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An integrated open-loop supply chain network configuration model with sustainable supplier selection: fuzzy multi-objective approach

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Abstract

Supply chain management is an essential part of most companies to reduce costs and increase the profit for a competitive business environment. The success of companies is directly connected to the performance of the supply chain. The sustainability of businesses broadly depends on the purchasing decision, and the process becomes more complex with resource limitations. In addition, relationships with suppliers are considered for optimal production and risk minimization in the supply chain network. This research aims to design and optimize a multi-echelon, multi-period, and multi-objective open-loop supply chain (OLSC) network design. The objectives are deemed to be the total costs, the minimization of environmental costs, and the maximization of the importance level of suppliers. Furthermore, a fuzzy approach based on interval type-2 fuzzy sets is designed to evaluate the supplier performance according to sustainable qualitative criteria. After determination of the weights of criteria and evaluation of suppliers by interval type-2 fuzzy numbers, the final score of suppliers is calculated based on the weights of criteria and evaluation scores. Three fuzzy programming approaches are utilized to solve the developed OLSC model. A numerical example is provided to validate the model and effectiveness of the proposed evaluation procedure. The results demonstrate the feasibility of the proposed evaluation procedure and the achievement of a compromise in the developed model.

Introduction

In recent years, interest in reverse logistics and closed-loop supply chains (CLSCs) has increased significantly by researchers and practitioners [1]. If end-of-life (EOL) products are collected from the end-use point through applying recovery processes to regain value, returning to the first production point and re-launching them to the market is defined as CLSC [2]. However, in an open-loop supply chain (OLSC) network, the products are not returned to the original manufacturers. Generally, the products or materials that have been processed under various operations are recovered by the parties who wish to reuse them [3]. Due to increased environmental awareness and legal requirements, both the public and private sectors are required to consider environmental issues in managing their operations and supply chains [4, 5].

CLSC network design is one of the most important long-term strategic decisions for businesses due to the activities of increasing globalization and the change in technology levels [6]. In recent years, the importance of reverse logistics, including reuse, repair, refurbishing, remanufacturing, and recycling of EOL, has been highlighted by researchers and businesses due to the changes that have taken place and the economic benefits of used products [7]. A comprehensive plan for collecting EOL is required if minimal environmental damage is desired [8]. Therefore, the CLSC network design can be the best solution for businesses that want to collect the EOL and add them to the re-evaluation process [9].

In supply chain management, many businesses have resorted to outsourcing and supplier selection in recent years because they seem more profitable and want to focus on their strategic activities [2]. Choosing the right supplier is a key issue in SCM, but it also has a major impact on the strategic and operational performance of a firm [10]. Considering that more than 50% of the production cost of enterprises consists of purchasing activities [11, 12], it is more coherent for businesses to purchase from many suppliers instead of linking them to the capacity and flexibility of a single supplier. In addition, suppliers should be evaluated by taking into consideration the cost of purchase as well as criteria such as late delivery, energy consumption and respect for human rights. For this reason, many organizations have recently started to incorporate environmental, social, and economic aspects of sustainability into their supplier selection processes [13].

Integrating sustainability into this selection process rather than the classical supplier selection has been the focus for businesses that provide a more competitive edge in global markets [14, 15]. Choosing the most appropriate one among the suppliers with different levels of capability and potential is a very demanding task that requires various criteria and different solution approaches [16]. In this context, multi-criteria decision-making (MCDM) methods can help solve the problem of supplier selection and find the optimal product quantity [17]. In the real world, linguistic terms may need to be used to evaluate alternatives and criteria, and words contain uncertainty [18]. To deal with linguistic uncertainty, it has been shown that deciding interval type-2 fuzzy sets (IT2FSs) are more effective than traditional decision-making tools [19].

Based on the discussions above, this paper presents a multi-objective model for OLSC network design and sustainable supplier selection problem. A new approach based on IT2FSs is proposed to find the best suppliers from a sustainability perspective. In the proposed approach, uncertain judgments of multiple decision-makers are captured by the IT2Fs; the importance levels of the evaluation criteria are determined by the interval type-2 analytic hierarchy process (AHP) method to take into account various and conflicting criteria. An integrated OLSC model is developed in a multi-period environment to address the importance of suppliers and to determine the number of products to be processed in each echelon. Finally, based on the opinions of experts, three fuzzy approaches are utilized in this paper to solve the multi-objective optimization problem in hand.

The rest of the study is organized as follows: In the second chapter, the related literature is reviewed, and the research is described. In the third chapter, the definition and formulation of the developed mathematical model are presented. The fourth section describes the proposed assessment procedure. The fifth section includes the analysis and interpretation of the numerical results obtained from the solution of the model. In the last section, the results of the article are presented, and future research suggestions are presented.

Literature review

Although CLSC and reverse logistics network designs have been a prevalent research topic of academic interest in recent years, OLSC network designs have not attracted much attention from researchers and practitioners. Gou et al. [20] presented a stochastic inventory model for an open-loop reverse supply chain to find the optimal economic delivery batch size for collection points. Ene and Öztürk [21] developed an open-loop reverse supply chain network design with a mixed-integer linear programming model. Özceylan [3] handled the CLSC and OLSC network simultaneously by developing a mixed-integer linear programming model. The model simulated with randomly using data and results showed that cost savings may be achieved by solving the integrated objective function instead of solving the individual objective functions. Kalverkamp and Young [22] tested three research questions by using three case examples and indicated that OLSCs may offer sustainable advantages to actors in supply chains.

To illustrate the difference of this study from others, a summary of the relevant literature is presented in Table 1 where the features of the proposed problem are presented in the last row. According to Table 1, although most of the studies have been discussed in economic and environmental objectives in CLSC objectives, the assessment of suppliers of OLSC is still in need of further research. Most publications have ignored the problem of supplier selection at the same time with the OLSC network design and with the identification of decision variables in a single period. Moreover, a limited number of studies have been performed with fuzzy sets in the OLSC network design, but the interval type-2 fuzzy sets have not been used yet. Thereby, the contributions of our study which aim to fill these gaps are explained in the following sections.

Table 1 Review of some articles

Problem definition

The structure of the OLSC discussed in this study is represented in Fig. 1. In the model, the forward supply chain consists of four levels: suppliers \(\left( i \right)\), plants \(\left( j \right)\), distribution centers \(\left( k \right)\), and customers \(\left( l \right)\). Similarly, the reverse supply chain includes six levels: collection centers \(\left( m \right)\), refurbishing centers \(\left( n \right)\), recycling centers \(\left( r \right)\), remanufacturing centers \(\left( q \right)\), secondary markets \(\left( s \right)\), and disposal.

Fig. 1
figure1

General structure of the OLSC network

Forward flow begins with suppliers who are responsible for the procurement of parts of the plants for each period, parts are transformed to final products in the plants, and the products are sent to the customers through distribution centers to meet the fixed customer demands. The flow of the final products does not end after the products are delivered to the customers. The reverse flow starts with the collection of the used products by the customers at a certain rate after a period of usage. In the next step, used products are shipped to collection/inspections centers. Used products are inspected at collection centers and transferred to refurbishing, recycling, remanufacturing, and disposal according to quality classification. The high quality of the products processed in these facilities can be sold as a second-hand product at a cheaper price in the secondary markets.

Model formulation

The OLSC network design considered in this study is a multi-echelon network consisting of ten layers: suppliers, plants, distribution centers, customers, collection centers, refurbishing centers, recycling centers, remanufacturing centers, secondary markets, and disposal. The indices, parameters, decision variables, objective functions, and constraints used in the formulation to solve the problem are presented below:

Indices

Symbol Description
i Set of suppliers, \(i = 1, 2 \ldots I\)
j Set of plants, \(j = 1, 2 \ldots J\)
k Set of distribution centers, \(k = 1, 2 \ldots K\)
l Set of customers, \(l = 1, 2 \ldots L\)
m Set of collection centers, \(m = 1, 2 \ldots M\)
n Set of refurbishing centers, \(n = 1, 2 \ldots N\)
r Set of recycling centers, \(r = 1, 2 \ldots R\)
q Set of remanufacturing centers, \(q = 1, 2 \ldots Q\)
s Set of secondary markets, \(s = 1, 2 \ldots S\)
v Set of type of vehicles, \(v = 1, 2 \ldots V\)
t Set of periods, \(t = 1, 2 \ldots T\)

Parameters

Symbol Description
\(d_{\mu \sigma }\) Distance from origin \(\mu\) to destination \(\sigma\); \(\mu ,\sigma \in \left\{ {I, J, K, L, M, N, R, Q, S} \right\}\) (km)
\(c_{v}\) Unit transportation cost by vehicle v (\(\$ / {\text{ton}}\,{\text{km}}\))
\(capt_{\vartheta t}\) Capacity of facility \(\vartheta\) in period t; \(\vartheta \in \left\{ {I, J, K, M, N, R, Q} \right\}\) (ton)
\(tal_{lt}\) Demand of customer l in period t (ton)
\(smf_{\tau }\) Fixed cost of facility \(\tau\); \(\tau \in \left\{ { J, M, N, R, Q} \right\}\) ($)
\(samt_{i}\), \(um_{j}\), \(top_{lm}\), \(yen_{n}\), \(grm_{r}\), \(yum_{q}\), \(dc\) Unit purchasing, production, collection, refurbishing, recycling, remanufacturing, disposal cost ($/ton)
\(tasyen_{n}\), \(tasgr_{r}\), \(tasyu_{q}\) Unit cost saving at refurbishing, recycling, remanufacturing center ($/ton)
\(cep_{j}\) The amount of \({\text{CO}}_{2}\) emission caused by the production of one ton of product at plant j (gr/ton)
\(C_{{CO_{2} }}\) Unit cost of \({\text{CO}}_{2}\) emission ($/gr)
\(cet_{v}\) Amount of \({\text{CO}}_{2}\) emission of one-ton load transported by vehicle v through one kilometer (gr/ton km)
\(u_{v}\) The maximum capacity of vehicle v (ton)
\(\pi\), \(\alpha\), \(\beta\), \(\theta\), \(\lambda\) Percentage of products collected by centers (%)
\(w_{i}\) Level of importance weight of supplier i for procured parts

Decision variables

Symbol Description
\(X_{ijvt}\), \(Y_{jkvt}\), \(Z_{klvt}\), \(U_{lmvt}\), \(A_{mnvt}\), \(B_{mrvt}\), \(C_{mqvt}\), \(D_{mvt}\), \(E_{nsvt}\), \(F_{rsvt}\), \(G_{qsvt}\) Quantity of parts/products between pair of nodes by vehicle v at time period t (ton)
\(XX_{ijvt}\), \(YY_{jkvt}\), \(ZZ_{klvt}\), \(UU_{lmvt}\), \(AA_{mnvt}\), \(BB_{mrvt}\), \(CC_{mqvt}\), \(DD_{mvt}\), \(EE_{nsvt}\), \(FF_{rsvt}\), \(GG_{qsvt}\) If parts are shipped between pair of nodes by vehicle v at time period t 1; otherwise, 0
\(XINT_{ijvt}\), \(YINT_{jkvt}\), \(ZINT_{klvt}\), \(UINT_{lmvt}\), \(AINT_{mnvt}\), \(BINT_{nvt}\), \(CINT_{nrvt}\), \(DINT_{rkvt}\), \(EINT_{nsvt}\), \(FINT_{rsvt}\), \(GINT_{qsvt}\) Number of tours between pair of nodes by vehicle v at time period t
\(OP_{j}\), \(OC_{m}\), \(OYEN_{n}\), \(OGR_{r}\), \(OYU_{q}\) If facility is opened at time period t, 1; otherwise, 0

Objective functions

The first objective function specifies the minimization of economic cost:

$${{Min}}\,Z_{1} = TC + POC + FOC$$
(1)
$$\begin{aligned} & TC \\ & = \mathop \sum \limits_{i} \mathop \sum \limits_{j} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{ij} \cdot X_{ijvt} + \mathop \sum \limits_{j} \mathop \sum \limits_{k} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{jk} \cdot Y_{jkvt} + \mathop \sum \limits_{k} \mathop \sum \limits_{l} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{kl} \cdot Z_{klvt} + \mathop \sum \limits_{l} \mathop \sum \limits_{m} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{lm} \cdot U_{lmvt} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{mn} \cdot A_{mnvt} + \mathop \sum \limits_{m} \mathop \sum \limits_{r} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{mr} \cdot B_{mrvt} + \mathop \sum \limits_{m} \mathop \sum \limits_{q} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{mq} \cdot C_{mqvt} + \mathop \sum \limits_{n} \mathop \sum \limits_{v} \mathop \sum \limits_{t} D_{mvt} \cdot d_{m} \\ & \quad + \mathop \sum \limits_{n} \mathop \sum \limits_{s} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{ns} \cdot E_{nsvt} + \mathop \sum \limits_{r} \mathop \sum \limits_{s} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{rs} \cdot F_{rsvt} + \mathop \sum \limits_{q} \mathop \sum \limits_{s} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{qs} \cdot G_{qsvt} \\ \end{aligned}$$
(2)
$$\begin{aligned} POC = & \mathop \sum \limits_{i} \mathop \sum \limits_{j} \mathop \sum \limits_{v} \mathop \sum \limits_{t} samt_{i} \cdot X_{ijvt} + \mathop \sum \limits_{j} \mathop \sum \limits_{k} \mathop \sum \limits_{v} \mathop \sum \limits_{t} um_{j} \cdot Y_{jkvt} + \mathop \sum \limits_{l} \mathop \sum \limits_{m} \mathop \sum \limits_{v} \mathop \sum \limits_{t} top_{lm} \cdot U_{lmvt} \\ & + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{v} \mathop \sum \limits_{t} yen_{n} \cdot A_{mnvt} + \mathop \sum \limits_{m} \mathop \sum \limits_{r} \mathop \sum \limits_{v} \mathop \sum \limits_{t} grm_{r} \cdot B_{mrvt} + \mathop \sum \limits_{m} \mathop \sum \limits_{q} \mathop \sum \limits_{v} \mathop \sum \limits_{t} yum_{q} \cdot C_{mqvt} \\ & + \mathop \sum \limits_{m} \mathop \sum \limits_{v} \mathop \sum \limits_{t} dc \cdot D_{mvt} - \mathop \sum \limits_{n} \mathop \sum \limits_{s} \mathop \sum \limits_{v} \mathop \sum \limits_{t} tasm_{r} \cdot E_{nsvt} - \mathop \sum \limits_{r} \mathop \sum \limits_{s} \mathop \sum \limits_{v} \mathop \sum \limits_{t} tasgr_{r} \cdot F_{rsvt} - \mathop \sum \limits_{q} \mathop \sum \limits_{s} \mathop \sum \limits_{v} \mathop \sum \limits_{t} tasyu_{q} \cdot G_{qsvt} \\ \end{aligned}$$
(3)
$$\begin{aligned} FOC = & \mathop \sum \limits_{j} smf_{j} \cdot OP_{j} + \mathop \sum \limits_{m} smto_{m} \cdot OC_{m} + \mathop \sum \limits_{n} smyen_{n} \cdot OYEN_{n} + \mathop \sum \limits_{r} smgr_{r} \cdot OGR_{r} \\ & + \mathop \sum \limits_{q} smyu_{q} \cdot OYU_{q} . \\ \end{aligned}$$
(4)

The first objective function has three components. The transportation cost between the facilities is the first component; the purchasing, production, collection, refurbishing, disposal costs, and cost saving owed to refurbished products are the second component; and the fixed-opening cost of facilities is the third component.

The second objective function (5) of the developed OLSC model minimizes the environmental impact. The generated \({\text{CO}}_{2}\) emission costs caused by transportation and manufacturing can be calculated as follows:

$$\begin{aligned} Min\,Z_{2} = & C_{{CO_{2} }} \cdot [cet_{v} \cdot \\ & \quad \left( {\mathop \sum \limits_{i} \mathop \sum \limits_{j} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{ij} \cdot X_{ijvt} } \right. + \mathop \sum \limits_{j} \mathop \sum \limits_{k} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{jk} \cdot Y_{jkvt} + \mathop \sum \limits_{k} \mathop \sum \limits_{l} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{kl} \cdot Z_{klvt} + \mathop \sum \limits_{l} \mathop \sum \limits_{m} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{lm} \cdot U_{lmvt} \\ & \quad \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{n} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{mn} \cdot A_{mnvt} + \mathop \sum \limits_{m} \mathop \sum \limits_{r} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{mr} \cdot B_{mrvt} + \mathop \sum \limits_{m} \mathop \sum \limits_{q} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{mq} \cdot C_{mqvt} + \mathop \sum \limits_{n} \mathop \sum \limits_{v} \mathop \sum \limits_{t} D_{mvt} \cdot d_{m} \\ \quad \quad + \mathop \sum \limits_{n} \mathop \sum \limits_{s} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{ns} \cdot E_{nsvt} + \mathop \sum \limits_{r} \mathop \sum \limits_{s} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{rs} \cdot F_{rsvt} \left. { + \mathop \sum \limits_{q} \mathop \sum \limits_{s} \mathop \sum \limits_{v} \mathop \sum \limits_{t} d_{qs} \cdot G_{qsvt} } \right) \\ \quad \quad + \left. {\mathop \sum \limits_{j} \mathop \sum \limits_{k} \mathop \sum \limits_{v} \mathop \sum \limits_{t} cep_{j} \cdot Y_{jkvt} } \right]. \\ \end{aligned}$$
(5)

The third objective function (6) expresses the maximum importance level of suppliers through the developed assessment approach:

$$Max Z_{3} = \mathop \sum \limits_{i} \mathop \sum \limits_{j} \mathop \sum \limits_{v} \mathop \sum \limits_{t} w_{i} \cdot X_{ijvt} .$$
(6)

Constraints

Constrains involved in the developed model are presented as follows:

$$\mathop \sum \limits_{j} \mathop \sum \limits_{v} X_{ijvt} \le capt_{it} \quad \forall_{i,t}$$
(7)
$$\mathop \sum \limits_{k} \mathop \sum \limits_{v} Y_{jkvt} \le capf_{jt} \cdot OP_{j} \quad \forall_{j,t}$$
(8)
$$\mathop \sum \limits_{l} \mathop \sum \limits_{v} Z_{klvt} \le capdm_{kt} \quad \forall_{k,t}$$
(9)
$$\mathop \sum \limits_{l} \mathop \sum \limits_{v} U_{lmvt} \le kapto_{mt} \cdot OC_{m} \quad \forall_{m,t}$$
(10)
$$\mathop \sum \limits_{m} \mathop \sum \limits_{v} A_{mnvt} \le capym_{nt} \cdot OYEN_{n} \quad \forall_{n,t}$$
(11)
$$\mathop \sum \limits_{m} \mathop \sum \limits_{v} B_{mrvt} \le capgr_{rt} \cdot OGR_{r} \quad \forall_{r,t}$$
(12)
$$\mathop \sum \limits_{m} \mathop \sum \limits_{v} C_{mqvt} \le capyu_{qt} \cdot OYU_{q} \quad \forall_{q,t}$$
(13)
$$\mathop \sum \limits_{k} \mathop \sum \limits_{v} Z_{klvt} \ge tal_{lt} \quad \forall_{l,t}$$
(14)
$$X_{ij1t} - M \cdot XX_{ij1t} \le 0\quad \forall_{i,j,t}$$
(15)
$$X_{ij1t} - M \cdot XX_{ij1t} \ge 0.001 - M\quad \forall_{i,j,t}$$
(16)
$$X_{ij2t} - M \cdot XX_{ij2t} \le 0\quad \forall_{i,j,t}$$
(17)
$$X_{ij2t} - M \cdot XX_{ij2t} \ge 10 - M\quad \forall_{i,j,t}$$
(18)
$$X_{ij3t} - M \cdot XX_{ij3t} \le 0\quad \forall_{i,j,t}$$
(19)
$$X_{ij3t} - M \cdot XX_{ij3t} \ge 20 - M\quad \forall_{i,j,t}$$
(20)
$$Y_{jk1t} - M \cdot YY_{jk1t} \le 0\quad \forall_{j,k,t}$$
(21)
$$Y_{jk1t} - M \cdot YY_{jk1t} \ge 0.001 - M\quad \forall_{j,k,t}$$
(22)
$$Y_{jk2t} - M \cdot YY_{jk2t} \le 0\quad \forall_{j,k,t}$$
(23)
$$Y_{jk2t} - M \cdot YY_{jk2t} \ge 10 - M\quad \forall_{j,k,t}$$
(24)
$$Y_{jk3t} - M \cdot YY_{jk3t} \le 0\quad \forall_{j,k,t}$$
(25)
$$Y_{jk3t} - M \cdot YY_{jk3t} \ge 20 - M\quad \forall_{j,k,t}$$
(26)
$$Z_{kl1t} - M \cdot ZZ_{kl1t} \le 0\quad \forall_{k,l,t}$$
(27)
$$Z_{kl1t} - M \cdot ZZ_{kl1t} \ge 0.001 - M\quad \forall_{k,l,t}$$
(28)
$$Z_{kl2t} - M \cdot ZZ_{kl2t} \le 0\quad \forall_{k,l,t}$$
(29)
$$Z_{kl2t} - M \cdot ZZ_{kl2t} \ge 10 - M\quad \forall_{k,l,t}$$
(30)
$$Z_{kl3t} - M \cdot ZZ_{kl3t} \le 0\quad \forall_{k,l,t}$$
(31)
$$Z_{kl3t} - M \cdot ZZ_{kl3t} \ge 20 - M\quad \forall_{k,l,t}$$
(32)
$$U_{lm1t} - M \cdot UU_{lm1t} \le 0\quad \forall_{l,m,t}$$
(33)
$$U_{lm1t} - M \cdot UU_{lm1t} \ge 0.001 - M\quad \forall_{l,m,t}$$
(34)
$$U_{lm2t} - M \cdot UU_{lm2t} \le 0\quad \forall_{l,m,t}$$
(35)
$$U_{lm2t} - M \cdot UU_{lm2t} \ge 10 - M\quad \forall_{l,m,t}$$
(36)
$$U_{lm3t} - M \cdot UU_{lm3t} \le 0\quad \forall_{l,m,t}$$
(37)
$$U_{lm3t} - M \cdot UU_{lm3t} \ge 20 - M\quad \forall_{l,m,t}$$
(38)
$$A_{mn1t} - M \cdot AA_{mn1t} \le 0\quad \forall_{m,n,t}$$
(39)
$$A_{mn1t} - M \cdot AA_{mn1t} \ge 0.001 - M\quad \forall_{m,n,t}$$
(40)
$$A_{mn2t} - M \cdot AA_{mn2t} \le 0\quad \forall_{m,n,t}$$
(41)
$$A_{mn2t} - M \cdot AA_{mn2t} \ge 10 - M\quad \forall_{m,n,t}$$
(42)
$$A_{mn3t} - M \cdot AA_{mn3t} \le 0\quad \forall_{m,n,t}$$
(43)
$$A_{mn3t} - M \cdot AA_{mn3t} \ge 20 - M\quad \forall_{m,n,t}$$
(44)
$$B_{mr1t} - M \cdot BB_{mr1t} \le 0\quad \forall_{m,r,t}$$
(45)
$$B_{mr1t} - M \cdot BB_{mr1t} \ge 0.001 - M\quad \forall_{m,r,t}$$
(46)
$$B_{mr2t} - M \cdot BB_{mr2t} \le 0\quad \forall_{m,r,t}$$
(47)
$$B_{mr2t} - M \cdot BB_{mr2t} \ge 10 - M\quad \forall_{m,r,t}$$
(48)
$$B_{mr3t} - M \cdot BB_{mr3t} \le 0\quad \forall_{m,r,t}$$
(49)
$$B_{mr3t} - M \cdot BB_{mr3t} \ge 20 - M\quad \forall_{m,r,t}$$
(50)
$$C_{mq1t} - M \cdot CC_{mq1t} \le 0\quad \forall_{m,q,t}$$
(51)
$$C_{mq1t} - M \cdot CC_{mq1t} \ge 0.001 - M\quad \forall_{m,q,t}$$
(52)
$$C_{mq2t} - M \cdot C_{mq2t} \le 0\quad \forall_{m,q,t}$$
(53)
$$C_{mq2t} - M \cdot CC_{mq2t} \ge 10 - M\quad \forall_{m,q,t}$$
(54)
$$C_{mq3t} - M \cdot CC_{mq3t} \le 0\quad \forall_{m,q,t}$$
(55)
$$C_{mq3t} - M \cdot CC_{mq3t} \ge 20 - M\quad \forall_{m,q,t}$$
(56)
$$D_{m1t} - M \cdot DD_{m1t} \le 0\quad \forall_{m,t}$$
(57)
$$D_{m1t} - M \cdot DD_{m1t} \ge 0.001 - M\quad \forall_{m,t}$$
(58)
$$D_{m2t} - M \cdot DD_{m2t} \le 0\quad \forall_{m,t}$$
(59)
$$D_{m2t} - M \cdot DD_{m2t} \ge 10 - M\quad \forall_{m,t}$$
(60)
$$D_{m3t} - M \cdot DD_{m3t} \le 0\quad \forall_{m,t}$$
(61)
$$D_{m3t} - M \cdot DD_{m3t} \ge 20 - M\quad \forall_{m,t}$$
(62)
$$E_{ns1t} - M \cdot EE_{ns1t} \le 0\quad \forall_{n,s,t}$$
(63)
$$E_{ns1t} - M \cdot EE_{ns1t} \ge 0.001 - M\quad \forall_{n,s,t}$$
(64)
$$E_{ns2t} - M \cdot EE_{ns2t} \le 0\quad \forall_{n,s,t}$$
(65)
$$E_{ns2t} - M \cdot EE_{ns2t} \ge 10 - M\quad \forall_{n,s,t}$$
(66)
$$E_{ns3t} - M \cdot EE_{ns3t} \le 0\quad \forall_{n,s,t}$$
(67)
$$E_{ns3t} - M \cdot EE_{ns3t} \ge 20 - M\quad \forall_{n,s,t}$$
(68)
$$F_{rs1t} - M \cdot FF_{rs1t} \le 0\quad \forall_{r,s,t}$$
(69)
$$F_{rs1t} - M \cdot FF_{rs1t} \ge 0.001 - M\quad \forall_{r,s,t}$$
(70)
$$F_{rs2t} - M \cdot FF_{rs2t} \le 0\quad \forall_{r,s,t}$$
(71)
$$F_{rs2t} - M \cdot FF_{rs2t} \ge 10 - M\quad \forall_{r,s,t}$$
(72)
$$F_{rs3t} - M \cdot FF_{rs3t} \le 0\quad \forall_{r,s,t}$$
(73)
$$F_{rs3t} - M \cdot FF_{rs3t} \ge 20 - M\quad \forall_{r,s,t}$$
(74)
$$G_{qs1t} - M \cdot GG_{qs1t} \le 0\quad \forall_{q,s,t}$$
(75)
$$G_{qs1t} - M \cdot GG_{qs1t} \ge 0.001 - M\quad \forall_{q,s,t}$$
(76)
$$G_{qs2t} - M \cdot GG_{qs2t} \le 0\quad \forall_{q,s,t}$$
(77)
$$G_{qs2t} - M \cdot GG_{qs2t} \ge 10 - M\quad \forall_{q,s,t}$$
(78)
$$G_{qs3t} - M \cdot GG_{qs3t} \le 0\quad \forall_{q,s,t}$$
(79)
$$G_{qqs3t} - M \cdot GG_{qs3t} \ge 20 - M\quad \forall_{q,s,t}$$
(80)
$$XINT_{ijvt} \ge \left( {X_{ijvt} /u_{v} } \right)\quad \forall_{i,j,v,t}$$
(81)
$$YINT_{jkvt} \ge \left( {Y_{jkvt} /u_{v} } \right)\quad \forall_{j,k,v,t}$$
(82)
$$ZINT_{klvt} \ge \left( {Z_{klvt} /u_{v} } \right)\quad \forall_{k,l,v,t}$$
(83)
$$UINT_{lmvt} \ge \left( {U_{lmvt} /u_{v} } \right)\quad \forall_{l,m,v,t}$$
(84)
$$AINT_{mnvt} \ge \left( {A_{mnvt} /u_{v} } \right)\quad \forall_{m,n,v,t}$$
(85)
$$BINT_{mrvt} \ge \left( {B_{mrvt} /u_{v} } \right)\quad \forall_{m,r,v,t}$$
(86)
$$CINT_{mqvt} \ge \left( {C_{mqvt} /u_{v} } \right)\quad \forall_{m,q,v,t}$$
(87)
$$DINT_{mvt} \ge \left( {D_{mvt} /u_{v} } \right)\quad \forall_{m,v,t}$$
(88)
$$EINT_{nsvt} \ge \left( {E_{nsvt} /u_{v} } \right)\quad \forall_{n,s,v,t}$$
(89)
$$FINT_{rsvt} \ge \left( {F_{rsvt} /u_{v} } \right)\quad \forall_{r,s,v,t}$$
(90)
$$GINT_{qsvt} \ge \left( {G_{qsvt} /u_{v} } \right)\quad \forall_{q,s,v,t}$$
(91)
$$\mathop \sum \limits_{i} \mathop \sum \limits_{v} X_{ijvt} - \mathop \sum \limits_{k} \mathop \sum \limits_{v} Y_{jkvt} = 0\quad \forall_{j,t}$$
(92)
$$\mathop \sum \limits_{j} \mathop \sum \limits_{v} Y_{jkvt} - \mathop \sum \limits_{l} \mathop \sum \limits_{v} Z_{klvt} = 0\quad \forall_{k,t}$$
(93)
$$\pi \cdot \left( {\mathop \sum \limits_{k} \mathop \sum \limits_{v} Z_{klvt} } \right) - \mathop \sum \limits_{m} \mathop \sum \limits_{v} U_{{lmv\left( {t + 1} \right)}} = 0\quad \forall_{l,t}$$
(94)
$$\mathop \sum \limits_{l} \mathop \sum \limits_{v} U_{lmvt} - \left( {\mathop \sum \limits_{n} \mathop \sum \limits_{v} A_{mnvt} + \mathop \sum \limits_{r} \mathop \sum \limits_{v} B_{mrvt} + \mathop \sum \limits_{q} \mathop \sum \limits_{v} C_{mqvt} + \mathop \sum \limits_{v} D_{mvt} } \right) = 0\quad \forall_{m,t}$$
(95)
$$\alpha \cdot \mathop \sum \limits_{l} \mathop \sum \limits_{v} U_{lmvt} - \mathop \sum \limits_{n} \mathop \sum \limits_{v} A_{mnvt} = 0\quad \forall_{m,t}$$
(96)
$$\beta \cdot \mathop \sum \limits_{l} \mathop \sum \limits_{v} U_{lmvt} - \mathop \sum \limits_{r} \mathop \sum \limits_{v} B_{mrvt} = 0\quad \forall_{m,t}$$
(97)
$$\theta \cdot \mathop \sum \limits_{l} \mathop \sum \limits_{v} U_{lmvt} - \mathop \sum \limits_{q} \mathop \sum \limits_{v} C_{mqvt} = 0\quad \forall_{m,t}$$
(98)
$$\lambda \cdot \mathop \sum \limits_{l} \mathop \sum \limits_{v} U_{lmvt} - \mathop \sum \limits_{v} D_{mvt} = 0\quad \forall_{m,t}$$
(99)
$$\mathop \sum \limits_{m} \mathop \sum \limits_{v} A_{mnvt} - \mathop \sum \limits_{s} \mathop \sum \limits_{v} E_{nsvt} = 0\quad \forall_{n,t}$$
(100)
$$\mathop \sum \limits_{m} \mathop \sum \limits_{v} B_{mrvt} - \mathop \sum \limits_{s} \mathop \sum \limits_{v} F_{rsvt} = 0\quad \forall_{r,t}$$
(101)
$$\mathop \sum \limits_{m} \mathop \sum \limits_{v} C_{mqvt} - \mathop \sum \limits_{s} \mathop \sum \limits_{v} G_{qsvt} = 0\quad \forall_{q,t}$$
(102)
$$X_{ijvt} ,Y_{jkvt} ,Z_{klvt} ,U_{lmvt} ,A_{mnvt} ,B_{mrvt} ,C_{mqvt} ,D_{mvt} ,E_{nsvt} ,F_{rsvt} ,G_{qsvt} , \ge 0\quad \forall_{i,j,k,l,m,n,r,q,n,s,v,t}$$
(103)
$$\begin{aligned} & OP_{j} ,OC_{m} ,OYEN_{n} ,OGR_{r} ,OYU_{q} ,XX_{ijvt} ,YY_{jkvt} ,ZZ_{klvt} ,UU_{lmvt} ,AA_{mnvt} , \\ & \quad BB_{mrvt} ,CC_{mqvt} ,DD_{mvt} ,EE_{nsvt} ,FF_{rsvt} ,GG_{qsvt} = \left\{ {0,1} \right\}\quad \forall_{i,j,k,l,m,n,r,q,n,s,v,t} . \\ \end{aligned}$$
(104)

Constraints (7)–(13) are the capacity constraints on facilities that the total amount of products sent from facilities to others should be equal to or less than the capacity of these facilities for any period. Constraint (14) ensures the demand of customers. Constraints (15)–(80) are the transportation constraints that provide the minimum level of load for each type of vehicle (M being a large number). Constraints (81)–(91) evaluate the number of tours at each arc regarding vehicle types. Constraints (92)–(102) are the balance equations for facilities. Constraint (103) preserves the nonnegativity restriction, and constraint (104) enforces the binary restriction of decision variables.

Proposed assessment procedure for sustainable supplier selection

In this study, a group decision-making approach based on IT2FSs is utilized which combines the judgments of the decision-makers according to the selected criteria [24,25,26]. To evaluate suppliers, a model based on triangular and/or trapezoidal fuzzy numbers and linguistic variables was developed in those studies. In this section, a method is suggested based on the proposed methods but using interval type-2 fuzzy numbers. The proposed method consists of three basic steps: (1) Determining the criteria and linguistic variables to be used in the model; (2) calculating the relative importance of the criteria; and (3) evaluating suppliers by decision-makers and calculating the evaluation scores. The decision-makers obtained the proposed solution by considering the following five steps:

  • Step 1 Qualitative and quantitative criteria to be used in the evaluation phase are determined.

  • Step 2 The weights of the criteria are calculated.

  • Step 3 Determine whether the opinions of the decision-makers are consistent.

  • Step 4 Under each criterion, suppliers are evaluated using interval type-2 fuzzy numbers and an assessment score is calculated for suppliers.

  • Step 5 The final scores of the suppliers are obtained by multiplying the weights obtained in the previous steps and the evaluation points.

The complexity of real-world problems and the optimization of multiple conflicting objectives that simultaneously under certain constraints have led to the development of multi-objective linear programming models. In order to solve the proposed multi-objective model, fuzzy programming approaches, such as Zimmermann [27], the ZIM method thereafter, Torabi and Hassini [28], the TH method thereafter, and F-WAMG approach [29], are utilized in this paper.

Computational experiment

Numerical example

To demonstrate the validity of the developed multi-objective linear programming model in supply chain network design and the effectiveness of the proposed solution methodology, a numerical experiment is performed in this section. It is assumed that there exist four suppliers, three plants, four distribution centers, four customers, three collection centers, two refurbishing centers, two recycling centers, two remanufacturing centers, two secondary markets, and three vehicles. All parameters for the proposed OLSC network are obtained through MATLAB software using a uniform distribution, and the relevant distributions are summarized in Table 2. In addition, three-period planning is considered in the OLSC. The other parameters defined are as follows: \(c_{v}\) 30, 20, and 10; \(dc = 50\); \(C_{{CO_{2} }} = 0.112\); \(cet_{v}\) 800, 1350, and 2450; \(u_{v}\) 10, 20, and 30; \(cep_{j}\) 60,331, 85,494, and 92,896; \(\pi = 0.80\); \(\alpha = 0.30\); \(\beta = 0.30\); \(\theta = 0.30\); \(\lambda = 0.10\). CPLEX 12.6, commonly used in optimization software to solve the developed OLSC mixed-integer linear programming model, is implemented on a computer with Intel Core i3 3.30 GHz and 4.00 GB RAM to solve the test problem.

Table 2 Model parameters

Results and analysis

To obtain the weights of suppliers in the third objective function, it aims to evaluate suppliers by considering the sustainability dimensions as well as the product flow in the network. With the help of the available literature [15, 30,31,32], an expert committee of three decision-makers is established for the evaluation of suppliers according to different criteria. The evaluation criteria are grouped under three dimensions as economic, social, and environmental and named as “main criteria.” Sub-criteria are then placed into dimensions corresponding to the main criteria. The main and sub-criteria are shown in Table 3.

Table 3 Evaluation criteria

At first, criteria weights are determined using the interval type-2 AHP method. Three experts evaluated the criteria of sustainable supplier selection by considering the scale in Table 4, and as a result of the pairwise comparison of the criteria, the evaluations for the main criteria and sub-criteria are acquired as in Tables 5 and 6. The fuzzy weights of the criteria are calculated using the interval type-2 fuzzy AHP methodology, and the results are presented in Table 7.

Table 4 Linguistic variables for the importance level of weight for each criterion
Table 5 Pairwise comparison matrix for main criteria
Table 6 Pairwise comparison matrix for sub-criteria
Table 7 Interval type-2 fuzzy weights for criteria

After determining the weights of the criteria, suppliers are evaluated using interval type-2 fuzzy numbers. The linguistic terms used to rank suppliers are given in Table 8. The linguistic assessments given by the decision-makers to each of the four suppliers under each sub-criterion are shown in Table 9. According to experts’ evaluation, the aggregated interval type-2 fuzzy numbers and assessment scores determined by gathering the opinions of the decision-makers are formed as in Table 10. The numerical data are then normalized and defuzzified using the center of area (CoA) method [33], and the crisp data are shown in Table 10. The weights obtained from the interval type-2 fuzzy AHP method are multiplied by the evaluation scores, and the final scores of the suppliers are calculated and indicated in Table 11.

Table 8 Linguistic variables for the assessments of suppliers
Table 9 Linguistic assessments for the four suppliers
Table 10 Aggregated assessment scores of both interval type-2 fuzzy and crisp values
Table 11 Final scores of suppliers

When the developed multi-objective OLSC model is solved separately for each objective, it is seen that the highest cost among the objective functions is the transportation cost. For example, when economic costs are minimized, 54% of the total cost belongs to the transportation cost; besides, if the objective function is maximized, 72% of the total cost is actualized by this cost. When environmental costs are minimized, transportation costs decrease to 47%, and when if it is maximized, it decreases to 69%. Considering environmental costs, the total cost can be reduced. When environmental costs are minimized, the total cost is 34,963,605.8 $, a 9.87% reduction compared to the highest cost.

After the final scores of the suppliers are determined, the fuzzy approaches are applied to solve the proposed multi-objective OLSC model. First, the upper (\(Z^{ + }\)) and the lower (\(Z^{ - }\)) bounds of each objective function are determined (Table 12). Then, a linear membership function is created for the purpose functions, and finally the multi-objective OLSC model is transformed into a single-objective model.

Table 12 Payoff table
$$\mu_{1} \left( {Z_{1} \left( x \right)} \right) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {Z_{1} \left( x \right) \le 17990123} \hfill \\ {\frac{{260743861 - Z_{1} \left( x \right)}}{260743861 - 17990123},} \hfill & {17990123 \le Z_{1} \left( x \right) \le 260743861 } \hfill \\ {0,} \hfill & {Z_{1} \left( x \right) \ge 260743861} \hfill \\ \end{array} } \right.$$
(105)
$$\mu_{2} \left( {Z_{2} \left( x \right)} \right) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {Z_{2} \left( x \right) \le 15693344.80} \hfill \\ {\frac{{101254381.76 - Z_{2} \left( x \right)}}{101254381.76 - 15693344.80},} \hfill & {15693344.80 \le Z_{2} \left( x \right) \le 101254381.76} \hfill \\ {0,} \hfill & {Z_{2} \left( x \right) \ge 101254381.76} \hfill \\ \end{array} } \right.$$
(106)
$$\mu_{3} \left( {Z_{3} \left( x \right)} \right) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {Z_{3} \left( x \right) \ge 458.37} \hfill \\ {\frac{{Z_{3} \left( x \right) - 1861 }}{458.37 - 1861},} \hfill & {1861 \le Z_{3} \left( x \right) \le 458.37} \hfill \\ {0,} \hfill & {Z_{3} \left( x \right) \le 1861 } \hfill \\ \end{array} } \right..$$
(107)

The formulation of the ZIM approach is as follows:

$${{Max}}\;\alpha$$
$$\alpha \le \mu_{1} \left( {Z_{1} \left( x \right)} \right) = \frac{{260743861 - Z_{1} \left( x \right)}}{260743861 - 17990123}$$
$$\alpha \le \mu_{2} \left( {Z_{2} \left( x \right)} \right) = \frac{{101254381.76 - Z_{2} \left( x \right)}}{101254381.76 - 15693344.80}$$
$$\alpha \le \mu_{3} \left( {Z_{3} \left( x \right)} \right) = \frac{{Z_{3} \left( x \right) - 1861 }}{458.37 - 1861}.$$

C7-C104

$$0 \le \alpha \le 1.$$
(108)

Then, the integrated OLSC model formulated by TH approach based on the weights of 0.383, 0.483, and 0.179 and \(\gamma = 0.6\) is as follows:

$$Max \gamma \lambda + \left( {1 - \gamma } \right)\left[ {0.383.\mu_{1} \left( {Z_{1} \left( x \right)} \right) + 0.483.\mu_{2} \left( {Z_{2} \left( x \right)} \right) + 0.179.\mu_{3} \left( {Z_{3} \left( x \right)} \right)} \right]$$
$$\lambda \le \mu_{1} \left( {Z_{1} \left( x \right)} \right) = \frac{{260743861 - {{Z}}_{1} \left( {{x}} \right)}}{260743861 - 17990123}$$
$$\lambda \le \mu_{2} \left( {Z_{2} \left( x \right)} \right) = \frac{{101254381.76 - {{Z}}_{2} \left( {{x}} \right)}}{101254381.76 - 15693344.80}$$
$$\lambda \le \mu_{3} \left( {Z_{3} \left( x \right)} \right) = \frac{{{{Z}}_{3} \left( {{x}} \right) - 1861 }}{458.37 - 1861}.$$

(C7-C104)

$$\gamma,\lambda \epsilon \left[{0,1} \right].$$
(109)

The results according to the different approaches are given in Table 13. Table 13 shows that objective function and satisfaction levels differ according to methods. The results indicate that satisfaction levels are increased with the use of weighted approaches and that decision-makers can choose the appropriate solutions according to their objectives.

Table 13 Satisfaction degree and order allocation of objectives

The share of suppliers on the basis of parts is shown in Fig. 2. For the problem being addressed, according to the consensus solution, in the ZIM approach, the Supplier 4 holds 67% of the total parts transferred to factories. Similarly, in the TH approach, the Supplier 1 delivers 66% of the parts, and in the F-WAMG approach, the Supplier 2 acquires 88% of the parts. Suppliers 1 and 2 are being found to be the most preferred suppliers in the proposed assessment approach. By using weighted approaches, it is determined that they have become effective in the distribution in-network and that these suppliers adequately assume the entire supply.

Fig. 2
figure2

Share of the parts procured at each supplier

Conclusion

In this study, a multi-objective OLSC model is proposed by considering sustainable supplier selection. The model presented has two features: First, environmental factors are included in the model, which includes environmental impacts from vehicles as well as different cost and traditional cost components. Secondly, the performances of the suppliers are evaluated considering the sustainability dimensions and included in the developed model. Initially, the relevant criteria and sub-criteria are determined according to the literature and the opinions of the experts. The interval type-2 fuzzy AHP method is then used to weight the selected criteria. Subsequently, the performance evaluation of suppliers under each sub-criterion is implemented and the final scores of the suppliers are calculated by considering the weights found in the previous step.

The multi-objective OLSC model has been transformed into a single-objective model by using three different fuzzy programming approaches to reach compromise solutions for the developed objectives. The changes in total cost, environmental cost, and weight of suppliers between objectives are analyzed. Using weighted solution approaches, it is determined that the majority of the parts sent to the factories are procured by the selected suppliers.

For future studies, the model can be adapted to real-world problems. At the same time, the model can be expanded taking into account the uncertainty in parameters. Also, the effectiveness of the proposed assessment approach can be tested with other methods using heuristic and meta-heuristic algorithms for multidimensional network design models.

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Çalık, A. An integrated open-loop supply chain network configuration model with sustainable supplier selection: fuzzy multi-objective approach. SN Appl. Sci. 2, 405 (2020). https://doi.org/10.1007/s42452-020-2200-y

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Keywords

  • Interval type-2 fuzzy sets
  • Multi-objective optimization
  • Open-loop supply chain
  • Sustainable supplier selection

Mathematical Subject Classification

  • 03E72
  • 90B10
  • 90B50
  • 90C29
  • 90C70

JEL Classification

  • C44
  • C61
  • D81
  • R41