A specific separation procedure, $(l,S)$ Inequalities, to solve the lot-sizing problem

Dear Abbas,

Thank you very much for contacting our community. Let me answer your questions in the same order:

• We could not find the second inequality in the paper of Laurence Wolsey. Most likely, it is either an aggregation of constraints in the original ULS formulation, so not a cut, or it is equivalent to to the (l,S) cut, in the sense that flow conservation equations can be added to obtain the one or the other.
• The mentioned Mosel code does indeed generate violated (l,S) inequalities. The produce variables are the $x$ variables and the setup variables are the $y$ variables in your expression. The if statement is responsible for defining the subset $S$ and its complement $L \backslash S$. That is, the summation in the Mosel expression represents the two summations in the (l,S) inequality.
• For an example of how to generate and add cuts in a callback using python, my colleagues Bruno Vieira and Susanne Heipke managed to translate the same Mosel model into similar python code, which I am attaching here (els.py). Notice that for this exemplary small problem instance, the Python callback overhead may offset the node reduction benefit from the cuts, so you may not see meaningful speedup until you scale to larger instances where node count dominates solve time.

Best regards,

Christian Raack