Optimal Control of Dynamic Bipartite Matching Models

TL;DR

This paper analyzes optimal matching policies in dynamic bipartite matching models, characterizing optimal strategies for specific graph structures and providing conditions for their extension to more complex graphs, with implications for cost minimization.

Contribution

It fully characterizes optimal policies for complete and N-shaped graphs and extends these results to certain other graph structures under specific cost assumptions.

Findings

Optimal policy for complete graphs is to match everything.

N-shaped graph policies prioritize extreme edges and use threshold strategies.

Simulations show that in W-shaped graphs, extreme edge prioritization is sometimes suboptimal.

Abstract

A dynamic bipartite matching model is given by a bipartite matching graph which determines the possible matchings between the various types of supply and demand items. Both supply and demand items arrive to the system according to a stochastic process. Matched pairs leave the system and the others wait in the queues, which induces a holding cost. We model this problem as a Markov Decision Process and study the discounted cost and the average cost problem. We fully characterize the optimal matching policy for complete matching graphs and for the N -shaped matching graph. In the former case, the optimal policy consists of matching everything and, in the latter case, it prioritizes the matchings in the extreme edges and is of threshold type for the diagonal edge. In addition, for the average cost problem, we compute the optimal threshold value. For more general graphs, we need to consider some assumptions on the cost of the nodes. For complete graphs minus one edge, we provide conditions on the cost of the nodes such that the optimal policy of the N-shaped matching graph extends to this case. For acyclic graphs, we show that, when the cost of the extreme edges is large, the optimal matching policy prioritizes the matchings in the extreme edges. We also study the W-shaped matching graph and, using simulations, we show that there are cases where it is not optimal to prioritize to matchings in the extreme edges.