Surgical scheduling via optimization and machine learning with long-tailed data

Abstract

Using data from cardiovascular surgery patients with long and highly variable post-surgical lengths of stay (LOS), we develop a modeling framework to reduce recovery unit congestion. We estimate the LOS and its probability distribution using machine learning models, schedule procedures on a rolling basis using a variety of optimization models, and estimate performance with simulation. The machine learning models achieved only modest LOS prediction accuracy, despite access to a very rich set of patient characteristics. Compared to the current paper-based system used in the hospital, most optimization models failed to reduce congestion without increasing wait times for surgery. A conservative stochastic optimization with sufficient sampling to capture the long tail of the LOS distribution outperformed the current manual process and other stochastic and robust optimization approaches. These results highlight the perils of using oversimplified distributional models of LOS for scheduling procedures and the importance of using optimization methods well-suited to dealing with long-tailed behavior.

Appendices

Appendix A: Mathematical formulations for $f(u_d),Q^{op}$

Solving the quadratic objective, $f(u_d) = u_d^2$ of the mixed-integer program can be slow. To reduce the runtime required to solve the problem, we implement the quadratic term in the objective function, $f(u_d) = u_d$, using a piece-wise linear approximation,

$$\begin{aligned} f(u_d)=e_1 u^{(1)}_d +e_2 u^{(2)}_d + e_3 u^{(3)}_d + e_4 u^{(4)}_d + e_5 u^{(5)}_d, \end{aligned}$$

(10)

and add additional constraints

$$\sum _{p\in P_b \cup P^{past}_b} y_{d,p}\le c + m - 1 + u^{(m)}_d \quad \forall d,m$$

$$u^{(m)}_d \ge 0\quad \forall d, m$$

In words, $u_d^{(m)}$ counts ICU overflow above $c+m-1$ for $m=1,\ldots ,5$. Here, $e_m$ are constant coefficients for the piecewise linear function.

In our formulation, we set $e_1 = 1, e_2=e_3=e_4=e_5=2$. Since $u_d$ only takes integer values, this coefficient choice leads to $f(u_d)\equiv u_d^2$ for any $u_d\le 5$. The piece-wise linear approximation is 20 times faster than the quadratic form in our numerical experiments.

The above approximation is used in all our algorithms. The formulations for stochastic and robust algorithms are analogous (e.g. copies of $u_{d}^{(m)}$ are created for different LOS realizations), and we thus omit the details here.

Next, we introduce the full mathematical formulation of $Q^{op}$ in constraint 1e for offline optimization below.

Sets

• $D = \{1,2,\ldots ,N_d\}$: index for days

• $P = \{1,2,\ldots ,N_p\}$: index for patients

• $P^{par} \subset P$: set of PAR patients

• K: set of surgeons

Parameters

• c: number of ICU beds reserved for elective patients, set to 8

• $q_p$: operation duration for $p \in P$.

• $l_p$: post-op length of stay in CVICU for $p \in P$

• $h_{d,k}$: number of hours surgeon k is available to perform surgery on day d, for $s \in S$. $h_{d,k} = 15$ if k operates on day d, 0 otherwise.

• $par\_day_{d,k}$: indicator variable for PAR days. $par\_day_{d,k} = 1$ if surgeon k can perform PAR surgeries on day d, 0 otherwise.

• $original\_date_{p} \in D$: the original date that a surgery is scheduled for for $p \in P$

• $actual\_date_{p} \in D$: the actual date that a surgery takes place for $p \in P$; can be different from $original\_date_{p}$ if the surgery was rescheduled.

• $arrived\_on_{p}$: the arrival date for $p \in P$, i.e., when a patient is first being scheduled for surgery; can be negative (i.e. outside the set D) if the surgery was scheduled before September 2018

• $lead_{p}$: the lead time of a surgery, i.e.

  $$\begin{aligned} original\_date_{p}- arrived\_on_{p} \end{aligned}$$

• $m_{k,p}$: binary, 1 if patient p is assigned to surgeon k and 0 otherwise

Decision Variables

• $u_d$: integer, the number of additional elective patients in the ICU on day d above c

• $x_{d,p}$: binary, $x_{d,p} = 1$ if patient p is scheduled to have her surgery on day d; otherwise 0

• $d_p$: the date when the surgery of patient p is scheduled by the model

• $y_{d,p}$: binary, $y_{d,p} = 1$ if patient p stays in the CVICU on day d; otherwise 0

• $z_{d,p}$: continuous, number of hours that the surgery of p lasts on day d. $z_{d,p} = q_p$ if $x_{d,p} = 1$, otherwise 0

Indicator Function

• Assumed feasible window where patient p is available for surgery:

  $$\begin{aligned} g(d,p) = \mathbbm {1}\{d^{min}_p \le d<d^{max}_p\} \end{aligned}$$

  where $d^{max}_p = \text {min}(actual\_date_p+90,N_d)$,

  $$\begin{aligned} d^{min}_p = {\left\{ \begin{array}{ll} \text {max}[1,original\_date_p- \frac{lead_p}{2}], &{} lead_p> 20\\ \text {max}[1,arrived\_on_{p}],&{}lead_p\le 20 \end{array}\right. } \end{aligned}$$

  (11)

Next we describe the optimization constraints 12-16 that are equivalent to $x\in Q^{op}.$

Constraint 12 ensures each patient is scheduled exactly once within that patient’s window of availability.

$$\begin{aligned} \sum _{d\in D} x_{d,p}\cdot g(d,p) = 1,\quad \forall p\in P \end{aligned}$$

(12)

Constraints 13-16 incorporate daily availability of surgeons. Constraint Eq. 13 includes number of hours a surgeon is available on each day for (non PAR) surgeries.

$$\begin{aligned} z_{d,p} = x_{d,p}q_p, \quad \forall d\in D, p\in P \end{aligned}$$

(13)

$$\begin{aligned} \sum _{p\in P}z_{d,p} m_{k,p} \le h_{d,k}, \quad \forall d\in D, k\in K \end{aligned}$$

(14)

Constraint 15 captures that PAR surgeries can only be done on pre-specified days.

$$\begin{aligned} \sum _{p\in P^{par}}x_{d,p} m_{k,p} \le par\_day_{d,k}, \quad \forall d\in D, k\in K \end{aligned}$$

(15)

Constraint 16 ensures each surgeon is not scheduled for PAR surgeries in subsequent days.

$$\begin{aligned} \begin{aligned} \sum _{p \in P^{par}}x_{d,p}m_{k,p} + \sum _{p \in P^{par}}&x_{d+1,p}m_{k,p} \le 1,\\&\forall d\in D\setminus \{N_d\},k\in K \end{aligned} \end{aligned}$$

(16)

1.1 Appendix A.1: $Q^{op}$ for batch optimization problem

When solving BOP for rolling scheduling, $Q^{op}$ is adjusted accordingly. In addition to the Sets and Parameters specified in Section A, we use the following sets and parameters.

Sets

• $P_b \subset P$: index of batch patients of period b

• $P^{past}_b = \bigcup _{k=1}^{b-1} P_k$: set of patients prior to period b

• $P_b^{par} = P^{par} \cap P_b$

Parameters

• $s_b$: start date/day of batch b

• $d^{min}_{p,b} = \max (d^{min}_p,s_b)$

• $x^*_{d,p}$: solutions obtained from previous periods

Since the definition of $d^{min}_{p,b}$ may restrict the original time window of availability for some patients, we also adjust the last available date, $d^{\max }_p$, to at least 90 days after the start of the period, i.e.,

$$\begin{aligned} d^{max}_{p,b} = \max (d^{max}_p,\min (s_b+90,N_d)) \end{aligned}$$

(17)

This adjustment allows flexible scheduling as described in Section 3.1. In practice, the definition of $d^{max}_{p,b}$ will not include $N_d$; we included it for our simulation runs. Although the latter adjustment could potentially increase wait time, it is necessary in ensuring that the set of feasible scheduling solutions is not too restricted, and any resultant increase in wait time will be penalized by the objective function.

Indicator Function

• Adjusted feasibility window for each patient.

$$\begin{aligned} g_b(d,p) = \mathbbm {1}\{d^{min}_{p,b} \le d< d^{max}_{p,b}\} \end{aligned}$$

Constraints 18-22 give the equivalent formulation of $x\in Q^{op}$ in all deterministic, stochastic and robust BOPs.

$$\begin{aligned} \sum _{d\in D} x_{d,p}g_b(d,p) = 1,\quad \forall p\in P_b \end{aligned}$$

(18)

$$\begin{aligned} z_{d,p} = x_{d,p}q_p, \quad \forall d\in D, p\in P_b \end{aligned}$$

(19)

$$\begin{aligned} \sum _{p\in P_b}z_{d,p} m_{s,p} \le h_{d,s}, \quad \forall d\in D, s\in S \end{aligned}$$

(20)

$$\begin{aligned} \sum _{p\in P^{par}_b}x_{d,p} m_{s,p} \le par\_day_{d,s}, \quad \forall d\in D, s\in S \end{aligned}$$

(21)

$$\begin{aligned} \begin{aligned} \sum _{p \in P^{par}_b}x_{d,p}m_{s,p} + \sum _{p \in P^{par}_b}&x_{d+1,p}m_{s,p} \le 1,\\&\forall d\in D\setminus \{N_d\},s\in S. \end{aligned} \end{aligned}$$

(22)

Appendix B: Solving AC &CG for robust BOP

In the following, we introduce the AC &CG algorithm used for solving the robust BOP.

1. Main problem. We start with a subset of traces in the uncertainty set, $\Omega ^t = \{\omega ^{(n)}:n= 1,\ldots ,|\Omega ^t|\}\subseteq \mathcal {U}_b$. Instead of minimizing the worst-case cost over the entire uncertainty set $\mathcal {U}$, the Main problem of AC &CG minimizes the worst-case cost over the subset, $\Omega ^t$:

$$\begin{aligned} \min _{x}\quad \sum _{p\in P_b} \sum _{d=s_b}^{N_d}(d - d^{min}_p)^+ x_{d,p} + \beta \cdot \theta \end{aligned}$$

(23a)

s.t.

$$\begin{aligned} x_{d,p} = \tilde{x}_{d,p}\quad \forall p\in P^{past}_b, d\in D \end{aligned}$$

(23b)

$$\begin{aligned} \sum _{d=s_b}^{N_d} x_{d,p} = 1,\quad \sum _{d=1}^{s_b -1} x_{d,p}=0 \quad \forall p\in P_b \end{aligned}$$

(23c)

$$\begin{aligned} \begin{aligned} y^{(n)} = \sum _{d'=\max (d-l^{(n)}_p+1,1)} x_{d',p}&\quad \forall p \in P_b\cup P^{past}_b \\ {}&\forall d\in D, n=1, \ldots ,|\Omega ^t| \end{aligned} \end{aligned}$$

(23d)

$$\begin{aligned} \sum _{p\in P_b \cup P^{past}_b} y^{(n)}_{d,p} \le c + u^{(n)}_d \quad \forall d\in D, n=1,\ldots ,|\Omega ^t| \end{aligned}$$

(23e)

$$\begin{aligned} x\in Q^{op}, x_{d,p}\in \{0,1\} \end{aligned}$$

(23f)

$$\begin{aligned} \theta \ge \sum _{d=s_b}^{N_d}f(u^{(n)}_d) \quad \forall n= 1,\ldots ,|\Omega ^t|. \end{aligned}$$

(23g)

Constraint 23g sets the variable $\theta$ to be equal to the maximum cost of overflow among all traces of LOS realization, $\omega ^{(n)}$. The objective 23a thus minimizes the weighted sum of total wait time and the maximum cost of overflow for all $\omega ^{(n)}\in \Omega ^t$. The remaining constraints mirror those for BSOP-SAA in optimization problem 6, only replacing the set of sampled traces with set $\Omega ^t$. The optimal objective value of the Main problem provides a lower bound to the optimal objective value of robust BSOP in optimization problem 7.

2. Recourse problem. At each iteration t, after the Main problem is solved with $\Omega ^t$, an optimal solution $x^{t*}$ is obtained. The Recourse problem aims to find a trace $\omega ^{t*}\in \mathcal {U}_b$ that maximizes the cost of ICU overflow under the given schedule $x^{t*}$. Since $x^{t*}$ and $\omega ^{t*}$ are feasible under the original BOP, solving the Recourse problem finds an upper bound of the objective value of robust BOP.

$$\begin{aligned} \max _{\omega = \{l_p: p\in P_b\cup P^{past}_b\}} \quad \min _{y,u} \sum ^{N_d}_{d=s_b} f(u_d) \end{aligned}$$

(24a)

s.t.

$$\begin{aligned} \begin{aligned} y_{d,p} = \sum _{d'=\max (d-l_p+1,1)}^d x^{t*}_{d',p} \,\, \forall p\in P_b\cup P^{past}_b, d\in D \\ \end{aligned} \end{aligned}$$

(24b)

$$\begin{aligned} \sum _{p\in P_b \cup P^{past}_b} y_{d,p} \le c + u_d \quad \forall d\in D \end{aligned}$$

(24c)

$$\begin{aligned} y _{d,p}\in \{0,1\}, u_d\ge 0. \end{aligned}$$

(24d)

$$\begin{aligned} l_p^{min} \le l_p\le l_p^{max}\quad \forall p \in P_b \cup P^{past}_b \end{aligned}$$

(24e)

$$\begin{aligned} \sum _{p\in A_b}\left[ \frac{l_p - l_p^{min}}{l^{max}_p - l_p^{min}}\right] \le \eta \cdot |A_b|. \end{aligned}$$

(24f)

This formulation follows from expressions 7e-7h, and constraints 24e and 24f use the definition of $\mathcal {U}_b$ in expressions 8 and 9.

One key difficulty of solving the Recourse problem in its current form is that the decision variable $l_p$ appears in the boundary of the summation in constraint Eq. 24b. We follow the reformulation approach in [22] and extend it to convex, piece-wise linear forms of $f(u_d)$. The reformulated Recourse problem is provided in Appendix B.1. We present the full AC &CG algorithm in Algorithm 5.

Algorithm 5

AC &CG for Robust BOP

1.1 Appendix B.1: Solving the recourse problem

The Recourse problem, denoted as $Q(x^{t*})$, involves constraints including $l_p$ decision variables in the boundary of summations. Here, we introduce our MIP reformulation that is readily solvable by Gurobi. We refer readers to [22] for more details and proofs of its validity.

As in [22] we define the variables $v_{d,p},w_{d,p} \in \{ 0,1\}$ for $d \in D, p \in P$. The variable $v_{d,p}$ is 1 only if patient p is admitted in the ICU by d. Given the temporary solution $x^{t*}$, $v_{d,p}$ are constant parameters determined by $x^{t*}$. The decision variable $w_{d,p}$ is 1 only if patient p leaves the ICU by day d. So, $y_{d,p} = v_{d,p} - w_{d,p}$.

The inner minimization problem of $Q(x^{t*})$ is

$$\min _{u\ge 0} \sum _{d\in D} e_1 u^{(1)}_d +e_2 u^{(2)}_d + e_3 u^{(3)}_d + e_4 u^{(4)}_d + e_5 u^{(5)}_d$$

$$\sum _{p\in P_b \cup P^{past}_b} (v_{d,p} - w_{d,p}) \le c + m - 1 + u^{(m)}_d \quad \forall d,m $$

where $e_1 = 1, e_2=e_3=e_4=e_5 = 2$.

We apply strong duality to reformulate the inner minimization as a maximization problem and also substitute for the definition of uncertainty set $\mathcal {U}_b$. Let $d_p$ denote the date of scheduled procedure for patient p according to $x^{t*}$, i.e., $d_p = \sum _{d\in D} d \cdot x^{t*}_{d,p}$. $Q(x^{t*})$ is reformulated below with decision variables $\lambda _d^{(m)}$ and $w_{d,p}$.

$$\begin{aligned} \max _{\lambda ,w} \sum _{d\in D}\sum _{m=1}^5\left[ \sum _{p\in P_b \cup P^{past}_b}(v_{d,p} \!-\! w_{d,p})\!-\!c\!-\!m\!+\!1)\right] \lambda _d^{(m)} \end{aligned}$$

$$\begin{aligned} d_p = \sum _{d\in D} d \cdot x^{t*}_{d,p}\quad \forall p \in P_b \cup P^{past}_b \end{aligned}$$

$$\begin{aligned} w_{d,p}\ge 1, \quad p\in P_b \cup P^{past}_b, d = d_p + l^{max}_p,\ldots ,T \end{aligned}$$

$$\begin{aligned} w_{d,p}\le 0, \quad p\in P_b \cup P^{past}_b, d = 0,\ldots ,d_p + l^{min}_p - 1 \end{aligned}$$

$$\begin{aligned} w_{d,p} \le w_{d+1,p}\quad \forall d\in D, p\in P_b \cup P^{past}_b \end{aligned}$$

$$\begin{aligned} \sum _{d\in D} (v_{d,p} - w_{d,p}) = l_p, \quad \forall p\in P_b\cup P_b^{past} \,\,\text {where}\,\, p \notin A_b \end{aligned}$$

$$\begin{aligned} \sum _{p\in A_b}\left[ \frac{\sum _{d\in D} (v_{d,p} - w_{d,p}) - l_p^{min}}{l^{max}_p - l_p^{min}}\right] \le \eta \cdot |A_b|. \end{aligned}$$

$$\begin{aligned} 0\le \lambda _d^{(m)}\le e_m \quad \forall d, m \end{aligned}$$

$$\begin{aligned} w_{d,p}\in \{0,1\},\quad \forall d,p \end{aligned}$$

For the optimal solution, we must have $\lambda ^{(m)}_d\in \{0,e_m\}$.

The formulation above involves a bilinear term, $w_{d,p}\lambda ^{(m)}_d$. Since $w_{d,p} \in \{0,1\}$, we can reformulate the problem by using $q^{(m)}_{d,p} = w_{d,p}\lambda ^{(m)}_d$ for all d, p, m. The final reformulation of the Recourse problem is the following.

$$\begin{aligned} Q(x^{t*}) =&\max _{\lambda ,w} \sum _{d\in D}\sum _{m=1}^5\sum _{p\in P_b \cup P^{past}_b} v_{d,p}\lambda ^{(m)}_d - \nonumber \\&\sum _{d\in D}\sum _{m=1}^5\sum _{p\in P_b \cup P^{past}_b} q^{(m)}_{d,p} - \nonumber \\&\sum _{d\in D}\sum _{m=1}^5 (c+m-1)\lambda _d^{(m)} \end{aligned}$$

$$\begin{aligned} d_p = \sum _{d\in D} d \cdot x^{t*}_{d,p}\quad \forall p \in P_b \cup P^{past}_b \end{aligned}$$

$$\begin{aligned} w_{d,p}\ge 1, \quad p\in P_b \cup P^{past}_b, d = d_p + l^{max}_p,\ldots ,T \end{aligned}$$

$$\begin{aligned} w_{d,p}\le 0, \quad p\in P_b \cup P^{past}_b, d = 0,\ldots ,d_p + l^{min}_p - 1 \end{aligned}$$

$$\begin{aligned} w_{d,p} \le w_{d+1,p}\quad \forall d\in D, p\in P_b \cup P^{past}_b \end{aligned}$$

$$\begin{aligned} \sum _{d\in D} (v_{d,p} - w_{d,p}) = l_p, \quad \forall p\in P_b\cup P_b^{past} \,\,\text {where}\,\, p \notin A_b \end{aligned}$$

$$\begin{aligned} \sum _{p\in A_b}\left[ \frac{\sum _{d\in D} (v_{d,p} - w_{d,p}) - l_p^{min}}{l^{max}_p - l_p^{min}}\right] \le \eta \cdot |A_b| \end{aligned}$$

$$\begin{aligned} q^{(m)}_{d,p} \ge \lambda ^{(m)}_t-e_m(1-w_{d,p}) \quad \forall d,p,m \end{aligned}$$

$$\begin{aligned} q^{(m)}_{d,p} \le e_m w_{d,p}, \,\, q^{(m)}_{d,p} \le \lambda ^{(m)}_d \quad \forall d,p,m \end{aligned}$$

$$\begin{aligned} q^{(m)}_{d,p} \ge 0, \lambda _d^{(m)} \in \{0, e_m\}, w_{d,p}\in \{0,1\} \quad \forall d,p,m. \end{aligned}$$

Appendix C: Additional numerical results

This section presents additional results on the optimality gap of AC &CG and biweekly scheduling.

Fig. 12

The optimality gap, $\frac{UB-LB}{LB}$, of most batches are below 1% after 10 iterations of BROP-AG &CG

Fig. 13

(Biweekly Scheduling) Performance trade-off of Standard-SRO between patient wait times and ICU congestion using different values of $\beta$ compared to the status quo. We also include greater values of $\beta$ to show that increasing $\beta$ further leads to longer wait times but insignificant reduction in ICU congestion

Fig. 14

(Biweekly Scheduling) Performance trade-off of Conservative-SRO between patient wait times and ICU congestion using different values of $\beta$ compared to the status quo

Fig. 15

(Biweekly Scheduling) Performance trade-off of RRO between patient wait times and ICU congestion using different values of $\beta ,\eta$ compared to the status quo