Optimal production-inventory policy for products with time-dependent demand pattern and backlogged shortages

In this work, a production-inventory system for items with a time-dependent demand pattern is studied. Shortages are allowed and these are met with the arrival of the next production run. Fluctuations of the net inventory level are analysed and the costs related with the production and inventory management are calculated. The aim is to maximise the profit per unit time. That profit is equal to the difference between sale revenue and production-inventory costs. In the analysis of this system, different scenarios can occur. All these possible cases are studied in detail and, for each scenario, the optimal production-inventory policy and the maximum profit per unit time are determined. Moreover, several numerical examples to illustrate the solution methodology are presented. Finally, a sensitivity analysis of the optimal inventory policy with respect to several parameters of the system and some useful managerial insights derived from the results are given.