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Stochastic Optimization Approaches for an Operating Room and Anesthesiologist Scheduling Problem

Man Yiu Tsang, Karmel S. Shehadeh, Frank E. Curtis, Beth Hochman, Tricia E. Brentjens

TL;DR

This work proposes computationally tractable stochastic programming and distributionally robust optimization methodologies for an integrated allocation, assignment, sequencing, and scheduling problem under uncertainty involving multiple ORs, anesthesiologists, and surgery types and demonstrates the computational efficiency of the proposed methodologies.

Abstract

We propose combined allocation, assignment, sequencing, and scheduling problems under uncertainty involving multiple operation rooms (ORs), anesthesiologists, and surgeries, as well as methodologies for solving such problems. Specifically, given sets of ORs, regular anesthesiologists, on-call anesthesiologists, and surgeries, our methodologies solve the following decision-making problems simultaneously: (1) an allocation problem that decides which ORs to open and which on-call anesthesiologists to call in, (2) an assignment problem that assigns an OR and an anesthesiologist to each surgery, and (3) a sequencing and scheduling problem that determines the order of surgeries and their scheduled start times in each OR. To address uncertainty of each surgery's duration, we propose and analyze stochastic programming (SP) and distributionally robust optimization (DRO) models with both risk-neutral and risk-averse objectives. We obtain near-optimal solutions of our SP models using sample average approximation and propose a computationally efficient column-and-constraint generation method to solve our DRO models. In addition, we derive symmetry-breaking constraints that improve the models' solvability. Using real-world, publicly available surgery data and a case study from a health system in New York, we conduct extensive computational experiments comparing the proposed methodologies empirically and theoretically, demonstrating where significant performance improvements can be gained. Additionally, we derive several managerial insights relevant to practice.

Stochastic Optimization Approaches for an Operating Room and Anesthesiologist Scheduling Problem

TL;DR

This work proposes computationally tractable stochastic programming and distributionally robust optimization methodologies for an integrated allocation, assignment, sequencing, and scheduling problem under uncertainty involving multiple ORs, anesthesiologists, and surgery types and demonstrates the computational efficiency of the proposed methodologies.

Abstract

We propose combined allocation, assignment, sequencing, and scheduling problems under uncertainty involving multiple operation rooms (ORs), anesthesiologists, and surgeries, as well as methodologies for solving such problems. Specifically, given sets of ORs, regular anesthesiologists, on-call anesthesiologists, and surgeries, our methodologies solve the following decision-making problems simultaneously: (1) an allocation problem that decides which ORs to open and which on-call anesthesiologists to call in, (2) an assignment problem that assigns an OR and an anesthesiologist to each surgery, and (3) a sequencing and scheduling problem that determines the order of surgeries and their scheduled start times in each OR. To address uncertainty of each surgery's duration, we propose and analyze stochastic programming (SP) and distributionally robust optimization (DRO) models with both risk-neutral and risk-averse objectives. We obtain near-optimal solutions of our SP models using sample average approximation and propose a computationally efficient column-and-constraint generation method to solve our DRO models. In addition, we derive symmetry-breaking constraints that improve the models' solvability. Using real-world, publicly available surgery data and a case study from a health system in New York, we conduct extensive computational experiments comparing the proposed methodologies empirically and theoretically, demonstrating where significant performance improvements can be gained. Additionally, we derive several managerial insights relevant to practice.
Paper Structure (71 sections, 15 theorems, 97 equations, 19 figures, 48 tables, 2 algorithms)

This paper contains 71 sections, 15 theorems, 97 equations, 19 figures, 48 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Structure of the Paper
  4. Literature Review
  5. Problem Setting
  6. Stochastic Programming Models
  7. Distributionally Robust Models
  8. DRO-E Model Reformulation
  9. DRO-CVaR Model Reformulation
  10. Solution Methods
  11. Valid Inequalities for the SP-E and SP-CVaR Models
  12. The C&CG Method for the DRO-E and DRO-CVaR Models
  13. Valid Inequalities for the DRO-E and DRO-CVaR Models
  14. Separability of the Models
  15. Symmetry-Breaking Constraints
  16. ...and 56 more sections

Key Result

Proposition 1

For $(x,y,z,v,u,s)$ satisfying eqn:1st_stage_con1-2--eqn:1st_stage_con20-21, the inner problem in eqn:mean_support_model, namely, to solve $\sup \limits_{\mathbb{P}\in\mathcal{P}(m,\mathcal{S})} \mathbb{E}_\mathbb{P}[Q(x,y,z,v,u,s,D)]$, is equivalent to

Figures (19)

  • Figure 1: Box plot of surgery duration (in minutes) for different surgery types
  • Figure 2: An illustration of the partition of an ORASP instance characterized by sets $(I,R,A)$ and a set of six surgery types $T=\{1,\dots,6\}$. The left panel shows the sets $(I,R,A)$. The right panel shows the partition $\{(I^g,R^g,A^g)\}_{g=1}^3$ of $(I,R,A)$ based on the partition $\{T_g\}_{g=1}^3$ of $T$.
  • Figure 3: Examples of equivalent solutions due to symmetries in OR opening order and load (solutions 1 and 2), and surgery-to-OR assignments (solutions 2 and 3).
  • Figure 4: Examples of equivalent solutions due to symmetry in surgery sequence.
  • Figure 5: Lower bound and relative gap over iteration with and without VIs in the DRO-E model (Instance 4)
  • ...and 14 more figures

Theorems & Definitions (39)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 2
  • Proposition 5
  • Example A.1
  • Lemma A.1
  • proof
  • ...and 29 more