Dynamic Lot Sizing with Batch Ordering and Truckload Discounts

Abstract

This paper studies two important variants of the dynamic economic lot-sizing problem that are applicable to a wide range of real-world situations. In the first model, production in each time period is restricted to a multiple of a constant batch size, where backlogging is allowed and all cost parameters are time varying. Several properties of the optimal solution are discussed. Based on these properties, an efficient dynamic programming algorithm is developed. The efficiency of the dynamic program is further improved through the use of Monge matrices. Using the results developed for the first model, an O(n3log n) algorithm is developed to solve the second model, which has a general form of product acquisition cost structure, including a fixed charge for each acquisition, a variable unit production cost, and a freight cost with a truckload discount. This algorithm can also be used to solve a more general problem with concave cost functions.

The single item uncapacitated lot-sizing problem with time-dependent batch sizes: NP-hard and polynomial cases

“…The author proposes an O(T 4 ) time algorithm. Li et al (2004) consider the more general problem both with time-dependent cost parameters ( f t /k t /p t /h t ) and with backlogging, and propose an O(T 3 ) time algorithm. Table 1 gives an overview of these different results.…”

Modeling industrial lot sizing problems: a review

“…Ben-Khedher and Yano (1994) assume that containers, which may be only partially filled, are assigned to trucks and there is a fixed charge for each truck used. Elmaghraby and Bawle (1972), Dorsey et al (1974), Van Vyve (2003) and Li et al (2004) impose that production is done in exact multiples of the batch size.…”

“…In the case of the incremental quantity discount (Diaby and Martel 1993, Chung et al 1996, the reduction is only valid for the amounts in a specific interval. A third alternative is the truckload discount scheme (Li et al 2004), where a less-than-truckload rate is charged until the total cost equal the truckload rate. If the total quantity is more than a truckload, this same scheme is applied for the excess quantity.…”

Optimal solutions for the economic lot-sizing problem with multiple suppliers and cost structures

“…Zhang et al [4] present a model with multi-break point all-unit quantity discount cost structure, and design a polynomial time DP algorithm. Indeed there are some other theoretical results for the ELS problem with other cost structures ( [5,6] for example). Other versions of the ELS problem concentrate on how to find the better approximation solutions.…”