Optimal inventory policy with partial backordering

This paper formulates an inventory stocking policy with partial backordering. An implicit assumption in most inventory control models is that when there is a shortage, either complete backordering or complete lost sales is assumed. Such an assumption is not always tenable in view of the different responses by customers during stockout. Others who assume a fixed ratio of backordering and lost sales encounter the complication of a non-convex cost function. When the cost function is non-convex, the equation formulated is unstable; hence the optimal value is rather tedious to find. In this paper the fraction of backordering and lost sales is constrained so that convexity can be assured by means of the Hessian matrix. In this way the search for the optimal solution is simplified.