On a General Solution of the Deterministic Lot Size Problem with Time-Proportional Demand

It is proved that for a givers number of replenishments m, there exists a unique vector of m time intervals that minimizes the total cost function and it is shown that the total carrying cost obtained after substituting the optimal value of that vector is a convex function of m.

Abstract

We reconsider the classical lot-size model with the assumption that demand rate is deterministic, starts at the origin, and linearly increases with time. Two relevant costs are involved: carrying cost and replenishment cost. The planning horizon is finite and known. The problem is to find the optimal schedule of replenishments, i.e., their number and the schedule of time intervals between consecutive orders. Mathematically, we have to find an integer m and the values of m continuous variables so as to minimize the total carrying and replenishments costs. We prove that for a givers number of replenishments m, there exists a unique vector of m time intervals that minimizes the total cost function. It is further shown that the total carrying cost obtained after substituting the optimal value of that vector is a convex function of m. The algorithm that determines the unique optimal value of m and the unique optimal scheduling of replenishments for m employs these mathematical results. Finally, we investigate the asymptotic properties of the model when the planning horizon is infinite.