Setup cost reduction in the dynamic lot-size model

Recently, one primary focus of Operations Management has turned to setup reduction because of the growth of Just-in-Time(JIT) manufacturing. Porteus [1] and Billington [2] have developed optimal policies for calculating investment in setup reduction and lot-size in the EOQ model when setup cost is some function of investment. Zangwill [3] has examined the effects of incremental setup cost reductions in the multi-factility dynamic demand environment on costs and zeroinventory facilities. His focus is on obtaining zero-inventory facilities and maximizing savings from lower setup costs without including the cost of such an event. As an extension of these developments, we model the Wagner-Whitin problem with a one-time oportunity to invest in setup reduction. Setup cost is treated as a policy variable and defined as a function of the decision variable representing the annual amortized investment in setup cost reduction. We use an exponential setup reduction function, but speculate that the results also hold for other functions as well. In contrast to Zangwill, we take a direct approach by explicitly finding the optimal investment in setup cost reduction while generating an optimal lot-sizing schedule. Solving a model that incorporates the trade-off between investment and savings results in more realistic solutions. We use a golden section search and the Wagner-Whitin algorithm to obtain solutions for lot-size, setup cost, and the investment in setup reduction. This model is also formulated as a network to better illustrate the interaction between the decision variables. The network formulation can also be exploited to solve the problem with linear programming or network techniques. Finally, we state and prove two theorems that hold for all setup reduction models whether they assume constant or dynamic demand. The first theorem asserts that optimal values for setupt cost and lot-sizes stay fixed over a particular range of holding costs. The second theorem states that the optimal setup cost is independent of initial setup cost.