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Combinatorial and Computational Insights about Patient-to-room Assignment under Consideration of Roommate Compatibility

Tabea Brandt, Christina Büsing, Felix Engelhardt

TL;DR

This paper tackles the patient-to-room assignment problem with roommate compatibility by formalizing PRA over a planning horizon and introducing a black-box roommate-fit function. It derives combinatorial insights showing that, for wards with two-bed rooms, the per-period roommate partition can be solved via minimum-weight perfect matching, enabling polynomial-time optimization under certain capacity constraints. The authors compare multiple IP formulations, with and without transfers, and evaluate five scoring functions to model roommate fit, demonstrating how scoring choices impact runtime more than formulation choice. They then integrate these approaches into a dynamic PRA framework (dpra), presenting a practical iterative algorithm that solves real-world-like instances within hours and, typically, seconds per iteration. The work provides a proof-of-concept that compatible roommate assignment can be efficiently integrated into dynamic PRA, offering actionable methods for improving patient experience and hospital operations.

Abstract

During a hospital stay, a roommate can significantly influence a patient's overall experience both positivly and negatively. Therefore, hospital staff tries to assign patients together to a room that are likely to be compatible. However, there are more conditions and objectives to be respected by the patient-to-room assignment (PRA), e.g., ensuring gender separated rooms and avoiding transfers. In this paper, we review the literature for reasons why roommate compatibility is important as well as for criteria that can help to increase the probability that two patients are suitable roommates. We further present combinatorial insights about computing patient-to-room assignments with optimal overall roommate compatibility. We then compare different IP-formulations for PRA as well as the influence of different scoring functions for patient compatibility on the runtime of PRA integer programming (IP) optimisation. Using these results and real-world data, we conclude this paper by developing and evaluating a fast IP-based solution approach for the dynamic PRA.

Combinatorial and Computational Insights about Patient-to-room Assignment under Consideration of Roommate Compatibility

TL;DR

This paper tackles the patient-to-room assignment problem with roommate compatibility by formalizing PRA over a planning horizon and introducing a black-box roommate-fit function. It derives combinatorial insights showing that, for wards with two-bed rooms, the per-period roommate partition can be solved via minimum-weight perfect matching, enabling polynomial-time optimization under certain capacity constraints. The authors compare multiple IP formulations, with and without transfers, and evaluate five scoring functions to model roommate fit, demonstrating how scoring choices impact runtime more than formulation choice. They then integrate these approaches into a dynamic PRA framework (dpra), presenting a practical iterative algorithm that solves real-world-like instances within hours and, typically, seconds per iteration. The work provides a proof-of-concept that compatible roommate assignment can be efficiently integrated into dynamic PRA, offering actionable methods for improving patient experience and hospital operations.

Abstract

During a hospital stay, a roommate can significantly influence a patient's overall experience both positivly and negatively. Therefore, hospital staff tries to assign patients together to a room that are likely to be compatible. However, there are more conditions and objectives to be respected by the patient-to-room assignment (PRA), e.g., ensuring gender separated rooms and avoiding transfers. In this paper, we review the literature for reasons why roommate compatibility is important as well as for criteria that can help to increase the probability that two patients are suitable roommates. We further present combinatorial insights about computing patient-to-room assignments with optimal overall roommate compatibility. We then compare different IP-formulations for PRA as well as the influence of different scoring functions for patient compatibility on the runtime of PRA integer programming (IP) optimisation. Using these results and real-world data, we conclude this paper by developing and evaluating a fast IP-based solution approach for the dynamic PRA.

Paper Structure

This paper contains 11 sections, 3 theorems, 47 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Literature
  3. Problem Definition
  4. Complexity
  5. Measuring the roommate fit
  6. Finding fitting roommates using linear programs
  7. General integer programming formulation
  8. Integer programming formulation without transfers
  9. Integrating the roommate fit into dynamic PRA
  10. Conclusion
  11. Acknowledgements

Key Result

Lemma 2

Let $\mathcal{I}$ be a feasible rmp instance with room capacities $c_r\in\{1,2\}$ and at least one double room. The graph $G_\mathcal{I}$ then contains a perfect matching.

Figures (6)

  • Figure 1: Graph $G_\mathcal{I}$
  • Figure 2: Comparison of IPs \ref{['IP:Q']} - \ref{['IP:S']} using 62 real-life instances, maximum runtime 4h
  • Figure 3: Comparison of IPs \ref{['IP:T']} - \ref{['IP:Va']} using 62 real-life instances, maximum runtime 4h
  • Figure 4: Algorithm for dynamic PRA optimization of single-room requests, patient compatibility and transfers
  • Figure 5: Runtime of algorithm for dynamic PRA with $T=365$
  • ...and 1 more figures

Theorems & Definitions (7)

  • Definition 1: rmp
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof