Algorithms for Single-Item Lot-Sizing Problems with Constant Batch Size

Abstract

The main result of this paper is an O(n3) algorithm for the single-item lot-sizing problem with constant batch size and backlogging. We consider a general number of installable batches, i.e., in each time period t we may produce up to mt batches, where the mt are given and time-dependent. This generalizes earlier results as we consider backlogging and a general number of maximum batches. We also give faster algorithms for three special cases of this general problem. When backlogging is not allowed and the costs satisfy the Wagner-Whitin property, the problem is solvable in O(n2 log n) time. When the production in each period is required to be either zero or equal to the installed capacity, it is possible to solve the problem with and without backlogging in O(n2) and O(n log n) time, respectively.

Two-phase algorithm for the lot-sizing problem with backlogging for stepwise transportation cost without speculative motives

“…Because the list α in the last row contains periods in nondecreasing order of Δ t;τ À 1 , the first element is period 1 and the second one is period 6 and so on. In this list, we see that the critical index a¼ 1; and b γ ¼ 2 since Δ τ;γ ¼ 7 and index 2 (α½2 ¼ 6) is the upper bound satisfying the condition in (17). With these critical indices, we have zones ½1; 1, ½2; 2 and ½3; 6.…”

“…As a crucial property of LTL productions, the following proposition says that each LTL period has no in-flows from the previous and the next periods [17].…”

“…The research extending Lippman's model includes Pochet and Wolsey [13] and Van Vyve [17]. In addition to the stepwise cost, Van Vyve [17] also considered the maximum bound on the number of cargos that can be used in each period.…”

“…The research extending Lippman's model includes Pochet and Wolsey [13] and Van Vyve [17]. In addition to the stepwise cost, Van Vyve [17] also considered the maximum bound on the number of cargos that can be used in each period. When only one cargo is allowed in each period, the problem reduces to the classical capacitated lot-sizing problem [6,4,16].…”

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

“…Recently Van Vyve (2007) proposes an O(T 3 ) algorithm for a general case with time-dependent costs, (−/k t /p t /h t ), time-dependent production capacity and allowing backlog. The time complexity of the algorithm reduces to O(T 2 log(T )) in the case without backlogging.…”