## Mathematical modeling and optimal blank generation in glass manufacturing.(English)Zbl 1442.90207

Summary: This paper discusses the stock size selection problem [M. L. Chambers and R. G. Dyson, “The cutting stock problem in the flat glass industry-selection of stock sizes”, Oper. Res. Quart. 27, No. 4, 949–957 (1976; doi:10.1057/jors.1976.187)], which is of relevance in the float glass industry. Given a fixed integer $$N$$, generally between 2 and 6 (but potentially larger), we find the $$N$$ best sizes for intermediate stock from which to cut a roster of orders. An objective function is formulated with the purpose of minimizing wastage, and the problem is phrased as a combinatorial optimization problem involving the selection of columns of a cost matrix. Some bounds and heuristics are developed, and two exact algorithms (depth-first search and branch-and-bound) are applied to the problem, as well as one approximate algorithm (NOMAD). It is found that wastage reduces dramatically as $$N$$ increases, but this trend becomes less pronounced for larger values of $$N$$ (beyond 6 or 7). For typical values of $$N$$, branch-and-bound is able to find the exact solution within a reasonable amount of time.

### MSC:

 90C90 Applications of mathematical programming 90C27 Combinatorial optimization 90C56 Derivative-free methods and methods using generalized derivatives 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut