This paper discusses the configuration of a space-effective rack cell for storing a given set of heterogeneous items. Rack cells are the primary components of rack storage areas. A rack cell configuration problem (RCCP) for heterogeneous storage is formulated as a combinatorial mathematical model. An effective heuristic for solving the RCCP in practical cases is presented. The proposed heuristic consists of multistage brute force searching of defined sets of feasible solutions and solving linear integer assignment problems by the branch-and-bound method. The developed algorithm was implemented and tested, and the rack cell obtained meets the modularity requirements in the design and operation of heterogeneous storage areas.

The fixed fleet heterogeneous open vehicle routing problem (HFFOVRP) is one of the most practical versions of the vehicle routing problem (VRP) defined because the use of rental vehicles reduces the cost of purchasing and routing for shipping companies nowadays. Also, applying a heterogeneous fleet is recommended due to the physical limitations of the streets and efforts to reduce the running costs of these companies. In this paper, a mixed-integer linear programming is proposed for HFFOVRP. Because this problem, like VRP, is related to NP-hard issues, it is not possible to use exact methods to solve real-world problems. Therefore, in this paper, a hybrid algorithm based on the ant colony algorithm called MACO is presented. This algorithm uses only global updating pheromones for a more efficient search of feasible space and considers a minimum value for pheromones on the edges. Also, pheromones of some best solutions obtained so far are updated, based on the quality of the solutions at each iteration, and three local search algorithms are used for the intensification mechanism. This method was tested on several standard instances, and the results were compared with other algorithms. The computational results show that the proposed algorithm performs better than these methods in cost and CPU time. Besides, not only has the algorithm been able to improve the quality of the best-known solutions in nine cases but also the high-quality solutions are obtained for other instances.