Heuristic Approaches for Using an Exact Model to Obtain the Placement of a Set of Rectangles on a Half-Strip

Abstract

There is considered the problem of the compact orthogonal placement, which consists in placing of the set of rectangles on the half-strip and its solution by constructing an optimization model that reduces to solving a high-dimensional linear partial Boolean programming problem. The main method of solving the problem is the algorithm of branches and boundaries, providing consistent fiXation of the Boolean variables. The direct use of this algorithm represents significant computational difficulties and is used only for the small size of the initial set of rectangles. In this paper are proposes approaches that reduce the computational complexity of the solution of this optimization model by using heuristic approaches to fix individual Boolean variables. Most of the variables responsible for the relative arrangement of all pairs of rectangles relative to each other are fixed on the basis of generating the permutation of rectangles and special code that determines the direction by which the intersection of the location of the rectangles should be monitored. Thus, the direct computing complexity of the method of branches and the boundaries of solving the problem becomes insignificant. To generate pre-fixation of the Boolean variables, genetic algorithms and local search algorithms are used. Computational experiments showed that the used techniques of fixing the Boolean variables allowed to reduce the volume of calculations and the quality of the resulting approximate solution, in general, it turns out acceptable.