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In this paper we study the pickup and delivery problem with multipletransportation modalities, and address the challenge of efficiently allocatingtransportation resources while price matching users with their desired deliverymodes. Precisely, we consider that orders are demanded by a heterogeneouspopulation of users with varying trade-offs between price and latency. Tocapture how prices affect the behavior of heterogeneous selfish users choosingbetween multiple delivery modes, we construct a congestion game taking placeover a star network with independent sub-networks composed of parallel linksconnecting users with their preferred delivery method. Using the uniquegeometry of this network we prove that one can define prices explicitly toinduce any desired network flow, i.e, given a desired allocation strategy wehave a closed-form solution for the delivery prices. In connection with priorworks that consider non-atomic congestion games, our result shows that one cansimplify the Linear Program formulations used to solve for edge prices by firstfinding the path prices combinatorially. We conclude by performing a case studyon a meal delivery problem with multiple courier modalities using data fromreal world instances.

Pricing for Multi-modal Pickup and Delivery Problems with Heterogeneous Users