On setup cost reduction in the economic lot-sizing model without speculative motives

An important special case of the economic lot-sizing problem is the one in which there are no speculative motives to hold inventory, i. e., the marginal cost of ]n'oducing one unit m some period plus the cost of holding it until some future period is at least the marginal production cost in the latter period. It is already known that this special case can be solved in linear time. In this paper we study the effects of reducing all setup costs by the same amount. It turns out that the optimal solution changes in a very structured way. This fact will be used to develop faster algorithms for several problems that can be reformulated as parametric lot ..,.sizing problems. One result, w07,th a separate mention, is an algorithm for the so called dynamic lot sizing problem with learning eifects in setups. This algorithm has a complexity that is of the same order as the fastest algorithm known so far, but it is valid for a more general class of models than usually considered, economic in reduction cost of algorithms, computational problem; Dynamic programming economic lot-sizing problem; Analysis lot-sizing parametric horizon: setup OR/MS subject classification: complexity: parametric economic /optimal control, applications: Inventory/production, planning lot-sizing model 1) Depart.ment of Mathemal.ics and Computing Science, Eindhoven University of Technology, P.O. [lox 513, 56UO MB Eindhoven, The Netherlands. 2) Economet.ric Tnstitul.e, Erasmns University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands; currently on leave at the Operations Research Center, Room E 40-16-1, Massach IIsetts Institute of Technology, Cambridge, MA 02139; financial support of the Netherlands Organization for Scientific Research (NWO) is gratefully acknowledged. o Introduction In 1958 Wagner and Whitin published their seminal paper on the "Dynamic Version of the Economic Lot Size Model", in which they showed how to solve the problem considered by a dynamic programming algorithm. It is well-known that the same approach also solves a slightly more general problem to which we will refer as the economic lot-sizing problem (ELS). Recently considerable improvements ha.ve been made with respect to the complexity of solving ELS and some of its special cases (see Aggarwal and Park, 1990, Federgruen and TzuI' , 1989, and Wagelmans, Van Hoesel and Kolen, 1992). Similar improvements can also be made for many extensions of ELS (see Van Hoesel, 1991). An important special case of ELS is the one in which there are no speculative motives to hold inventory, i.e., the marginal cost of producing one unit in some period plus the cost of holding it until some future period is at least the margina.l production cost in the la.tter period. It is already known that this special case can be solved in linear time. In this paper we study the effects of reducing all setup costs by the same amount. It turns out that the optimal solution changes ill a very structured way. This fact will be used to develop faster algorithms for several problems that can be reformulated as parametric lot-sizing problems. One result, worth a separate mention, is all algorithm for the so-called dynamic lot-sizing problem with learning effects in setups. This algorithm has a complexity that is of the same order as the fastest algorithm known so far, but it is valid for a more general class of models than usua.lly considered. The pa.per is orga.nized a.s rollows. In Section 1 we introduce the economic lot-sizing problclll without speculative motives and describe briefly a linear time algorithm to solve it. Section 2 deals with the parametric version of the problem in which all setup costs are reduced by the same amount. We will characterize how the optimal solution changes and present a linear time algorithm to calculate the reduction for which the change actually occurs. In Section 3 we discuss applications of the results of Section 2. Finally, Section 4 contains some concluding remarks. 1 The economic lot-sizing problem without speculative motives In the economic lot sizing problem (ELS) one is asked to satisfy at minimum cost the known demands for a specific commodity in a number of consecutive periods (the planning horizon). It is possible to store units of the commodity