Improving the efficiency of the operating room environment with an optimization and machine learning model

Abstract

The operating room is a major cost and revenue center for most hospitals. Thus, more effective operating room management and scheduling can provide significant benefits. In many hospitals, the post-anesthesia care unit (PACU), where patients recover after their surgical procedures, is a bottleneck. If the PACU reaches capacity, patients must wait in the operating room until the PACU has available space, leading to delays and possible cancellations for subsequent operating room procedures. We develop a generalizable optimization and machine learning approach to sequence operating room procedures to minimize delays caused by PACU unavailability. Specifically, we use machine learning to estimate the required PACU time for each type of surgical procedure, we develop and solve two integer programming models to schedule procedures in the operating rooms to minimize maximum PACU occupancy, and we use discrete event simulation to compare our optimized schedule to the existing schedule. Using data from Lucile Packard Children’s Hospital Stanford, we show that the scheduling system can significantly reduce operating room delays caused by PACU congestion while still keeping operating room utilization high: simulation of the second half of 2016 shows that our model could have reduced total PACU holds by 76% without decreasing operating room utilization. We are currently working on implementing the scheduling system at the hospital.

Appendix: Reformulation of IP2

We make IP2 linear by representing the objective function as a piecewise linear function using auxiliary variables, δ_ti.

$$\begin{array}{@{}rcl@{}} && \text{IP2}: \min \quad \sum\limits_{t \in T} \sum\limits_{i = 1}^{C} (2i - 1)\delta_{ti} \end{array} $$

(31)

$$\begin{array}{@{}rcl@{}} && \text{subject to} \\ && z_{t} = \sum\limits_{i = 1}^{C} \delta_{ti} {\kern55pt} (\forall t \in T) \end{array} $$

(32)

$$\begin{array}{@{}rcl@{}} && \delta_{ti} \geq \delta_{t(i + 1)} {\kern50pt}\begin{array}{l} (\forall t \in T)\\ \quad(\forall i \in \{1, 2, \ldots, C-1\}) \end{array} \end{array} $$

(33)

$$\begin{array}{@{}rcl@{}} && f_{r} \leq f_{r}^{*} + \hat{f}{\kern50pt} (\forall r \in R) \end{array} $$

(34)

$$\begin{array}{@{}rcl@{}} && z_{t} ={} \underset{t^{*} \in T :h(t^{*},t,p) = 1}{\sum\limits_{p \in P_{p}}} {}x_{pt^{*}} {\kern38pt}(\forall t \in T) \end{array} $$

(35)

$$\begin{array}{@{}rcl@{}} && z_{t} \leq C {\kern75pt}(\forall t \in T) \end{array} $$

(36)

$$\begin{array}{@{}rcl@{}} && 1 = \sum\limits_{t \in T} x_{pt}{\kern55pt}(\forall p \in P) \end{array} $$

(37)

$$\begin{array}{@{}rcl@{}} && 0 ={} \sum\limits_{t^{*} \in T:b(t^{*},p) = 1} {}x_{pt^{*}}{\kern42pt}(\forall p \in P) \end{array} $$

(38)

$$\begin{array}{@{}rcl@{}} && f_{r} \geq \sum\limits_{t \in T} (t+s_{p}+d_{p}+c_{p}) x_{pt}\\&&{\kern100pt}(\forall r \in R)(\forall p \in P_{r} \setminus P_{f}) \end{array} $$

(39)

$$\begin{array}{@{}rcl@{}} && 1 \geq{} \underset{t^{*} \in T:a(t^{*},t,p) = 1}{\sum\limits_{p \in P_{r}}} {}x_{pt^{*}} {\kern39pt}(\forall t \in T)(\forall r \in R) \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} && 1 \geq{} \underset{t^{*} \in T:a(t^{*},t,p) = 1}{\sum\limits_{p \in P_{e}}x_{pt^{*}}}{\kern39pt}(\forall t \in T)(\forall e \in E)\end{array} $$

(41)

$$\begin{array}{@{}rcl@{}} && 1 = x_{pk_{p}}{\kern65pt}(\forall p \in P_{q}) \end{array} $$

(42)

$$\begin{array}{@{}rcl@{}} && (1 - x_{pt})M \geq{} \underset{t^{*} \in T: t^{*} < t}{\sum\limits_{v \in P_{r(p)},v\neq p}} {}x_{st^{*}}{\kern13pt}(\forall t \in T)(\forall p \in P_{f}) \end{array} $$

(43)

$$\begin{array}{@{}rcl@{}} && (1 - x_{pt})M \geq{} \underset{t^{*} \in T: t^{*} > t}{\sum\limits_{v \in P_{r(p)},v\neq p}} {}x_{st^{*}}{\kern15pt}(\forall t \in T)(\forall p \in P_{l}) \end{array} $$

(44)

$$\begin{array}{@{}rcl@{}} && x_{pt} \in \{0, 1\}{\kern55pt}(\forall p \in P)(\forall t \in T) \end{array} $$

(45)

$$\begin{array}{@{}rcl@{}} && \delta_{ti} \in \{0, 1\}{\kern56pt}\begin{array}{l} (\forall t \in T)\\ \quad(\forall i \in \{1, 2, \ldots, C\}) \end{array} \end{array} $$

(46)

Equation 32 calculates the PACU occupancy as the sum of the auxiliary variables, δ_ti, and Eq. 33 ensures that a δ_ti, variable is 1 if δ_t(i+ 1) is 1. Together, these constraints make δ_it equal to 1 when the PACU occupancy is at least i.