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Integer programming regards optimization problems, where one seeks to find integer values for a set of unknowns, that optimizes the objective function. A common subset of this type of problems are integer linear programming problems, where all inequalities, equalities and the objective function are linear in the unknowns.

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Lot sizing problem: how to add these cuts efficiently

I think you forgot the demand in the sum. The lower bound for storage is too low in your cuts. Whether fixing $k$ and $l$ helps you depends on your problem and how often you produce in the optimal so …
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