An optimal algorithm for the quantity discount problem
Authors: Chung Chia‐Shin , Chiang Dalen T. , Lu Cheng‐Yi 
Published: October 2002

The quantity discount problem determines order quantities in a dynamic environment where demand rate changes over time, replenishments are made periodically, and discounts are available for quantity purchases. The undiscounted case, which is known as the dynamic lot‐sizing problem, has been studied extensively in the literature. In particular, the algorithm of Wagner and Whitin [17] gives an optimal order policy for the undiscounted problem. However, for the discounted problem, the only optimal algorithm that the authors know of in the literature is the mixed integer programming model presented by Callerman and Whybark [4], which requires excessive computation time even for problems with only twelve periods. The purpose of this article is to present an efficient algorithm to find an optimal order policy when all‐units quantity discounts are available.

For the undiscounted lot‐sizing problem, an optimal order policy exists with the property that no order will be placed in a period that has positive inventory. Consequently, lot sizes that need to be considered are those that equal the total demand of a number of successive periods. This property no longer holds when there are quantity discounts, where lot sizes at quantity discount levels must also be considered in search of an optimal policy. The authors show that an optimal order policy exists with the property that between two consecutive periods with zero ending inventory, there is at most one order with lot size not equal to any discount level, and this order is the last one between these two periods. A tree search procedure is then implemented to construct sequences of orders that satisfy this property. The total cost of each of these sequences is evaluated to determine the optimal solution using dynamic programming technique.

Based on the framework set up by Callerman and Whybark [4], 3600 test problems were randomly generated according to different combinations of four control factors and one discount level. The four factors are the number of periods in the planning horizon, the coefficient of variation of demand, the ratio of the discount quantity to the average demand, and the percentage of the discount. The computation result shows that the algorithm presented in this paper is very efficient. Compared with the twelve‐period mixed integer programming model of Callerman and Whybark [4], which took 97.46 seconds on CDC 6500, the average computation time for twelve‐period problems using the authors' algorithm took only 0.0195 seconds on IBM 370/3033. For the case of multiple discount levels, the 400 twelve‐period demand patterns, which are generated for the single discount level with different coefficients of variation, are used to investigate the effect of the number of discount levels on the computation time. The result shows that the computation time increases significantly as the number of discount levels increases.

The computation experiment also shows that for twelve‐period problems, the Wagner and Whitin algorithm achieves optimal solution for the quantity discount problem less than 50% of the time. For longer planning horizon problems, this percentage gets even smaller. This result implies that all heuristics based on setting the lot size of an order equal to the total demand of a number of successive periods will not be robust for the quantity discount problem with a longer planning horizon.