A diving heuristic for planning and scheduling surgical cases in the operating room department with nurse re-rostering

Abstract

The decisions in the operating room scheduling process related to the case mix planning, the master surgery schedule and the nurse roster are based on the expected demand, predicted by historical data. Patients are only scheduled in the operational phase when the actual demand is known. However, the actual patient demand may differ from the expected demand. In this paper, we integrate the surgical case planning and scheduling problem and include the nurse re-rostering decision and nurse assignment to specific patients in order to utilise the operating room department as efficiently as possible and maximise the operating room profit. We propose a two-phase heuristic that uses the LP solution generated via column generation to construct a high-quality feasible solution. Computational experiments have been conducted on a diverse artificial data set generated in a controlled and structured manner and real-life data from the Sina Hospital (Tehran, Iran). We show that the presented approach is able to produce (near-)optimal solutions and benchmark the procedure with other optimisation strategies and solution methodologies.

A General MIP formulation

We defined the general mathematical formulation for SCPS-NRRSA problem in the following:

Additional decision variables

$e{rd}^{}$:

1, if OR r is open on day d; 0, otherwise

$nx_{ndv}^{}$:

1, if nurse n is assigned to shift v, day d; 0, otherwise

$npx_{ni}^{}$:

1, if nurse n is assigned to surgical case i; 0, otherwise

$Sn_{nd}^{}$:

Duty start time of nurse n on day d

$Cn_{nd}^{}$:

Duty finish time of nurse n on day d

$DD_{i\bar{i}}$:

1, if surgical case i and $\bar{i}$ have been performed in a same day ; 0, otherwise

$$\begin{aligned} \hbox {Max} \ Z&= \sum _{itrd}^{} Rev_{i} \times px_{itrd} - \sum _{r} CoR_{r} \times e_{i}\nonumber \\&- \sum _{ndv} CoV_{v} \times nx_{ndv} \end{aligned}$$

(33)

$$\begin{aligned}&\sum _{tdr}^{}px_{itrd}^{} \le 1 \quad \forall i \end{aligned}$$

(34)

$$\begin{aligned}&\sum _{i}^{}px_{itrd}^{} \le 1 \quad \forall t,r,d \end{aligned}$$

(35)

$$\begin{aligned}&\sum _{\bar{i}}^{} \sum _{\bar{t}=t+1}^{\bar{t} = t + t_i} px_{\bar{i}\bar{t}rd}^{} \le BM(1-px_{itrd}^{}) \quad \forall i,t,r,d \end{aligned}$$

(36)

$$\begin{aligned}&\sum _{it}^{}px_{itrd}^{} \le BM \times e_{rd} \quad \forall r,d \end{aligned}$$

(37)

$$\begin{aligned}&S_i = \sum _{trd}^{} t \times px_{itrd}^{} \quad \forall i, t, d \end{aligned}$$

(38)

$$\begin{aligned}&C_i = S_i + t_i \times \sum _{trd}^{} px_{itrd}^{} \quad \forall i, t, d \end{aligned}$$

(39)

$$\begin{aligned}&\sum _n npx_{ni} = nu_i \times \sum _{rdt} px_{itrd} \quad \forall i \end{aligned}$$

(40)

$$\begin{aligned}&Cn_{nd} \ge C_i - BM \times (1-npx_{ni})\nonumber \\ {}&\quad - BM \times (1-\sum _rt px_{itrd}) \quad \forall n, i, d\end{aligned}$$

(41)

$$\begin{aligned}&Sn_{nd} \le S_i + BM \times (1-npx_{ni}) \nonumber \\ {}&\quad + BM \times (1-\sum _rt px_{itrd}) \quad \forall n, i, d\end{aligned}$$

(42)

$$\begin{aligned}&Cn_{nd} \le \sum _v Et^V_v \times nx_{ndv} \quad \forall n, d\end{aligned}$$

(43)

$$\begin{aligned}&Sn_{nd} \ge \sum _v St^V_v \times nx_{ndv} \quad \forall n, d\end{aligned}$$

(44)

$$\begin{aligned}&\sum _{vd} nx_{ndv} \le MA \quad \forall n \end{aligned}$$

(45)

$$\begin{aligned}&\sum _{v} nx_{ndv} \le 1 \quad \forall n,d \end{aligned}$$

(46)

$$\begin{aligned}&\sum _{rt}(px_{itrd}+ px_{\bar{i}trd}) - 1 \le DD_{i \bar{i}} \quad \forall i , \bar{i},d \end{aligned}$$

(47)

$$\begin{aligned}&S_i \ge S_{\bar{i}} - BM \times (1 - w_{i \bar{i}}) \quad \forall i , \bar{i} \end{aligned}$$

(48)

$$\begin{aligned}&(S_i - C_{\bar{i}}) + BM \times P_{i \bar{i}} \nonumber \\ {}&\quad \quad + BM \times w_{i \bar{i}} + BM \times (2 - npx_{ni} - npx_{n\bar{i}})\nonumber \\&+ BM \times (1-DD_{i\bar{i}})\ge 0 \quad \forall n, i, \bar{i} \end{aligned}$$

(49)

$$\begin{aligned}&t_i \times px_{itrd} \le \sum _{\bar{t}=t}^{\bar{t}=t+t_i} A_{s\bar{t}rd} \quad \forall s, i \in \varphi _s , t, d \end{aligned}$$

(50)

$$\begin{aligned}&nx_{ndv} \le ROS_{ndv} + Prf_{ndv} \quad \forall n,v,d\end{aligned}$$

(51)

$$\begin{aligned}&px_{itrd}^{},nx_{ndv},npx_{nu},DD_{i \bar{i}}, w_{i \bar{i}}, P_{i \bar{i}}, ad_{id}^{}, e_{r} \in \{0,1\} \nonumber \\ {}&\quad \quad \forall i,\bar{i},n, v, t, r, d \end{aligned}$$

(52)

$$\begin{aligned}&S_i, C_i,Sn_{nd},Cn_{nd} \ge 0 \quad \forall i \end{aligned}$$

(53)

For the description of the objective function (Eq. 33) and the constraints (Eqs. 34–39), we refer to the manuscript (see Eqs. 1), 10–16). Constraint (40) assigns nurses to particular surgeries based on the nurse requirements per surgery. Constraints (15) and (16) calculate the start and completion time of each surgery. Constraints (41) and (42) define the time window during which a nurse has to be on duty. The total number of time slots is a relevant upper bound in the right-hand side of the constraint, i.e. $BM = |T|$. Based on this time window, predefined shifts are assigned to nurses (constraints (43) and (44)). Constraint (45) stipulates that a nurse cannot be assigned to more than MA shifts during the planning horizon. Constraint (46) states that a nurse cannot have more than one shift per day.

Constraint (47) checks if two surgical cases are conducted on the same day. Constraint (48) concerns the order of two surgical cases. Constraint (49) stipulates that two overlapping surgery should not be assigned to the same nurse. This constraint is only relevant when both surgeries are conducted on the same day ($DD_{i\bar{i}} =1$). Constraint (50) implies that a surgical case can only take place according to the availability of its related surgeon. Constraint (51) imposes that nurses can only be assigned to shift duties conform to their original roster or nurse preferences. Constraints (52) and (53) state the domain of the variables.