Planning Horizons for the Dynamic Lot Size Model: Zabel vs. Protective Procedures and Computational Results

Abstract

Forward algorithms that solve successively longer finite horizon problems and that possess good stopping rules (such as a planning horizon) seem better suited to the needs of a manager facing a partial information environment than the more common procedure of selecting a horizon [0, T] in advance. In this light, the Wagner and Whitin forward algorithm with a planning horizon procedure for the dynamic lot size model goes far beyond computational savings. Building on additional results due to Zabel, we develop new planning horizon procedures and near planning horizon procedures. (A brief sketch of how to develop similar results for the production smoothing problem suggests that the basic methodology developed possesses some generality.) We present an extensive empirical study that reports that Wagner-Whitin planning horizons were found for a reasonably small subset of problems within 500 periods, while planning horizons, or at least near planning horizons, were found universally by the modified procedure. (The Zabel procedure was intermediate in power.) The number of periods until a “near” horizon seems to be given empirically by [Formula: see text] for the linear stationary case, where K is the setup cost, h the holding cost, and D the average demand rate. Extensions useful for a broad subset of the general concave cost case are also given.

The dynamic lot-sizing problem with convex economic production costs and setups

“…They established that, in an optimal plan with positive fixed setup costs and linear production and holding costs, production is done in a period only if its net demand (actual demand less inventories) is positive, and a period's demand is satisfied entirely by production in a single period (that is, integrality of demand is preserved.) For linear production costs, extensions include Zangwill (1966), Blackburn and Kunreuther (1974), Lundin and Morton (1975), Federgruen and Tzur (1991), Wagelmans et al (1992), Aggarwal and Park (1993), Azaron et al (2009), Ganas and Papachristos (2005), Okhrin and Richter (2011) and Toy and Berk (2013). The fundamental properties of the optimal plans for linear costs hold for piecewise linear and concave cost structures, as well.…”

A novel decomposition approach for on-line lot-sizing

“…Here, one may exploit the fact that afterwards it is always easy to determine what decision would have been optimal. If a planning horizon (Lundin & Morton [1975]) exists, only a limited number of future demands suffices to locate the optimal next regeneration point. This feature provides the basis for a learning or a statistical approach.…”

Algorithm for turnpike policies in the dynamic lot size model

“…A horizon was formally defined first in a little known but precursory paper of Loś [10] as well as in the papers of Hinderer and Hübner [8], Lundin and Morton [12], Lee and Denardo [9] for the deterministic case and Bes and Sethi [3] for the stochastic case. The ideas presented here can be combined with their results.…”

“…Because Algorithm 1 enables us to find all optimal finite stage policies we can solve the problem of the existence of planning and forecast horizons (introduced in Loś [10] and Lundin and Morton [12]). We can now formulate: Corollary 2.…”