Designing a closed-loop supply chain network considering multi-task sales agencies and multi-mode transportation

Abstract

Current national and international regulations, along with growing environmental concerns, have deeply influenced the design of supply chain networks. These decisions stem from the fact that decision-makers try to design the supply chain network to align with their economic and environmental objectives. In this paper, a new closed-loop supply chain network with sales agency and customers is formulated. The proposed model has four echelons in the forward direction and five echelons in the backwards direction. The model not only considers several constraints from previous studies, but also addresses new constraints in order to better explore real-life problems that employ different transportation modes and that rely on sale agency centers. The objective function is to maximize the total profit. In addition, this study firstly considers distinct cluster of customers based on the product life cycle. These customers are utilized in different levels of the proposed network in order to purchase the final products, returned products, and recycled products. The structure of the model is based on linear mixed-integer programming, and the proposed model has been investigated through a case study regarding the manufacturing industry. To verify the model efficiency, a set of metaheuristics and hybrid algorithm are applied in various test problems along with a data from a real-world case study in a building construction industry. The findings of the proposed network illustrated that using the attributes of sale agency centers and clusters of customers both increase the problem total revenue and the number of the collected returned products.

1 Introduction

The prevention of failure, survival, permanence, competition, growth, and achieving more market share are the strategic purposes of decision-makers (DMs) as they confront different situations (Amiri et al. 2020; Liao et al. 2020; Govindan et al. 2015). In order to achieve these objectives, DMs understand they have to control financial processes (including costs and revenues) and increase their adaptations; therefore, they design and manage supply chain based on situations, purposes, and conditions (Amiri et al. 2020).

Supply chains integrate products and service flows from suppliers to customers in a way that not only contributes to the production planning and inventory control, but also manages distribution and logistics processes (Beamon 1998; Bowersox et al. 2007). In this regard, DMs design their supply chain network by considering decision variables that include supplier or buyer relationships, facility location, production planning, inventory control, and logistics to minimize cost and to increase revenue (Beamon 1998).

Globalization, changes in the economy, and progresses in technology have led organizations to realize that if they want to survive and preserve their industries, they should pay attention to the customer’s satisfaction (Ramezani et al. 2014a). Hence, both industries and decision-makers should pay great attention toward customer’s needs since these needs are constantly changing over time. Identifying correct and timely needs of the target markets includes learning about customers’ geography, culture, customs, expectations, and so forth; it is one of the most important objectives of each company (Cheraghalipour et al. 2019). Managing product life cycle, increase in product costs, and decrease in resources have forced companies to utilize strategic decisions. Designing a supply chain (SC) network as a strategic decision has a considerable and long effect on tactical and operational decisions (Beamon 1998). However, due to pressure from the government’s Environmental Protection Agency (EPA) policies regarding the protection of primary resources, recycling, the management of product waste, and with enhanced customer requirements for green products, specific considerations to reverse logistics in the supply chain have dramatically increased (Srivastava 2007). Therefore, an effective plan is needed especially for managers to both control the product flow and gain the customers satisfaction.

The products and service flows in reverse supply chain (RSC) are designed from customers to suppliers; companies often prefer to design RSC as a competitive step to decrease environmental problems (Cheraghalipour et al. 2018). In 2018, Braz et al. (2018) illustrated that environmental problems that cause an increase in emissions, waste, and consumption of natural resources can be solved by enhancing the rate of returned products. All supply chain participants (from suppliers to customers) and their behavior play a significant role in RSC performance and the value of returned products (Alshamsi and Diabat 2015; Mokhtar et al. 2019). Therefore, a supply chain should consider all participants in the product flow and product return method (Alshamsi and Diabat 2015). Shaharudin et al. (2017) showed the importance of utilizing closed-loop supply for manufacturing facilities, addressed the role of implementing such a network for industry profitability, and highlighted the benefits of returned products. A closed-loop supply chain enables us to optimize forward and backward directions, simultaneously (Beamon 1998; Mokhtar et al. 2019; Pishvaee and Torabi 2010). Hence, addressing numerous and also newly needs of various industries is directly involved with an efficient closed-loop supply chain.

The prosperity in supply chain entails the need for consideration of numerous criteria. Adoption of investment strategy in the field of improving environmental operation has its own advantages and benefits, such as enhanced characteristic of suppliers, resources consumption, organizing returned products, addressing the potential demands of customers in backward and forward flow, and in general, it emphasizes promotion and efficiency for organizations (Golmohamadi et al. 2017). As a consequence, many companies started to produce, manage, and observe product flow due to customers’ expectations and obtain their diverse benefits. Abdi et al. (2020), Shaharudin et al. (2019) considered the effects of using closed-loop supply chain network on green capabilities and on product recovery. According to Boons (2002), there are three basic steps for considering products flow: seeing the production chain as a network of actors, surveying the options available to reduce the ecological impact of a product, and validating assumptions about the behavior of actors in the production chain.

The supplier’s major role in making products or services has led companies to follow strict decisions for supplier selection and the procurement of raw materials (Mokhtar et al. 2019). Another factor impacting decision-making is the non-exclusivity of many raw materials. A common objective is to increase production quality and raw material procurement services to lead to higher quality products (Navid Akbarpour et al. 2020a, b; Ramezani et al. 2014b). On the other hand, other producers seek to reduce quality costs appropriately, manage the needs and expectations of the target market, and introduce flexibility and competition in the chain. Thus, cost-driven supply chain networks largely depend on customer demands from products and services (Hajiaghaei-Keshteli and Sajadifar 2010; Hajiaghaei-Keshteli et al. 2011; Yadegari et al. 2012; Yazdani et al. 2014).

Contributing to the products life cycle and their return process usually increases the overall costs in a relatively short time (Fathollahi-Fard et al. 2019). As a result, managers and industries who are committed to product life cycle in their strategic decisions to reduce cost in their organization and also improve their processes, should consider customers as a vital participant of the supply chain to satisfy their potential demands. Customers need to recognize the value of return products and the special features of the reverse direction (Devika et al. 2014; Fathollahi-Fard et al. 2020). Besides, many customers buy a product according to the brand, background, value, and reliability of the product. That’s why the benefits of closed-loop supply chain network increase accordingly through recovery and reselling new products (Schenkel et al. 2019). As a result, both the characteristic and the benefits of different opened centers must be identified before assigning any monetary resources.

Usually, organizations with various customers and more demands are more flexible in the manufacturing and production of different products because of their presence in different markets (Diabat and Govindan 2011). These organizations improve the production process and produce cheaper products because of the advantages they have. Therefore, other companies with fewer customers should understand the customer’s basic needs and focus on some products so that they can survive in competition with both existing and new companies. All companies should improve both their processes of production and CRM methods in order to remain successful in the marketplace. In this regard, using techniques such as designing an effective supply chain network alongside with attention to the product life cycle help to retain loyal customers and to win the trust of new customers. With increased confidence and a strong reputation with customers and society, the company’s commitment to these processes is more achievable and the increased cost of production is more comprehensible (Amin and Zhang 2013; Diabat and Govindan 2011). The contribution of the current work is as follows:

• Designing an innovative closed-loop supply chain network to address different cluster of customers and optimize the total profit,

• Categorizing customers based on real life cycle of the products,

• Considering multi-task centers and multi-mode transportation,

• Applying old and newly invented metaheuristics and hybrid algorithm.

The remainder of this paper is organized as follows. In Sect. 2, a pertinent literature review about designing a supply chain network is presented. Section 3 describes the problem and provides a comprehensive closed-loop supply chain model for integration of forward and reverse flows; it employs the approach of a trade-off between environmental protection and increasing profit. Solution approach along with encoding and decoding plan and utilized metaheuristics are defined in Sect. 4. Section 5 provides computational results for considered metaheuristics. A case study on a building construction supply chain in Iran’s Mazandaran province is presented and compared with a study by Ramezani et al. (2014b) in Sect. 6. Section 7 represents a thorough sensitivity analysis and discussion on various parameters. Finally, Sect. 8 concludes the paper with a discussion on potential opportunities for future research.

2 Literature review

Generally, existing models in the field of a closed-loop supply chain (CLSC) design have cost minimization or profit maximization as their primary objective functions differ according to the product type services and the priorities of the decision-makers. The following subsections provide the most relative previous works in this manner.

2.1 CLSC network with the aim of cost minimization

Amin and Zhang (2013) presented a multi-objective closed-loop supply chain network under certain demand. Their proposed model is designed for multiple plants, demand markets, collection centers, and products, and it uses mixed-integer programming with the aim of minimizing the total cost. Özceylan et al. (2014) presented a nonlinear model for designing a closed-loop supply chain in a deterministic environment for balance isolation and supply lines. The model has four forward levels (including suppliers, assembly, retailers, and final customers) and four backward levels (including collection centers, refining, separation, and disposal). Devika et al. (2014) introduced a mixed linear model for designing resistant closed-loop supply chain in a deterministic environment. The model has four levels containing supply, production, distribution, and the customers in a forward direction, and it has six levels of collection, recycling, rebuilding, refinement, disposal, and secondary customers. For solving the model, the combination of three metaheuristic algorithms is used. The algorithms are neighborhood search, imperialistic competitive algorithm, and new hybrid algorithm. Pishvaee and Torabi (2010) suggested a model having two objective functions for a closed-loop supply chain network in a nondeterministic environment. The first objective function minimizes costs, and the second one minimizes the delivery time of products and services. The model has three levels of production, distribution, and customers in the forward direction, and four levels of collection, rebuilding, recycling, and secondary customers. Giri and Sharma (2015) presented two multi-period and single-period closed-loop supply chain models considering production and reproduction based on inventory lot and reordering point. The objective of their models is minimizing costs. Pishvaee et al. (2012) suggested a model with two objective functions for a closed-loop supply chain network in a nondeterministic environment. The first objective function minimizes costs, and the second one minimizes the amount of carbon emission; however, the model has three levels containing production, distribution, and customers. Fahimnia et al. (2013) proposed a closed-loop supply chain model considering levels of production and distribution in a forward direction and collection, recycling, and disposal levels in reverse direction. In this model, the effects of the carbon tax on the economic and environmental situation in a closed-loop supply chain are discussed. A bi-objective closed-loop optimization model is proposed by Zhen et al. (2019). The model considers not only sustainability but also greenness within a proposed network. Using a closed-loop supply chain that considers remanufacturing of products leads to minimizing the total system costs. A closed-loop multi-objective supply chain network proposed by Fathollahi-Fard and Hajiaghaei-Keshteli (2018) simultaneously minimizes supply chain costs and environmental effects. Ruiz-Torres et al. 2019) considered this fact and developed a new model that considers returns from multiple sources. A new optimization model considering reverse logistics was proposed by Kim et al. 2018). The developed closed-loop supply chain network considers not only production planning but also recycling to avoid environmental pollution. Nayeri et al. (2020) developed multi-objective model with the aim of minimizing the total cost and maximizing the social impacts. The considered network optimized the number of opened facilities and the products total flow. Accorsi et al. (2020) considered the objective function of minimization of the total cost. This paper considered the aspects of the strategical and tactical decisions. By developing multi-scenario, the authors concluded that the proposed network can significantly improve the profitability by applying on real packaging case problem. By reviewing the literature mentioned in this section, many studies have worked in the field of the reverse supply chain. Some studies have considered two or several levels in forward or backward direction, and other studies include multi-objective functions. All seek to reduce the cost of the supply chain network defined in their model.

2.2 CLSC network with the aim of maximizing profit

Yadegari et al. (2012) presented a multi-product, multi-echelon model for designing the integrated closed-loop network in a deterministic environment. The objective function of the problem is maximizing the total profit of the chain; the mixed-integer linear programming model uses combined centers of distribution and collection for reducing costs. Yazdani et al. (2014) presented a multi-product, multi-echelon model for designing a closed-loop supply chain in a deterministic environment. In their study, producers are divided into two groups of reliable and unreliable distribution centers. As a result, in case of shortages, the distribution centers must resolve their needs from other reliable distribution centers. A multi-objective model for closed-loop supply chain optimization is presented by Rezaee et al. (2017). The authors used a discount policy to encourage customers to buy products and to maximize the total profit of the supply chain network. Garg et al. (2014) suggested two objective, multi-echelon, multi-period models for designing a closed-loop supply chain in a deterministic environment. The first objective function seeks profit maximization, and the second one minimizes carbon emissions. Ramezani et al. (2014b) presented a model for designing a closed-loop, multi-product, and multi-period in a fuzzy environment. The model has the levels of suppliers, manufacturers, distributors, and customers in the forward direction and levels of recycling, collection centers, and disposal centers in the backward direction. The model has three objective functions of maximizing the total profit of the supply chain, maximizing customer service through minimizing transportation time, and maximizing the quality level through maximizing the sigma level. Bottani et al. (2015) examined a model with multi-objective optimization of CLSA asset management including a pallet supplier, manufacturer, and seven retailers based on economic order quantity (EOQ) policy. Govindan and Popiuc (2014) studied a closed-loop supply chain having two- and three-echelon reverse logistics in computer technology, and they optimized both two- and three-echelon models using a subscription of profit to profit. Cardoso et al. (2013) presented a closed-loop supply chain considering step and forward arrangement in both forward and reverse directions. The proposed network has levels of production, assembly, distribution, and the market in the forward direction and levels of de-assembly, redistribution, and disposal in the backward direction. They improved the inventory level to maximize the total profit of the chain. In this regard, a reverse recycling supply chain network for waste cooking oil with a discount contract was considered by Yang et al. (2018). Wang et al. (2019) developed a closed-loop manufacturer–retailer supply chain network that considers not only product recovery but also recycling. In addition, they analyzed the effect of donations on operations management. (Heydari and Ghasemi (2018) adopted a revenue-sharing contract to develop a two-echelon reverse supply chain that included a single remanufacturer and a single collector. They focused on returned products and the proposed coordination model. Bhattacharya et al. (2018) considered using remanufacturing and returned products in their proposed multistage closed-loop supply chain network. The objective function was to maximize the profitability for the entire network. To deal with the informal collection of waste which interrupts the normal order of end-of-life products, Li et al. (2017) proposed using an efficient recycling center in a reverse supply chain network. Diabat and Jebali presented multi-product, multi-period CLSC network under take-back legislation. They proposed a MIP model in order to consider durable products in their given lifespan. They showed that utilizing the proposed approach can result an increase in the total profit of the supply chain network and collected return product. In order to increase the total profit of supply chain, Samuel et al. (2020) introduced CLSC network considering presorting, return quality, and environmental factors. By considering such a network, the authors decided on the best policy to improve the model profitability.

In the previous section, the structures of several studies were briefly presented. As it has been pointed out, most models in the field of designing closed-loop supply chain are mixed linear programming and they have utilized supply chain structures in both deterministic and nondeterministic environments. Some studies have objectives not only of minimizing the costs or maximizing the profit, but also of reducing environmental impacts caused by production or transportation, reducing social effects, or improving delivery service and reducing transportation costs (Brandenburg et al. 2014; Rezapour et al. 2015; Schenkel et al. 2019; Srivastava 2007). In addition to the mentioned topics, Table 12 (see “Appendix”) shows a useful summary of the various studies using the checklist. In this detailed checklist, it is easy to compare the application and applied approach of diverse studies in the field of CLSC.

According to the works cited in the introduction and literature review, although many previous works have addressed designing CLSC network, the attention toward both multi-task centers and distinct clusters of customers is quite limited. To the best of our knowledge, there is no study to consider these issues simultaneously. What difference the distinct clusters of customers is make it to the proposed problem is that, based on the product life cycle, different types of customers can purchase their desire products due to their requirements. Considering this supposition in the proposed network not only involve more potential customers, but also make it closer to the real-world problems. In addition, there is a gap between these works and real-world problems. Another point is identified as different functions that exist with each supply chain partner which means that utilizing multi-task centers turn old previous centers to a more effective and capable one. The most contributive benefits of such a center are:

• Decreasing the opening facility cost,

• Decreasing the number of workers,

• Balancing the transportation,

• Decreasing the overall costs of the supply chain by opening one multi-task center instead of opening multiple centers.

Hence, this study tries to design a closed-loop supply chain model that not only applies to unstable economic conditions, but also considers levels for recycling and remanufacturing to increase the total profit of the supply chain network. This increase in total profit arises from selling returned products based on their quality to each type of customer. In a nutshell, most studies have considered returned products (backward direction) as a function of cost in designing a supply chain network. Accordingly, in this paper, returned products are firstly considered as a competitive advantage for profitability and an untapped presence in new markets. Therefore, different types of customers have been considered. In addition, we focus on spare parts and supply chain capability for the collection of returned products. That’s why sales agency centers have been utilized with the aim of marketing, customer service, spare parts sale market, and returned products collection. To the best of our knowledge, there is no research in this area that has considered multi-echelon, multi-raw material, multi-product, multi-transportation type, and multi-period concerns simultaneously. Furthermore, the review of literature, including the recent literature review papers by Govindan et al. (2013), Schenkel et al. (2019), demonstrates that ample studies have implemented their proposed supply chain network in an economically stable environment, but, in this research study, the proposed model is implemented in unstable economic conditions.

3 Problem statement

The proposed problem presents an objective function to maximize the total profit of the supply chain. This supply chain includes suppliers, production and distribution centers, customers, and sales agencies in the forward direction; the reverse direction includes collection and inspection, recycling, disposal centers, and secondary customers.

The structure of the closed-loop logistics network is shown in Fig. 1. In the forward direction, suppliers are responsible to provide raw materials for manufacturers. The manufacturers produce the final products and spare parts and distribute them accordingly to customers and sales agency centers through distribution centers (production and distribution centers are considered as one echelon). In order to reduce the complexity of the problem, the relationship between spare parts and final products is not considered. Therefore, all spare parts and finished products are considered as a cluster of products.

Fig. 1

Structure of proposed closed-loop logistic network

In the reverse direction, returned products of primary customers and sales centers are collected and inspected through collection centers (which are considered with production and distribution centers at the same center). After inspection, resalable items are sold to secondary customers. Then, some items (which can be rebuilt) are sent to manufacturing centers for remanufacturing and recyclable parts are sent to recycling centers. Consequently, recycled products are sold to the suppliers and to third-party customers, and at the end, the unusable products are transferred to disposal centers. Using this strategy, excessive transportation of recursive products is controlled and returned products are transferred directly to the facility as well.

In this paper, the supply chain network is assumed to have the following features:

• Multi-product, multi-period, and multi-echelon scenarios are considered in the proposed network.

• Manufacturers produce and distribute products.

• Manufacturers also collect and redistribute returned products.

• There are different modes of transportation.

• The network has sale agencies, disposal, and recycling centers.

• Different clusters of customers are considered in the proposed model.

Due to the multiplicity of the model’s characteristics, the problem is complex and requires more effort to analyze in both types of forward and reverse chains. According to the geographical conditions (closeness of target points to each other) and to decrease the model’s complexity, we place the production process, distribution, and collection centers in the same echelon as the manufacturing, distribution, and collection center (MDCC). Understanding the effects of different transportation modes on the total profit for the proposed model, we consider different modes between each echelon.

According to Fig. 1, due to the values of products before being residual and to have more business profitability, the model is assumed to have three clusters of customers. The classification of customers is based on the quality of products during their life cycle, sale price, geographical features, and revenues of each customer type.

In the present model, the sale agencies are responsible for selling spare parts, fixing products, and collecting returned products. In addition, because of the features of aboriginal clients and the cost of attractive advertisements, these centers are also in charge of face-to-face marketing in order to gain customers’ trust and to build sustainable relationships.

3.1 Model suppositions

The assumptions of the model are as follows:

• The number and capacity of potential centers of supply, production, distribution, collection, sale, disposal, and recycling are predetermined.

• Demand of the periods for each type of customers is predicted and should be calculated at the end of each period. Shortage is not allowed.

• In all production, distribution, collection, sales, and recycling centers, we have a warehousing system. The capacity of each warehouse is limited, and the cost is considered for the inventory that remains at the end of each period.

• To better serve and to increase total profit of supply chain, different transportation modes are considered. Between each center, only one of the modes is allowed.

• Supply chain management is considered as centralized (Giannoccaro 2018).

A practical value of the present study lies in the development of decision-making models in setting up an efficient logistic system, one that is designed for organizations and start-ups that want both profitability and greenness. It should be noted that because this paper contains a general and generic model in the definition of the problem, it may be used in various industries such as electronics, digital, transport, and manufacturing. To describe the abovementioned logistics network, Table 1 lists the indices used in the model.

Table 1 Indices

All parameters and variables are on indicated Tables 2 and 3.

Table 2 Parameters of the proposed model

Table 3 Variables o the proposed model

3.2 The problem formulation

In terms of the abovementioned notations and assumptions, the multi-echelon, multi-product, multi-raw material, multi-period, closed-loop model for the logistic network design problem is formulated as follows:

3.2.1 Objective function

The objective function maximizes the net present value (NPV) of all in the planned horizon (Žižlavský 2014; Zore et al. 2018). It counts NPV by calculating the difference between income, type of costs, and interest rate in each period.

$$ {\text{Max}}\quad \quad {\text{NPV}} = \mathop \sum \limits_{t} \frac{{{\text{Income}}_{t} - {\text{Cost}}_{t} }}{{\left( {1 + ir } \right)^{t - 1} }} $$

(1)

3.2.1.1 Income objective function

Income in each period t is derived from the sum of selling the products to customers, selling spare parts, fixing products, selling the collected products to the second-hand customer, selling the recycled products to the suppliers and third parties, as mentioned in (2):

$$ \begin{aligned} {\text{Income}}_{t} & = \mathop \sum \limits_{m} \mathop \sum \limits_{c} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mcpl}^{t} PM_{mcp}^{t} + \mathop \sum \limits_{g} \mathop \sum \limits_{p} QS_{gp}^{t} PS_{gp}^{t} + \mathop \sum \limits_{g} \mathop \sum \limits_{p} QF_{gp}^{t} PF_{gp}^{t} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{e} \mathop \sum \limits_{p} QQ_{mep}^{t} PM_{mep}^{t} + \mathop \sum \limits_{b} \mathop \sum \limits_{s} \mathop \sum \limits_{p} QQ_{bsp}^{t} PB_{bsp}^{t} + \mathop \sum \limits_{b} \mathop \sum \limits_{h} \mathop \sum \limits_{p} QQ_{bhp}^{t} PB_{bhp}^{t} \\ \end{aligned} $$

(2)

3.2.1.2 Cost objective function

The cost in each period is derived from the sum of facilities setup costs, ordering costs, building and rebuilding products cost, operational costs in each complex, collecting and inspection costs, maintenance costs, transportation costs, as mentioned in (3):

$$ \begin{aligned} {\text{Cost}}_{t} & = {\text{Cost}}_{{{\text{opening}}}} + {\text{Cost}}_{{{\text{order}}}} + {\text{Cost}}_{{{\text{product}}}} + {\text{Cost}}_{{{\text{operation}}}} + {\text{Cost}}_{{{\text{collection}}}} \\ & \quad + {\text{Cost}}_{{{\text{holding}}}} + {\text{Cost}}_{{{\text{transportion}}}} \\ \end{aligned} $$

(3)

Setup cost entails the cost of opening the production and distribution and collection centers, sales agency centers, recycling centers, and disposal centers as follows.

$$ \begin{aligned} {\text{Cost}}_{{{\text{opening}}}} & = \mathop \sum \limits_{m} FX_{m}^{t} \left( { XI_{m}^{t} - XI_{m}^{t - 1} } \right) + \mathop \sum \limits_{g} FX_{g}^{t} \left( { XI_{g}^{t} - XI_{g}^{t - 1} } \right) \\ & \quad + \mathop \sum \limits_{b} FX_{b}^{t} \left( { XI_{b}^{t} - XI_{b}^{t - 1} } \right) + \mathop \sum \limits_{k} FX_{k}^{t} \left( { XI_{k}^{t} - XI_{k}^{t - 1} } \right) \\ \end{aligned} $$

(4)

According to the cost of selecting and supplying raw materials from suppliers, the cost function of the ordering process is as follows.

$$ {\text{Cost}}_{{{\text{order}}}} = \mathop \sum \limits_{s} FX_{s}^{t} w_{s}^{t} + \mathop \sum \limits_{s} \mathop \sum \limits_{m} \mathop \sum \limits_{r} \mathop \sum \limits_{l} QQ_{smrl}^{t} S_{sr}^{t} $$

(5)

Production cost function includes the cost of production of final products in production centers.

$$ {\text{Cost}}_{{{\text{product}}}} = \mathop \sum \limits_{m} \mathop \sum \limits_{p} QM_{mp}^{t} MP_{mp}^{t} $$

(6)

According to the repair costs at sales agency centers, remanufacturing products, recycling products, and disposal cost, the operational cost function is as follows.

$$ \begin{aligned} {\text{Cost}}_{{{\text{operation}}}} & = \mathop \sum \limits_{g} \mathop \sum \limits_{p} QF_{gp}^{t} OP_{gp}^{t} + \mathop \sum \limits_{m} \mathop \sum \limits_{e} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mepl}^{t} OP_{mp}^{t} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{b} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mbpl}^{t} OP_{bp}^{t} + \mathop \sum \limits_{m} \mathop \sum \limits_{k} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mkpl}^{t} OP_{kp}^{t} \\ \end{aligned} $$

(7)

Collection cost function includes the cost of collecting the products from sales agency centers and customers and the cost of the inspection.

$$ \begin{aligned} {\text{Cost}}_{{{\text{collection}}}} & = \mathop \sum \limits_{g} \mathop \sum \limits_{m} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{gmpl}^{t} (SP_{gp}^{t} + IP_{mp}^{t} ) \\ & \quad + \mathop \sum \limits_{c} \mathop \sum \limits_{m} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{cmpl}^{t} \left( {SP_{cp}^{t} + IP_{mp}^{t} } \right) \\ \end{aligned} $$

(8)

Transportation cost includes the cost of shipping products from suppliers to MDCCs, MDCCs to customers and sales agency centers in the forward flow. It also includes the cost of shipping products from sales agency centers and customers to MDCCs, MDCCs to secondary customers, recycling centers, and disposal centers, recycling centers to suppliers and third-party customers in the backward flow.

$$ \begin{aligned} {\text{Cost}}_{{{\text{transportion}}}} & = \mathop \sum \limits_{s} \mathop \sum \limits_{m} \mathop \sum \limits_{r} \mathop \sum \limits_{l} QQ_{smrl}^{t} TC_{smrl}^{t} + \mathop \sum \limits_{m} \mathop \sum \limits_{c} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mcpl}^{t} TC_{mcpl}^{t} \\ & \quad + \mathop \sum \limits_{m} \mathop \sum \limits_{g} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mgpl}^{t} TC_{mgpl}^{t} + \mathop \sum \limits_{g} \mathop \sum \limits_{m} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{gmpl}^{t} TC_{gmpl}^{t} \\ & \quad + \mathop \sum \limits_{c} \mathop \sum \limits_{m} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{cmpl}^{t} TC_{cmpl}^{t} + \mathop \sum \limits_{m} \mathop \sum \limits_{e} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mepl}^{t} TC_{mepl}^{t} \\ \mathop \sum \limits_{m} \mathop \sum \limits_{b} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mbpl}^{t} TC_{mbpl}^{t} + \mathop \sum \limits_{m} \mathop \sum \limits_{k} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mkpl}^{t} TC_{mkpl}^{t} \\ \mathop \sum \limits_{b} \mathop \sum \limits_{s} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{bspl}^{t} TC_{bspl}^{t} + \mathop \sum \limits_{b} \mathop \sum \limits_{h} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{bhpl}^{t} TC_{bhpl}^{t} \\ \end{aligned} $$

(9)

Maintenance cost includes holding inventory cost of products at the MDCCs, sales agency centers, and recycling centers, as mentioned in (10):

$$ Cost_{holding} = \mathop \sum \limits_{m} \mathop \sum \limits_{p} IV_{mp}^{t} HP_{mp}^{t} + \mathop \sum \limits_{g} \mathop \sum \limits_{p} IV_{gp}^{t} HP_{gp}^{t} + \mathop \sum \limits_{b} \mathop \sum \limits_{p} IV_{bp}^{t} HP_{bp}^{t} $$

(10)

3.2.2 Constraints of the model

Constraints of the model are partitioned in 5 categories that are, respectively, balance, capacity, transportation method, amount of shipments, and binary constraints.

3.2.2.1 Balance constraints

$$ \mathop \sum \limits_{p} nf_{rp} \times QM_{mp}^{t} \le \mathop \sum \limits_{s} \mathop \sum \limits_{l} QQ_{smrl}^{t}\quad \forall m,r,t $$

(11)

$$ \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{mcpl}^{t} \ge D_{cp}^{t}\quad \forall c,p,t $$

(12)

$$ \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{mgpl}^{t} \ge D_{gp}^{t}\quad \forall g,p,t $$

(13)

$$ \mathop \sum \limits_{c} \mathop \sum \limits_{l} QQ_{mcpl}^{t} + \mathop \sum \limits_{g} \mathop \sum \limits_{l} QQ_{mgpl}^{t} \le QM_{mp}^{t}\quad \forall m,p,t $$

(14)

$$ \mathop \sum \limits_{c} \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{cmpl}^{t} + \mathop \sum \limits_{g} \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{gmpl}^{t} \le RR_{p} \times D_{cp}^{t}\quad \forall p,t $$

(15)

$$ \mathop \sum \limits_{e} QQ_{mep}^{t} = \mathop \sum \limits_{c} \mathop \sum \limits_{l} QQ_{cmpl}^{t} \times RE_{p} + \mathop \sum \limits_{g} \mathop \sum \limits_{l} QQ_{gmpl}^{t} \times RE_{p}\quad \forall m,p,t $$

(16)

$$ \mathop \sum \limits_{k} \mathop \sum \limits_{l} QQ_{mkpl}^{t} = \mathop \sum \limits_{c} \mathop \sum \limits_{l} QQ_{cmpl}^{t} \times RK_{p} + \mathop \sum \limits_{g} \mathop \sum \limits_{l} QQ_{gmpl}^{t} \times RK_{p}\quad \forall m,p,t $$

(17)

$$ \mathop \sum \limits_{b} \mathop \sum \limits_{l} QQ_{mbpl}^{t} = \mathop \sum \limits_{c} \mathop \sum \limits_{l} QQ_{cmpl}^{t} \times RB_{p} + \mathop \sum \limits_{g} \mathop \sum \limits_{l} QQ_{gmpl}^{t} \times RB_{p}\quad \forall m,p,t $$

(18)

$$ \mathop \sum \limits_{s} QQ_{bsp}^{t} + \mathop \sum \limits_{h} QQ_{bhp}^{t} \le \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{mbpl}^{t}\quad \forall b,p,t $$

(19)

$$ \mathop \sum \limits_{s} QQ_{bsp}^{t} \le \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{mbpl}^{t} \times RS_{p}\quad \forall b,p,t{ } $$

(20)

$$ \mathop \sum \limits_{h} QQ_{bhp}^{t} \le \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{mbpl}^{t} \times RH_{p}\quad \forall b,p,t $$

(21)

$$ ng_{gp} \times QF_{gp}^{t} + QS_{gp}^{t} \le D_{gp}^{t}\quad \forall g,p,t $$

(22)

$$ IV_{mp}^{t} - IV_{mp}^{t - 1} \le \mathop \sum \limits_{n} QM_{mp}^{t} - \mathop \sum \limits_{c} \mathop \sum \limits_{l} QQ_{mcpl}^{t} - \mathop \sum \limits_{g} \mathop \sum \limits_{l} QQ_{mgpl}^{t}\quad \forall m,p,t $$

(23)

$$ IV_{gp}^{t} - IV_{gp}^{t - 1} \le \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{mgpl}^{t} - ng_{gp} QF_{gp}^{t} - QS_{gp}^{t}\quad \forall g,p,t $$

(24)

$$ IV_{bp}^{t} - IV_{bp}^{t - 1} \le \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{mbpl}^{t} - \mathop \sum \limits_{s} QQ_{bsp}^{t} - \mathop \sum \limits_{h} QQ_{bhp}^{t}\quad \forall b,p,t $$

(25)

Constraint (11) ensures that the present flow during the use of any material in each period, and at each, the production center is, at most, equal to the sum of the raw material input from the supplier. Constraint (12) expresses that the amount of outgoing flow of finished product from production and distribution center should be at least equal to the demand for the finished products. Constraint (13) shows that the outgoing flow of spare parts from production and distribution center to sales agency center should be at least equal to the demand for the spare parts. Constraint (14) illustrates that the outgoing flow of products from production and distribution center to customers and the sales agency center should be at least equal to the amount of production. Constraint (15) states the utmost equality of returned products flow with the demand percentage. Constraints (16)– (18) show the equality of input flow to the production centers, second-hand customers, disposal and recycling with the output flow of collection center to each one, respectively. Constraint (19) ensures that the input flow of the recycling center should be at most equal to its output flow. Constraints (20) and (21) indicate that the output flow from the recycling center should be at most equal to the exchangeable rate of recycled products for both useable and unusable products of suppliers. Constraint (22) expresses that the demand at sales agencies should be at most equal to the need for the product in each period. Constraints (23)–(25) illustrate utmost equality of in-stock inventory with the input and output flows in each production and distribution, sales agencies, collection and recycling centers.

3.2.2.2 Capacity constraints

$$ \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{smrl}^{t} \le CS_{sr}^{t} \times OS_{sr}^{t} \quad \forall s,r,t $$

(26)

$$ QM_{mp}^{t} \le \left( {CFIP_{mp}^{t} } \right) \times XI_{m}^{t} \quad \forall m,p,t $$

(27)

$$ \mathop \sum \limits_{p} m_{p} IV_{mp}^{t} \le \left( {CH_{m}^{t} } \right) \times XI_{m}^{t} \quad \forall m,t $$

(28)

$$ QF_{gp}^{t} \le \left( {CG_{gp}^{t} } \right) \times XI_{g}^{t} \quad \forall g,p,t $$

(29)

$$ \mathop \sum \limits_{p} m_{p} IV_{gp}^{t} \le \left( {CH_{g}^{t} } \right) \times XI_{g}^{t} \quad \forall g,t $$

(30)

$$ \mathop \sum \limits_{c} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{cmpl}^{t} + \mathop \sum \limits_{g} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{gmpl}^{t} \le \left( {CC_{m}^{t} } \right) \times XI_{m}^{t} \quad \forall m,t $$

(31)

$$ \mathop \sum \limits_{m} \mathop \sum \limits_{l} QQ_{mbpl}^{t} \le \left( {CB_{bp}^{t} } \right) \times XI_{b}^{t} \quad \forall b,p,t $$

(32)

$$ \mathop \sum \limits_{p} m_{p} IV_{bp}^{t} \le \left( {CH_{b}^{t} } \right) \times XI_{b}^{t} \quad \forall b,t $$

(33)

$$ \mathop \sum \limits_{m} \mathop \sum \limits_{p} \mathop \sum \limits_{l} QQ_{mkpl}^{t} \le \left( {CK_{k}^{t} } \right) \times XI_{k}^{t} \quad \forall k,t $$

(34)

Constraint (26) ensures that the output of raw materials from a supplier should not exceed the capacity of it in each period. Constraint (27) shows that the amount of production in each production and distribution center should not surpass the production capacity. Constraint (28) indicates that the amount of inventory in each production and distribution center should not be more than the capacity in stock. Constraint (29) and (30) illustrate the repair capacity in each sales agency center and the inventory level should not exceed the capacity in each sales agency center, respectively. Constraint (31) shows the collecting capacity in each MDCCs should not surpass the capacity in stock in the collection center. Constraints (32) and (33) ensure that the recycling capacity and the inventory level should be less or equal to their capacity each recycling center, respectively. Constraint (34) shows the disposing capacity in each disposal center.

3.2.2.3 Transportation method constraints

$$ \mathop \sum \limits_{l} AL_{sml}^{t} \le \frac{1}{2}\left( {W_{S}^{t} + XI_{m}^{t} } \right) \quad \forall s,m,t $$

(35)

$$ \mathop \sum \limits_{l} AL_{mcl}^{t} \le XI_{m}^{t} \quad \forall m,c,t $$

(36)

$$ \mathop \sum \limits_{l} AL_{mgl}^{t} \le \frac{1}{2}\left( {XI_{m}^{t} + XI_{g}^{t} } \right) \quad \forall m,g,t $$

(37)

$$ \mathop \sum \limits_{l} AL_{cml}^{t} \le XI_{m}^{t} \quad \forall c,m,t $$

(38)

$$ \mathop \sum \limits_{l} AL_{gml}^{t} \le \frac{1}{2}\left( {XI_{m}^{t} + XI_{g}^{t} } \right) \quad \forall g,m,t $$

(39)

$$ \mathop \sum \limits_{l} AL_{mbl}^{t} \le \frac{1}{2}\left( {XI_{m}^{t} + XI_{b}^{t} } \right) \quad \forall m,b,t $$

(40)

$$ \mathop \sum \limits_{l} AL_{mkl}^{t} \le \frac{1}{2}\left( {XI_{m}^{t} + XI_{k}^{t} } \right) \quad \forall m,k,t $$

(41)

$$ \mathop \sum \limits_{l} AL_{bsl}^{t} \le \frac{1}{2}\left( {W_{S}^{t} + XI_{b}^{t} } \right) \quad \forall b,s,t $$

(42)

$$ \mathop \sum \limits_{l} AL_{bhl}^{t} \le XI_{b}^{t} \quad \forall b,h,t $$

(43)

Constraints (35)–(43) indicate that merely one transportation modes should be selected for transporting products from one center to the other.

3.2.2.4 Amount of shipments constraints

$$ \mathop \sum \limits_{r} QQ_{smrl}^{t} \le BM.AL_{sml}^{t} \quad \forall s,m,l,t $$

(44)

$$ \mathop \sum \limits_{p} QQ_{mcpl}^{t} \le BM.AL_{mcl}^{t} \quad \forall m,c,l,t $$

(45)

$$ \mathop \sum \limits_{p} QQ_{mgpl}^{t} \le BM.AL_{mgl}^{t} \quad \forall m,g,l,t $$

(46)

$$ \mathop \sum \limits_{p} QQ_{cmpl}^{t} \le BM.AL_{cml}^{t} \quad \forall c,m,l,t $$

(47)

$$ \mathop \sum \limits_{p} QQ_{gmpl}^{t} \le BM.AL_{gml}^{t} \quad \forall g,m,l,t $$

(48)

$$ \mathop \sum \limits_{p} QQ_{mbpl}^{t} \le BM.AL_{mbl}^{t} \quad \forall m,b,l,t $$

(49)

$$ \mathop \sum \limits_{p} QQ_{mkpl}^{t} \le BM.AL_{mkl}^{t} \quad \forall m,k,l,t $$

(50)

$$ \mathop \sum \limits_{p} QQ_{bspl}^{t} \le BM.AL_{bsl}^{t} \quad \forall b,s,l,t $$

(51)

$$ \mathop \sum \limits_{p} QQ_{bhpl}^{t} \le BM.AL_{bhl}^{t} \quad \forall b,h,l,t $$

(52)

Constraints (44)–(52) illustrate that if there is any transportation method between centers, the products could be carried between them.

3.2.3 Location constraints

$$ XI_{m}^{t - 1} \le XI_{m}^{t} \quad \forall m,t $$

(53)

$$ XI_{g}^{t - 1} \le XI_{g}^{t} \quad \forall g,t $$

(54)

$$ XI_{b}^{t - 1} \le XI_{b}^{t} \quad \forall b,t $$

(55)

$$ XI_{k}^{t - 1} \le XI_{k}^{t} \quad \forall k,t $$

(56)

Constraints (53)–(56) show that after the establishment of a facility, it is in use until the next period.

4 Solution approaches

In this section, solution approach along with the considered encoding and decoding plan is presented. Large-scale problems are always difficult to deal with, and when it comes to the solving approach, it increases its complexity as the dimensions of the problem grow larger. Besides, real-case problems are usually large-sized and this is another factor which increase the complexity rate of the problem. As aforementioned, the proposed model has been formulated as a mix integer linear programing with one objective function. This problem decides on opening new facilities and choosing best transportation method which makes the decision variables quite large. Therefore, using exact solutions to the deal with such problems is not time efficient. In this regard, we utilized a problem with number of metaheuristics and hybrid algorithm. Here, we implemented simulated annealing (SA), genetic algorithm (GA), Keshtel algorithm (KA), and hybrid KA and GA (KAGA) to evaluate the model solutions.

4.1 Programming encoding and decoding

A proper encoding and decoding plan is needed to illustrated how a model behavior is verified and also satisfied the proposed constraints. Hence, for the small-sized example, the chromosome matrix of the considered problem with their levels is exemplified in Fig. 2. For a small-scale problem, we assume that the number of considered centers is 2,4,4,2,2,2,2,2, and 2. Each column occupied by a level or the combination of levels and random solutions between $\left( {0,1} \right)$ is generated in the first place and then sorted for each section. As elaborated in Gholian-Jouybari et al. (2016), Yadollahinia et al. (2018), Chouhan et al. (2020), this method guaranties both the optimum quantity and satisfies the considered constraints. The considered decoding plan is presented in Fig. 3.

Fig. 2

Proposed chromosome design

Fig. 3

Schematic representation for encoding scheme for Sect. 5

The structure of the considered metaheuristics and hybrid algorithm is presented below:

4.2 The proposed metaheuristics

In this section, the structure of the considered metaheuristics and hybrid algorithm is presented. Simulated annealing (SA) and genetic algorithm (GA) are among the most prominent algorithms which are applied on countless previous studies and modeling. Each algorithm starts by generation a random feasible solution and then evaluates the objective function. In the next step by some correction such as setting the heat in SA or mutation in GA, in each iteration, they optimize the solution while maintaining the feasibility of the problem. Each best solution replaces with the current best until the stopping condition of the algorithms is met. The elaborated description of SA and GA is detailed in Akbarpour et al. (2020a, b), Hajiaghaei-Keshteli et al. (2014), Salimi (2015) and Delavar et al. (2010), Goldberg (2006), Golshahi-Roudbaneh et al. (2017).

Another considered metaheuristic is Keshtel algorithm (KA) which is recently proposed by Hajiaghaei-Keshteli and Aminnayeri (2013, 2014). The application of the algorithm was based on the feeding behavior of ducks when they search for the best food sources. When a good feeding source was found, all the other neighbor ducks swirl together and gather in a circle. This current best feeding source would replace the other best source when it found by other same species and this swirling goes on until the source runs out. For the details of KA, the authors could kindly refer to Hajiaghaei-Keshteli and Aminnayeri (2014).

Besides using GA, SA, and KA, here we utilized a hybrid GAKA algorithm. Combination of various metaheuristics enables the new algorithm to benefit from the best characteristic of the both algorithms. Here, the proposed GAKA starts by generating random solutions as it is in GA. The difference occurs in the mutation section where instead of general mutation in the GA, we utilize KA algorithm in order to make the generated results of GA even better using steps in KA algorithm. This method ensures better solutions in each iteration of the algorithm.

GA has two main operators for intensification and diversification, mutation and crossover, respectively. To achieve a better trade-off between two phases, the parameters should be tuned. Besides, to boost and enhance the performance of each operator, researchers usually combine or hybrid the algorithms. The mutation operator in GA accepts one chromosome and, after local search on this chromosome, gives a better or worst solution. Also, this search is accomplished blindly without a greedy strategy. On the other hand, in KA, the local search in the intensification phase is done by swirling around the local optimum with a greedy strategy not only about solution quality, but also about solution time. This issue motivated us to firstly develop this hybrid metaheuristic in this paper. The pseudocode of GAKA is shown in Fig. 4.

Fig. 4

Pseudocode of proposed GAKA

5 Computational results

In order to ensure the efficiency of the proposed methodologies, an intelligent experiment is needed to evolve best settings for each algorithm. Therefore, Taguchi methodology is applied here to set the algorithm’s parameter and further evaluation about the tuned parameter of each algorithm. Taguchi experiments guarantee the best results according to the formulation of the problem for each algorithm. The suggested values are then used to achieve the optimum results (Hajiaghaei-Keshteli and Fard 2019; Sahebjamnia et al. 2020).

5.1 Generating data

In order to evaluate the effectiveness of the considered metaheuristics, one proper method is to consider various problems in different sizes. Hence, in this work, the test problems are divided from small to large sizes. According to proposed model, suppliers (s), MDCCs (m), customers (c), sales agency centers (g), recycling centers (b), disposal center (k), secondary customers (e), third-party customers (h), products (p), raw materials (r), and periods (t) are considered. Table 4 illustrates parameter values for the considered test problems. It should be noted that three test problems including SP2, SP4, and MP6 are derived from real case study. Further description about the case study is provided in Sect. 7.

Table 4 Problem classification

The considered parameters to initiate the model is described in Table 5.

Table 5 Values of parameters

The utilized data in Table 6 were obtained from private and governmental organizations such as Islamic Republic of Iran Ministry of Industry, Mine & Trade,¹ TGJU,² Market Panorama,³ Shakesban,⁴ Marketban,⁵ Alomsar group,⁶ Alstar group.⁷

Table 6 The proposed metaheuristic algorithm’s factors and their levels

5.2 Parameter setting: setting model parameter

To get the best results when dealing with metaheuristics, a key factor which is highly helpful is parameter tuning. Tuning the parameters helps the considered metaheuristic to reach its optimum value in relatively shorter time and also with more quality outputs. Therefore, using Taguchi approach, some levels are considered here for parameter tuning (Gholian-Jouybari et al. 2018). The parameters and their description are provided in Table 7. Due to the objective function of maximization, the concept of “larger is better” is applied to choose the best tuned parameters. To achieve this, the definition of signal to noise ratio is applied here and presented in Eq. (57) (Fard et al. 2017; Hajiaghaei-Keshteli et al. 2018).

$$ S/N = - 10\log \left( {\frac{{\mathop \sum \nolimits_{i = 1}^{n} Y_{i}^{2} }}{n}} \right) $$

(57)

Table 7 Tuned parameters of algorithms

In this equation, $Y$ and $n$ are the observed data and number of experiments, respectively. According to the applied method, the utilized metaheuristics and their levels are depicted in Table 6.

Using the defined 15 test problems, the problem has run for 40 times for different considered classifications. Since the problem dimensions are different, it cannot directly be solved. This issue could be solved using the relative percentage deviation (RPD) (Abdi et al. 2020; Hajiaghaei-Keshteli and Fathollahi-Fard 2018). The basic formulation of RPD is represented in Eq. (58).

$$ {\text{RPD}} = \frac{{\left| {{\text{Alg}}_{{{\text{sol}}}} - {\text{Min}}_{{{\text{sol}}}} } \right|}}{{{\text{Min}}_{{{\text{sol}}}} }} $$

(58)

In RPD equation, ${\text{Alg}}_{{{\text{sol}}}}$ shows the objective value in each trial and ${\text{Min}}_{{{\text{sol}}}}$ shows the best answer among trials (Abdi et al. 2020; Hajiaghaei-Keshteli and Fathollahi-Fard 2018). After changing these values with RPD value, using Taguchi method, it would turn to $S/N$ ratio and then averaged. To decrease the number of trials, Taguchi method suggests some orthogonal arrays. In this problem, L16 design is selected for SA, GA, and KA while L27 was applied for designing GAKA experiments. Using aforementioned, the results of the parameter tuning are depicted in Table 7.

5.3 Experimental results

After calculation of the tuned parameters, we need to evaluate their performance for model solving. In this regard, two criteria RPD and convergency rate are utilized to examine algorithm’s efficiency. To meet the randomization nature of proposed metaheuristics, the achieved results turned into RPD value. The mean RPDs are depicted in Fig. 5.

Fig. 5

Mean RPD of each algorithm

As depicted in Fig. 5, KA is having the best results among the proposed algorithms. Meaning that the deviation of its results is much less than other algorithms while maintaining good solutions. Among the proposed algorithms, SA showed least quality solutions while GAKA and GA had the better results, respectively. It should be noted that GAKA was able to reach KA results in some point, and even in test problem three it had the better result compared with KA. Based on average RPD, KA was the best among others.

The second evaluation criterion which is applied on the problem was the convergency rate. For three test problems, the convergency rate is developed for 100 iterations. Then we stopped the algorithms and compared the results (See Fig. 6). The figure is developed for SP4, MP8, and LP12, respectively. In three illustrated problems, KA showed its capability to reach its best answer much sooner than other algorithms. However, as the iteration of the problem rises, GAKA showed best final answers compared with other algorithms. In addition, in bigger test problems, GA was able to reach the better answers than KA.

Fig. 6

Convergence behavior of the proposed algorithms in various test problems

In addition to the considered metaheuristics, it is worth to compare these results with the results from exact solution using GAMS software. The overall results of this comparison are illustrated in Table 8 with thousand iteration for each metaheuristic. It should be noted that, due to the increased complexity of the problem in larger problem sizes, GAMS was unable to reach answer after MP8.

Table 8 The output results ${ }\left( {{\text{value}} \times 10^{8} {\text{ and CPU time based on second}}} \right)$

6 Case study

In this section, the case study, the model’s implementation, and a comparison of the proposed model with the study by Ramezani et al. (2014b) are indicated based on SP₄. The structure of this section is as below:

6.1 Introducing case study

Current research has been conducted in a production company in Mazandaran province, Iran. The main activity performed at this corporation is building aluminum and UPVC (unplasticized polyvinyl chloride) doors and windows. Aluminum, UPVC, and glass comprise the primary raw materials in this company and they comprise the basic determiner of the company’s prices. In recent years, a significant increase in raw material prices (see Figs. 7 and 8), greater customer demands for green products, and Iran’s unstable economy has caused an inordinate increase in basic costs. Therefore, the company’s working capital decreased and customer demands for products decreased, leading the market to shift to returned products.

Fig. 7

source link: http://pubdocs.worldbank.org)

Average aluminum prices from 2012 to 2018 (Sources: LME; World Bank,

Fig. 8

Aluminum, UPVC, and Glass prices in Iran in 2018 (Based on company documentary)

Due to the low capacity investment restrictions, this company started to capture more market share by continuous improvement, centralization in some products, and by attracting a certain type of customer. Centralization in production has resulted in better production cost control, and attracting the special type of customer has led to use low and multi-purpose facilities and applying low-price marketing methods.

According to the very high costs of attractive advertising, this company has started cooperating with small and popular firms, near to the target markets. The responsibility of each of these firms is to attract customers, fix the products, sell spare parts, and collect used products. For the sake of this research, these firms are called the “sales agency.”

This paper examines the impact of added levels to the supply chain network. Hence, the demands are examined just for the areas with sales service facilities. The demand data for each area are mentioned in Fig. 9.

Fig. 9

Demand trend of each customer area

The interest rate of return is equal to the combined interest rate from average monthly inflation rate and the average bank interest rate. This rate is extracted from the central bank of the Islamic Republic of Iran (2018 statistics)⁸ in Table 9.

Table 9 Interest rate in 2018

6.2 Model implementation

As mentioned before, due to increase in production costs and also compete with similar rival companies, the decision-maker decided to utilize the sales agencies to achieve a higher market share. The main responsibilities of the sales agencies are as follows:

• To gain customers’ trust and retain them,

• To sell spare parts,

• To fix (repair) sold products,

• To collect returned products.

After opening the sale agency centers, there is a growing trend in the demand rates in each zone (Sari, Qaemshahr, Babol, Neka). The demand data for each customer area are depicted in Fig. 9. Although GDP growth in the mining industry⁹ has been -5.4% and -2.9% in 2017 and 2018, respectively, but which can be clearly seen in Fig. 9 is the continual growth of demand for each zone.

In this regard, company’s revenue shows an almost ascendant trend. Due to the situation of recession in Iran and considering the rise in manufacturing and raw material costs, the company’s profits demonstrate a growing trend (See Table 10 and Fig. 10).

Table 10 The values of variables ($\times 10^{4}$)

Fig. 10

Trend of variables in each period

After solving the model (See Table 10), (1) some variables including ordering, production, operation, collection, and transportation costs are expected to grow due to increasing material and energy costs (See Fig. 6 and 7). (2) Regarding high inventory costs at any period, the company keeps no inventory for the next period. (3) Since the only goal of the objective function in this corporation is to maximize the profit, the transportation mode used takes benefit of its cheapness. (4) What is the interesting in this table is the phenomenal growth of total income and profit. These amounts are normally expected not to display such growth because of economic conditions and increasing costs.

After obtaining the cash flow in each period, the trend of variables value is shown in Fig. 10. According to the interest rate, and assuming that all costs and incomes are received at the beginning of each period, the total profit for the supply chain is calculated as follows:

Total profit of supply chain

5,481,054,727.9323

6.3 Comparison of the proposed model with Ramezani et al. (2014b)

As mentioned in the literature review, a closed-loop supply chain network is highly varied. The most effective causes are varieties of products, customer’s value and expectations, current economic conditions on target markets, governmental rules, etc. Ramezani et al. (2014b) presented a general model for designing a closed-loop supply chain. Like the work of Ramezani et al. (2014b), this paper has a multi-echelon, multi-period, multi-product model and the differences are manufacturers and sale agencies in the forward direction, and secondary customers and third-party levels in the backward flow. Targeting to increase product sales in the target markets, the model considers not only primary customers (main customers), but also sale agencies centers. These centers are responsible for marketing and providing after-sales services.

The proposed model considers not only primary products but also considers primary customers, spare parts sale, product repair, sale returned product to the secondary customers, sale recycled products to suppliers and third parties to earn money, compared with Ramezani et al. (2014b) model. As a result, according to Fig. 11a (dividing the amount of each period revenue to the amount of sale in each period) a significant increase in sale area can be detected. In Fig. 11a, b, and c, the indicator of the income per product (I/N) is considered to compare the proposed model with the Ramezani et al. (2014b). This is also applied to Fig. 15, Fig. 16, and Fig. 18.

Fig. 11

Comparisons the proposed model with (2014b) (a) Income indicator (b) Ordering cost indicator (c) Production cost indicator (d) Trend of holding inventory cost $\left( { \times 10^{4} } \right)$

Increased sales in each period cause more productivity compared with Ramezani et al. (2014b) model. As a result, for high productivity (to satisfy customers' demand), the primary product order has been increased. The indicator of ordering costs (C/N) is very close to each other and even the proposed model has a lower cost in period three. Stability in ordering costs and its corresponding indicator is another advantage of the proposed model (Fig. 11b).

Like ordering cost, with increased sale, the production cost has been increased compared with Ramezani et al. (2014b) model. But according to Fig. 11c, the production cost indicator (C/N) is less than Ramezani et al. (2014b) model in most periods. We can conclude that in the present model, in addition to increased production, costs that are relative to production have dropped. On the other hand, stability in indicator of production cost (see Fig. 11c) and ordering costs (see Fig. 11b) in each period shows the model being consistent with the economic conditions and regional markets according to the 25.3 percent inflation rate and poor economic conditions in Iran in 2017–2018.

Other proper aspects of the proposed model in comparison with Ramezani et al. (2014b) model are a significant reduction of maintenance costs. The proposed model does not consider any inventory in any period. As a result, demand in each period is estimated over the same period (See Fig. 11d).

In order to improve the environmental performance, in the proposed model, the purchase price of returned products in sales agency centers is slightly higher in the model to focus on the collecting points to return the products. This explains why the returned products are only collected through sale agencies. The collection cost in each period is higher in Ramezani et al. (2014b) model, but the indexes for the two models are identical. (See Fig. 12).

Fig. 12

Collection cost indicator

By increasing the number of demand, production, and raw material, the transportation cost in comparison with Ramezani et al. (2014b) model has increased in some periods, but, by determination of the transportation cost indicator (C/N), we can observe that transportation index is much lower in the proposed model than Ramezani et al. (2014b) model. On the other hand, stability in transportation cost is another positive aspect of the proposed model (See Fig. 13).

Fig. 13

Transportation cost indicator

Operating cost in the Ramezani et al. (2014b) model is derived from two types of operating costs in distribution centers and disposal centers. In the proposed model, however, operating costs are derived from costs of repair in sales agencies, recycling, and disposal centers. Hence, an actual comparison of the two costs is not possible due to their different parameters. By separating the cost of repair production in sales agencies and obtaining an indicator (C/N), we can observe that the operational costs of the two models are very close to each other (See Fig. 14).

Fig. 14

Operation cost indicator

As shown earlier, using continuous sale for different types of customers and using sale agencies in the proposed model caused increasing income in each period compared with Ramezani et al. (2014b) model. Other costs such as production costs, ordering costs, and operational costs have been increased due to the increase in sales. But by improving the relationship and model, in most periods in addition to stability, the profit of each period is greater and positive. In short, the total profit of the supply chain in the proposed model has improved five times better than Ramezani et al. (2014b) model (See Fig. 15).

Fig. 15

Trend of supply chain profit $\left( { \times 10^{4} } \right)$

In addition to total profit improvement and environmental performance of the supply chain, improved modeling creates better solutions than Ramezani et al. (2014b) model. These changes are:

• Correction between balance constraints,

• Using production and transferring variables in production centers to other facilities separately,

• Correction of capacity constraint using binary variables,

• Using a place constraint for correction of problem solutions.

Some of the facilities have not been established in Ramezani et al. (2014b) model, but capacity has been allocated to them. For example, the distribution and collection centers facilities have not been set up; so far, setup costs have not been considered either in Ramezani et al. (2014b) model. But the capacity has been allocated to them and the amount of production transportation and collected products in distribution centers has been positive. The other advantages of the proposed model compared with Ramezani et al. (2014b) model are improved modeling and setup objective function using binary constraints. According to Fig. 16, we can observe that if incorrect values are used, the binary value of the fourth period for opening cost is negative.

Fig. 16

Trend of opening cost $\left( { \times 10^{4} } \right)$

In an overview, the proposed model offers many more factors to provide after-sales services for customers, and these factors make the model closer to real-world models. The result of considering these factors is the increase in the demand for so-called goods which increases sales profit in different periods of consumption. On the other hand, the increase in demand has led to an increase in production, which ultimately has led to a decrease in the cost of goods in the company. In total, the presented model has shown itself to be efficient and operational.

7 Sensitivity analysis and discussion

As mentioned before, a closed-loop supply chain network as a mixed integer linear program is developed. In order to study more on the proposed model, several scenarios are proposed. The first scenario considers the change in transportation costs while the second one tries to investigate the change in the amount of returned products (Alizadeh et al. 2020; Braz et al. 2018; Özceylan et al. 2014; Ramezani et al. 2014b).

As the transportation costs increase (from one to four times, as shown in horizontal axis), the behaviors of the proposed model are exemplified in Figs. 17, 18, 19.

Fig. 17

Trend of ordering cost vs total profit

Fig. 18

Opening, operation, and collection cost

Fig. 19

Trend of opening cost vs production cost

According to Fig. 17, the more the transportation costs, the more the ordering costs. Closer inspection of this figure shows that ordering cost has exactly opposite behavior in comparison to the total profit. In addition, it also shows the independency of transportation cost in total profit.

Interestingly, the main finding in Fig. 18 is the behavior of opening cost. With successive increases in transportation cost, the total cost moves higher with successive steps, so the model is sensitive to some values in transportation costs. When the transportation cost reaches a particular threshold, the model decides to establish and open new facilities, and then we face some steps in opening cost. Therefore, the costs of operation and collection increase a little in initial steps and then decrease.

Figure 19 illustrates that as transportation costs increase, the proposed model decides to create new centers to avoid excessive shipping costs and maximize the total profit. With the launch of new production and distribution centers, production costs in some of these centers decrease, and this leads to have lower overall production costs.

The other scenario to evaluate the performance of the proposed model is to change the returned product amount. With this change (from 0.6 to 2.8 times, as shown in horizontal axis), the model behavior is shown in Figs. 20, 21.

Fig. 20

Trend of opening, operation, collection, and transportation costs

Fig. 21

Trend of ordering and operation costs vs total income and total profit

As seen in Fig. 20, the transportation, collection, inspection, and operation costs increase by extending the returned products, and with a decrease in the returned products, these costs also decrease. Moreover, with the change in returned products, opening costs are increased in some values to establish new capacities at these centers for the recycling process.

In Fig. 21, by increasing the returned products, ordering and production costs remain flat, indicating that these variables are independent of the returned products. But, as sales turnaround, the revenue increases and the total profit of the supply chain grows.

7.1 Managerial insights

As aforementioned, in this paper, a closed-loop supply chain network is designed with the aim of maximizing the total supply chain profitability. In this section, the results and the proposed model from a managerial perspective are reviewed and analyzed as follows:

• The customers are categorized based on the product life cycle: primary customers (customers of final product), secondary customers (customers of returned products), and third parties (customers of recycled product). As shown in Figs. 10, 11a and Table 11, considering different clusters of customers in the product life cycle increase both the share of target markets and the profitability of the total supply chain. Hence, the products are sold based on their quality. As a result, in order to increase sales and profitability, managers should pay attention to not only the final product markets, but also to the return products markets.

• As mentioned, in the proposed model, the sales agency centers are considered with the aims of selling spare parts, repairing products, collecting returned products, marketing and gaining market share, and also maintaining the market. In fact, these centers serve as a bridge between the production center and customers. According to Fig. 9, the sales agency centers increased the demand of products in the four studied areas. In addition, due to Fig. 10, the total profit of supply chain has been increased. As a consequence, in areas where the target market is large or the cost of opening production, distribution, or store centers is high, managers should utilize sales agency centers to gain more market share. On the other hand, the amount of collection of returned products based on Figs. 10 and 12 has increased. Most returned products are collected by sales agency centers. This indicates that customers are more willing to return returned products through sales agency centers than an independent center called a collection center (more familiar with the center). Moreover, according to Fig. 12, it is obvious that the cost of collecting each unit of returned product compared to the one in Ramezani et al. (2014b) (without sales agency centers) is almost the same. Therefore, managers should use sales agency centers instead of opening multiple collection centers.

• The multi-task centers (combined) were considered in the network structure. Figures 13 and 16 illustrate that opening and transportation costs have been reduced. On the other hand, Fig. 17 shows that the more increase in transportation costs, the more increase in opening cost in an ascending manner. Therefore, the use of multi-task centers is more suitable for small, local, and national organizations rather than international organizations. Thus, in designing the network structure, managers should pay attention to the opening costs, the tasks of the centers, and the transportation costs. If the transportation cost is much less than the opening costs, utilizing multi-task centers is a good option to reduce the costs and increase the profitability.

Table 11 The total profit of compared supply chains

8 Conclusions and further study

This study seeks to establish a multi-echelon, multi-period, multi-product supply chain network model in order to optimize net present profit value. The structure of the proposed model is a closed-loop supply chain, and it considers suppliers, manufacturers, distributors, sale agencies, and primary customers in the forward direction. Further, it considers product collection, recycling, disposal, secondary customers, and third person in the backward direction. This paper not only discusses the reasons for using returned products, but also shows the effect of returned products on the revenue supply chain.

Although the proposed model has more echelons in comparison with other similar ones (based on configuration and network structure), the findings of this research illustrate the improvement in costs and profit by modifying relations between the facilities and the use of multi-task facilities. The research has also shown that using binary variables in the structure of the capacity, location, and transportation constraints in addition to maintaining the linearity of the model causes the capacity to be assigned only to the launched facilities. Utilizing and embedding the sales agency center are another significant assumption. The results of the investigation on the case study show that demand districts increased by using face-to-face marketing, making stable relationships among customers, and attracting new ones in addition to the product repair, spare parts sale, and returned product collection. The study contributes to our understanding of returned products value. Therefore, customers were divided into three groups (main customers, secondary customers, and third persons) due to the value of goods at each stage of product life, culture, each region’s ability to purchase, and the type of demand. The results of this research support this idea and show that the total profits of the proposed model increase by using continuous sales in each stage of product life. Although this study focuses on profitability, the findings may well have a bearing on environmental performance by collection and redistribution, recycling, and disposal returned products. This work provides insights for increasing profit and improving environmental performance, simultaneously. Other advantages are improvements in transportation modes and corresponding costs, and inventory system and stability in costs and revenues in each period. On the other hand, the proposed model is implemented during unstable economic conditions in a company in Iran. In Sects. 4.2 and 4.3, the adaption of the proposed model in the previously cited economic conditions and model profitability has been indicated.

From a practical point of view, the proposed network is thought of as a general closed-loop supply chain network. Therefore, the model can be applied in companies with regional production and distribution dimensions, distribution-oriented firms, electronic markets and e-stores, and companies interested in the direct distribution of their product without any intermediate. Further, according to the implementation of the model in a production network and construction product supply in a country, this model can be used for small manufacturing and service companies that are looking to increase sales using greenness. Based on the outcomes of this research, several issues raised by this study include global warming concerns, energy savings, and uncertain environments. Therefore, further research should determine environmental performance based on rates of carbon emissions. Further research could also be conducted to consider more factors and criteria in transportation modes. More broadly, research is also needed to consider uncertainty in the proposed model. The most important question to be pursued through further study is how renewable energies can answer the threat of unstable energies in terms of condition and profitability. Further research should also be undertaken to solve the model by other solution approaches, such as utilizing other exact methods like Benders decomposition or different metaheuristic algorithms including invasive weed. Also, the current work can be utilized in other industries such as electronic and automotive industries.

Appendix

See Table

Table 12 A brief overview of the relevant state-of-the-art publications

12.