(33)dF1=minVF1≥F2,F3,F4=min1,1,1=1(34)dF2=minVF2≥F1,F3,F4=min0.89,1,1=0.89(35)dF3=minVF3≥F1,F2,F4=min0.65,0.77,1=0.65(36)dF4=minVF4≥F1,F2,F3=min0.17,0.34,59=0.1737)W′=1,0.89,0.65,0.17T=0.37,0.33,0.24,0.06.We can conclude from the analysis of the above fuzzy AHP results that, the manufacturing enterprise considers the cost of products as the most important selection criteria in supplier selection, followed by product quality, supplier service level, and product carbon emission, with weights of 0.37, 0.33, 0.24, and 0.06, respectively. We also note that the weight of carbon emission standard of products is relatively low, which shows that although the enterprise has included carbon emission into the selection criteria, it still has a certain gap compared with the traditional selection criteria. We calculate the relative importance of each index based on the judgment matrix of experts and present the results in Tables3–6.Table 3
Fuzzy judgment matrix of product cost.
Product priceA1Transportation costA2WeightProduct priceA1(1, 1, 1)(1, 3, 10/3)0.74Transportation costA2(3/10, 1/3, 1)(1, 1, 1)0.26Table 4
Fuzzy judgment matrix of product quality.
Reject rateA3Quality assuranceA4Remedies to quality problemsA5WeightReject rateA3(1, 1, 1)(1, 2, 9/4)(5/4, 2, 5/2)0.56Quality assuranceA4(4/9, 1/2, 1)(1, 1, 1)(3/2, 2, 9/4)0.36Remedies to quality problemsA5(2/5, 1/2, 4/5)(4/9, 1/2, 2/3)(1, 1, 1)0.08Table 5
Fuzzy judgment matrix of supplier service quality.
Lead timeA6Timely deliveryA7Technical supportA8Response efficiencyA9WeightLead timeA6(1, 1, 1)(1/2, 1, 5/4)(1, 2, 9/4)(5/2, 3, 13/4)0.39Timely deliveryA7(4/5, 1, 2)(1, 1, 1)(1/2, 1, 5/4)(1, 2, 5/2)0.29Technical supportA8(4/9, 1/2, 2)(4/5, 1, 2)(1, 1, 1)(1, 2, 9/4)0.26Response efficiencyA9(4/13, 1/3, 2/5)(2/5, 1/2, 1)(4/9, 1/2, 1)(1, 1, 1)0.06Table 6
Fuzzy judgment matrix of product carbon emission.
Carbon footprint verificationA10Transport carbon emissionA11WeightCarbon footprint verificationA10(1, 1, 1)(3/2, 4, 9/2)0.88Transport carbon emissionA11(3/2, 4, 9/2)(1, 1, 1)0.12


### 4.2. Solution of Fuzzy GP Model

The supplier selection model in this case includes four potential supplier objects to choose from. The specific selection criteria include product cost, product quality, service quality, and product carbon emission. The supply capacity of suppliers and the product demand of enterprises are the constraints of fuzzy GP. The data of product cost, product quality, service quality, and carbon emission are fuzzy, and their values and supplier supply capacity are shown in Table7. The product demand of the enterprise is about 1000.Table 7
Suppliers information.
piqisigiUiSupplier 113980.8400Supplier 28671.4300Supplier 310861.1400Supplier 4117101.5250According to the relevant information of the above suppliers and the weight of the selection criteria obtained in 3.1, the multi-objective fuzzy programming is further constructed. Among them, targetZ1 is to minimize the purchase cost of products, target Z2 is to maximize the product quality, target Z3 is to maximize the service quality, and target Z4 is to minimize the carbon emission of purchased products.(38)Z1=13x1+8x2+10x3+11x4(39)Z2=9x1+6x2+8x3+7x4(40)Z3=8x1+7x2+6x3+10x4(41)Z4=0.8x1+1.4x2+1.1x3+1.5x4(42)s.t.x1+x2+x3+x4=1000(43)x1≤400,x2≤300,x3≤400,x4≤250(44)xi≥0,i=1,2,3,4.Calculate the upper and lower bounds of objective functionsZ1∼Z4 based on equations (12), (17), and (18), the results are shown in Table 8.Table 8
Upper and lower bounds of objective function.
μ=0μ=1μ=0Z1—980011450Z272008200—Z369008100—Z4—10401275According to equation (13), we adopt the weighted maximum minimum model to transform the multi-objective fuzzy programming into a single objective linear programming. The weights of fuzzy objectives are determined by fuzzy AHP method. According to the results obtained in 3.1, the weights of product cost, product quality, service quality, and product carbon emission objectives in this model are 0.37, 0.33, 0.24, and 0.06, respectively.(45)maxλ(46)s.t.0.37λ≤11450−13x1+8x2+10x3+11x41650(47)0.33λ≤9x1+6x2+8x3+7x4−72001000(48)0.24λ≤8x1+7x2+6x3+10x4−69001200(49)0.06λ≤1275−0.8x1+1.4x2+1.1x3+1.5x4235(50)x1+x2+x3+x4=1000(51)x1≤400,x2≤300,x3≤400,x4≤25052)xi≥0,i=1,2,3,4.We use the LINGO platform to solve this integer linear programming, and obtain the results as follows:(53)x1=201,x2=270,x3=400,x4=12954)Z1=10192,Z2=7532,Z3=7088,Z4=1172.3.According to the method provided by [25], we calculate the optimal solution of the fuzzy programming is under the condition that different objectives have the same weight and compare the results with the results of the weighted maximum minimum method. The results are reported in Table 9.Table 9
Comparison of optimal solutions of different methods.
Weighted maximum minimization methodZimmermann methodx1201257x2270176x3400364x4129203Z11019210622Z275327702Z371887502Z41172.31156.9From Table9 we can see that, the value of Z1∼Z3 decreases and the value of Z4 increases in the results of weighted maximum minimization method compared with the results from Zimmermann method. This appearance can be explained as: the weight of target Z1 in the weighted maximum minimum method model is high, so it is more inclined to reduce the product cost in this model, and the completion of this goal is at the expense of other targets (Z2∼Z4), that is, Z1∼Z3 decreases and Z4 increases.Through the analysis above, it is easy to see the advantages and disadvantages of the alternative green suppliers and the field and direction that 4 suppliers need to improve. Based on the concept of continuous improvement, the selected enterprise can continue to improve its competitiveness and profitability in low-carbon supply chain.


## 4.1. Determination of the Weight of Selection Criteria

This paper takes a manufacturing enterprise as an example to illustrate the effectiveness of the method discussed above. In addition to the general concerns of supplier selection criteria when purchasing a product, such as the quality, cost, and service, the enterprise also takes into account the environmental friendliness of the product (i.e., the carbon emission information of the products), expecting to improve the environmental efficiency and economy simultaneous during the product purchase process. After establishing the supplier selection criteria (as shown in Figure2), we use the way of questionnaire to collect the evaluations of the relative importance of each criteria and index from purchasing managers, and then use the fuzzy AHP method to calculate the weight of the criteria and index. The fuzzy judgment matrix of experts is shown in Table 2.Table 2
Fuzzy judgment matrix of expert evaluation criteria.
Product cost (C1)Product quality (C2)Service quality (C3)Product carbon emission (C4)Product cost (C1)(1, 1, 1)(1/2, 1, 5/4)(1, 2, 5/2)(7/3, 3, 13/4)Product quality (C2)(4/5, 1, 2)(1, 1, 1)(1/2, 2, 9/4)(5/4, 2, 5/2)Service quality (C3)(2/5, 1/2, 1)(4/9, 1/2, 2)(1, 1, 1)(1, 2, 9/4)Product carbon emission (C4)(4/13, 1/3, 3/7)(2/5, 1/2, 4/5)(4/9, 1/2, 1)(1, 1, 1)According to equations (1)–(5), the relative importance of supplier selection criteria is calculated as follows:(21)∑i=1n∑j=1nMgij=1,1,1+0.5,1,1.25+⋯+1,1,1=13.37,19.33,25.23(22)∑i=1n∑j=1nMgij−1=0.0396,0.0517,0.0748(23)F1=∑i=1n∑j=1nMgij⊗∑i=1n∑j=1nMgij−1=4.83,7,8⊗0.0396,0.0517,0.0748=0.19,0.36,0.60(24)F2=3.55,6,7.75⊗0.0396,0.0517,0.0748=0.14,0.31,0.58(25)F3=2.84,4,6.25⊗0.0396,0.0517,0.0748=0.11,0.21,0.47(26)F4=2.15,2.33,3.23⊗0.0396,0.0517,0.0748=0.09,0.12,0.24(27)VF1≥F2=0.19−0.580.31−0.58−0.36−0.19=0.89,VF2≥F3,VF2≥F4=1(28)VF3≥F1=0.19−0.470.21−0.47−0.36−0.19=0.65(29)VF3≥F2=0.14−0.470.21−0.47−0.31−0.14=0.77,VF3≥F4=1(30)VF4≥F2=0.19−0.240.12−0.24−0.36−0.19=0.17(31)VF4≥F2=0.14−0.240.12−0.24−0.31−0.14=0.34(32)VF4≥F3=0.11−0.240.12−0.24−0.21−0.11=0.59(33)dF1=minVF1≥F2,F3,F4=min1,1,1=1(34)dF2=minVF2≥F1,F3,F4=min0.89,1,1=0.89(35)dF3=minVF3≥F1,F2,F4=min0.65,0.77,1=0.65(36)dF4=minVF4≥F1,F2,F3=min0.17,0.34,59=0.1737)W′=1,0.89,0.65,0.17T=0.37,0.33,0.24,0.06.We can conclude from the analysis of the above fuzzy AHP results that, the manufacturing enterprise considers the cost of products as the most important selection criteria in supplier selection, followed by product quality, supplier service level, and product carbon emission, with weights of 0.37, 0.33, 0.24, and 0.06, respectively. We also note that the weight of carbon emission standard of products is relatively low, which shows that although the enterprise has included carbon emission into the selection criteria, it still has a certain gap compared with the traditional selection criteria. We calculate the relative importance of each index based on the judgment matrix of experts and present the results in Tables3–6.Table 3
Fuzzy judgment matrix of product cost.
Product priceA1Transportation costA2WeightProduct priceA1(1, 1, 1)(1, 3, 10/3)0.74Transportation costA2(3/10, 1/3, 1)(1, 1, 1)0.26Table 4
Fuzzy judgment matrix of product quality.
Reject rateA3Quality assuranceA4Remedies to quality problemsA5WeightReject rateA3(1, 1, 1)(1, 2, 9/4)(5/4, 2, 5/2)0.56Quality assuranceA4(4/9, 1/2, 1)(1, 1, 1)(3/2, 2, 9/4)0.36Remedies to quality problemsA5(2/5, 1/2, 4/5)(4/9, 1/2, 2/3)(1, 1, 1)0.08Table 5
Fuzzy judgment matrix of supplier service quality.
Lead timeA6Timely deliveryA7Technical supportA8Response efficiencyA9WeightLead timeA6(1, 1, 1)(1/2, 1, 5/4)(1, 2, 9/4)(5/2, 3, 13/4)0.39Timely deliveryA7(4/5, 1, 2)(1, 1, 1)(1/2, 1, 5/4)(1, 2, 5/2)0.29Technical supportA8(4/9, 1/2, 2)(4/5, 1, 2)(1, 1, 1)(1, 2, 9/4)0.26Response efficiencyA9(4/13, 1/3, 2/5)(2/5, 1/2, 1)(4/9, 1/2, 1)(1, 1, 1)0.06Table 6
Fuzzy judgment matrix of product carbon emission.
Carbon footprint verificationA10Transport carbon emissionA11WeightCarbon footprint verificationA10(1, 1, 1)(3/2, 4, 9/2)0.88Transport carbon emissionA11(3/2, 4, 9/2)(1, 1, 1)0.12


## 4.2. Solution of Fuzzy GP Model

The supplier selection model in this case includes four potential supplier objects to choose from. The specific selection criteria include product cost, product quality, service quality, and product carbon emission. The supply capacity of suppliers and the product demand of enterprises are the constraints of fuzzy GP. The data of product cost, product quality, service quality, and carbon emission are fuzzy, and their values and supplier supply capacity are shown in Table7. The product demand of the enterprise is about 1000.Table 7
Suppliers information.
piqisigiUiSupplier 113980.8400Supplier 28671.4300Supplier 310861.1400Supplier 4117101.5250According to the relevant information of the above suppliers and the weight of the selection criteria obtained in 3.1, the multi-objective fuzzy programming is further constructed. Among them, targetZ1 is to minimize the purchase cost of products, target Z2 is to maximize the product quality, target Z3 is to maximize the service quality, and target Z4 is to minimize the carbon emission of purchased products.(38)Z1=13x1+8x2+10x3+11x4(39)Z2=9x1+6x2+8x3+7x4(40)Z3=8x1+7x2+6x3+10x4(41)Z4=0.8x1+1.4x2+1.1x3+1.5x4(42)s.t.x1+x2+x3+x4=1000(43)x1≤400,x2≤300,x3≤400,x4≤250(44)xi≥0,i=1,2,3,4.Calculate the upper and lower bounds of objective functionsZ1∼Z4 based on equations (12), (17), and (18), the results are shown in Table 8.Table 8
Upper and lower bounds of objective function.
μ=0μ=1μ=0Z1—980011450Z272008200—Z369008100—Z4—10401275According to equation (13), we adopt the weighted maximum minimum model to transform the multi-objective fuzzy programming into a single objective linear programming. The weights of fuzzy objectives are determined by fuzzy AHP method. According to the results obtained in 3.1, the weights of product cost, product quality, service quality, and product carbon emission objectives in this model are 0.37, 0.33, 0.24, and 0.06, respectively.(45)maxλ(46)s.t.0.37λ≤11450−13x1+8x2+10x3+11x41650(47)0.33λ≤9x1+6x2+8x3+7x4−72001000(48)0.24λ≤8x1+7x2+6x3+10x4−69001200(49)0.06λ≤1275−0.8x1+1.4x2+1.1x3+1.5x4235(50)x1+x2+x3+x4=1000(51)x1≤400,x2≤300,x3≤400,x4≤25052)xi≥0,i=1,2,3,4.We use the LINGO platform to solve this integer linear programming, and obtain the results as follows:(53)x1=201,x2=270,x3=400,x4=12954)Z1=10192,Z2=7532,Z3=7088,Z4=1172.3.According to the method provided by [25], we calculate the optimal solution of the fuzzy programming is under the condition that different objectives have the same weight and compare the results with the results of the weighted maximum minimum method. The results are reported in Table 9.Table 9
Comparison of optimal solutions of different methods.
Weighted maximum minimization methodZimmermann methodx1201257x2270176x3400364x4129203Z11019210622Z275327702Z371887502Z41172.31156.9From Table9 we can see that, the value of Z1∼Z3 decreases and the value of Z4 increases in the results of weighted maximum minimization method compared with the results from Zimmermann method. This appearance can be explained as: the weight of target Z1 in the weighted maximum minimum method model is high, so it is more inclined to reduce the product cost in this model, and the completion of this goal is at the expense of other targets (Z2∼Z4), that is, Z1∼Z3 decreases and Z4 increases.Through the analysis above, it is easy to see the advantages and disadvantages of the alternative green suppliers and the field and direction that 4 suppliers need to improve. Based on the concept of continuous improvement, the selected enterprise can continue to improve its competitiveness and profitability in low-carbon supply chain.


## 5. Conclusion

In this paper, we propose a fuzzy AHP-GP multicriteria decision-making approach for green supplier selection in low-carbon supply chain. According to the demand of enterprises in low-carbon supply chain, 4 main criteria and 11 subcriteria are established for green supplier selection. A method integrating fuzzy analytic hierarchy process and fuzzy goal programming is also proposed. This method can make the localization of individual green supplier more objectively and more accurately in the same trade, and it is easier for us to find the problems in green supplier selection. A typical steel manufacturing enterprise is studied to verify the scientificity and feasibility of the proposed criteria and method for green supplier selection. The result shows that the proposed criteria and method have good applicability in a practical situation. To some extent, the proposed method can be widely used as a structural model for green supplier selection. For further research, a fuzzy-based questionnaire can be used for data collection in order to prevent information bias. We are also interested in comparing the performance of the approach used in this paper with some alternative methods, such as the fuzzy FUCOM, fuzzy SWARA, or Fuzzy BWM.



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