Multi-item lot-sizing with joint set-up costs

• Published in: Mathematical programming Vol. 119; no. 1; pp. 79 - 94
• Main Authors: Anily, Shoshana, Tzur, Michal, Wolsey, Laurence A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-06-2009; Springer; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Costs; Exact sciences and technology; Full Length Paper; Integer programming; Inventory control, production control. Distribution; Linear programming; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Packing problem; Production costs; Production planning; Studies; Theoretical; Variables; Mixed integer programming; Convex hull; Multi-item lot-sizing; Extended formulation; Joint set-up cost; Backorder; Unit production; Linear programming; Lot sizing; In-process inventory; Upper bound; Storage; Multi item management; Batch process; Batch production; Production cost; Production unit; Mathematical programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-007-0202-9

We consider a multi-item lot-sizing problem with joint set-up costs and constant capacities. Apart from the usual per unit production and storage costs for each item, a set-up cost is incurred for each batch of production, where a batch consists of up to
C
units of any mix of the items. In addition, an upper bound on the number of batches may be imposed. Under widely applicable conditions on the storage costs, namely that the production and storage costs are nonspeculative, and for any two items the one that has a higher storage cost in one period has a higher storage cost in every period, we show that there is a tight linear program with
O
(
mT
2
) constraints and variables that solves the joint set-up multi-item lot-sizing problem, where
m
is the number of items and
T
is the number of time periods. This establishes that under the above storage cost conditions this problem is polynomially solvable. For the problem with backlogging, a similar linear programming result is described for the uncapacitated case under very restrictive conditions on the storage and backlogging costs. Computational results are presented to test the effectiveness of using these tight linear programs in strengthening the basic mixed integer programming formulations of the joint set-up problem both when the storage cost conditions are satisfied, and also when they are violated.