-- Question about cutting-stock problem

A recent posting in the forum 'sci.op-research' shows an integer
programming (IP) formulation to a cutting-stock problem. It suggested
to me a point that I have not seen addressed. First: we know that for
getting linear programming (LP) solutions to realistic problems, the
way to do it is via Gilmore-Gomory column generation, NOT by
formulating the humungous complete LP that has all possible cutting
patterns in it. However, if we want /integer/ solutions, much of the
"theory" underlying the column-generation method is inapplicable, due
in part to the absence of a genuine duality concept for IP. So, my
question is: can column-generation still be justified rigorously (not
just as a heuristic) for the IP formulation? The Wiki article
http://en.wikipedia.org/wiki/Cutting_Stock_Problem says:

"A limitation of the Gilmore and Gomory method is that it does not
handle integrality, so the solution may contain fractions, e.g. a
particular pattern should be produced 1.67 times. Rounding to the
nearest integer often does not work, in the sense that it may lead to
a sub-optimal solution and/or under- or over-produce some of the
orders. These limitations are overcome in modern algorithms, which can
solve to optimality very large instances of the problem (generally
larger than encountered in practice)."

This suggests that appropriate methods exist, but the article does not
give appropriate references.

R.G. Vickson

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