Chemotherapy appointment scheduling and daily outpatient–nurse assignment

Abstract

Chemotherapy planning and patient–nurse assignment problems are complex multiobjective decision problems. Schedulers must make upstream decisions that affect daily operations. To improve productivity, we propose a two-stage procedure to schedule treatments for new patients, to plan nurse requirements, and to assign the daily patient mix to available nurses. We develop a mathematical formulation that uses a waiting list to take advantage of last-minute cancellations. In the first stage, we assign appointments to the new patients at the end of each day, we estimate the daily requirement for nurses, and we generate the waiting list. The second stage assigns patients to nurses while minimizing the number of nurses required. We test the procedure on realistically sized problems to demonstrate the impact on the cost effectiveness of the clinic.

Appendices

Appendix A: Measure of hourly workload (${G^{p}_{i}}$)

The CTC prepared a list of rules related to workload balancing. We use it to develop two parameters to facilitate the estimation of workload. The first, ${G^{p}_{i}}$ (Table 9), represents the hourly workload that the patient requires.

Table 9 Hourly workload associated with each type of task, ${G^{p}_{i}}$

Appendix B: Linearization of the formulation

Constraints (1l) and (1m) are both nonlinear. We use a big-M approach to linearize them: Constraints (4a) and (4b) replace (1l), and Constraints (4c) and (4d) replace (1m). The parameter ${\Theta }_{i^{p}}$ plays the role of the big M; it is tightened as much as possible in each case.

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \left( \sum\limits_{j \in J} j x^{p}_{(i-1)j}+a^{p}_{(i-1),i}\right)\\ -\sum\limits_{j \in J} j x^{p}_{ij}\leq \left( 1-{y^{p}_{i}}\right){{\Theta}^{p}_{i}} \\ \quad\forall i \in \{I^{p}| i > 1\}, \forall p\in P \end{array} \end{array} $$

(4a)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{j \in J} j x^{p}_{ij} \leq \sum\limits_{j \in J} j x^{p}_{(i-1)j}+a^{p}_{(i-1),i}\\ \quad \forall i \in \{I^{p}| i > 1\}, \forall p\in P \end{array} \end{array} $$

(4b)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{h \in H} h \sum\limits_{j\in J}u^{p}_{(i-1)jh}-\sum\limits_{h\in H} h \sum\limits_{j \in J}u^{p}_{ijh}\leq \\ \left( 1-{y^{p}_{i}}\right)|H| \quad \forall i \in \{I^{p}| i > 1\}, \forall p\in P \end{array} \end{array} $$

(4c)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{h \in H} h \sum\limits_{j \in J}u^{p}_{ijh} \leq \sum\limits_{h \in H} h \sum\limits_{j \in J}u^{p}_{(i-1)jh}\\ \quad \forall i \in \{I^{p}| i > 1\}, \forall p\in P \end{array} \end{array} $$

(4d)

Appendix C: Formulation of offline model

Variables

$x^{p}_{ifjh}$::

1 if treatment i^p of patient p is assigned to nurse f on day j in time slot h; 0 otherwise

${y^{p}_{i}}$::

1 if treatment i^p of patient p is assigned; 0 otherwise

z_fj::

1 if nurse f is assigned; 0 otherwise

v_fjh::

1 if nurse f is on break in time slot h; 0 otherwise

τ_fj::

Integer variable: number of tasks handled, {0...E}

σ_fj::

Integer variable: first overflow level, {0...B}

ι_fj::

Integer variable: second overflow level, {0...C}

Model

$$ \begin{array}{lllllllllll} min \sum\limits_{f\in F\setminus\{virtual\}}\sum\limits_{j\in J}\eta z_{fj}+\alpha \tau_{fj} +\beta \sigma_{fj} + \gamma \iota_{fj} \\ + \epsilon \sum\limits_{p\in P}\sum\limits_{i^{p}\in i^{p}}\sum\limits_{j\in J}\sum\limits_{h\in H} x^{p}_{i(virtual)jh} \end{array} $$

(5a)

s.t.

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{p\in P}\sum\limits_{i\in I^{p}}\sum\limits_{f\in F}\sum\limits_{h^{\prime}=max(1,h + 1 -{D^{p}_{i}})}^{min(h,H + 1-{D^{p}_{i}})}L^{p}_{li} x^{p}_{ifjh^{\prime}}\leq \\ V_{l} \quad \forall j\in PP, \forall h \in H, \forall l \in L \end{array} \end{array} $$

(5b)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{f\in F}\sum\limits_{j\in PP}\sum\limits_{h\in H} h x^{p}_{ifjh}+{D^{p}_{i}}\leq |H| \\ \quad \forall i\in I^{p}, \forall p \in P \end{array} \end{array} $$

(5c)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{p\in P}\sum\limits_{i\in i^{p}}\sum\limits_{h^{\prime}=max(1,h + 1-{D^{p}_{i}})}^{min(h,H + 1-{D^{p}_{i}})} {G^{p}_{i}} x^{p}_{ijfh^{\prime}}\leq N \\ \quad \forall f \in F\setminus\{virtual\}, \forall j \in PP, \forall h \in H \end{array} \end{array} $$

(5d)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{p\in P}\sum\limits_{i\in I^{p}}\sum\limits_{h \in H} {W^{p}_{i}} x^{p}_{ifjh^{\prime}}\leq W \\ \quad \forall f \in F, \forall j\in PP, \forall h \in H \end{array} \end{array} $$

(5e)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{p\in P} \sum\limits_{i\in I^{p}} x^{p}_{ifjh}\leq 1\\ \quad \forall f \in F\setminus\{virtual\},\forall j \in PP, \forall h \in H \end{array} \end{array} $$

(5f)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{p\in P}\sum\limits_{i\in i^{p}}\sum\limits_{h\in H} P^{p}_{ai} x^{p}_{ifjh}\leq M_{a} \\ \quad \forall f \in F\setminus\{virtual\},\forall j\in PP, \forall a \in A \end{array} \end{array} $$

(5g)

$$\begin{array}{@{}rcl@{}} a^{p} \leq \sum\limits_{j\in PP} \sum\limits_{h \in H}j x^{p}_{1fjh}\leq b^{p}\quad \forall p\in P \end{array} $$

(5h)

$$\begin{array}{@{}rcl@{}} {y^{p}_{i}} \leq y^{p}_{(i-1)} \quad \forall i \in \{I^{p}| i > 1\}, \forall p\in P \end{array} $$

(5i)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} {y^{p}_{i}} \left( \sum\limits_{j\in PP} j x^{p}_{ifjh} -\sum\limits_{j\in PP} j x^{p}_{(i-1)fjh}\right)\\ = {y^{p}_{i}} a^{p}_{(i-1),i} \quad \forall i \in \{I^{p}| i > 1\}, \forall p\in P \end{array} \end{array} $$

(5j)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} v_{fjh}+v_{fj(h + 2)}+v_{fj(h + 4)} = z_{fj} \\ \quad h={7},\forall f \in F\setminus\{virtual\}, \forall j \in PP \end{array} \end{array} $$

(5k)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} v_{fjh}+v_{fj(h + 2)}+v_{fj(h + 4)} = z_{fj} \\ \quad h={8},\forall f \in F\setminus\{virtual\},\forall j \in PP \end{array} \end{array} $$

(5l)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} h v_{fjh}+(h + 2) v_{fj(h + 2)}\\ +(h + 4) v_{fj(h + 4)} \leq (h + 1) v_{fj(h + 1)} \\+(h + 3) v_{fj(h + 3)}+(h + 5) v_{fj(h + 5)} \quad h={7}, \\ \forall f \in F\setminus\{virtual\},\forall j \in PP \end{array} \end{array} $$

(5m)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} 0 \leq v_{fj(h-1)}-v_{fjh}+v_{fj(h + 1)} \leq1 \\ \quad \forall f \in F\setminus\{virtual\},\forall j \in PP, \forall h \in H \end{array} \end{array} $$

(5n)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} 0 \leq h v_{fjh} + \left( h + 1\right) v_{fj(h + 1)} \leq 1 \\ \quad \forall f \in F\setminus\{virtual\},\forall j \in PP, \forall h \in H \end{array} \end{array} $$

(5o)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{f \in F} v_{fjh} \leq \frac{1}{3} \sum\limits_{f\in F \setminus\{virtual\}} z_{fj} \\ \quad \forall h =\{7,9,11\} \end{array} \end{array} $$

(5p)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} v_{fjh}+x^{p}_{ifjh} \leq 1 \quad \forall p \in P,\forall \in I^{p},\\ \forall f \in F\setminus\{virtual\},\forall j \in PP, \forall h = \{7,8,9,10,11,12\} \end{array} \end{array} $$

(5q)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{f\in F}\sum\limits_{j\in PP}\sum\limits_{h\in H} x^{p}_{ifjh} = {y^{p}_{i}} \\ \quad\forall i \in I^{p}, \forall p \in P \end{array} \end{array} $$

(5r)

$$\begin{array}{@{}rcl@{}} \begin{array}{lllllllllll} \sum\limits_{p\in P}\sum\limits_{i\in I^{p}}\sum\limits_{h\in H} x^{p}_{ifjh} = \tau_{fj} +\sigma_{fj} +\iota_{fj} \\ \quad\forall f\in F\setminus\{virtual\}, \forall \in PP \end{array} \end{array} $$

(5s)

$$\begin{array}{@{}rcl@{}} x^{p}_{ifjh},{y^{p}_{i}},z_{fjh},v_{fjh} \in \{0,1\} \end{array} $$

(5t)

$$\begin{array}{@{}rcl@{}} \tau_{fj},\sigma_{fj}, \iota_{fj} \in \mathbb{N} \end{array} $$

(5u)

Appendix D: Outline of the procedure to generate cancellations and absences

Fig. 6

Simulation of patient cancellations

Fig. 7

Simulation of nurse absences