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

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 \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$t = 5\sqrt{2K/hD}$\end{document} 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.