Multi-objective Optimization Models for Many-to-one Matching Problems

Abstract

This paper is concerned with many-to-one matching problems for assigning resident physicians (residents) to hospitals according to their preferences. The stable matching model aims at finding a stable matching, and the assignment game model involves maximizing the total utility. These two objectives however are generally incompatible. We focus on a case involving predetermined groups of residents who want to be matched in groups. To pursue these conflicting objectives simultaneously, we propose several multi-objective optimization models for many-to-one matching problems. We first derive a bi-objective optimization model for maximizing the total utility while minimizing the number of blocking pairs to promote stability. We next introduce a small-subgroup penalty, which will be minimized as the third objective for the purpose of matching in groups. Our multi-objective optimization models are formulated by means of the ε-constraint method as scalar objective mixed-integer optimization problems, which can be solved to optimality by using optimization software. The efficacy of our method is assessed through simulation experiments via comparison with the outcomes of two common matching algorithms: the deferred acceptance algorithm and Gale's top trading cycles algorithm. Our results highlight the potential of optimization models for computing good-quality solutions to a variety of difficult matching problems.

Performance Assessment of a Rehabilitation Transportation Reservation Matching Service with Market Design Mechanisms

“…Equation (4) ensures that the number of matches would be less or equal to the total capacity provided by the rehabilitation bus schedule. Equation (5) refers to Shimada et al [46] proof of disruptions in the restricted model regarding the matching of resident physicians and hospitals. Where j ≻ d s , it means that, after the matching process, patient d prefers shift j more than the one assigned to them, shift s. Also, i ≻ s d implies that, after the matching process, the rehabilitation bus prefers patient i more than the one assigned to them, patient d. When ω ds � 0, it indicates no disruptions, and the matching is stable.…”

Matching Algorithms: Fundamentals, Applications and Challenges

“…Typical examples of many-to-one matching arise in the person-institution matching (shown in Fig. 2) problems such as student-college or doctor-hospital matching [96]. The rule behind this kind of matching is that agents on one side (e.g., institutions) can provide many of the same positions for the agents on the other side (e.g., students), but the reverse is not valid.…”