Cumulative staircase considerations for dynamic lotsizing when backlogging is allowed

Abstract

The dynamic lotsizing problem concerns the determination of optimal batch quantities, when given required amounts appear at discrete points in time. The standard formulation assumes that no shortages are allowed and that replenishments are made instantaneously.For the case when no shortage is allowed, previously it has been demonstrated that the inner-corner condition for an optimal production plan in continuous time reduces the number of possible replenishment times to a finite set of given points at which either a replenishment is made, or not. The problem is thus turned into choosing from a set of zero/one decisions with 2n−1 alternatives, of which at least one solution must be optimal, where n is the number of requirement events. Recently, the instantaneous replenishment assumption has been replaced by allowing for a finite production rate, which turned the inner-corner condition into a condition of tangency between the cumulative demand staircase and cumulative production.In this paper we investigate relationships between optimal cumulative production and cumulative demand, when backlogging is permitted. The production rate is assumed constant and cumulative production will then be a set of consecutive ramps. Cumulative demand is a given staircase function. The net present value (NPV) principle is applied, assuming a fixed setup cost for each ramp, a unit production cost for each item produced and a unit revenue for each item sold at the time it is delivered.Among other results, it is shown that optimal cumulative production necessarily intersects the demand staircase. Instead of having 2n−1 production staircases as candidates for optimality, there are 2n−1 production structures as candidates. These are made up of sequences of batches, of which the set of batches may be optimised individually. Also is shown that the NPV of each batch has a unique timing maximum and behaves initially in a concave way and ends as convex.Results for the average cost approach are obtained from a zeroth/first order approximation of the objective function (NPV).