An approach to determine optimal policies for production—inventory systems†

A production—inventory problem is studied in a form amenable to the discrete-time optimal control theory. First, a mathematical model as a first-order approximation of the system is derived and a set of linear difference equations with time-varying coefficients are obtained. A quadratic cost function for the system is optimized with respect to a decision variable using dynamic programming. Then, a more general mathematical model is presented for the production-inventory problem. The discrete-time maximum principle is applied to optimize the system performance. A discussion is also presented on determining the effects of possible errors occurring in the optimal performance of the system, if the optimal continuous-state results must be quantized to integer values. Examples for a production-inventory system are presented. An interpretation of the results is given in a form which can serve as a guideline for decisions by management.