Online stochastic optimization of radiotherapy patient scheduling

Abstract

The effective management of a cancer treatment facility for radiation therapy depends mainly on optimizing the use of the linear accelerators. In this project, we schedule patients on these machines taking into account their priority for treatment, the maximum waiting time before the first treatment, and the treatment duration. We collaborate with the Centre Intégré de Cancérologie de Laval to determine the best scheduling policy. Furthermore, we integrate the uncertainty related to the arrival of patients at the center. We develop a hybrid method combining stochastic optimization and online optimization to better meet the needs of central planning. We use information on the future arrivals of patients to provide an accurate picture of the expected utilization of resources. Results based on real data show that our method outperforms the policies typically used in treatment centers.

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Acknowledgments

We would like to thanks Dr Bruno Carrozzo and Ms Julie Heon for their great help to understand CICL operations and to obtain their data.

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Correspondence to Antoine Legrain.

Additional information

Research funded in part by the Natural Sciences and Engineering Research Council of Canada and Elekta

Appendix A: The online stochastic algorithm

Appendix A: The online stochastic algorithm

We present here the online stochastic algorithm introduced by Legrain and Jaillet [21]. We rewrite this algorithm for our application. Recall the stochastic optimization formulation (2):

$$\begin{array}{*{20}l} & \min \sum\limits_{i \in S_{j}} c_{ij} x_{ij} \,+\, \mathbb{E}_{\omega \in \Omega_{j}}\left[\sum\limits_{l \in \mathcal{P}^{\omega}} \sum\limits_{i\in S_{l}} c_{il} y^{\omega}_{il} \,+\, \sum\limits_{k \in H} \sum\limits_{m \in M} c^{o} z^{\omega}_{mk}\right]\\ &\text{subject to:}\\ &\sum\limits_{i \in S_{j}} x_{ij} = 1\\ &\sum\limits_{i \in S_{l}} y^{\omega}_{il} = 1, \qquad \forall \omega \in \Omega_{j}, \forall l \in \mathcal{P}^{\omega}\\ &\sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij} + \sum\limits_{l \in \mathcal{P}^{\omega}} \sum\limits_{i \in S_{l}} a^{m}_{ilk} y^{\omega}_{il} \leq {F^{m}_{k}} + z^{\omega}_{mk},\\ & \qquad\qquad\forall m \in M, \forall k \in H, \forall \omega \in \Omega_{j}\\ &\mathbb{1}_{\mathcal{P}_{p}}(j)\sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij} + \sum\limits_{l \in \mathcal{P}_{p}^{\omega}} \sum\limits_{i \in S_{l}} a^{m}_{ilk} y^{\omega}_{il} \geq z^{\omega}_{mk},\\ & \qquad\qquad\forall m \in M, \forall k \in H, \forall \omega \in \Omega_{j}\\ &\sum\limits_{k=b}^{b+4} z^{\omega}_{mk} \leq O_{week}\qquad \forall m \in M, \forall b \in \mathcal{B}, \forall \omega \in \Omega_{j}\\ &z^{\omega}_{mk} \in [0, O_{day}], \qquad \forall m \in M, \forall k \in H, \forall \omega \in \Omega_{j}\\ &x_{ij} \in \{0,1\}, \qquad \forall i \in S_{j}\\ &y^{\omega}_{il} \in \{0,1\}, \qquad \forall l \in \mathcal{P}^{\omega}, \forall i \in S_{l}, \forall \omega \in \Omega_{j} \end{array} $$

A.1 L-shaped method

The L-Shaped method [4] enables us to solve formulation (2). This technique is based on Benders decomposition [3]. Formulation (2) is first decomposed into a master problem (3) and slave problems (4).

$$\begin{array}{*{20}l} & \text{(Master problem)} ~ \min \sum\limits_{i \in S_{j}} c_{ij} x_{ij} + \mathbb{E}_{\omega \in \Omega_{j}}[Q(x,\omega)]\notag\\ &\text{subject to:}\notag\\ &\sum_{i \in S_{j}} x_{ij} = 1 \notag\\ &x_{ij} \in \{0,1\}, \qquad \forall i \in S_{j} \end{array} $$
(3)

The function Q(x,ω) is called the recourse; it deals with the stochastic part of the objective. The goal of the slave problems (4) is to compute the value of this function for each x and each ω. The variables inside the parentheses (.) are the dual variables associated with the constraint.

$$\begin{array}{*{20}l} & \text{(Slave problems)} \notag\\ & Q(x,\omega) ~ = ~ \min \sum\limits_{l \in \mathcal{P}^{\omega}} \sum\limits_{i\in S_{l}} c_{il} y^{\omega}_{il} + \sum\limits_{k \in H} \sum\limits_{m \in M} c^{o} z^{\omega}_{mk}\notag\\ &\text{subject to:}\notag\\ &\sum\limits_{i \in S_{l}} y^{\omega}_{il} = 1, \qquad \forall l \in \mathcal{P}^{\omega} ~ (\alpha^{\omega}_{l})\notag\\ &\sum\limits_{l \in \mathcal{P}^{\omega}} \sum\limits_{i \in S_{l}} a^{m}_{ilk} y^{\omega}_{il} \leq {F^{m}_{k}} + z^{\omega}_{mk} - \sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij},\notag\\ & \hspace{4cm} \forall m \in M, \forall k \in H ~ (\beta^{\omega}_{mk})\notag\\ &\sum\limits_{l \in \mathcal{P}_{p}^{\omega}} \sum\limits_{i \in S_{l}} a^{m}_{ilk} y^{\omega}_{il} \geq z^{\omega}_{mk} - \mathbb{1}_{\mathcal{P}_{p}}(j)\sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij},\notag\\ &\hspace{4cm} \forall m \in M, \forall k \in H ~ (\gamma^{\omega}_{mk})\notag\\ &\sum\limits_{k=b}^{b+4} z^{\omega}_{mk} \leq O_{week}, \qquad \forall m \in M, \forall b \in \mathcal{B} ~ (\pi^{1\omega}_{mb})\notag\\ &z^{\omega}_{mk} \in [0, O_{day}], \qquad \forall m \in M, \forall k \in H ~ (\pi^{2\omega}_{mk})\notag\\ &y^{\omega}_{il} \in \{0,1\}, \qquad \forall l \in \mathcal{P}^{\omega}, \forall i \in S_{l} ~ (\pi^{3\omega}_{mk}) \end{array} $$
(4)

Since the recourse function Q has to be computed for each value of its variables, an approximation of Q is built. First, we relax the integrality constraint on the variables \( y^{\omega }_{il} \) and then we use the duals (5) of the slave problems.

$$\begin{array}{*{20}l} & \text{(Dual slave problems)} \notag\\ & \max \sum\limits_{l \in \mathcal{P}^{\omega}} \alpha^{\omega}_{l}\notag\\ &{\kern10pt} -\sum\limits_{m \in M} \left\{ \sum\limits_{k \in H} \left[\left({F^{m}_{k}} - \sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij}\right) \beta^{\omega}_{mk} + \mathbb{1}_{\mathcal{P}_{p}}(j)\right.\right.\notag\\ &{\kern5.9pc}\left.\times\!\left(\sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij}\right) \gamma^{\omega}_{mk} \,+\, O_{day} \pi^{2\omega}_{mk} \,+\, \pi^{3\omega}_{mk} \right] \notag\\ &{\kern4pc}\left.+\sum\limits_{b \in \mathcal{B}} O_{week} \pi^{1\omega}_{mb} \vphantom{\left({F^{m}_{k}} - \sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij}\right)}\right\}\notag\\ &\text{subject to:}\notag\\ &\alpha^{\omega}_{l} + \sum\limits_{m \in M} \sum\limits_{k \in H} a^{m}_{ilk} (\gamma^{\omega}_{mk} - \beta^{\omega}_{mk}) - \pi^{3\omega}_{mk} \geq c_{il}, \notag\\ & \hspace{4cm} \forall l \in \mathcal{P}^{\omega}, \forall i \in S_{l}\notag\\ &\beta^{\omega}_{mk} - \gamma^{\omega}_{mk} - \pi^{1\omega}_{mb(k)} - \pi^{2\omega}_{mk} \geq c^{o}, \qquad \forall m \in M, \forall k \in H\notag\\ &\beta^{\omega}_{mk},~\gamma^{\omega}_{mk},~\pi^{2\omega}_{mk},~\pi^{3\omega}_{mk} \geq 0, \qquad \forall m \in M, \forall k \in H\notag\\ &\pi^{1\omega}_{mb} \geq 0, \qquad \forall m \in M, \forall b \in \mathcal{B} \end{array} $$
(5)

where b(k) is the index of the Monday of the same week as the working day indexed by k. These problems are always feasible; furthermore they are bounded when there are enough free slots on the linacs for the chosen pattern i (x ij = 1). There always exists such a pattern, because the first day of treatment can be sufficiently far from the current day to ensure that there are enough free slots.

Thus, the weak duality theorem gives an approximation (a cut) of the recourse function Q:

$$\begin{array}{*{20}l} \forall i \in S_{j} ,& \forall \omega \in \Omega_{j},\notag\\ Q(x,\omega) & \geq \sum\limits_{l \in \mathcal{P}^{\omega}} \alpha^{\omega}_{l}\notag\\ & - \sum\limits_{m \in M} \left\{ \sum\limits_{k \in H} \left[{F^{m}_{k}} \beta^{\omega}_{mk} + \sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij} \right.\right.\notag\\ &{\kern5.5pc}\times\left.(- \beta^{\omega}_{mk} +\mathbb{1}_{\mathcal{P}_{p}}(j) \gamma^{\omega}_{mk})\vphantom{ \sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij}}\right]\notag\\ &{\kern3pc} + O_{day} \pi^{2\omega}_{mk} + \pi^{3\omega}_{mk} \notag\\ &{\kern3pc}\left. + \sum\limits_{b \in \mathcal{B}} O_{week} \pi^{1\omega}_{mb} \vphantom{ \sum\limits_{i \in S_{j}} a^{m}_{ijk} x_{ij}}\right\} \end{array} $$
(6)

For the qth cut for scenario ω, approximation (6) can be simplified. All the constants of this cut can be gathered into one constant \(C^{\omega }_{q}\):

$$\begin{array}{@{}rcl@{}} C^{\omega}_{q} &=& \sum\limits_{l \in \mathcal{P}^{\omega}} \alpha^{\omega}_{l} - \sum\limits_{m \in M} \left\{ \sum\limits_{k \in H} \left [{F^{m}_{k}} \beta^{\omega}_{mk} + O_{day} \pi^{2\omega}_{mk} + \pi^{3\omega}_{mk}\right] \right.\\ &&{\kern6pc} +\left.\sum\limits_{b \in \mathcal{B}} O_{week} \pi^{1\omega}_{mb} \right\} \end{array} $$

Each time the slave problems are solved, we obtain a cut for the master problem (3). The master problem is then transformed as follows:

$$\begin{array}{*{20}l} &\text{(Master problem)} \qquad ~ \min \sum\limits_{i \in S_{j}} c_{ij} x_{ij} + \mathbb{E}_{\omega \in \Omega_{j}}[\theta^{\omega}] \end{array} $$
(7a)
$$\begin{array}{*{20}l} &\text{subject to:}\\ &\sum\limits_{i \in S_{j}} x_{ij} = 1 \\ &\theta^{\omega} \geq C^{\omega}_{q} + \sum\limits_{i \in S_{j}} \left(\sum\limits_{m \in M} \sum\limits_{k \in H} a^{m}_{ijk} \left(\beta^{\omega}_{mk} - \mathbb{1}_{\mathcal{P}_{p}}(j) \gamma^{\omega}_{mk}\right)\right) x_{ij},\\ & \qquad\qquad\qquad\qquad \forall \omega \in \Omega_{j}, \forall q \end{array} $$
(7b)
$$\begin{array}{*{20}l} &x_{ij} \in \{0,1\}, \qquad \forall i \in S_{j} \end{array} $$

Benders Decomposition is also known as row generation. The duals (5) of the slave problems generate cuts that are inserted into the master problem (7a, 7b). The L-Shaped procedure (Algorithm 6) uses this decomposition.

Algorithm 6 L-shaped procedure
q = 0, x q = 0, x = 1
while x qx do
x = x q
q + +
for all ω ∈ Ω j do
   solve the dual slave problem with the value x for the event ω
   add the qth cut to the master problem (7a, 7b)
end for
solve the master problem (7a, 7b) and save the solution in x q
end while

When the L-Shaped procedure stops, the optimum is reached. Indeed, when the dual slave problems are solved for a value x, the recourse function Q is equal to the cut in this point thanks to the strong duality theorem. Then, if the master problem (7a, 7b) finds the same solution x, it has to be the optimum, because there is no approximation at this point and all the other points are underestimated.

A.2 Stochastic primal-dual algorithm

Legrain and Jaillet [21] make a simplification: the master problem is solved only once. Then, for each event ω, there is only one cut inserted into the master problem (7a, 7b). Thus, the constraints (7b) become equalities due to the minimization. The variables 𝜃 are therefore replaced in the objective (7a) by their expressions. The objective (7a) becomes: \(\displaystyle \min \sum\limits _{i \in S_{j}} c_{ij} x_{ij} + \mathbb {E}_{\omega \in \Omega _{j}}\!\left [\!C^{\omega }_{q} \,+\, \sum\limits _{i \in S_{j}} \left (\sum\limits _{m \in M} \sum\limits _{k \in H} a^{m}_{ijk} \left (\beta ^{\omega }_{mk} \,-\, \mathbb {1}_{\mathcal {P}_{p}}(j) \gamma ^{\omega }_{mk}\right )\right ) x_{ij}\!\right ]\).

We can also remove the constants \(C^{\omega }_{0}\) from the objective to obtain the final master problem (8):

$$\begin{array}{*{20}l} & \text{(Master problem)} \notag\\ & \min \sum\limits_{i \in S_{j}} \left(\!c_{ij} \,+\, \sum\limits_{m \in M} \sum\limits_{k \in H} \mathbb{E}_{\omega \in \Omega_{j}}\left[\beta^{\omega}_{mk} \,-\, \mathbb{1}_{\mathcal{P}_{p}}(j)\gamma^{\omega}_{mk}\right] a^{m}_{ijk}\!\right) x_{ij}\notag\\ &\text{subject to:}\notag\\ & \sum\limits_{i \in S_{j}} x_{ij} = 1\notag\\ & x_{ij} \in \{0,1\}, \qquad \forall i \in S_{j} \end{array} $$
(8)

Finally, the master problem (8) just chooses the pattern with the minimum objective. We can link our procedure with the primal-dual algorithm. The two algorithms are similar; the only difference is the computation of the variables r i . If we define the variables \( \displaystyle u_{ij} = \sum\limits _{m \in M} \sum\limits _{k \in H} \mathbb {E}_{\omega \in \Omega _{j}}[\beta ^{\omega }_{mk} - \mathbb {1}_{\mathcal {P}_{p}}(j)\gamma ^{\omega }_{mk}] a^{m}_{ijk}\), we obtain the stochastic primal-dual algorithm (4).

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Legrain, A., Fortin, M., Lahrichi, N. et al. Online stochastic optimization of radiotherapy patient scheduling. Health Care Manag Sci 18, 110–123 (2015). https://doi.org/10.1007/s10729-014-9270-6

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Keywords

  • Patient scheduling
  • Radiotherapy booking
  • Stochastic optimization
  • Online optimization
  • Dynamic scheduling

Mathematics Subject Classifications (2010)

  • MSC 90B36
  • MSC 90C15
  • MSC 90C39