# A Test of Wagner’s Heuristics for the Spare Parts Inventory Control Problem

Ibrahim S. Kurtulus (School of Business, Virginia Commonwealth University, Richmond, VA, USA)
DOI: 10.4018/joris.2012100106
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## Abstract

Over the years, Wagner’s (1975) heuristic rules appealed to the practitioners because they had simple data requirements, were easy to understand and hence easy to apply. In his algorithmic solution, Wagner assumes that the exact distribution of demand (during lead time) is known. If such a distribution is not available, he recommends using the normal distribution. The author’s purpose is to compare the cost of the solutions provided by Wagner’s (1975) heuristics to optimal (Archibald & Silver, 1978) and determine their quality. Their second goal is to perform sensitivity analysis on the results with respect to demand’s skewness, the ratio of ordering cost to carrying cost. The author’s third goal is to use as much actual data as they possibly can.
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## Introduction

The problem studied is a type of inventory control system (Hadley & Whitin, 1963) for a single item, which is subject to sporadic demand that is also highly variable in size. Item replenishment occurs after a fixed period of time (lead time) and there is a fixed ordering (or setup) cost plus a unit cost. Units carried in excess of demand incur a holding cost of per unit and demand not satisfied is backordered and incurs a penalty cost of per unit. The objective function is to minimize the expected value of the sum of carrying, backordering and ordering costs. Although most problems of this nature can be solved optimally, when mean demand and its frequency of occurrence are small and backordering costs are high, it takes a very long time (days) to find the optimal solution, indicating that we still need heuristics. The goals of this research - none of which have been addressed before - are as follows:

• 1.

Using the expected total cost as the criterion, compare the performance of two popular heuristic lot sizing rules (Wagner, 1975) with the corresponding optimal solutions.

• 2.

Perform sensitivity analysis on ordering costs, carrying costs, and the symmetry of the distribution of demand. Show how each affects the performance of the heuristics.

• 3.

Use as much actual data as possible.

For the problem discussed, when the ordering decisions are restricted to demand occurrences, a policy of type will minimize the undiscounted expected cost over an infinite horizon (Beckmann, 1961; Vienott & Wagner, 1965; Iglehart, 1963; Archibald & Silver, 1978; Ehrhard, 1979). Accordingly, when inventory on hand plus on order minus backorders, is less than or equal to, a replenishment order of size is placed (Silver & Peterson, 1985).

In studies dealing with the spare parts inventory control problem, various assumptions have been made with respect to demand. However, the most commonly used were the Poisson distribution (Schultz, 1987; Gelders & Van Looy, 1978; Hadley & Whitin, 1963) and the normal distribution (Croston, 1972; Bartakke, 1981; Porteus, 1985; Vereecke & Verstraeten, 1994; Sani & Kingsman, 1997).

A few (Dunsmuir & Snyder, 1989; Segerstedt, 1994; Yeh, 1997) have used the gamma distribution, for demand size and occurrence, after observing that most spare parts inventories typically possess positively skewed frequency profiles with a large spike at zero.

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