Two effective methods for the irregular Knapsack problem

Two methods are developed for a two-dimensional cutting problem with irregular shaped items. The concepts of inner-fit raster and no-fit raster are used to search for a feasible positioning of items on a rectangular container. The first method is a Biased Random Key Genetic Algorithm, which is a population method, while the other is a Variable Neighborhood Search, which is a single trajectory method. In the proposed methods, a solution is represented by a vector of items, and the positioning of items is achieved with three rules inspired by the bottom-left strategy. When positioning items, feasible positions can be skipped as a strategy to diversify the search and escape from local optima solutions. Numerical experiments performed on literature instances show that the methods are better than the current state-of-the-art method since they obtained equal or better solutions for all the instances. On average, the occupied area increased 6.44%, and the known optimal solution was obtained for 60% of the instances. The population-based method performed better overall, obtaining solutions with better-occupied areas.