A fuzzy mathematical model for hybrid inventory and purchase optimization in a reverse logistics system considering shortage and warehouse capacity.

Making decisions about the design and implementation of a logistics network is crucial as it has long-term impacts. However, it is important to consider that demand factors and the number of returned items by customers may change over time. Therefore, it is necessary to design a logistics network that can adapt to various demand fluctuations. The main goal of this study is to calculate the quantity of products that should be sent at different times in a supply chain network to minimize the overall cost of reverse logistics and tardiness time. Accordingly, a multi-objective mathematical model is proposed that aims to optimize the total cost and the amount of delay in sending customer orders in a three-level logistics network, assuming that some parameters are uncertain. Additionally, the minimization of waiting time, considering the level of delay in sending, is applied as the second objective function. To handle the uncertainty in the reverse logistics network, a fuzzy approach is implemented, and the proposed model is solved using GAMS software. Furthermore, to solve the mathematical model in large dimensions, the Cuckoo Optimization Algorithm (COA) is applied in MATLAB software, and the results are compared to the global optimal solution. The outcomes show that the proposed algorithm has a desirable performance, as the total values sent to the manufacturer are equal to those obtained from the exact solution, and the objective function value decreases as the number of repetitions increases.