Optimal Control of Complementary and Substitute Items in a Production System for Infinite Time Horizon

The objective of this paper is to formulate an optimal control problem under infinite time horizon for complementary and substitute items. Here the advertisement rate is unknown which considered as a control variable and the stock level is taken as state variable. The demand is controlled by stock as well as advertisement rate. The unit production cost is a function of production rate and also dependent on raw material cost, development cost due to reliability and wear-tear cost. In the production process, reliability plays an important role to improve the quality of products and to decrease the defective rate. It is formulated to optimize the advertisement rate and maximize the optimal profit. Finally, total profit, which consists of the revenue, production cost, holding cost, and advertisement cost, is formulated as optimal control in the steady state with infinite time horizon. The effect of time value of money is taken into consideration. The model is formulated analytically for substitute and complementary items. The optimum results are illustrated and solved by Lyapunov Stability Theory and Pontryagin’s Maximum Principle accordingly. A sensitivity analysis is also presented with respect to time value of money.