Nested evolutionary algorithms for solving a bi-level warehouse location problem that considers inventory decisions

Warehouse location problems play a significant role in the efficiency of supply chains. These strategic problems involve stakeholders determining the optimal locations for warehouses to ensure effective distribution of commodities. However, other stakeholders are also involved in the supply chain. Hence, combining warehouse location decisions with operational decisions, such as distribution, inventory management or production planning, among others, allows for an integrated approach to supply chain management. In this study, we are focusing on stakeholders at two different levels: those in charge of warehouse locations and the managers of each warehouse, who are responsible for inventory decisions. Usually, these stakeholders are not aligned since each of them has their own interests. It is worth emphasizing that an obvious hierarchy among stakeholders exists. The decisions of the warehouse locator (leader) restrict the decision space of each warehouse manager (follower). Moreover, the inventory-related decisions made by each follower have an impact on the total cost of the supply chain. This hierarchical relationship between leader and followers can be modeled as a bi-level programming problem with multiple independent followers. To solve this complex problem, two nested evolutionary algorithms are proposed. The first one follows a traditional evolutionary scheme, and the second one is innovative, since similar individuals are grouped into sub-populations and a migration process is included to enhance the convergence of the algorithm. The nested component of the proposed algorithms is obtained by optimally solving each follower's problem using the Lagrangian multipliers method. Computational experimentation is conducted over a set of benchmark instances to validate the effectiveness and efficiency of the proposed nested evolutionary algorithms. Additionally, a case study is presented, and a sensitivity analysis regarding the number of warehouses to be opened is conducted. Interesting managerial insights are revealed, highlighting an evident trade-off between costs and the number of orders placed. Finally, conclusions drawn from the computational experimentation and suggestions for further research directions are outlined.