On multi-item inventory

Abstract

The paper gives a new approach towards a two––item inventory model for deteriorating items with a linear stock––dependent demand rate. In fact, for the first time, the interacting terms showing the mutual increase in the demand of one commodity due to the presence of the other is accommodated in the model. Again, from the linear demand rate, it follows that more is the inventory, more is the demand. So a control parameter is introduced, such that it maintains the continuous supply to the inventory. Next an objective function is formed to calculate the net profit with respect to all possible profits and all possible loss (taken with negative sign). The paper obtains a necessary criterion for the steady state optimal control problem for optimizing the objective function subjected to the constraints given by the ordinary differential equations of the inventory. It also considers a particular choice of parameters satisfying the above necessary conditions. Under this choice, the optimal values of control parameters are calculated; also the optimal amount of inventories is found out. Finally, with respect to these optimal values of control parameters and those of the optimal inventories, the optimal value of the objective function is determined. Next another choice of parameters is considered for which the aforesaid necessary conditions do not hold. Obviously, in that case the steady state solution is non-optimal. In such a case a suboptimal problem is considered corresponding to the more profitable inventory. It is shown that such suboptimal steady state solution fails to exist in this case.