Outpatient clinic scheduling with limited waiting area capacity

Figures & data

Figure 1. Schematic depiction of a patient trajectory and timeline.

Table 1. Sets, parameters and variables.

Sets
$\mathcal{S}=\{1,\dots ,S\},$ indexed by s	stages
${\mathcal{I}}_s=\{1,\dots ,I_s\},$ indexed by is	available resources at stage s
${\mathcal{J}}_s=\{1,\dots ,J_s\},$ indexed by js	appointment types at stage s
$\mathcal{J}{\prime }_s\subseteq {\mathcal{J}}_s$	appointment types at stage s that can take place in-person or digitally
${\tilde{\mathcal{J}}}_s={\mathcal{J}}_s\setminus \mathcal{J}{\prime }_s$	appointment types at stage s that can only take place in-person
$\mathcal{P}\subseteq {\mathcal{J}}_1\times \cdots \times {\mathcal{J}}_S,$ indexed by $(j_1,\dots ,j_S)$	patient trajectories consisting of an appointment of type js at stage s, $s=1,\dots ,S$
${\mathcal{P}}_s\subseteq {\mathcal{J}}_s$	patient trajectory consisting only of a single appointment of type js at stage s
$\mathcal{T}=\{1,\dots ,T\},$ indexed by t	time slots
$\mathcal{A}=\{1,\dots ,A\},$ indexed by a	shared waiting areas
${\mathcal{S}}_a\subseteq \mathcal{S}$	stages of which patients wait in shared waiting area a before their appointment
Parameters
$c(j_s)$	demand of appointment type js
$d(j_s)$	duration of appointment js, expressed in number of time slots
$b(j_{s-1},j{\prime }_s)$	minimum bridging time between appointment $j_{s-1}$ at Stage 1 and $j{\prime }_s$ at stage 2
$\tilde{b}(j_s)$	early arrival time for appointment $j_s\in {\mathcal{P}}_s,$ i.e. the time a patient coming from home spends in the waiting area of stage s to have appointment js
$w_{s,t}$	maximum number of patients allowed in the waiting area of Stage s in time slot t
$w_{a,t}$	maximum number of patients allowed in the shared waiting area a in time slot t
$r(j_s)$	reward for planning appointment js
M	big-M
Variables
$x_s(i_s,j_s,t)\in \{0,1\}$	$x_s(i_s,j_s,t)=1$ if, at stage s, appointment js is scheduled in-person at the start of time slot t using resource is, $x_s(i_s,j_s,t)=0$ otherwise
$y_s(i_s,j_s,t)\in \{0,1\}$	$y_s(i_s,j_s,t)=1$ if, at stage s, appointment js is scheduled digitally at the start of time slot t using resource is, $y_s(i_s,j_s,t)=0$ otherwise

Table 2. Appointment types rheumatology clinic SMK. Each appointment type is accompanied with its stage and duration (in minutes). The colour of a trajectory corresponds to the blueprint schedule from Figure 2(a).

Appointment type		Stage	Duration
	New patient	3 (consultation with physician)	30
	Follow-up type I	3 (consultation with physician)	15
	Follow-up type II	3 (consultation with physician)	15
	New patient - PA	3 (consultation with PA)	60
	Follow-up type I - PA	3 (consultation with PA)	15
	Follow-up type II - PA	3 (consultation with PA)	15
	New patient	2 (consultation with nurse)	60
	Follow-up	2 (consultation with nurse)	15

Table 3. Trajectories rheumatology clinic SMK. For each trajectory the appointment type per stage is denoted together with the option to replace a consultation with a digital consultation, the minimum bridging times (in minutes) and the number of occurrences required in the blueprint schedule. The duration of every appointment type can be found in Table 2. The colour of a trajectory corresponds to the blueprint schedules from Figures 2(b) – 7(a).

Trajectory		Stage 1	Stage 2	Stage 3	Stage 4	Digital consultation possible?	Bridging time before Stage 2	Bridging time before Stage 3	Bridging time before Stage 4	Number of occurrences in blueprint
	A			New patient		No	–	–	–	28
	B			Follow-up type I		Yes	–	–	–	28
	C			Follow-up type II	Pharmacy	No	–	–	15	26
	D			Follow-up type II	Blood test	No	–	–	15	34
	E			Follow-up type II		Yes	–	–	–	28
	F	Blood test	Follow-up	Follow-up type II		No	0	30	–	12
	G	Blood test	Follow-up	Follow-up type II	Pharmacy	No	0	30	15	26
	A-PA			New patient - PA		No	–	–	–	6
	B-PA			Follow-up type I - PA		Yes	–	–	–	6
	C-PA			Follow-up type II - PA	Pharmacy	No	–	–	15	12
	D-PA			Follow-up type II - PA	Blood test	No	–	–	15	16
	E-PA			Follow-up type II - PA		Yes	–	–	–	14
	F-PA	Blood test	Follow-up	Follow-up type II - PA		No	0	30	–	12
	G-PA	Blood test	Follow-up	Follow-up type II - PA	Pharmacy	No	0	30	15	6
	H		New patient			No	–	–	–	6
	I		Follow-up			Yes	–	–	–	4

Figure 2. The pre-COVID-19 blueprint schedule (a) and its best-case realisation (b) for the rheumatology clinic of SMK. In both (a) and (b), at the top is the blueprint schedule for nurses (Stage 2) and at the bottom is the blueprint schedule for physicians and PAs (Stage 3). Along the x-axis is the time and along the y-axis is the resource (at the bottom resources 1 – 7 are the physicians and 8 – 10 the PAs). In (c) the waiting room occupancy corresponding to the blueprint schedule from (b) is depicted over time. Here the grey bars denote waiting area occupancy levels taking into account bridging and average early arrival times (result from the ILP) and the blue line, together with its 95% confidence interval as the shaded area, depicts the waiting area occupancy including randomness in early arrival and consultation times (result from the MCS).

Table 4. Trajectories medical oncology & haematology clinic UMCU. For each trajectory the appointment type per stage is denoted together with the option to replace a consultation with a digital consultation, the minimum bridging times (in minutes) and the number of occurrences required in the blueprint schedule. The colour of a trajectory corresponds to the blueprint schedules in Figure 8(a) – 9(b).

Trajectory		Stage 1	Stage 2	Stage 3	Digital consultation possible?	Bridging time before Stage 2	Bridging time before Stage 3	Number of occurrences in blueprint
	A – oncology		New patient		Yes	–	–	10
	B – oncology		Follow-up		Yes	–	–	28
	C – oncology	Blood test	Follow-up		No	60	–	40
	D	Blood test	Follow-up	Chemo 60 min.	No	60	60	6
	E	Blood test	Follow-up	Chemo 90 min.	No	60	60	11
	F	Blood test	Follow-up	Chemo 120 min.	No	60	60	10
	G	Blood test	Follow-up	Chemo 210 min.	No	60	60	3
	A – haematology		New patient		Yes	–	–	8
	B – haematology		Follow-up		Yes	–	–	21
	C – haematology	Blood test	Follow-up		No	60	–	30
	H	Blood test	Follow-up	Chemo 30 min.	No	60	45	10
	I	Blood test	Follow-up	Chemo 60 min.	No	60	45	10
	J	Blood test	Follow-up	Chemo 120 min.	No	60	45	4
	K	Blood test	Follow-up	Chemo 180 min.	No	60	45	3
	L			Chemo 30 min.	No	–		3
	M			Chemo 60 min.	No	–	–	3
	N			Chemo 120 min.	No	–	–	3
	O			Chemo 210 min.	No	–	–	2

Figure 3. Optimal blueprint schedule (a) for the rheumatology clinic of SMK when only the number of in-person consultations is maximised; and the corresponding waiting room occupancy (b) for three variability settings. We refer to the caption of Figure 2 for a more detailed explanation of the figure.

Figure 4. Optimal blueprint schedule (a) and waiting room occupancy (b) for the rheumatology clinic of SMK when the number of in-person consultations is maximised; the waiting room occupancy is levelled; and appointment types are evenly distributed over all resources. We refer to the caption of Figure 2 for a more detailed explanation of the figure.

Figure 5. Optimal blueprint schedule (a) and waiting room occupancy (b) for the rheumatology clinic of SMK when the waiting room capacity is set to its minimum necessary to schedule all required occurrence for each trajectory. We refer to the caption of Figure 2 for a more detailed explanation of the figure.

Figure 6. Optimal blueprint schedule (a) and waiting room occupancy (b) for the rheumatology clinic of SMK with time-dependent restrictions on the number of patients simultaneously allowed in the waiting room. We refer to the caption of Figure 2 for a more detailed explanation of the figure.

Figure 7. Optimal blueprint schedule (a) and waiting room occupancy (b) for the rheumatology clinic of SMK when the number of trajectories is reduced to 90% of the total case mix. We refer to the caption of Figure 2 for a more detailed explanation of the figure.

Figure 8. Optimal blueprint schedule of the outpatient clinic (a) and day-care department (b) for the medical oncology & haematology clinic in UMCU when only the number of in-person consultations is maximised. Along the x-axis is the time and along the y-axis is the resource (in (a) resources 1 – 4 and 5 represent the oncologists and oncology PA, resp., and resources 6 – 8 and 9 represent the haematologists and haematology PA, resp.). In (c) and (d) the waiting room occupancy for the outpatient clinic and day-care department, resp., is depicted over time. Here the grey bars denote waiting room occupancy levels taking into account bridging and average early arrival times (result from the ILP) and the blue line, together with its 95% confidence interval as the shaded area, depicts the waiting room occupancy including randomness in early arrival and consultation times (result form the MCS).

Figure 9. Optimal blueprint schedule of the outpatient clinic (a) and day-care department (b) for the medical oncology & haematology clinic in UMCU with time-dependent restrictions on the number of patients simultaneously allowed in the waiting rooms. The corresponding waiting room occupancy is shown in (c) and (d), resp. We refer to the caption of Figure 8 for a more detailed explanation of the figure.

Table

Algorithm 1: Monte Carlo simulation to evaluate waiting area occupancy of blueprint schedule