A Max–Min Ant System for the split delivery weighted vehicle routing problem

• This paper proposes the split-delivery weighted vehicle routing problem (SDWVRP).
• A Max–Min Ant System (MMAS-S) algorithm to solve the SDWVRP is developed.
• Large sets of benchmark instances are tested to demonstrate the efficiency of the model.
• SDWVRP are compared with SDVRP and WVRP.

In real-word cargo transportation practice, such as the deliveries of perishable food and hazardous materials, neglecting the cargo weight in a typical vehicle routing problem (VRP) may prevent the routes from being the most cost effective. Thus, this paper proposes the split-delivery weighted vehicle routing problem (SDWVRP), which consists of constructing the optimal routes, with respect to the constraints on vehicle capacity and cargo weight, to serve a given set of customers with the minimum cost. A Max–Min Ant System (MMAS-SD) algorithm to solve the SDWVRP is developed and a set of theorems and corollaries are proposed to provide an easy approach for route splitting in a typical Weighted VRP (WVRP). The benefit of Split-Delivery for WVRP, as compared to that of SDVRP, primarily lies in its impact on the geographic position and loading weight feature. Large sets of benchmark instances, which are classified into cluster, random and mix types of the three different distribution types, are calculated to demonstrate the effectiveness of the SDWVRP modeling. In addition, the comparison between SDWVRP and WVRP is also carried out via analysis of vehicle numbers, total cost-savings, and the impact of weight variance and mean weight on the ratio of cost-savings and related vehicle numbers of SDWVRP over WVRP to demonstrate the superiority of SDWVRP in determining optimal routes and resulting in substantial cost savings.