The minimum raster set problem and its application to the $$d$$-dimensional orthogonal packing problem.(English)Zbl 1403.90574

Summary: We consider the well-known $$d$$-dimensional orthogonal packing problem (OPP-$$d$$). Using the toolset of conservative scales introduced by S. P. Fekete and J. Schepers [Math. Oper. Res. 29, No. 2, 353–368 (2004; Zbl 1082.90095); Math. Methods Oper. Res. 60, No. 2, 311–329 (2004; Zbl 1076.90049)], we are able to change items’ sizes of the initial instance to obtain an equivalent instance with the same solution. In this paper, we present an efficient algorithm for building equivalent instances with certain properties. We also consider the so-called raster model for OPP-$$d$$ introduced by G. Belov et al. [Int. Trans. Oper. Res. 16, No. 6, 745–766 (2009; Zbl 1179.90273); OR Spectrum 35, No. 2, 505–542 (2013; Zbl 1263.90069)]. It is a 0/1 ILP model in which the number of variables and constraints depends on the total number of raster points over all dimensions. Using our algorithm, we construct equivalent instances with a reduced number of raster points. We also present an algorithm to find a lower bound on the minimum possible number of raster points over all equivalent instances. Numerical results are presented.

MSC:

 90C27 Combinatorial optimization 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry) 05B40 Combinatorial aspects of packing and covering 90C10 Integer programming 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
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