Closing the loop of a global supply chain through a robust optimal decentralized decision support system

This paper presents a novel decentralized decision support system to optimally design a general global closed-loop supply chain. This is done through an original risk-based robust mixed-integer linear programming that is formulated based on an initial uncertain bi-level programming. Addressing the decision-maker’s (DM’s) attitude toward risk, a scenario-based conditional value-at-risk is used to deal with demand and return uncertainty. Also, the Karush–Kuhn–Tucker (KKT) conditions are employed to transform the model into its single-level counterpart. The results obtained from solving a numerical example through the proposed framework are compared with those of the corresponding centralized system, which is formulated through deterministic multi-objective programming and solved by the Lp-metric method. The results show that the use of the proposed framework improves the robustness of profit, income, and cost by about 28%, 34%, and 36% on average. However, a more conservative DM faces a larger cost of robustness than an optimistic DM while experiencing a more significant improvement in the system responsiveness. Using the proposed framework, the manager can measure the advantages, disadvantages, and consequences of their decisions before their actual implementation. This is because the model is capable of establishing fundamental trade-offs among risk, cost, profit, income, robustness, and responsiveness according to the DM’s attitude toward risk.

Binary variable to define whether distribution center b is opened Z g Binary variable to define whether disposal center g is opened

Dependent variables:
Income System's total income Ψ 1 Facilities' total opening cost Ψ 2 Products' total purchasing, transportation, and customs duties cost Ψ 3 Total production and logistics cost from manufacturers to distribution centers Ψ 4 Returned products' total cost of transportation to collection centers Ψ 5 Products' total disposal and the relative transportation cost Ψ 6 Sum of saved and transportation costs of returned products to manufacturers Z leader Z leader * Leader's objective function/optimal objective function value Z follower Z follower * Follower's objective function/ optimal objective function value The unit purchasing cost of product p from supplier s in country n ($) r cd bspn Customs duty rate of product p from supplier s in country n c tr (1) psfn Unit transportation cost of product p from supplier s to manufacturer f in country n ($) c tr (2) pfb Unit transportation cost of product p from manufacturer f to distribution center b ($) c tr (3) pbm Unit transportation cost of product p from distribution center b to market center m ($) c tr (4) pmg Unit transportation cost of product p from market center m to collection center g ($) c tr (5) pgo Unit transportation cost of product p from collection center g to disposal center o ($) c tr (6) pgf Unit transportation cost of product p from collection center g to manufacturer f ($) c pr p The unit production cost of product p ($) c o p Unit disposal cost of product p ($) d (1) sfn Distance between supplier s and manufacturer f in country n (× 100 km) d (2) fb Distance between manufacturer f and distribution center b (× 100 km) d (3) bm Distance between distribution center b and market center m (× 100 km) d (4) mg Distance between market m and collection center g (× 100 km) d (5) go Distance between collection center g and disposal center o ( x 100 km) d (6) gf Distance between collection center g and manufacturer f (× 100 km) a p Cost-saving of product p due to its recovery ($) o p Minimum disposal fraction of product p v (1) fp The capacity of manufacturer f for product p v (2) bp The capacity of distribution center b for product p v (3) gp The capacity of collection center g for product p r ex n Introduction Today, the importance of sustainability issues has shifted the conventional supply chain (SC) configuration from reverse logistics and green SC models to closed-loop supply chain (CLSC) models (Golpîra and Javanmardan 2022). CLSC is generally an environmentally friendly approach that can potentially reduce environmental impacts and achieve sustainable development of society and economy (Guan et al. 2020). With an emphasis on environmentally sustainable end-of-life product treatment (Yu et al. 2022), it successfully reflects not only sustainability but also a circular economy, taking into account both the forward and backward material flows (Lotfi et al., 2022). Covering the reverse effects of logistics, including treatments such as waste recycling, disposal, and landfilling, has further motivated a range of manufacturing companies to initiate the transition from conventional SC to CLSC (Magazzino et al. 2021). This is due to the high environmental results and significant improvements that are expected to positively affect the effectiveness and competitiveness of companies and SCs Ruimin et al. 2016).
Despite its advantages, it is difficult to deal with closedloop supply chain network design (CLSCND) problem and achieve acceptable results . In addition to the complexity of designing such a multiactor structure with conflicting goals, coupled with the multiplicity of variables, constraints, and data uncertainty, its flexibility and economic profitability are still questionable. These issues have driven research, like the research at hand, into end-of-life product recovery processes for reuse, recycling, and remanufacturing Rubio and Corominas 2008) to optimize costs, lead time, etc.
Optimizing multiple conflicting objectives, in different areas such as energy management (Golpîra 2020b) and maintenance scheduling (Golpîra and Tirkolaee 2019), is usually done through multi-objective optimization . However, CLSC networks may include multiple players and more than one decision-maker (DM) in a non-cooperative game structure who are influenced by each other's decisions. Such a decentralized decision-making process can be modeled through a multi-level optimization approach. Bi-level programming (BLP) is a special type of multi-level programming in which the problem has two hierarchical objectives along with some constraints. The first/second problem, along with the first/second objective function, models the first/second actor called leader/follower (Jalil et al. 2018a). Although such a structure makes it difficult to solve the problem (Li 2015), some researchers, e.g., Rezapour et al. (2015), Golpîra and Javanmardan (2021), and Cheraghalipour and Roghanian (2022), have confirmed its application to design CLSCs.
The networks the aforementioned researchers studied are all non-global and include domestic components. Nonetheless, the elements of SCs can be placed in different countries to form a global closed-loop supply chain (GCLSC) (Amin and Baki 2017). More specifically, in some less-developed countries, e.g., Iran, India, etc., the operations of such manufacturers as automobile companies depend on some multinational suppliers in their SCs, while the other components of their SCs are located in the same countries. Given such cases and the importance of paying attention to global logistics as an important part of the growing global economy, the need to move towards globalization in the case of CLSC is undeniable. Globalization may be accompanied by the transfer of many products from their SCs in developed countries to developing countries . The poor logistics infrastructure of less-developed/developing countries, compared with developed countries, poses far greater challenges to global logistics . This is why this research mainly focuses on the global closed-loop supply chain network design (GCLSCND) to provide a more realistic decentralized decision support system (DSS) for it.
How to select/locate international suppliers/facilities and how to consider the factors of globalization in an integrated optimal framework are some of the challenges in designing GCLSCs. Given the negative impact of globalization on delivery time, one of the most critical factors that keep suppliers competitive in the global market is timely delivery (Singh and Kumar 2020). The importance of on-time delivery in the global market is so much that the manufacturer, even if some of the suppliers are disrupted, still tries to guarantee the on-time delivery of the orders even by adopting an emergency procurement strategy (Chen et al. 2022). Despite this level of importance, most studies often neglect it and have emphasized a limited set of factors, including cost, distance, and variety of supplies (Amin and Zhang 2012;Guan et al. 2022;Liu et al. 2022).

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Timely delivery becomes even more important under uncertainty, especially in supply and demand. This is while such mathematical optimizations as BLP, although reducing the computational time, traditionally work with certain input data (Golpîra 2020a). Also, taking into account the uncertainty makes them more complicated. This is because in the real world, by changing one uncertain parameter, many constraints may be violated and the model thereby resulting in infeasibility. The greater vulnerability of the chain to this uncertainty in the global scenario makes GCLSCND even more complicated (Singh et al. 2012).
Since robust optimization (RO) is based on determining some possible scenarios for uncertain parameters without considering any probability distribution function, it is a good idea in response to the uncertainty of input data. Among the OR approaches, conditional value-at-risk (CVaR) simply handles the DM's risk-aversion by indirectly embedding additional materials to the problem's objective function and constraints (Golpîra 2017b). However, despite the rich literature on CLSC, there is no unified theory and integrated approach that adapts the CVaR to provide a satisfactory answer to the following research questions (RQs) in the field of the GCLSCND problem. (RQ1) How does the risk-taking of a DM affect its willingness to invest in GCLSC? (RQ2) How does it affect the responsiveness of the network? (RQ3) How do the cost, profit, and income of a GCLSC change in proportion to the DM's risk-taking level?
This paper aims to mathematically answer the RQs by presenting an original, decentralized framework through a novel risk-based robust linear BLP to optimize the GCLS-CND problem. The first level of the corresponding problem involves the subsystem dedicated to the production, distribution, sale, and recovery of products, and the second level includes suppliers. In the first level, the total cost is minimized, while in the second level, on-time delivery is optimized as an important factor for the manufacturer in selecting multinational suppliers. From a theoretical and structural point of view, the network designed in this research is unique and novel. This is because the manufacturer subsystem plays the role of leader and imposes supplier competition based on on-time delivery to the follower subsystem to minimize costs. Such a novel structure is much closer to reality in the global market while increasing the responsiveness of the network. The results indicate the superiority of the framework not only in terms of SC responsiveness but also in solution robustness. This is while globalization increases the uncertainty in the demand of the SC. Also, the closed-loop nature of the network also causes more uncertainty due to the dependence of the product's rate of return on the uncertain demand. Such uncertainty greatly increases the supplier's on-time delivery risk. Therefore, CVaR is chosen as a suitable RO method to not only mitigate the uncertainty of demand and product returns but also deal with the arising risk. As a modeling novelty of the research, this is the first to design a GCLSCN through a bi-level decentralized DSS in such a way that both the uncertainty and the resulting risk are handled in an optimal integrated robust structure. The results show the superiority of the proposed BLP model in establishing original risk-cost, risk-responsiveness, and profit-income-robustness trade-offs considering the DM's attitude toward risk. By this means, the framework enables managers to predict and measure the advantages, disadvantages, and consequences of their decisions before their actual implementation.
After formulating the resulting convex BLP model, the Karush-Kuhn-Tucker (KKT) conditions are used to transform it into a single-level mixed integer linear programming (MILP) to obtain an exact global optimal solution. Finally, the results obtained from the proposed scenario-based robust MILP problem are compared with those of the corresponding centralized bi-objective MILP problem, which is solved by the Lp-metric. Although there is a rich literature on CLSCND, there is relatively little information about the GCLSCND. More specifically, no research can be found that directly investigates the use of the BLP to the GCLSCND through the CVaR to obtain robustness in a risky environment. In addition, the relative literature reveals that it is still a challenging issue to cover uncertainties caused by random input data while optimizing the BLP in the economy. The contributions of the research can then be summarized as follows: (a) From a theoretical and structural point of view, a unique decentralized DSS is presented to design a GCLSC, which is especially suitable for the conditions of lessdeveloped countries such as Iran and India. Manufacturers in such countries, while requiring multinational suppliers, produce and sell most of their products domestically. In this sense, the manufacturer subsystem has the role of a leader so that selecting the supplier in the follower problem is based on the decision it has made. In this way, the model for the first time ties the GCLSC configuration to the competition between suppliers, based on the manufacturer's needs. (b) This is the first research that compares centralized and decentralized DSSs in a GCLSCN. In doing so, the theoretical structure is joined with the mathematical model of the GCLSCN utilizing BLP. (c) The main mathematical contribution of the research is to formulate a robust GCLSCND problem by integrating the BLP and the CVaR to cope with the uncertainty of both the demand and the number of returned products while handling the resulting risk. (d) Given the global nature of the network and the possibility of replacing suppliers in the absence of timely delivery, the on-time delivery factor, which is often overlooked in relative literature, is modeled so that the responsiveness of the SC is not neglected. This means the model can deal with supplier selection problems based on on-time delivery along with the GCLSCN configuration. (e) The framework is the first to perform good trade-offs between DM's risk-aversion levels and economic concerns as well as the network's responsiveness in a robust manner.
The rest of the paper is organized as follows. A literature review is reported in Section 2. The problem is defined in Section 3 and formulated in Section 4. Model reformulation and solution approach are provided in Section 5 and the computational results, discussions, and solution analysis are reported in Section 6. The managerial/policy implications of the study are highlighted in Section 7. Some interesting theoretical, structural, mathematical, and managerial insights are mentioned in Section 8, and the research limitations and future potential directions are reported in Section 9.

Literature review
The last decades of the twentieth century witnessed a considerable expansion of SCs into international locations, especially in the apparel, automobile, and computer industries (Dornier et al. 2008). Today, despite enjoying the benefits of globalization, SC management has identified it as one of its five biggest challenges (i.e., risk management, increasing customer demands, cost containment, SC visibility, and globalization) (Butner 2010). The GSC also suffers from some weaknesses such as delays, supply and demand unfulfillment, and labor shortages (Duong et al. 2022). Economic benefits and reliability improvement, due to conforming to customs duties and trade concessions, access to cheap labor, overseas markets, and global suppliers, benefiting from subsidies, and logistics cost reduction in foreign markets, are, on the other hand, the drivers of the SC toward globalization (Cagliano et al. 2008;Cohen and Lee 2020;Manuj and Mentzer 2008).
The big issue is how to balance such benefits and the corresponding challenges. It may require the use of stateof-the-art technologies such as the Internet-of-Things Rahmanzadeh et al. 2022) or optimization techniques. Optimization approaches require the evaluation of a few designs and therefore significantly reduce the computational time required to achieve an optimal design (Golpîra 2020a). However, GSC optimal design is more complex than domestic SC management (Cohen and Lee 2020;Du et al. 2022). This is because considering the globalization factors in SC design may lead to an increase in outsourcing and the need for decentralized decision-making processes (Cohen and Lee 2020;Meixell and Gargeya 2005), especially in facing uncertainties (Golpîra and Javanmardan 2021).
Among the challenges in SC globalization, timely delivery is more crucial due to its significant effect on the SC configuration (Hammami and Frein 2013;. It may increase inventory costs (Zeng and Rossetti 2003) and reduce the SC efficiency (Cagliano et al. 2008) and leanness (Caniato et al. 2013) due to the technological (Quintens et al. 2006), physical, and cultural distances (Caniato et al. 2013), and infrastructural deficiencies in less-developed countries (Meixell and Gargeya 2005). Nonetheless, from the literature, it is not yet clear whether timely delivery can be a driver (Gupta et al. 2021) or a barrier (Mahmud et al. 2021;Tan et al. 2006) to GSCs. Also, in general, few models take this factor into account. Nagao et al. (2021) formulated a GSCN design problem with customs duties and supply disruption scenarios through a MILP model to minimize the network's total cost without considering the delivery time. Hasani (2021) claimed the formulation of the GSC using a bi-objective non-linear programming problem to optimize the cost and the lead-time, without considering globalization factors, addressed in Table 1. Zhang et al. (2013) addressed trade-offs between SCs cost and leadtime in a bi-objective MILP model by taking into account the exchange rate. Urata et al. (2017a) formulated the same GSC regardless of customs duty. This is while the effect of the customs duty on the GSC configuration has been proved by Tsiakis and Papageorgiou (2008). Accordingly, Nakamura et al. (2018) reformulated the model by considering customs duty. Nakamura et al. (2019) further suggested reconfiguration in such GSCs as Toyota and Panasonic by extending their model to address the effects of the new customs duty rate and lead-time that may result from Brexit. Reich et al. (2021) contributed to the literature on the hybrid analytical hierarchy process-based MILP approaches, introduced in Irawan et al. (2018) and Validi et al. (2014), for GSCN design by considering the shipping tariffs to optimize cost and delivery time.
Although the abovementioned SCs are globalized, none of them are closed-loop. However, the transition from a forward SC to a CLSC has become an ideal that many companies seek to achieve in the long run (Choi et al. 2012). The essential question in this regard is how to close the loop (Agrawal et al. 2019) in such a way that the design of the resulting network does not become an NP-hard problem. In response to this question, Fleischmann et al. (2001) were among the first to investigate the CLSCND. Since then, several modeling approaches have been developed that are reviewed by Battini et al. (2017), Amin et al. (2020), MahmoumGonbadi et al. (2021), and Oliveira and Machado (2021). Nevertheless, there are a limited number of research papers on the GCLSCND.         (2013) proposed a linear model to design a GCLSC by considering transfer price and net profit between two countries. Panchal et al. (2022) minimized the gross cost of a GSCN considering the impacts of barriers to resource accessibility and external economic decisions on infrastructure through a MILP model. Not only these research works, but all the aforementioned studies, except for what Hasani (2021) has done, ignore the uncertainty, despite its importance (Saraeian et al. 2019). Uncertainty can be in supplying materials and product demand (Zeballos et al. 2014), the number of returned products (Salema et al. 2007), exchange rates (Li and Wang 2010), economic instability (Kwak et al. 2018), political instability (Ras and Vermeulen 2009), and changes in the regulatory environment, especially about carbon emissions and green energy (Srinivasan et al. 2021). Accordingly, Goh et al. (2007) presented a multi-echelon stochastic model to maximize the profit of GSCs, while minimizing the risk of uncertainty in supply, demand, exchange, and disruption. Although the model is written under uncertainty, the corresponding network is not a CLSC. Using a sample approximation approach, Cheng et al. (2018) maximized the total after-tax profit in a GSCN considering transfer price and customs duty through a MILP problem under the uncertainty of transportation costs and customer demands. Amin and Baki (2017) proposed a centralized bi-objective MILP to model a GCLSC under the fuzzy demand. However, there are more reliable approaches such as RO to deal with uncertainty. They are more scalable compared to the fuzzybased models (Golpîra 2018), and more suitable compared to deterministic methods due to achieving a significant improvement in the reliability of the results, despite being slightly more expensive (Golpîra and Javanmardan 2021). Hasani and Khosrojerdi (2016) proposed a robust mixed-integer non-linear model to design a resilient non-closed-loop GSC. Through a MILP model, Xu et al. (2017) examined a robust approach to tackle the uncertainty of exchange rate, maritime transportation costs, and mixed waste collection in a reverse, but non-closed-loop, GSCN.  maximized the net profit of a GCLSC considering the distance from markets, resource availability, exchange and tax rates, import tariffs, and trade regulations. They modeled interval uncertainty of demand and transfer price through an RO model and solved it by Memetic algorithm. Although these models are robust, the DM's attitude toward risk has been completely neglected. However, it is a crucial parameter in decisionmaking under uncertainty (Golpîra 2018). This is because the DM's approach in dealing with uncertainty depends entirely on its degree of risk-taking. Rahimi and Ghezavati (2018) maximized the profit and social impacts while minimizing the environmental impacts of a reverse SC through two-stage stochastic programming. They addressed the degree of the DM's riskaversion in their model by using the CVaR approach. The corresponding network is neither global nor closed-loop. Using CVaR, Ma et al. (2020) developed a new RO model for a nonglobal CLSCND, in which the distributions of transportation cost, demand, and returned products are only partially known. Golpîra and Javanmardan (2022) formulated an optimal sustainable non-global CLSC by considering several carbon emission policies through a CVaR-based robust MILP model. Yousefi and Pishvaee (2018) developed a mathematical model to integrate the financial and physical flows in a GSC. They employed the economic value-added index to evaluate a nonclosed-loop SC's financial performance and the fuzzy CVaR to cope with the uncertainty of the exchange rate. Yousefi et al. (2021) extended the approach to design a GCLSC. All the aforementioned research works concentrate on centralized mono-or multi-objective approaches, while the use of the BLP to move toward a decentralized decisionmaking strategy has been less studied (Amirtaheri et al. 2017;Chalmardi and Camacho-Vallejo 2018). Nevertheless, the use of a decentralized approach in the CLSCND problem, despite being slightly more expensive, is preferable for cost-sensitive DM due to the much higher reliability of the results (Golpîra and Javanmardan 2021). Therefore, introducing ports and their cooperation as the key elements of the GSC, Asadabadi and Miller-Hooks (2018) provided a BLP model aiming at enhancing global port network resiliency. The model is non-closed-loop and deep uncertainty of the parameters is completely neglected in its formulation. Considering random disruptions, Ghomi-Avili et al. (2018) presented a fuzzy bi-objective BLP to formulate a non-global CLSCND taking into account a price-dependent demand. Through a BLP model, Hassanpour et al. (2018) used Min-Max regret scenario-based RO approach to design a CLSC under the governmental legislative decisions. Finally, Table 1 provides detailed information about some more relevant papers as well as the approach introduced in this paper from a comparative perspective. As shown in the table, there is no research in the literature to formulate the GCLSCND problem through a BLP by using CVaR to address risk and uncertainty in an integrated robust manner.

Problem description
Although the operational cost has traditionally been highly regarded as an important concern of CLSCND problems, on-time delivery has received little attention. This is while its importance is very high in response to the inherent uncertainty of today's global and competitive business environment. Since the manufacturer is ultimately responsible for the on-time delivery of products, it prefers suppliers with higher timely delivery performance. Such preference of the manufacturer causes a kind of competition between suppliers to reduce the delivery time. The dependence of manufacturers in less-developed countries on international suppliers, in the conditions that other processes such as production, distribution, and retailing are carried out domestically, reinforces the existence of such competition between suppliers.
As illustrated in Fig. 1, manufacturing in less-developed countries mostly requires multinational suppliers, while most of their products are produced and retailed domestically. Distributors receive the products from manufacturers and send them to market. Nonetheless, customers do not necessarily accept all the products purchased from the markets, and some return them to collection centers to be returned to the manufacturers for reprocessing after being separated from those discarded. As the demand for products is uncertain, the number of returned products will also be uncertain. Also, the manufacturer, as the primary DM, places an order from the supplier in such an uncertain and risky situation. However, the order quantity, offered by the manufacturer, affects not only its costs but also the supplier's delivery. The supplier should respond to the manufacturer's requirement with the least delay if it wishes to stay in the market. However, the uncertainty and the manufacturer's attitude toward the resulting risk will lead to fluctuations in the manufacturer's demand and supply disruption. In this sense, delay in delivery is inevitable, and this delay will negatively affect the manufacturer and ultimately the network's responsiveness. As shown in Fig. 2, independent decisions under supply and production-distribution subsystems establish a noncooperative game that can be modeled using BLP.
The leader's subsystem comprises manufacturers and distribution, market, collection, and disposal centers, while suppliers are included in the follower subsystem. According to the literature (Garg et al. 2015), the CLSC network is essentially classified into two main categories, i.e., forward and reverse SCs. As reported in the paper at hand, reverse flow needs to be separated from the forward flow to achieve greater efficiency (Garg et al. 2015;Rogers and Tibben-Lembke 2001;Rogers 2009), reduce task conflicts, prevent double handling, and improve its inferior role than the forward logistics (Autry 2005). Forward SC includes suppliers, manufacturers, and distribution and market centers, while reverse SC includes collection and disposal centers. Forward SC begins with the material procurement and continues with the manufacturing process. Hereafter, the final products move to the end-users through distribution centers. Reverse flow, on the other hand, begins with the collection of end-of-life products. Returned products are collected in regional collection centers and then they undergo a quality check by the manufacturer to be further re-manufactured. The following assumptions are also adopted.
(a) The number of returned products is considered as a fixed ratio of the total uncertain demand. (b) The phase lag between the forward and reverse SC is constant and can be set to zero without loss of generality. In a general sense, the GCLSCND includes strategic decisions that may result in long-term impacts on the performance of the network. By this means, compared with the planning period, which is usually measured by seasons or a year, the phase lag between the forward and backward SCs is very short and can be neglected. (c) Following the same logic as discussed in assumption (b), the inventory level is also constant and set to zero along the chain, so the model is designed as a static model. (d) In less-developed countries, there is a great deal of confusion about how to manage the exchange (Phetsuksiri et al. 2018) and customs duties (Khattry and Rao 2002) rates. Since, from Table 1, these are also key factors in the design of GSCNs, they are used for modeling in this paper. (e) The number of returned products is considered uncertain because it is a function of the uncertain demand.

Problem formulation
The initial uncertain BLP can be formulated through Eqs. (1) to (22).
(1) Fig. 1 The schematic structure of the proposed CLSC network (1) includes two terms: the first is the retailer/market net profit, defined by Eq.
(2), and the latter is the total cost of the system, defined through Eqs. (3)-(8). Using some well-defined binary variables, Eq. (3) defines the total opening cost of selected manufacturers and centers. Equation (4) is devoted to the total purchasing, transportation, and customs duty costs. Since the suppliers are in different countries, while the other facilities are in the same country, the customs duties are only intended for suppliers. Equation (5) is the total cost of production and transportation from manufacturers to distribution centers, while Eq. (6) is the transportation cost of returned products. Equation (7) is the sum of the disposal and relative transportation costs of returned products. The cost-saving obtained from returned products is calculated by Eq. (8) based on the amounts of returned products denoted by q (6) gfpt . Constraint set (9) defines that the products that are distributed to market centers should be as much as the primary products. Constraint set (10) defines the uncertain demand as the upper bound of the distributed products. Constraint set (11) enforces returned products to be equal to or less than the initial amount of distributed products. Constraint set (12) defines the sum of returned products as the constant proportion of the uncertain demand. This constraint set establishes a relationship between the uncertain demand and the number of returned products, which will be uncertain, subsequently. Constraint set (13) is to explain the fact that some of the returned products will be discarded. Constraint set (14) strikes a rational balance between the number of returned products and the total number of discarded ones and those returned to the manufacturers. This constraint is quite reasonable. This is because the returned products, whatever they are, have no choice but to discard or return to the manufacturer. Besides, the capacity of disposal centers is not limited to a specific capacity. Constraints (15)-(17) are capacity constraints, and constraints (18) and (19) are devoted to defining the type of variables. As the objective function of the model's lower level, Eq. (20) outlines that the on-time delivery of materials should be maximized. Constraint set (21) defines that distributed products consist of those produced directly by manufacturers and those remanufactured, and constraint set (22) is to define the type of variables q (1) sfpt .

Model reformulation and solution approach
The model developed in Problem formulation is initially formulated as an uncertain BLP. Due to the complexity of the bilevel coordination, the model is first transformed into a singlelevel uncertain problem using KKT conditions. A linearization method is then used to deal with the non-linearity arising from the complementary slackness conditions. Finally, the CVaR approach is employed to handle the uncertainty of demand and the number of returned products to provide a final robust single-level linear problem that can be simply solved by commercial software.

The KKT conditions and the relaxation approach
There are some approaches for solving linear BLP problems. Under an appropriate constraint qualification, the lower-level problem can be replaced by its KKT conditions to obtain an equivalent (single-level) mathematical program (Vicente and Calamai 1994). Then, it can be easily solved by commercial software. Accordingly, the lower-level model, denoted by Eqs.
In the equations, fpt and sfpt ≥ 0∀s, fp, t are the KKT multipliers associated with Eqs. (21) and (22) to define Eq. (26) based on the Lagrangian method, in which (1) is the optimal vector for (1) .
According to the non-linearity of Eq. (24), the problem is not easy to solve. The equation is linearized before trying to solve the problem. In doing so, given the binary variables sfpt = {0, 1}∀s, f , p, t , the non-linear constraint is substituted by two linear ones and, therefore, relaxed as follows:

Dealing with uncertainty
There are generally two categories of uncertainty: (1) epistemic or so-called subjective or internal uncertainty, and (2) aleatory or so-called objective, stochastic, or inherent uncertainty. The former is due to the imperfection of human knowledge, while the latter arises from the natural variability of the physical world. Therefore, even without considering the limitations of human knowledge, the existence of uncertainty is undeniable. In such a situation, sfpt = {0, 1}∀s, f , p, t recognizing and dealing with uncertainty becomes very important to mitigate its effects in the decision-making process. Hence, since the early days of introducing mathematical programming when stochastic programming was introduced by Dantzig (1955), it has received more attention day by day.
While aleatory uncertainty can be modeled by the probability theory, epistemic uncertainty can be modeled by both probability and non-probability approaches such as RO (Golpîra and Javanmardan 2021). The goal of the RO is to obtain a robust solution that can guarantee all the scenarios at near optimal levels (Mirzapour Al-E-Hashem et al., 2011). The first RO models were developed by Soyster (1973), Nemirovski (1998), andEl Ghaoui andLebret (1997). Since then, many variations of the RO have emerged over the years. Although, from Fig. 3, some approaches, such as value-at-risk (VaR), negatively affect the convexity of the initial problem, some others, such as CVaR, retain the convexity (Bertsimas and Brown 2009), while capturing the DMs attitude toward risk as an important parameter in decisionmaking (Golpîra, 2018). Fig. 3 reveals the nice mathematical characteristics of the CVaR such as convexity, convenient calculation, and better tractability. Since CVaR can be equivalent to a special mean-risk criterion, it not only takes into account the downside risk of the VaR, shown in Fig. 4, but also captures profit (Choi and RuszczyńSki 2008) and is therefore widely used in finance (Wu et al. 2013). It could maximize the average profit of the profit falling below a certain quantile level, i.e., VaR, which is defined as the maximum profit at a specified DMs risk-aversion level (Wu et al. 2013). For a risk-averse DM, the expected value of a random variable is preferred to the variable itself. Since, from the figures and Eq. (30), CVaR is the expected value of loss exceeding α − VaR, it involves the VaR and has more integrity.
In the figures and the equation, VaR represents the − percentile of loss distribution as the smallest value such that the probability that losses exceed or are equal to this value is (30) CVaR = VaR + (1 − )CVaR + , 0 ≤ ≤ 1 greater than or equal to .CVaR + , or so-called mean excess loss and expected shortfall, is the expected loss given that the loss is strictly exceeding VaR , and CVaR − or so-called Tail VaR is the expected loss given that the loss is weakly exceeding VaR , i.e., the expected loss which is equal to or exceeds VaR (Rockafellar and Uryasev 2002).
Although Wu et al. (2013) claim that a common assumption of all VaR and CVaR literature is to consider demand as the only source of uncertainty, the framework presented in this paper addresses the uncertainty of not only demand but also returned products. The approach The solution approach introduced in this paper used in this paper to tackle the uncertainty of the demand is "data-driven." As claimed by Bertsimas and Brown (2009), in this approach, the finite set of sampled vectors 1 , 2 , ..., s is the only information on the uncertain vector . So, the approach is appropriate for practical settings in which the realization of uncertain parameters is the only information available to avoid complex distribution assumptions. In this sense, all available realizations of the demand are defined over scenarios as mpt1 , mpt2 , ..., mptS . If S defines the number of cases remaining after trimming to the level S = S × (1 − ) + ≈ S × (1 − ) and mpt(s) is the s th smallest component of, yielding Finally, to provide a better theoretical insight into the approach introduced in this research to deal with the initial uncertain bi-level model, the introduced reformulation and solution method is completely depicted in Fig. 5.

Numerical example and solution results analysis
This section is devoted to computational results, discussions, and solution analysis on the proposed MILP problem that is solved in GAMS v.24.7.1 using CPLEX 11.1.1. All simulations are run on an Intel Core i7-6700HQ CPU @ 2.60 GHz computer with 16 GB memory.

Numerical example definitions
As shown in Fig. 6, the model is tested on a network with four international suppliers located in two countries that provide raw materials for the manufacturers that are located in three locations to produce two types of products. The products produced by the manufacturers are transported to 11 markets through 4 distribution centers, and they are sold to customers at time t over 12 months. Some products are returned to 4 collection centers to move to 10 disposal centers or returned to be remanufactured. Following those reported in Problem formulation , the modeling approach and the solution procedure used in this paper are quite different from those discussed in individual research papers in the GCLSCND literature. The input data, then, cannot be provided using a single source. More specifically, the model introduced in Golpîra (2017a), although addresses the demand uncertainty through the CVaR and a robust BLP, is neither global nor closed-loop. So, corresponding data and parameters should be gathered from the relevant literature such as Amin and Baki (2017), which of course is neither decentralized bi-level nor robust and risk-based. By this means, some corrections are needed to increase the homogeneity of the collected data. However, due to the similarity of the data used for both the proposed and conventional models, such corrections do not have a negative effect on the reliability of the results. Mentioning the conventional

Suppliers
Plant  (2017), is only to make it possible to compare the results and is not the contribution of the paper at hand. However, the use of their selected methods to deal with bi-objectivity and uncertainty, i.e., Corley method and fuzzy programming, makes this comparison impossible. Given the wide range of uncertainty considered in this paper, the use of fuzzy theory may dramatically increase the dimension Domestictransportation cost ratio to import cost=%10 International transportation cost ratio to import cost=%25 ⇒ c tr(4) pmg = c tr (5)   , it is not necessary to use it because it is beyond the contribution of the paper. This is also the case in the use of the Corley method because by using this method, the number of the parameters to be tuned has been multiplied by two, hence it is harder for the DM to express its preferences by adjusting the whole set of parameters (Collette and Siarry 2013). However, unlike the approach used in Amin and Baki (2017), the DM's attitude toward risk is a key parameter in the approach introduced in the paper at hand. Besides, the Corley method is fundamentally designed through non-linear programming (Fan et al. 2010). However, the proposed model is essentially formulated to be linear. Also, the literature has shown that the results obtained from non-linear theories can be significantly different from those obtained from linear counterparts (Crowley et al. 1982). Accordingly, the Corley method is also not appropriate in this paper. Therefore, among the existing methods in dealing with multi-objective programming, this paper uses the Lp-metric method, which eventually leads to the following model, where r is set to 1 to keep the linearity. Eqs.
(2)-(19), (21), (22) The input data are included in Table 2. In agreement with a theorem called reduced robust problem (Thiele 2004), the simulations are run based on 100 scenarios. The theorem defines that if the DM keeps the S worst cases among the realizations, it should observe a number S ≥ S , which can be very large; however, it only needs to consider S + 1 scenarios. Accordingly, although any number of scenarios are allowed, it is set to S + 1 , without loss of generality. This is because, although the greater number of scenarios has no significant effect on the results, it affects the simulation run time. Accordingly, the problem contains about 6833 single variables, 42 discrete variables, and 68,404 non-zero coefficients. The mean required time for obtaining the results is approximately 10 s.

Solution results and discussions
Solution robustness is the focus of this research. As mentioned by Huang and Goetschalckx (2014), Altmann and Bogaschewsky (2014), and Golpîra and Javanmardan (2021), it can be evaluated directly by measuring the variability of the objective function value over the different scenarios.
After proving the superiority of the model in terms of robustness, since this model is designed to be risk based, it is necessary to analyze the results obtained from different DMs' risk-aversion levels. For this purpose, the average optimal values of cost, income, and profit obtained from the proposed framework are outlined in Fig. 10.
From Fig. 10, the lower the α-value, the higher the cost of making more profit and income. This is because, more like what is concluded in Krug et al. (2021) and Mao et al. (2022), an optimistic DM, i.e., a DM who has a smaller α-value, is ready to pay more for higher expected profit and income due to the greater expected demand shown in Figs. 7 and 8. Mathematically, regarding Eqs. (3)-(8), an increase in demand and, consequently, an increase in returned products lead to an increase in all components of the system's cost. More specifically, a fully optimistic DM with α = 0.01, compared to a completely pessimistic DM with α = 0.99, expects to face a higher cost of about 161%. However, the expected profit and income will increase by almost the same amount. Therefore, it can be said that the higher costs imposed on the risk-taking DM are considered an investment. Also, in this way, the DM may be motivated to reduce his risk-aversion level so that, despite bearing more costs, he expects more income and profit, in proportion to the cost incurred. More specifically, a decrease in the α-value from 0.99 to 0.90 causes an increase in income, profit, and cost by about 15%. However, a similar increase in the level of risk-taking of an optimistic DM does not lead to the same financial results. Also, reducing the α-value from 0.10 to 0.01 results in only a 6% increase in profit, cost, and income. By this means, a similar increase in the pessimistic DM's risk-taking is associated with more positive effects than an optimistic one. In other words, the sensitivity of the income, profit, and cost of the system to the degree of risk aversion of the pessimistic DM is much higher than that of the optimistic one. This is the reason why a pessimistic DM tends to invest more and reduce his risk aversion level because it has brought him more income than he expected and he sees that the more expenses he spends can be compensated. However, in any case, the ratio of the cost increase is equal to the ratio of increase in income and profit. Such economic outcomes together with the resulting higher standard deviations, shown in Fig. 11, establish a constructive trade-off between the standard deviation, as a measure of the solution robustness, and the financial outcomes against α-values.
From Fig. 10, a pessimistic DM who tends to earn about 15% more profit should decrease his level of riskaversion from 99 to 90% and endures about 70% more risk of profit, income, and cost volatility shown in Fig. 11. At the same time, an optimistic DM who tends to earn about 6% more profit should decrease his degree of conservatism from 10 to 1% and endure about 20% more financial risk. More specifically, an increase in the level of optimism of an optimistic DM has much less than one unit of effect on the economic elements. On the contrary, an increase in the level of optimism of a pessimistic DM affects economic elements more than 1.5 times while increasing their volatility by more than 7.5 times. Finally, a DM who tends to earn 115% more profit shown in Fig. 10 should decrease its level of risk-aversion from 90 to 10% and endure about 70% more risk of volatility in income, profit, and cost shown in Fig. 11. The estimated cost of such an increase in profit may be an incentive for the DM to increase its risk-taking. Furthermore, the risk of income, profit, and cost volatility for a DM who turns from complete cynicism to complete optimism is about 245%, shown in Fig. 11. The DM should then decide whether to accept such amounts of volatility risk to achieve a potential profit of 161%, shown in Fig. 10.   Fig. 13 The differences between the profit-to-cost ratios, income-to-cost ratios, and optimal on-time deliveries of the conventional and the proposed models (%) As one can see, in almost all cases, the ratio of the cost increase is equal to the ratio of increase in income and profit. In agreement with these results, as outlined in Fig. 12, the amount of profit-to-cost and income-to-cost ratios are not depending on the risk aversion level of the DM. At all levels of the DM's risk aversion, the values of the incometo-cost and the profit-to-cost ratios obtained from the proposed approach, although low, are higher than those of the conventional approach. In other words, the amount of profit obtained from each unit of the cost of implementing the decentralized system presented in this research is higher than its amount in the conventional centralized system. Meanwhile, it is impossible to compare the average costs of the two systems due to their difference in considering the DM's attitude toward risk in the decentralized system. However, the corresponding data are reported in Table 3, Table 4, and Table 5. Correspondingly, the differences between the income-to-cost and profit-to-cost ratios, as well as on-timedelivery values obtained from the proposed and the conventional models are also reported in Fig. 13.
The dotted lines and the dashed lines in the figures claim that achieving more robustness is costly, regardless of the DM's optimism or pessimism. This is what is also claimed in Ben-Tal et al. (2010) and further investigated in some other relevant research such as the research done by Golpîra and Javanmardan (2022). Most DMs are willing to sacrifice a part of their profits and incomes in exchange for a higher level of confidence. Therefore, a 34% reduction in income risk, as well as a 28% reduction in profit risk, could be a good incentive for them to bear about a 3.5% reduction in income and a 12% reduction in profit per unit of cost. Also, this is a good trade-off established between risk and cost, obtained from the proposed model.
In addition, the continuous line in Fig. 13, drawn based on the information mentioned in Table 6, outlines that in any case, the framework proposed in this research results in a significant improvement in an on-time-delivery factor. More specifically, a fully optimistic DM faces about 36% improvement in the on-time-delivery factor, while a pessimistic DM with α = 99% faces about 73% improvement. In other words, the proposed model not only results in a significant reduction in the risk of volatility of all the results but also improves the responsiveness of the network, especially for a pessimistic DM.

Managerial policy implications
The current findings can help managers, economists, researchers, policymakers, and firms' owners to design more effective and appropriate GCLSC strategies aiming at increasing their income and profit, trade volume, responsiveness, and SC configuration reliability and robustness, especially in developing/less-developed countries. It focuses on the concerns including costeffectiveness, simplicity of generalizing, and competency of providing timely delivery to acquire the GCLSC along with a decentralized DSS. Accordingly, the managerial policy implications of the study can be highlighted as follows: (a) The risk-averse DM of a GCLSC is suggested to reduce his risk aversion in favor of more income, although at higher costs. The more risk-averse the DM is, the more confident this suggestion is given. (b) It is expected that by increasing the level of risk-taking, the DM will bear more financial risk while experiencing more profit. Therefore, it is suggested to the DM, at any level of risk aversion, to make a risk-cost tradeoff to change his attitude toward risk. However, the importance of this analysis is much higher for the conservative DM than for the risk-taking DM. Meanwhile, the moderate DM, compared to them, can more easily decide to reduce his risk-taking. (c) The cost efficiency of the proposed decentralized DSS is higher compared to the traditional centralized system. So, the manager of the GCLSC is suggested to use it, especially for the GCLSCs that are located in developing countries. (d) Since the higher responsiveness of the proposed system has been confirmed, it is suggested to use the proposed decentralized approach in situations where one of the priorities of decision-making systems is timely delivery.

Conclusion
This paper presents a decentralized DSS through a novel risk-based robust linear BLP to optimally design a general GCLSC. The first level of the problem is dedicated to the subsystem of production, distribution, retailing, and recovery of products and the second level includes suppliers. At the first level, the total cost is minimized, while at the second level, on-time delivery is optimized as an important factor in the selection of multinational suppliers. From a theoretical and structural point of view, the network designed in this research is unique and novel in that the subsystem of the manufacturer plays the role of the leader. Accordingly, in the follower subsystem, competition is established between the suppliers based on timely delivery. Therefore, while optimizing the entire configuration of the GSCN, competition is formed among the suppliers, which will ultimately lead to the achievement of responsiveness in the entire global network. Globalization, in turn, increases uncertainty in demand. The closed-loop nature of the network also causes more uncertainty due to the dependence of the number of the returned product on uncertain demand. Such uncertainty greatly increases the supplier's on-time delivery risk. Therefore, CVaR is employed as a suitable RO method to not only reduce the uncertainty of demand and product returns but also to deal with the risk caused by such uncertainty factors. At the same time, addressing the DM's attitude toward risk, the research also aims to establish essential trade-offs among cost, profit, income, responsiveness, and robustness of the GCLSCN.
Although there is rich literature on CLSCND, there is relatively little information on GCLSCND, and no research can be found that directly examines the use of BLP in GCLSCND through CVaR to gain robustness in a highrisk environment. After formulating the initial convex BLP model, the KKT conditions are used to transform it into a MILP to obtain an exact global optimal solution. Finally, the results obtained from the robust MILP problem based on the proposed scenario-based approach are compared with the results obtained from the corresponding deterministic biobjective MILP problem, which is solved by the Lp-metric method.
The major theoretical contributions that emerged from the research are as follows: (a) From the theoretical and structural point of view, providing a unique decentralized DSS for GCLSCND is the most important contribution of this research. The framework is especially suitable for the conditions of less-developed countries such as Iran. This is because manufacturers in such countries, while needing mul-tinational suppliers, produce and sell most of their products domestically. In this sense, the manufacturer subsystem plays the role of the leader so that selecting the supplier in the follower problem is based on the decision made by this subsystem. In this way, the proposed model for the first time links the configuration of GCLSC based on the manufacturer's needs to the competition between suppliers. (b) This is the first research to compare centralized and decentralized DSSs in a GCLSCN. Therefore, it gives this view to managers and DMs in different situations so that they can predict and measure the advantages, disadvantages, and consequences of their decisions before their actual implementation. (c) The main mathematical contribution of the research is to formulate a robust GCLSCND problem by integrating BLP and CVaR to deal with the uncertainty of demand and the number of returned products while managing the risk arising from them. In this sense, the research formulates a novel mathematical framework to simply handle the uncertainty of demand and returned products. Despite being close to the real world, this modeling approach, due to its linearity and convexity, makes it possible to easily reach the solution using existing commercial software. (d) Due to the global nature of the network and the possibility of replacing suppliers in the case of delivery delays, the factor of on-time delivery, which is often neglected in the literature, is addressed in this research. Directly considering the on-time-delivery factor as the objective function of the follower problem improves the responsiveness of the network, especially for a more pessimistic DM. (e) Establishing original risk-cost, risk-responsiveness, and profit-income-robustness trade-offs in a decentralized GCLSC considering the DM's attitude toward risk is the other contribution of the research. In this sense, proposing multidimensional relationships between DM's risk-taking and the system's average and standard deviation of profit, income, and cost makes the research unique. (f) Significantly improving the solution robustness is the other contribution of the paper. Using the proposed framework, the standard deviations of the profit, income, and cost are reduced by about 28%, 34%, and 36% on average.
Several managerial insights that can be drawn from the current research are as follows: (a) The higher expected profit and income of an optimistic DM is a considerable incentive for a pessimistic DM to reduce its level of conservatism.
(b) Although the increase in the optimism level of the optimistic DM is associated with less economic effects than the pessimistic DM, it also affects the reliability and robustness of the solution less. (c) Both for the optimistic and the pessimistic DMs, the amount of money spent to increase profit seems justified. (d) The use of the proposed decentralized decision-making framework significantly improves responsiveness, especially for a pessimistic DM. (e) Achieving more robustness is costly, regardless of the DM's optimism or pessimism.

Research limitations and future potential directions
Despite the above findings, this research includes some limitations that can be considered for future research.
(a) It is worth noting that waste transportation also leads to significant global carbon emissions (Mele et al. 2022;Mishra et al. 2020). In addition, economic growth in such developing countries as India generates pollution (Mele and Magazzino 2021). However, the research at hand has not considered the existing carbon emission policies that are studied by Golpîra and Javanmardan (2022). Therefore, as a suitable potential for future research, it is suggested to study the effect of different carbon emission policies in GCLSCN management to achieve greater sustainability. (b) CLSC has a complex operating system that includes several participants. One of the issues in CLSC management involves the distribution of benefits to the participants, which is so-called fairness. Unlike classical economics, where the theory is based on the complete rationality of the participant, fairness appears in the theory of social preference. In the literature, researchers often incorporate fairness concerns in the SC configuration using game-theoretic models and evaluate the impact of fairness concerns on channel participants' decisions and profits in CLSC . Ignoring this issue in the proposed model is another limitation of the research at hand. However, the use of Stackelberg's game theory in this research has made it not difficult to include fairness in the proposed model. Therefore, including fairness in the proposed model can be a direction for future research in this field. (c) The model investigated in this paper is developed regardless of the inventory aspects as a static model. This is while inventory management in CLSC is complicated due to the additional uncertainties enforced by reverse material flow in addition to the regular uncer-tainties of the forward flow. Obtaining dynamic recovery information of the end-of-life products, by using some advanced information technologies such as Internet-of-Things Yang et al. 2018), may monitor recovery status and provides a reliable estimation of return quantity. Thus, it can be considered another direction for future research.
Author contribution The author confirms sole responsibility for the study conception and design, data collection, analysis and interpretation of results, and manuscript preparation.
Data availability I confirm that all relevant data are included in the article and information file.

Declarations
Ethical approval The manuscript has not been submitted to any other publication for simultaneous consideration. The submitted work is original and it has not been published elsewhere in any form or language (partially or in full), unless the new work concerns an expansion of previous work. A single study has not been split up into several parts to increase the number of submissions and submitted to various publications or one publication over time. Results have been presented clearly, honestly, and without fabrication, falsification, or inappropriate data manipulation. I adhere to discipline-specific rules for acquiring, selecting, and processing data. No data, text, or theories by others are presented as if they were the author's own ("plagiarism").
Conflict of interest I wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.