Dynamic lot-sizing with setup cost reduction

One of the fundamental tenets of the Just-in-Time (JIT) manufacturing philosophy is that reduction or even elimination of inventory conserves valuable resources and reduces wasteful spending. In many cases, to achieve inventory reductions requires investment in reduction of setup costs. For this reason, certain proposals for incorporating means for reducing setup costs into classical production-inventory models have been offered in recent years. This article considers a dynamic lot-sizing model M where the values of the setup costs can be reduced by various amounts depending upon the level of funds R committed to this reduction. We show that for each fixed value of R, the model can be represented as a shortest path problem. By minimizing the optimal value function V(R) of the shortest path problem over R, model M can, in theory, be solved. In practice, the viability of this approach depends crucially upon the properties of the function V. Since these properties depend upon the nature of the setup cost function K used in model M, we analyze how V varies as K varies. This allows us to propose two exact, finite algorithms for solving model M, one for the case when K is a concave function, the other for the case when K is convex. Computational results for the convex case are presented. The problems solved demonstrate that, in practice, setup cost reductions chosen according to model M have the potential to significantly reduce both inventory levels and total costs.