Decentralized Decision System for Closed-Loop Supply Chain: A Bi-Level Multi-Objective Risk-Based Robust Optimization Approach

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Highlight

  • A decentralized Bi-level model is designed for a closed-loop supply chain.

  • The robust multi-objective bi-level model is compared with its single-level variant.

  • The deterministic models are compared with the robust models.

  • Cost, work damage, and job opportunity are linked to DMs risk averseness.

  • The DMs attitude toward risk is carefully considered in the model formulation.

Abstract

This paper proposes a novel risk-based robust mixed-integer linear programming to design a decentralized closed-loop supply chain. The model is formulated as an uncertain bi-level multi-objective programming with multiple suppliers, manufacturers, and distributors, as the leader, and recovery, recycling, and disposal centers as the follower. A Scenario-based Conditional Value-at-Risk is employed to capture the demand uncertainty. The Karush-Kuhn-Tucker approach, εconstraint, and LP-metric are leveraged to deal with the complexity of the bi-level coordination, and the multi-objectivity of the leader and follower. The performance of the model is compared with the performance of the deterministic decentralized model and the corresponding multi-objective model designed for the centralized system in both the robust and deterministic modes. Results indicate better performance of robust approaches compared to deterministic approaches. The decentralized approach provides better performance for the cost-sensitive decision-maker, especially the optimistic one, and those who are sensitive to social parameters prefer the centralized approach.

Introduction

Today, increasing pressure from environmental regulations has forced companies to collect and recycle their end-of-life products through Closed-Loop Supply Chain (CLSC) networks, which include both forward and backward logistics (Govindan et al., 2015; Khan et al., 2020). The final products are transferred directly from the supplier to the customer through the forward/traditional Supply Chain (SC) and return to the network through backward/reverse logistics, usually at the end of their life cycle (Haghighi et al., 2016).

Despite the significant efforts of developed countries to optimize the performance of their forward and reverse SCs (Krikke et al., 2004), manufacturers in developing countries, despite advances in green SC management, are still in the early stages of their CLSC management practices (Zhu & Sarkis, 2004). This may be because, despite the complexity of designing such a multi-actor structure with conflicting goals, the multiplicity of decision variables and constraints, and the uncertainty of the data (Mawandiya et al., 2017; Soleimani & Kannan, 2015), its flexibility and economic profitability are still in question (Soleimani & Govindan, 2014; Stindt & Sahamie, 2014). This is while, in such developing countries like Iran, Mexico, and China, due to the availability of cheap labor, the focus is more on recovering and disposing of end-of-life products through CLSC networks instead of preventing waste and emissions through green SCs (Shaharudin et al., 2019; Zhu et al., 2008).

In the widespread presence of manual labor, the aspect of occupational health and safety will also be of particular importance as a social responsibility to achieve sustainability. Thus, improving job opportunities and reducing work-related damages, which are among the main goals of social responsibilities, become more important (Fathollahi-Fard et al., 2018b; Hutchins & Sutherland, 2008; Khan et al., 2020). Accordingly, it is vital to establish a good trade-off between environmental, economic, and social issues.

Multi-objective optimization is a popular mathematical modeling technique in dealing with such situations, which has been used not only in the field of SC management (Wu et al., 2017) and reverse logistics management (Tosarkani et al., 2020) but also in the areas of energy management (Golpîra, 2020b), maintenance management (Golpîra & Tirkolaee, 2019), etc. Such a modeling approach has also been widely used in the area of CLSC network design (Abdolazimi et al., 2020; Garai & Roy, 2020; Nayeri et al., 2020). However, CLSC networks include multiple players and more than one DM, which, despite being influenced by each other, do not have a direct impact on each other's decision-making process. Such a decentralized decision-making process can be mathematically constructed through a multi-level optimization approach, which has been widely employed in not only forward SC (Golpîra, 2017a), reverse SC (Muneeb et al., 2018), relife SC (Haeri et al., 2020), and green opportunistic SC (Golpîra et al., 2017a) management, but energy management (Golpîra et al., 2020), maintenance management (Mazidi et al., 2018), etc. Several empirical studies have also been presented that have used multi-level programming in dealing with CLSC network design problems. Using case studies in Iran, Fard and Hajaghaei-Keshteli (2018) and Fathollahi-Fard et al. (2018a) have employed tri-level programming, and Ghomi-Avili et al. (2018) have used Bi-Level Programming (BLP) to cope with the CLSC network design in the glass, tire, and filter industries. And Sun et al. (2018) have used the BLP to deal with the reverse condition of the end-of-life vehicles in China.

BLP, as a special type of multi-level programming, consists of two hierarchical levels, i.e. upper level (leader) and the nested lower level (follower), with their owned objectives and constraints. The ultimate decision of the BLP is the result of a non-cooperative game between these two levels through a decentralized decision-making process called the Stackelberg game (Haque et al., 2020). It is an asymmetric game in which the leader has full knowledge of the follower's optimal recourse decision (Briest et al., 2012). Creating such a game between the government and enterprises in CLSC networks, especially in developing countries such as China, could lead to a balance in the environmental, economic, and social aspects (Tiew et al., 2019). Following this idea, Hassanpour et al. (2019) have designed a CLSC network through a min-max regret robust approach. In their model, the government as a leader tends to collect end-of-life products while ensuring a minimum distribution ratio, and the CLSC as a follower tries to maximize its net profit value. Jalil et al. (2019) have also proposed a fuzzy bi-level multi-objective CLSC network design problem by considering three objective functions to minimize the total cost under the control of the leader and the cost of inventory and defects under the control of the follower. Despite emphasizing the importance of considering uncertainty, these research works have not considered the risk of uncertainty and the Decision-Maker's (DM's) attitude towards it. However, they are very important in making decisions under uncertainty (Golpîra, 2018).

In general, although optimization techniques make a considerable contribution and deliver practical plans in a reasonable time (Golpîra, 2020a; Golpîra et al., 2021; Jalil et al., 2018a), it is still challenging to cover uncertainties and risks caused by random data while optimizing multi-level programming in the economy. They have traditionally assumed input data to be certain, while changing an uncertain parameter may, in the real world, violate many constraints and make the model infeasible and the SC more vulnerable. The idea of Robust Optimization (RO) is developed to deal with this challenge by defining some possible scenarios for non-deterministic parameters to obtain a robust solution that can guarantee all scenarios set at near optimal levels (Mirzapour Al-E-Hashem et al., 2011). Among the extensive literature on RO, in line with the growing interest of academia and industry, Bertsimas and Brown (2009) have identified the main features of uncertainty sets-based robust linear optimization through Conditional Value-at-Risk (CVaR) by capturing the DM's attitude towards risk. Since the convexity of the original problem is not negatively affected by the CVaR, it becomes an efficient approach to provide optimal solutions, especially for initially linear models, which is the case in the research at hand.

This research seeks to design a robust multi-objective mathematical model considering the economic and social dimensions of a decentralized CLSC under uncertainty. In this study, not only cost minimization is considered as the first objective function but also such social issues as maximizing job opportunities and minimizing work-related damages are addressed at both the leader and follower levels of the initial multi-objective BLP model. The main idea stems from the fact that the responsibility for such social issues in developing countries cannot be placed directly on manufacturers. Few available studies on corporate social responsibility in these countries show overwhelming concerns about profitability and lower priority for social responsibility due to more urgent survival issues (Fülöp et al., 2000; Jamali, 2007). Thus, managing the social responsibilities in these countries has usually been entrusted to public governmental sectors (Medina, 2008; Salehi et al., 2019), such as municipalities, which are also responsible for municipal solid waste management (Aleluia & Ferrão, 2017). In Colombia, for instance, local municipalities provide equipment and infrastructure for waste pickers who provide manual labor to do recycling (Medina, 2008; Sekerka & Stimel, 2014). In Iran, as the other representative example, municipal solid waste management is done through an organization called the Municipal Waste Management Organization that has been established in most centers of Iranian provinces in recent years (Esmaeilizadeh et al., 2020).

In such cases, the manufacturer manages the forward SC as the leader to minimize the cost. The product reaches the customer and eventually turns into municipal waste at the end of its life cycle. At the beginning of the reverse SC, it is collected by the public sector to be recycled or disposed of. Since efficient disposal is usually costly (Kartam et al., 2004) for the government and creates future environmental hazards (Narayana, 2009), it is willing to provide incentives for the manufacturer to recycle the collected used goods. As a rule, the manufacturer's decision on the amount of production, which is made based on the uncertain market demand, affects the amount of waste at the end of the SC, and the public sector determines the incentives accordingly. Based on incentives, the manufacturer produces and recycles, and the game continues between the manufacturer as the leader in the forward chain and the government as the follower in the reverse SC. This non-cooperative Stackelberg game, in which both players' decisions affect each other but they have no direct control over each other's decisions, has been mathematically formulated via BLP. During the game, given that the recycling process in developing countries requires more manual labor (Chi et al., 2011), the public sector has been able to not only provide more employment but also exercise more oversight of workplace safety by providing incentives. In the game, the goal of the DM at the first level is to reduce costs while benefiting as much as possible from government incentives for recycling, reduce work-related damages, and increase employment. And the second-level DM, while creating financial incentives for the manufacturer to cooperate in recycling, concerns about its cost and tries to achieve social goals by setting standard job opportunities and work-related damages. Due to the importance of trading between three goals for the leader, the LP-metric method is used to deal with the multi-objectivity of the problem at the upper level. At the same time, the public sector at the lower level is not looking for such trading, but rather for formulating and moving towards meeting the standards of social issues. Therefore, at this level, the ε constraint method is used to deal with the multi objectivity of the problem. Afterward, the classical Karush-Kuhn-Tucker (KKT) approach is used to cope with the bi-level nature of the problem to obtain an exact optimal solution by transforming the BLP model into its relative single-level counterpart. Finally, the results obtained from the proposed decentralized scenario-based robust Mixed-Integer Linear Programming (MILP) problem are compared with the results achieved from the corresponding conventional centralized multi-objective MILP problem in both deterministic and robust modes as well as the results of the corresponding deterministic bi-level model. Accordingly, a comprehensive sensitivity analysis is done to better outline the superiority, usefulness, and reliability of the proposed framework and its results and presenting some practical implementations for practitioners. The contributions of the research can be summarized as follows:

  • a)

    Proposing a robust approach to a decentralized CLSC network design using the combination of multi-objective programming, BLP, and CVaR to simultaneously optimize costs and social issues while coping with the demand uncertainty is one of the main contributions of the research.

  • b)

    This is the first to present a decentralized decision-making process for CLSC networks of developing countries in which the manufacturer is considered the leader and the government the follower. This is the unique or rather the novelty of the structure of the proposed CLSC network. The framework contributes to the literature in such a way that the public sector (follower) while guaranteeing minimum standards for social issues, considers financial incentives to involve the private sector (leader) in these activities as much as possible.

  • c)

    Establishing a logical and multidimensional relationship between the DM's risk-aversion level and the system's economic and social responsibilities is the other contribution of the paper, which is not addressed previously in the relative literature.

  • d)

    This is the first to compare the performance of the centralized CLSC with its corresponding decentralized variant in both the deterministic and non-deterministic modes.

The rest of the paper is organized as follows. A literature review is proposed in Section 2 followed by the problem definition, which is discussed in Section 3, and its formulation in Section 4. The computational results, discussion, and solution analysis are reported in Section 5. Finally, some interesting theoretical and managerial insights, as well as some future directions, are reported in Section 6.

Section snippets

Literature Review

This paper comparatively combines robust, bi-level, and multi-objective optimization approaches concerning demand uncertainty, risk, and the DM's attitude towards risk in not only a novel decentralized structure but the corresponding traditional centralized one for the CLSC network. Therefore, in the following, the CLSC network design literature is studied by considering all these dimensions focusing on different mathematical modeling approaches.

Forward SC is traditionally established to meet

Problem Statement

Since reverse logistics separation from the forward chain flow improves the CLSC efficiency (Fu et al., 2021; Garg et al., 2015; Rogers & Tibben‐Lembke, 2001; Rogers, 2009) while reducing task conflicts, double handling, and the inferior role of reverse logistics compared to the forward one (Autry, 2005), this research configures the network at two main levels: forward SC and reverse SC. From Fig. 1, the upper level is devoted to forward SC including suppliers, manufacturers, and distributors

Problem Formulation

In this section, the proposed uncertain bi-level model is fully formulated in Section 4.1 and the transformation approach to achieve the final robust single-level MILP model is outlined in Section 4.2.

Numerical Example and Solution Results Analysis

As aforementioned, the performance of the proposed Robust Bi-Level (RBL) model should be compared with the performance of not only its corresponding Deterministic Bi-Level (DBL) model, but the corresponding Robust Multi-Objective (RMO) model and Deterministic Multi-Objective (DMO) model. The corresponding uncertain multi-objective model can be defined as follows:minCost=Costl+CostfmaxJob=Jobl+JobfminWork=Workl+Workf

Eqs. (4)-(15) and (19)-(29)

The reformulation and the solution approach can then

Problem Definition

As aforementioned, the final proposed models are MILP problems and can be solved in GAMS v.24.7.1 using CPLEX 11.1.1. Simulations are run on an Intel(R) Core (TM) i7-6700HQ CPU @ 2.60 GHz with 16 GB memory. The schematic of the problem is shown in Fig. 3, and the corresponding input data are provided in Table 2.

For the robust models that use the CVaR method, i.e. RBL and RMO models, in full agreement with a theorem called Reduced Robust Problem (Thiele, 2004) the simulations are run based on

Solution Results

Results obtained for the leader and the follower of the proposed RBL model concerning various risk-aversion levels of optimistic DMs, i.e. α=0.01,0.05,0.10, moderate DMs, i.e. α=0.50, and pessimistic DMs, i.e. α=0.99,0.95,0.90, are provided in Table 3. The total results obtained from all the model formulation variants of the proposed framework, i.e. RBL, DBL, DMO, and RMO, are also provided in Table 4 in a comparative perspective. Besides, the networks that are optimally designed for

Conclusion

This research has mainly been proposed to fill this research gap. Following the linearization done to deal with the non-linearity of the corresponding complementary slackness conditions, the KKT conditions are used to deal with the complexity of the bi-level coordination. And the scenario-based CVaR has been leveraged to cope with the uncertainty of the demand in a robust manner. Finally, the results obtained from the proposed model are compared with those provided from the corresponding

CRediT authorship contribution statement

Hêriş Golpîra: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing. Ahvan Javanmardan: Data curation, Formal analysis, Software, Visualization.

Declaration of Competing Interest

We 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.

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