Relax-and-Fix and Fix-and-Optimise algorithms to solve an integrated network design problem for closing a supply chain with hybrid retailers/collection centres

This paper studies a multi-echelon closed-loop supply chain network design problem that is characterised by a set of hybrid retailers/collection centres in a multi-period setting. This problem is motivated by the return-to-retail approach currently prevalent in the retail industry under the deposit return scheme. This paper proposes a mathematical programming model that integrates strategic decisions regarding the number and location of hybrid retailer/collection centre facilities, with dynamic decisions pertaining to manufacturing and remanufacturing/recycling, inventory level, and fleet size across the network. This optimisation problem is formulated as a mixed integer linear programming model to fulfil customers’ demands while minimising the total network costs. To solve the problem, a matheuristic solution approach is devised, incorporating Relax-and-Fix and Fix-and-Optimise heuristics augmented by novel relaxation and fixing strategies. We introduce an integrality test which accounts for possible rounding-off errors allowing a user-defined integer feasibility tolerance. Moreover, a variable partitioning is applied to shrink the problem’s dimensions and to shorten the search space. The latter is then iteratively updated to explore neighbourhoods within a given search radius size. To evaluate the validity and efficiency of the proposed model and the solution approach, 90 instances are generated using a case study within the geographical scope limited to the network of a retail chain in France. Numerical results show that the proposed solution method provides near-optimal solutions for small- and medium-size instances in a reasonable computational time and outperforms the commercial solver for large- and extra large-size problems. Managerial insights derived from the computational experiments regarding key performance indicators of the problem are presented and discussed.