The split-delivery vehicle routing problem (SDVRP) is a natural extension of the classical vehicle routing problem (VRP) that allows the same customer to be served by more than one vehicle. This problem is a very challenging combinatorial optimization problem and has attracted much academic attention. To solve it, most of the literature articles adopted heuristic approaches in which the solution is represented by a set of delivery patterns, and the search operators were derived from the traditional VRP operators. Differently, our approach employs the combination of a set of routes and a forest to represent the solution. Several forest-based operators are accordingly introduced. We integrate the new operators into a simple tabu search framework and then demonstrate the efficiency of our approach by conducting experiments on existing benchmark instances.

The feasible solutions of the traveling salesman problem with pickup and delivery (TSPPD) are represented by vertex lists in existing literature. However, when the TSPPD requires that the loading and unloading operations must be performed in a last-in-first-out (LIFO) manner, we show that its feasible solutions can be represented by trees. Consequently, we develop a variable neighbourhood search (VNS) heuristic for the TSPPD with last-in-first-out loading (TSPPDL) involving several search operators based on the tree data structure. Experiments show that our VNS heuristic is superior to the current best heuristics for TSPPDL in terms of both solution quality and computing time.