> How can you tell that the model is degenerate?

Because the objective is not changed after a number of simplex
iterations.

> (and what does
> degenerate mean?)

See, for example, the following:
http://optlab.mcmaster.ca/~feng/4O03/LP.Degeneracy.pdf

> Also, how do I use the cutting plane options?

You can try to specify options '--mir' or '--gomory' for glpsol.



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