An efficient computational method for large scale surgery scheduling problems with chance constraints

Abstract

We propose an efficient solution method based on a decomposition of set-partitioning formulation of an integrated surgery planning and scheduling problem with chance constraints. The studied problem is characterized by a set of identical operating rooms (ORs), a set of surgeries with uncertain durations, a set of surgeons, and surgery dependent turnover times. The decision variables include the number of ORs to open, assignments of surgeries and surgeons to ORs in admissible periods, and the sequence of surgeries to be performed in a period. The objective is to minimize the cost of opening ORs and the penalties associated with turnover times.In the proposed formulation, the column generation subproblem is decomposed over ORs and time periods. The structure of the subproblem is further exploited and transformed to a shortest path problem. A search algorithm has been devised to efficiently solve the resulting subproblem, subject to some optimality and feasibility conditions. The proposed computational method outperforms the standard chance constrained model and reduces the solution time significantly. Furthermore, extensive simulation experiments have been carried out to compare the performance of three variants of the underlying formulations and evaluate the sensitivity of the decisions to the probability of exceeding a session length.

Appendix: Standard chance constrained model for the stochastic surgery scheduling problem

Here, we present the standard chance constrained equivalent model using the popular scenario based chance constrained programming [14]. The notation for column generation (6) are used to define the standard model. The additional binary variables $z_r^t$ and $q_k^{r,t}$ indicate if OR r is used in period t, and if surgeon k is allocated to OR r in period t.

$$\begin{aligned}&\min \sum _{r\in R}\sum _{t\in T}c_Rz_r^t+\sum _{i\in S}\sum _{j\in I}\sum _{l\in I}c_{ij}\tau _{ij}y^l_{ij}+\sum _{i\in I}\sum _{t\in T_i}\sum _{r\in R}c_{i}\bar{\tau }_iw_i^{rt} \end{aligned}$$

(10a)

$$\begin{aligned}&\text {s.t.: }\sum _{l\in I}\sum _{r\in R}\sum _{t\in T_i}x_{i,l}^{r,t}=1, \ \forall i\in I \end{aligned}$$

(10b)

$$\begin{aligned}&{\qquad }\sum _{l\in I}x_{il}^{rt}\le z_r^t,\ \forall i\in l\in I, r\in R,t\in T \end{aligned}$$

(10c)

$$\begin{aligned}&{\qquad }\sum _{i\in I_k}\sum _{l\in I}x_{il}^{rt}\le Mq_k^{rt},\ \forall k\in K, \ r\in R, t\in T \end{aligned}$$

(10d)

$$\begin{aligned}&{\qquad }\sum _{r\in R}q_{k}^{rt}\le 1,\ \forall k\in K, \ t\in T \end{aligned}$$

(10e)

$$\begin{aligned}&{\qquad }\sum _{i\in I}x_{il}^{rt}\ge \sum _{i\in I}x_{i,l+1}^{rt},\ \forall l\in I, \ r\in R, \ t\in T \end{aligned}$$

(10f)

$$\begin{aligned}&{\qquad }x_{il}^{rt}\le \sum _{j\in I}x_{j,l+1}^{rt}+w_i^{rt},\ \forall i, l\in I, \ r\in R, \ t\in T \end{aligned}$$

(10g)

$$\begin{aligned}&{\qquad }w_{i}^{rt}\le \sum _{l\in I}x_{il}^{rt},\ \forall i\in I, \ r\in R, \ t\in T \end{aligned}$$

(10h)

$$\begin{aligned}&{\qquad }\sum _{t\in T}x_{il}^{rt}+\sum _{t\in T}x_{j,l+1}^{rt}-1\le y_{ij}^l\ \forall i, l\in I, \ r\in R, \ t\in T \end{aligned}$$

(10i)

$$\begin{aligned}&{\qquad }y_{ii}\ge x_{i0}^{rt}\ \forall i\in I, \ r\in R, \ t\in T \end{aligned}$$

(10j)

$$\begin{aligned}&{\qquad }\sum _{i,l\in I}d_i^\varsigma x_{il}^{rt}+\sum _{i,j\in I}\tau _{ij}y_{ij}^l+\sum _{i\in I}\bar{\tau }_{i}\delta _i^{rt}\le \text {L}+M(1-\pi _\varsigma ^{rt}),\ \forall r\in R,\ t\in T,\varsigma \in \mathcal {S} \end{aligned}$$

(10k)

$$\begin{aligned}&{\qquad }\sum _{\varsigma \in \mathcal {S}}p_\varsigma \pi _\varsigma ^{rt}\ge 1-\alpha , \ \forall r\in R,\ t\in T \end{aligned}$$

(10l)

$$\begin{aligned}&{\qquad }\sum _{r=1}^{\min \{k,|R|\}}q_k^{r,t}\le \sum _{r=1}^{\min \{k,|R|\}}z_r^{t}, \ \forall k\in K, \ t\in T \end{aligned}$$

(10m)

$$\begin{aligned}&{\qquad }x_{il}^{rt},y_{ij}^{l}, z_r^t,w_i^{rt},q_{k}^{rt},\pi _\varsigma ^{rt}: \text {Binary}\ \forall i,l, j\in I, \varsigma \in \mathcal {S}, r\in R,\ t\in T . \end{aligned}$$

(10n)

The objective function computes the opening cost and the penalty cost. Constraint (10b) corresponds with the first constraint of SP ensuring every surgery is assigned to a OR at some period. Note that $v_{it}=\sum _{l\in I}\sum _{r\in R}x_{i,l}^{r,t}$. Constraint (10c) enforces the limitation of the available ORs at each period i.e., $z_r^t$ to take a value of 1 if at least one surgery is assigned to OR r in period t. This constraint is associated with constraint (1e). Constraints (10d and 10e) are related to constraint (6e) forbidding the assignment of a surgeon to more than one OR in each period. Constraints (10f–10j) determine the arrangement of each sequence. In SP, this is presented by variable $u_{s_t}$ through the column generation problem. Finally, constraint (10k–10l) impose the probabilistic condition on the OR available length via a scenario-based chance constraint. These constraints are equivalents of the column generation feasibility condition (9). Constraint (10m) is a symmetry breaking constraint and implies that surgeon 1 is assigned to OR 1, surgeon 2 can be assigned to OR 1 and OR 2 and so on. Finally, Constraints (10n) enforce the integrality conditions on the decision variables.