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

Dear Christian,

Many thanks for your comments as well as the provided link. I agree with you that it is either an aggregation of constraints in the original ULS formulation, so not a cut. I also could not find any further information, maybe a bit more on the `production planning by mixed integer programming book`.

Now, let me explain more regarding the issue I have encountered. Based on what the authors mentioned, the first or second inequality, besides the original problem constraints, I mean flow/inventory and setup constraints, should be able to describe a complete linear combination of the original problem convex hull. I totally talk about linear programming, not a MIP. Please, correct me if my inference from the references is being wrong.

Also, I modified the provided code to accommodate my problem. (The Wolsey example in PPBYMIP). Please, see the attached Basic_file. In the MIP model, the objective is 736000, and its dual bound (linear relaxation) would be 712188. I have tried solving your provided example by relaxing the integrality constraint and observed that the model and its separation procedure cannot reconstruct the original solution space. While I expected the code to do that. Actually, it cannot separate any new cuts and add them to the model. The same behaviour goes to my problem, see the attached file example_PPBYMIP, and this is where I am getting stuck in understanding how this cut can be added to the model.

As a second attempt, I managed to use the MIP mechanism of the solver to add these kinds of cuts by changing the value of the cutdepth parameter and adding an integrality constraint. However, the result did not change. Also, I tried to use the first form of the proposed cut and add them to the model in the original solution space as a constraint in the compact formulation. In this case, everything goes well, but there are actually an exponential number of such cuts even for my tiny example!!!

I was wondering if, how I can fix the issue.

Best regards

Abbas