Optimal Chaining of Vehicle Plans with Time Windows

TL;DR

This work tackles the problem of plan chaining in Mobility-on-Demand (MoD) systems under time-window constraints, aiming to connect short vehicle plans into longer chains to reduce fleet size and total distance. It introduces a two-step method: generating a compact set of delayed plan variants and solving the chaining problem via a constrained min-cost flow formulation, with proven optimality guarantees. The approach is demonstrated on large-scale static DARP instances by partitioning demand into batches solved optimally and then chained, showing improved solution quality over two heuristics in many cases while maintaining practical computational requirements. The results indicate that time-window aware plan chaining is both theoretically solid and practically applicable for real-world MoD planning and fleet-sizing scenarios. The methodology offers a scalable tool for dispatching and fleet management in ridesharing and MoD contexts, enabling better utilization of real-time and scheduled plans.

Abstract

For solving problems from the domain of Mobility-on-Demand (MoD), we often need to connect vehicle plans into plans spanning longer time, a process we call plan chaining. As we show in this work, chaining of the plans can be used to reduce the size of MoD providers' fleet (fleet-sizing problem) but also to reduce the total driven distance by providing high-quality vehicle dispatching solutions in MoD systems. Recently, a solution that uses this principle has been proposed to solve the fleet-sizing problem. The method does not consider the time flexibility of the plans. Instead, plans are fixed in time and cannot be delayed. However, time flexibility is an essential property of all vehicle problems with time windows. This work presents a new plan chaining formulation that considers delays as allowed by the time windows and a solution method for solving it. Moreover, we prove that the proposed plan chaining method is optimal, and we analyze its complexity. Finally, we list some practical applications and perform a demonstration for one of them: a new heuristic vehicle dispatching method for solving the static dial-a-ride problem. The demonstration results show that our proposed method provides a better solution than the two heuristic baselines for the majority of instances that cannot be solved optimally. At the same time, our method does not have the largest computational time requirements compared to the baselines. Therefore, we conclude that the proposed optimal chaining method provides not only theoretically sound results but is also practically applicable.

Figures (7)

• Figure 1: Example of using the minimum path cover to solve the fleet-sizing problem. On the left, there is a vehicle shareability graph. An arrow between any two plans signals that these plans can be served sequentially. On the right, there is the minimum path cover. Each color represents a chain of plans to be served by one vehicle. The connections (arrows) between plans in the chain are bold.
• Figure 2: Example showing how the min path cover can be converted to the maximum bipartite matching. We use the same example as in Figure \ref{['fig:min_path_cover']}; plans are now numbered for clarity. Each color represents a chain of plans to be served by one vehicle. The connections (arrows) between plans in the chain are bold.
• Figure 3: An example of plan chaining formulated as an assignment problem is a min-weight matching of a specific cardinality. The cardinality is given by the number of vehicles and plans: here, we have two cars available (below, in the circle) and three plans, resulting in cardinality 1 (at least one connection between plans). The numbers on the arcs determine the travel cost between the plans. Each color represents a chain of plans to be served by one vehicle. The connection (arrow) between plans in the chain is bold.
• Figure 4: Example of the final chaining formulation formulated as an MCFP. The vertices are, from the left: the source, then left plan vertices for each plan, left variant vertices for each variant (e.g., 2B translates to plan 2, variant B), and vehicle vertices, then right plan and variant vertices, and finally, the sink. If a plan has no delayed variants, there is no variant vertex. The travel cost is expressed by the number over the arc. Arcs without numbers have zero cost. The inflow of the source is equal to the number of plans, and the outflow of the sink is reversed to that. In the bottom image, there is the solution marked by bold arcs (used arcs with active flow). For readability, vertices between solution arcs are painted blue.
• Figure 5: The scheme of the proposed method.

Theorems & Definitions (8)

• Definition 2.1
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem \ref{thm:variant-generation}
• proof : Proof of Theorem \ref{['thm:variant-generation']}