A branch-and-cut algorithm for the pallet-loading vehicle routing problem considering load balance of semi-trailer trucks

The work presented in this paper is motivated by some practical applications, introducing and solving a new problem called the split delivery vehicle routing problem with time windows and two-dimensional loading constraints considering load balance (2L-SDVRPTW-LB). This problem expands the integrated routing and loading problem, in which the required pallet orders are delivered to customers by a fleet of homogeneous semi-trailer trucks. Firstly, with the goal to minimize the total transportation cost, a mixed-integer linear programming model for the 2L-SDVRPTW-LB is proposed. Secondly, an exact algorithm framework based on the branch-and-cut algorithm is designed to solve the problem. In the algorithm framework, a metaheuristic algorithm is designed to obtain the upper bound. Subsequently, various classes of valid inequalities are introduced to strengthen the formulation, while the constraints ensuring load balance are applied in a lazy manner to optimize computational efficiency. Numerical experiments based on modified benchmark instances demonstrate that the proposed branch-and-cut algorithm effectively solves the 2L-SDVRPTW-LB and its variant of excluding split delivery. This study provides valuable insights for addressing the pallet-loading vehicle routing problem considering load balance of semi-trailer trucks.