This paper introduces the single item lot sizing problem with inventory gains. This problem is a generalization of the classical single item capacitated lot sizing problem to one in which stock is not conserved. That is, the stock in inventory undergoes a transformation in each period that is independent of the period in which the item was produced. A 0--1 mixed integer programming formulation of the problem is given. It is observed, that by projecting the demand in each period to a distinguished period, that an instance of this problem can be polynomially transformed into an instance of the classical problem. As a result, existing results in the literature can be applied to the problem with inventory gains. The implications of this transformation for problems involving different production capacity limitations as well as backlogging and multilevel production are discussed. In particular, it is shown that the polynomially solvable classical constant capacity problems become NP-hard when stock is not conserved.

## Citation

Technical report, Department of Engineering Science, The University of Auckland, April 2006