Tight big-M values

It is well known large big-M values can cause big headaches for MIP solvers. There are many examples where something like M=9999999999 is used, leading to very wrong results. In some cases we can use a MIP model to calculate tight values for big-M constants. In this case I have a model where the MIP model to calculate the tightest bound is actually using the big-M itself:

This is of course not a useful approach. Luckily there are a few alternatives:

• Keep (1) and most of (2), and we end up with a case where it is possible to find a bound by just looking at the data and so some simple calculations (no MIP is needed). From the problem (a design problem) we know this is a really good bound.
• More sophisticated we can relax only equations (3), and solve a MIP to find a slightly tighter bound. We need to do this for several y_k's. But the models are small and solve fast (we can even solve them in parallel). That is the approach I am going to take.

In GAMS we can write something like:

eq3(k)$(not relax(k)).. y(k) =L= <expr>;

to turn on or off an equation based on set relax. This allows us to re-use this equation unaltered in both this preprocessing step, in the real model and in some reporting step.