An optimal algorithm for the quantity discount problem

Abstract

EXECUTIVE SUMMARYThe 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 [ 171 givesThe 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. Journal of Operations Management 165For 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.

An efficient optimal algorithm for the quantity discount problem in material requirement planning

“…There are three properties of an optimal order policy, which are used in this paper. Properties 1 and 2 have been proved by Chung et al [7] and Wagner and Within [5], respectively. Property 3 is proved in this paper.…”

“…This section demonstrates the computational efficiency of our algorithm in comparison with CCL algorithm [7]. For this purpose, the average CPU time and the peak memory usage (PMU) required to solve the randomly generated problems are compared.…”

“…, m, is the ith discount level, and it is assumed p 1 > p 2 > · · · > p m > p m+1 . The problem of finding an order policy with the above-mentioned assumptions that minimize the sum of ordering cost, inventory holding cost, and purchasing cost over a multiperiod planning horizon is called the QDP in the literature [7].…”

“…Hemhill and Whybark [9] and Benton and Whybark [10] have compared these heuristics with McLaren's order moment [11] for longer planning horizon. Bregman [12] has used Chung et al [7] optimal (CCL) algorithm to test the above-mentioned six heuristics in various environments of material requirement planning framework. The result of this study suggests that the choice among these lot-sizing procedures is a function of the environment.…”

“…But, when variety of items should be procured from several suppliers in a single-period planning horizon, the QDP will change into the total quantity discount (TQD) problem, which is NP-hard, and also there is no polynomial-time approximation algorithm with a constant ratio (unless P = NP) (Goosens et al [14]). Chung et al [7] have presented an algorithm for the QDP, the computational time of which in the number of periods is polynomial when there is only one discount level and becomes exponential in multi-discount level instances.…”