The goal of this paper is to set a constraint programming framework to solve
lot-sizing problems. More specifically, we consider a single-item lot-sizing
problem with time-varying lower and upper bounds for production and inventory.
The cost structure includes time-varying holding costs, unitary production
costs and setup costs. We establish a new lower bound for this problem by using
a subtle time decomposition. We formulate this NP-hard problem as a global
constraint and show that bound consistency can be achieved in pseudo-polynomial
time and when not including the costs, in polynomial time. We develop filtering
rules based on existing dynamic programming algorithms, exploiting the above
mentioned time decomposition for difficult instances. In a numerical study, we
compare several formulations of the problem: mixed integer linear programming,
constraint programming and dynamic programming. We show that our global
constraint is able to find solutions, unlike the decomposed constraint
programming model and that constraint programming can be competitive, in
particular when adding combinatorial side constraints.

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