The parallel machine lot-sizing problem consists of finding the optimal timing and level of production and the best allocation of products to machines. With multiple identical machines there are many alternative optimal solutions, which can be created by renumbering the machines. We propose new constraints to break this symmetry.

The production planning and scheduling research has been considering Economic Lot Scheduling Problem (ELSP) for more than 40-year period. In most cases ELSP is found NP-hard, therefore most of the researchers have focused on development of efficient heuristic approaches to the problem. In this paper, We consider the time-varying lot sizing approach to solve the ELSP. A Simulated Annealing (SA) algorithm is developed. A general factorial design of experiment is carried out to study the e®ect of SA algorithm control parameters on its performance. The computational experience shows that the SA algorithm outperforms the best known Dobson's heuristic and Hybrid Genetic algorithm to the problem.

We consider two lot-sizing problems in which demand is a function of price. In the first problem, demand functions may differ from period to period but one single price has to be set for the whole planning horizon. The usual costs are involved and the objective is to maximize total profit. The second problem is similar, but for each period a different price may be set. Both problems have been studied in the literature, but no polynomial algorithms were given. We show how these two problems can be solved in polynomial time.