A Two-Warehouse Lot Sizing Problem for Defective Items with a Completely Backlogged Shortage Under Limited Storage Capacity for Rented Warehouses

Determining an economic order quantity across inventory management is vital for any production or distribution company. There is an assumption in classical lot-sizing problems that the inventories are stored in a single warehouse with unlimited capacity. However, the mentioned assumption may be unrealistic, and in many markets, owned warehouses (OW) have limited capacity, therefore, in some cases, a rented warehouse (RW) is considered for storing goods to create more flexibility in terms of space capacity. In this study, we proposed a more realistic extension of a two-warehouse system, which intends to investigate an optimal EOQ considering limited storage for both OW and RW. This paper contributes to considering multiple RWs with limited capacity. For this purpose, a two-objective MINLP lot-sizing problem with a discreet period and finite time horizon is taken into account. The complexity of the model is to determine if there is a need to rent one or multiple warehouses regarding limited storage on OWs and RWs as well as budget restrictions to hire one or multiple warehouses. Also, all parameters are considered to be stochastic which means they follow a specific probability distribution. As the capacity of RWs and the total available budget to rent them are random parameters, the constraint on budget, as well as RW's capacity in a period, can be formulated as chance constraints. Finally, four multi-objective decision-making methods namely Goal programming, Lp-Metric, Goal Attainment, and Augmented E-Constraint are utilized to solve the concerned problem. Then, the best method is chosen based on graphical, statistical, and TOPSIS analysis. Moreover, the Lagrangian Relaxation method is applied since solving this optimization model in large-size problems requires a considerable amount of time. In the end, the effect of changing the rates of three parameters on objective function values is evaluated through sensitivity analysis.