A multi-objective closed-loop supply chain under uncertainty: An efficient Lagrangian relaxation reformulation using a neighborhood-based algorithm

Population growth and unprecedented industrialization have contributed to a significant surge in global waste generation, underscoring the need for efficient waste management practices such as recycling, disassembling, restoration, and disposal. Closed-loop supply chain (CLSC) networks offer a solution for effective waste management by enabling the recycling, reassembly, and reuse of waste products. However, designing an optimal CLSC network presents challenges due to uncertainties associated with various parameters, including customer demand, sustainable development goals, and the complexity of network design optimization. This paper addresses these challenges by proposing a comprehensive CLSC network that optimizes environmental, economic, and social footprints through a multi-objective optimization approach. To account for parameter uncertainties, we employ scenario generation using a scenario-based stochastic programming procedure. To solve the proposed model, we transform the multi-objective formulation into a single-objective model using a weighted sum method considering a set of problem-specific heuristics. Furthermore, we leverage Lagrangian relaxation theory to formulate various problem reformulations, aimed at attaining an optimal lower bound for the CLSC problem. We complement this with a neighborhood-based algorithm that helps identify a feasible upper bound. By utilizing a sub-gradient framework, we iteratively reoptimize both the selected reformulation and the neighborhood-based algorithm, culminating in the achievement of an optimal solution. To exemplify the practical applicability of our research, we present a case study focusing on the light engineering industry in Bangladesh. Through rigorous analyses, we showcase that our proposed algorithm can deliver an optimal solution with an optimality gap of 0.06. By offering insights into the successful implementation of our CLSC network in a real-world context, we aim to provide valuable guidance for practitioners seeking to establish efficient and sustainable waste management systems.