Two-period production planning and inventory control

We study a single product two-period production/inventory model, in which the demands at each period are independent random variables. To optimally satisfy these random demands, quantities can be produced at the beginning of each period using slow or fast production mode, under capacity constraints. In addition to the usual decision variables for such models, we consider that a certain quantity can be salvaged at the beginning of each period. Such salvage processes are useful if the initial inventory of a period is considered to be too high. The unsatisfied demands for each period are backlogged to be satisfied during the next periods. After the end of the second period, a last quantity is produced in order to satisfy remaining orders and to avoid lost sales. The remaining inventory, if any, is salvaged. We formulate this model using a dynamic programming approach. We prove the concavity of the global objective function and we establish the closed-form expression of the second period optimal policy. Then, via a numerical solution approach, we solve the first period problem and exhibit the structure of the corresponding optimal policy. We provide insights, via numerical examples, that characterize the basic properties of our model and the effect of some significant parameters such as costs, demand variabilities or capacity constraints.