A new approach to the solution of a modified single item continuous production inventory problem

Abstract

Abstract The paper studies critically different types of single item continuous Production Inventory models with regard to the existence of their solutions. The aim is to calculate the minimum value of the objective function based on the total cost involved in the whole process. As the problems are solved by Pontryagin’s maximum principle, so the solution reduces to the solution of a pair of ordinary differential equations. One corresponds to the state dynamics and the other one corresponds to the co-state dynamics; the latter one is to be solved in reverse time. Therefore, the solution is not possible to obtain, in general. But it is known that the models of standard continuous linear quadratic regulators are almost similar to those of continuous production inventory models and at the same time they can be solved completely under a proper choice of the transition matrix. So, the present paper first finds out the production inventory model, which is the most general amongst all such existing inventory models. Next it makes suitable modifications of this general existing model of continuous production inventory in the exact form of linear quadratic regulator in order to assure the existence of its solutions. Once the solution is assured, the paper tries to solve the model completely in the form of obtaining the optimal production rate and optimal inventory level. In this connection, some additional restrictions are imposed on the model, so that the necessary transition matrix be available to get the final solution. Finally, the whole analysis is subjected to some special choice of the parametric functions involved in the system. The resulting solutions in the form of optimal production rate and optimal inventory level are shown graphically and they are interpreted properly from their graphs.