We consider integer optimization models for finding stable solutions to many-to-one, utility-weighted matching problems with incomplete preference lists and ties. Whereas traditional algorithmic approaches for the stable many-to-one matching problem, such as the deferred acceptance algorithm, offer efficient performance for the strict problem setting, adaptation to alternative settings often requires careful customization. Optimization-based approaches are free of the need to create customized algorithms for each unique context and can readily accommodate such extensions as (incomplete) preference lists with ties, alternative and nontraditional objective functions, and side constraints including those that ensure stable matching outcomes free of waste. We explore the flexibility of optimization-based approaches in several ways. First, we introduce four new constraint sets that prevent justified envy and a new system of constraints that prevents waste; taken together, they jointly ensure stable matching outcomes. Second, we create two algorithms to accelerate the generation of our proposed constraints. Third, we construct aggregate objective functions to reflect multiple hierarchical emphases by imposing a strict lexicographical order on the individual components. Fourth, we conduct comprehensive experiments to study the computational performance of our proposed optimization models and compare them with models from the extant literature under a variety of problem attributes. Our experiments reveal the circumstances under which each stability representation excels in terms of optimality criteria and computational efficiency on a variety of real and synthetic data sets. One such setting in which our proposed stability representations excel includes the important context of when sufficient seats exist for applicants, such as school choice problems and hospital residency matching. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [ https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1237 ] or is available from the IJOC GitHub software repository ( https://github.com/INFORMSJoC ) at [ http://dx.doi.org/10.5281/zenodo.6892615 ].

In recent years, several universities have adopted an algorithmic approach to the allocation of seats in courses, for which students place bids (typically by ordering or scoring desirable courses), and then seats are awarded according to a predetermined procedure or mechanism. Designing the appropriate mechanism for translating bids into student schedules has received attention in the literature, but there is currently no consensus on the best mechanism in practice. In this paper, we introduce five new algorithms for this course-allocation problem, using various combinations of matching algorithms, second-price concepts, and optimization, and compare our new methods with the natural benchmarks from the literature: the (proxy) draft mechanism and the (greedy) bidding-point mechanism. Using simulation, we compare the algorithms on metrics of fairness, efficiency, and incentive compatibility, measuring their ability to encourage truth telling among boundedly rational agents. We find good results for all of our methods and that a two-stage, full-market optimization performs best in measures of fairness and efficiency but with slightly worse incentives to act strategically compared with the best of the mechanisms. We also find generally negative results for the bidding-point mechanism, which performs poorly in all categories. These results can help guide the decision of selecting a mechanism for course allocation or for similar assignment problems, such as project team assignments or sports drafts, for example, in which efficiency and fairness are of utmost importance but incentives must also be considered. Additional robustness checks and comparisons are provided in the online supplement.