7

Replace each undirected edge with two directed arcs in opposite directions. The original capacity constraint on each edge now applies to the sum of the arcs in both directions. If the costs are positive, every optimal solution will have flow in at most one of the two directions.


6

That IPOPT message means that IPOPT could not find a feasible solution to your problem. The reason could be either that: Once you set that value below 30, IPOPT can no longer find that basin of attraction (or that basin vanishes). Another feasible solution might exist (unless your problem is convex), but IPOPT can't find it. Your problem actually becomes ...


5

Traditionally, the objective of the multicommodity flow problem is to minimize cost. The usual situation that the costs are positive naturally avoids cycles. The same idea arises with the shortest path problem: if the costs are positive, there are no negative cycles and an LP solver returns a path rather than a path plus cycles.


5

The general approach, whether you are using a mixed integer linear programming model or a constraint programming model, would be to have (nonnegative) variables representing the inventory of different materials at different times, plus flow constraints saying that the inventory at the end of each period is the starting inventory plus any production of the ...


4

"ON THE COMPLEXITY OF TIMETABLE AND MULTI-COMMODITY FLOW PROBLEMS" talks about integral flow while "Multicommodity flows over time: Efficient algorithms and complexity" does not. The integrality constraint (flows being integer valued) is important see LP vs ILP.


4

What you are looking for sounds like, combining the job-shop scheduling problem with material requirement planning or Multi-Periods Job-Shop Scheduling Problem. I am not aware how you would like to use that in the paper or real situation, but in the second one, applying mixed-integer programming (without boosting from the special algorithm like column ...


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