Improved algorithms for a lot‐sizing problem with inventory bounds and backlogging

Abstract

This paper considers a dynamic lot-sizing problem with storage capacity limitation in which backlogging is allowed. For general concave production and inventory costs, we present an O(T 2 ) dynamic programming algorithm where is the length of the planning horizon. Furthermore, for fixed-charge and nonspeculative costs, we provide O(Tlog T) and O(T) algorithms, respectively. This paper therefore concludes that the time complexity to solve the bounded inventory lot-sizing problem with backlogging is the same as the complexity to solve the uncapacitated lot-sizing problem for the commonly used cost structures.

Monopoly price discrimination and privacy: The hidden cost of hiding

“…where M is a large number with M ≥ d 1n . Here 1are the balance constraints, (2) ensures the stock upper bound and the overload, 3and 4fix the ranges of the continuous variables, (5) ensures that y t = 1 if x t > 0 , (6) ensures that σ t = 1 if s t > 0 and (7) indicates that y t and σ t are binary variables.…”

“…For lot-sizing with both stock upper bounds and stock fixed costs Atamtürk and Küçükyavuz [1] present several valid inequalities as well as computational experience, and in [2] they present an O(n 2 ) algorithm. Fast algorithms for stock upper bounds and backlogging are presented in Hwang and van den Heuvel [7]. Lot-sizing with production time windows was examined by Brahimi [3], treating both inclusive and non-inclusive cases.…”

Two-level lot-sizing with inventory bounds

“…supplier) level. The supplier demand is the amount ordered at the retailer level at each period t. Constraints (4) and (5) force the setup variables to be equal to 1 if there is an order, i.e. if x R t > 0 or x S t > 0 respectively.…”

“…Similarly, parameter M S t can be replaced by min (5). The mathematical formulation of the 2ULS-IB SR problem is obtained by adding the constraints (8) and (9) to the mathematical formulation of the 2ULS problem.…”

“…Toczylowski [4] addresses this problem by solving a shortest path problem in O(T 2 ) time. More recently, Hwang and van den Heuvel [5] propose an O(T 2 ) dynamic programming algorithm to solve the ULS-IB problem with backlogging and a concave cost structure by exploiting the so-called Monge property. Gutiérrez et al [6] improved the time complexity by developing an algorithm that runs in O(TlogT) using the geometric technique of Wagelmans and van Hoesel [7].…”