Local search algorithms for the rectangle packing problem with general spatial costs

We propose local search algorithms for the rectangle packing problem to minimize a general spatial cost associated with the locations of rectangles. The problem is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. Each rectangle has a set of modes, where each mode specifies the width and height of the rectangle and its spatial cost function. The spatial costs are general piecewise linear functions which can be non-convex and discontinuous. Various types of packing problems and scheduling problems can be formulated in this form. To represent a solution of this problem, a pair of permutations of n rectangles is used to determine their horizontal and vertical partial orders, respectively. We show that, under the constraint specified by such a pair of permutations, optimal locations of the rectangles can be efficiently determined by using dynamic programming. The search for finding good pairs of permutations is conducted by local search and metaheuristic algorithms. We report computational results on various implementations using different neighborhoods, and compare their performance. We also compare our algorithms with other existing heuristic algorithms for the rectangle packing problem and scheduling problem. These computational results exhibit good prospects of the proposed algorithms.