Optimal 1-Request Insertion for the Pickup and Delivery Problem with Transfers and Time Horizon

In this paper, we deal with a subproblem of the Pickup and Delivery Problem with Transfers (PDPT) where a finite set of transportation requests has been assigned to a homogeneous fleet of limited capacity vehicles, while satisfying some constraints imposed by a set of pre-scheduled tours. Then, a new transportation request appears, and we also have to serve it. For that, we can modify the current tours by performing a finite sequence of changes allowing to pick up, transport, transfer or deliver this new request. The resulting tours must serve all the requests and satisfy the original constraints, but within a given time horizon. To solve this problem, we present an empirical Dijkstra algorithm that computes tentative solutions whose consistence is checked through constraint propagation, and an exact algorithm which is an adaptation of the well-known A* algorithm for robot planning, that performs an exhaustive search in a tree of partial solutions and reduces the combinatorial explosion by pruning some unfeasible/redundant tree nodes. We conclude by comparing the performance of both algorithms.