A Minimum-Risk Dynamic Assignment Mechanism Along with an Approximation, Heuristics, and Extension from Single to Batch Assignments

In the classic linear assignment problem, items must be assigned to agents in a manner that minimizes the sum of the costs for each item-agent assignment, where the costs of all possible item-agent pairings are observed in advance. This is a well-known and well-characterized problem, and algorithms exist to attain the solution. In contrast, less attention has been given to the dynamic version of this problem where each item must be assigned to an agent sequentially upon arrival without knowledge of the future items to arrive. This study proposes an assignment mechanism that combines linear assignment programming solutions with stochastic programming methods to minimize the expected loss when assignments must be made in this dynamic sequential fashion, and offers an algorithm for implementing the mechanism. The study also presents an approximate version of the mechanism and accompanying algorithm that is more computationally efficient, along with even more efficient heuristic alternatives. In addition, the study provides an extension to dynamic batch assignment, where items arrive and must be assigned sequentially in groups. An application on assigning refugees to geographic areas in the United States is presented to illustrate the methods.