Container packing problem for stochastic inventory and optimal ordering through integer programming

A supplier to a major manufacturing company is trying to cope with a probabilistic inventory management problem that has stochasticity in demand, material consumption rate, delivery times and other logistics-related parameters. Based on the specialized nature of the finished goods, the input materials are imported from Asia and carried in refrigerated containers that are subject to size and weight limits. The material should be used within a few weeks of receipt; otherwise the rate of manufacturing faults increases drastically. Unsatisfied demand (stockout) can be recovered expensively through the use of air transportation. The stochastic inventory problem with variable demand and lead time along with the optimal packing of containers are modelled as a stochastic optimisation problem. The solution to this problem expresses the optimal order size for raw materials considering all the stochasticities and other logistic limitations by minimizing the expected cost of stockout and over-supply. In addition, the optimal packing of pallets of raw materials into the containers is calculated simultaneously. In this paper, along with the modeling of the probabilistic inventory problem, we develop integer programming models for the optimal container loading and compare the numerical outputs.