A multistart variable neighborhood descent metaheuristic for the board packing problem

The Board Packing Problem (BoPP) considers a rectangular board divided in cells with m rows and n columns. In this problem, a subset from a set of rectangles with different costs may be allocated on the cells, and in turns, each cell has an associated revenue obtained if a rectangle is placed on it. The objective of the BoPP is to allocate rectangles on the board, covering cells in order to maximize the total profit, measured as the revenues of the selected cells where the rectangle is placed minus the cost of purchasing such rectangles. The revenue of a cell is collected only once, and only if a rectangle is covering the cell. We propose a Variable Neighborhood Descent (VND) approach for solving the BoPP. Two constructive procedures are proposed for generating the initial solution for the VND: a totally greedy approach and a greedy randomized method to favor diversity. The experimental comparison analyses the contribution of each component of the final algorithm and then performs a competitive testing to evaluate the performance of the algorithm when comparing it with the best method found in the state of the art. The superiority of the proposal is supported by non-parametric statistical tests.