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Citation: Lyons, J.S.F.; Begen, M.A.;
Bell, P.C. Surgery Scheduling and
Perioperative Care: Smoothing and
Visualizing Elective Surgery and
Recovery Patient Flow. Analytics
2023,2, 656–675. https://doi.org/
10.3390/analytics2030036
Received: 27 April 2023
Revised: 1 August 2023
Accepted: 11 August 2023
Published: 21 August 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
Surgery Scheduling and Perioperative Care: Smoothing and
Visualizing Elective Surgery and Recovery Patient Flow
John S. F. Lyons 1, Mehmet A. Begen 2, * and Peter C. Bell 2
1Kings University College, Western University, London, ON N6A 2M3, Canada; jlyons24@uwo.ca
2Ivey Business School, Western University, London, ON N6G 0N1, Canada; pbell@ivey.ca
*Correspondence: mbegen@ivey.uwo.ca; Tel.: +1-519-661-4146
Abstract:
This paper addresses the practical problem of scheduling operating room (OR) elective
surgeries to minimize the likelihood of surgical delays caused by the unavailability of capacity for
patient recovery in a central post-anesthesia care unit (PACU). We segregate patients according to their
patterns of flow through a multi-stage perioperative system and use characteristics of surgery type
and surgeon booking times to predict time intervals for patient procedures and subsequent recoveries.
Working with a hospital in which 50+ procedures are performed in 15+ ORs most weekdays, we
develop a constraint programming (CP) model that takes the hospital’s elective surgery pre-schedule
as input and produces a recommended alternate schedule designed to minimize the expected peak
number of patients in the PACU over the course of the day. Our model was developed from the
hospital’s data and evaluated through its application to daily schedules during a testing period.
Schedules generated by our model indicated the potential to reduce the peak PACU load substantially,
20-30% during most days in our study period, or alternatively reduce average patient flow time by
up to 15% given the same PACU peak load. We also developed tools for schedule visualization that
can be used to aid management both before and after surgery day; plan PACU resources; propose
critical schedule changes; identify the timing, location, and root causes of delay; and to discern the
differences in surgical specialty case mixes and their potential impacts on the system. This work
is especially timely given high surgical wait times in Ontario which even got worse due to the
COVID-19 pandemic.
Keywords: OR scheduling; constraint programming; load levelling
1. Introduction
1.1. Motivation
Demographic changes and political economic conditions have intensified the need and
demand for more efficient health care operations, including a call to reduce elective surgery
wait-times. For example, Health Quality Ontario, an organization established to advise
the province regarding the performance of its $55 billion annual health care expenditures,
maintains an up-to-date public Internet dashboard listing of surgical wait-times for six key
categories of procedures, not only at the provincial level but also by region and individual
hospital [
1
]. More recently, the province of Ontario has been trying to find ways, including
the privatization of healthcare, to reduce high and chronic surgical wait times, which have
gotten much worse over the years and the pandemic [
2
,
3
]. The situation is similar for
the rest of the country, and Canadian federal government and provinces reached a new
funding deal to improve healthcare over the next 10 years [4].
Our partner hospital is a 600-bed regional tertiary care hospital that has been exploring
opportunities to increase surgical throughput by means of establishing some operating
room (OR) schedule blocks as Rapid and Standardized Operating Rooms (called ‘RASTOR
rooms”). Their aim is to reduce wait-times in services where they exceed the provincial
benchmarks, by more than double in some service categories. While these RASTOR rooms
Analytics 2023,2, 656–675. https://doi.org/10.3390/analytics2030036 https://www.mdpi.com/journal/analytics
Analytics 2023,2657
involve more numerous and shorter procedures than other ORs, they operate within the
same perioperative system which includes a centralized Post-Anesthesia Care Unit (PACU).
More rapid patient procedures in these RASTOR rooms create an imperative that these
ORs can function without obstruction from external processes and conditions, as happens
when a patient whose surgery has been completed cannot be moved from the OR due
to the PACU being at full capacity. This delays the subsequent patient, causes lost OR
time for the surgeon, and is a waste of utilities and staff time to support open ORs, often
re-incurring added costs later at higher overtime rates. Meanwhile, some of the PACU bed
capacity may be occupied inappropriately, due to a downstream patient destination being
unable to receive a patient into the next stage of care [
5
,
6
], either a hospital ward or PACU2
(step-down recovery, just prior to discharge from the hospital).
This problem already exists with our partner hospital’s current daily volumes of sched-
uled patients, and it is likely to worsen if volumes are increased to address the wait-time
performance issue. We note that some recent developments in anesthesia practices [
7
] can
eliminate the need for PACU recovery in certain cases. However, general anesthesia remains
dominant and leads to similar recovery times for most RASTOR surgeries, notwithstanding
that the procedures themselves are being made shorter and more rapid and frequent. One
consideration is whether additional PACU physical capacity is required to accommodate
greater patient volumes. Our research considers whether case scheduling might be better
coordinated across ORs, and perhaps integrated with PACU bed management, to improve
patient flow.
We addressed these issues through developing a model to translate daily patient
rosters (i.e., existing schedules) into individual OR sequences aimed collectively at min-
imizing the peak PACU patient load. We run our model iteratively, with each iteration
seeking a schedule to meet a progressively lower peak PACU load target, and within
that context, to choose a solution that minimizes total patient flow time and thereby the
ORs’ makespans. Flow time is defined by [
8
] as the total time a job (patient) spends in a
shop
(i.e., preparation
, OR, and recovery), using the simplifying, common, and justifiable
assumption that all patients arrive at the start of their respective OR block opening times.
We used information available on days following surgery to evaluate our model and to
visualize events (including delays) across time, location, and status of patients, ORs, and
PACU beds. Our methods take the existing ‘pre-schedule’ and respond with the best
alternative sequencing of procedures for the ORs. We then extract and transform data
collected during schedule execution into a visual flow format, with complimentary metrics,
for surgeons and perioperative management to monitor and refine their scheduling and
staffing decisions.
To the best of our knowledge, we are the first to address the general problem pre-
sented in this paper using constraint programming (CP) with real data and to demonstrate
strengths of this form of programming in terms of its intuitiveness and flexibility. In effect,
we develop a platform which can be easily implemented to suggest beneficial schedule
changes based on a wide range of factors that can be captured and conditions that might be
imposed in a complex setting. Notwithstanding such extensions, our analysis shows that a
collective and calculated approach to scheduling in multiple ORs can have a significant
positive impact, reducing by 20% or more the expected peak patient loads in the PACU.
Our research is unique in a variety of ways. It focuses on the PACU as a flow-limiting
stage of the OR process using constraint programming to establish surgery sequences
across multiple ORs to achieve maximum average utilization and to minimize peak load in
the PACU.
1.2. Background Literature
The body of research concerned with operating room scheduling is substantial.
Refs. [9–12]
provide extensive literature reviews. During the eight years spanning the final three of
these reviews, the number of papers considered expanded from 115 to 170, or roughly
one every two months, as a testament both to the heightened importance of these issues
Analytics 2023,2658
to health care managers and to the variety of specific problems and operating contexts
involved. Refs. [
10
,
11
] share a similar structure which classifies papers according to six
dimensions: patient-case characteristics, performance criteria and measurement, decision
delineation or planning context, research method or solution technique, handling of uncer-
tainty, and applicability of research. Some other related references to patient scheduling
include [13–15].
Patient-case characteristics refer to whether the problem considered includes both
elective (pre-scheduled) and/or urgent or emergent (unscheduled) surgeries, and whether
distinctions are drawn between inpatients and outpatients. Our research focuses on elective
surgeries, as do nearly two-thirds of the papers reviewed in [
12
]. This is warranted by
three levels of capacity allocation the hospital has for unscheduled, emergent, and urgent
cases with dedicated OR time blocks reserved for priority cases to be inserted into the
surgery day (real time allocation). In our partner hospital, one OR is fully dedicated to
priority 1 emergencies as they arrive. Finally, a scheduled elective case may be pre-empted,
if necessary, to accommodate an emergent case if other options are not available. As for
elective cases, our research considers and distinguishes between outpatients and inpatients
and also the-same-day admissions that share a common routing inbound and outbound
with outpatients and inpatients.
The dominant performance criterium applied in the OR scheduling literature is uti-
lization (especially of ORs, surgeons, and nursing teams), followed by waiting times (of
patients, surgeons, or as a measure of system throughput), overtime, and load levelling,
e.g., [
9
,
11
,
15
]. However, roughly 40% of papers reviewed in [
12
] included other and/or
multiple criteria, as a reflection of many different perspectives about what outcomes are
most/least desirable in a given hospital setting. Our model’s objective function falls in the
category of throughput measure, but indirectly also OR utilization, as through minimizing
average patient flow time we are also minimizing the sum of individual OR makespans
given their specified workloads. Our iterative approach, on the other hand, falls in the
category of load levelling, as it explicitly seeks to do that for the PACU.
Decision delineation or planning context incorporates but extends traditional clas-
sifications of planning: strategic, e.g., case mix planning [
10
]; tactical, e.g., block time
allocation [
5
,
6
]; and operational, e.g., case sequencing [
16
]. Ref. [
10
] considers the matter in
two dimensions, which represent whether decisions are to be made concerning date, time,
room, or capacity allocation and whether the operational unit of concern is individual or
a grouping of surgeons, patients, or other units such as the PACU. Refs. [
17
,
18
] combine
constraint programming and goal programming to assign surgical teams and block times
with the goal of maximizing OR utilization while effectively balancing workload. Our
research takes an integrated approach in assuming daily surgeon-patient-room assignments
as given (by the hospital’s pre-schedule) and seeking the best sequence of these procedures
across and within the ORs such that they can be performed unhampered by downstream
capacity limitations, in the PACU, and/or wards.
Regarding research methodology and solution techniques, [
12
] identified thirteen
categories dominated by mixed integer linear programming (MILP), heuristic algorithms,
simulation, and integer programming (IP). Only four out of 170 papers employ a CP
solution technique, as does our research, the oldest of these being from 2010. Of special
interest is [
19
], which provides a comparison of MILP to CP for scheduling problems in
operating theatres. Our research exploits certain advantages of CP such as enabling a
compact and intuitive model formulation, the ease with which expressing multiple and
complex constraints can be added, and computational efficiencies.
Whereas other mathematical programming techniques, such as mixed integer pro-
gramming (MIP), can lead to exact ‘optimal’ solutions, or improved approximations thereof,
we argue that optimality is not the ultimate promise in surgery scheduling, as the process
involves many entities (patients, surgeons, support staff, and equipment) with uncertainty
at every stage from preparation through procedure and recovery [
5
]. About two-thirds of
papers reviewed in [
12
] employ deterministic models, while acknowledging uncertainty
Analytics 2023,2659
and generally accommodating it with some manner of time buffering. We take a similar
approach, firstly, through using only specified allowable start times, and secondly, through
hedging long on procedure and recovery duration estimates. To the extent that procedures
are not on the long side of their estimates, cleanup time does not present a problem, and
where procedures turn out to be longer than expected, cleanup can be accelerated relatively
easily with the help of floating staff in the system.
In classifying reviewed articles according to the ‘applicability of research’, [
12
] iden-
tified a near-equal division of studies that used a theoretical data set versus those based
on real data. Our model is parameterized from actual hospital data and is executed with
a daily schedule as input using only data available to the scheduler prior to the surgery
day. We also measure our model’s performance retrospectively, using actual procedure
and recovery durations that become available on the day following surgery, whereupon
we compare the day’s actual PACU load to what would have occurred with the actual task
durations but following case sequences proposed by our model.
We note that 6 out of 115 articles reviewed in [
10
] address load levelling of the PACU,
but in different planning contexts and settings than those we study, generally using different
solution techniques and/or applying theoretical data rather than actual data. The articles
that are most proximate to our research include [
20
–
22
]. The first of these applies theoretical
data and a deterministic MIP for a master surgery-scheduling problem that considers the
possibility of next-day surgical blocking by the PACU given the current state of occupancy
in various units of the hospital. We instead use actual daily case mixes and seek to avoid the
blocking of surgeries by the PACU indirectly, without specific knowledge of downstream
unit occupancies. Retrospectively, we provide a tool to identify the frequency, timing, and
location of these events to help characterize and remediate delays, according to what else
had happened in situ or was happening elsewhere in the system to cause the delay. Ref. [
21
]
employs a MIP and two-phase heuristic to determine surgeon-to-OR assignments followed
by surgical case sequencing. We study a context where surgeons have pre-assigned OR time
blocks and a pre-specified list of patient procedures to perform within those blocks. The
work of Ref. [
22
] is similar to this article in subject matter and setting but uses MIP to level
daily PACU loads, placing special emphasis on a machine learning model for more accurate
prediction of procedure and recovery durations. We have much less about information
available for such accurate predictions and instead emphasize the potential of intuitive
and adaptable CP optimization, combined with process visualization, disaggregation, and
iterative refinement, in spite of the vast uncertainties in perioperative care.
1.3. Problem Description
Figure 1provides a global view of the perioperative system through which three basic
categories of patients flow: one-day stay (ODS), same-day admission (SDA), and in-patients
(IPs). At the highest level, ODS and SDA patients enter the system from outside the hospital
through admission and preparation (A), whereas IP patients arrive from a hospital ward
(B). As for departures, ODS patients leave through Day Surgery from whence they arrived
(C and F), whereas SDA patients are transferred into a hospital ward (E), through which
IP patients also return to their ward, although either may be transferred instead to an
intensive care unit (ICU) bed.
At a more detailed level within the perioperative process, ODS patients follow one
of three different paths involving recovery in either or both the PACU and Day Surgery,
the latter referred to as PACU2 (D) in this context. SDA patients normally recover in the
PACU before being transferred to a ward bed when one becomes available. IP patients
often recover in the PACU but occasionally bypass it, returning directly to their ward bed.
Similarly, some ODS patients may bypass the PACU if they require only minor recovery,
which they can undergo in PACU2. Because our model addresses only elective cases, we
make a conservative assumption that all surgeries will be followed by a PACU stay, as is
most often the case. Some patients may be diverted rarely on a given day, but these cases
Analytics 2023,2660
will serve as a counterbalance to some of the added PACU load from unplanned emergent
and urgent cases that arise.
Analytics 2023, 2, FOR PEER REVIEW 5
make a conservative assumption that all surgeries will be followed by a PACU stay, as is
most often the case. Some patients may be diverted rarely on a given day, but these cases
will serve as a counterbalance to some of the added PACU load from unplanned emergent
and urgent cases that arise.
Admission/
Preparation
Day
Surgery
Recovery
(PACU 2)
OR 1 OR 2 …. OR R
Wards
PACU
C
B
A
E
FD
Recovery
(PACU 2)
Figure 1. Perioperative system flow.
The stage of perioperative flow of greatest concern to management (point ‘C’ in Fig-
ure 1) lies on the most common path followed by patients immediately after surgery. If
this pathway is blocked, due to a PACU hold, a patient must remain under anesthesiolo-
gist surveillance in the OR, preventing room turnover and delaying the start of the next
surgery. In these situations, the hospital incurs the opportunity costs of an idle and ex-
pensive room and OR team. Additional costs are incurred if overtime is required at the
end of the day to make up for the delay or if surgeries have to be postponed. Surgeons
lose some of their limited OR block time to perform revenue-generating (and backlog-
reducing) activities. The health care system and patients awaiting treatment suffer, as
fewer elective surgeries performed in a given day, week, month, and year translate into
larger queues and longer wait-times.
Figure 2 shows several examples of patient flow. The characters in the ‘Flow’ and
‘Delay’ columns correspond to critical points in the process as depicted in Figure 1, that
is, the path through which each example patient flowed and the location(s) at which they
encountered any delays. The paerns within the timelines correspond to the various loca-
tions in Figure 1. We are concerned with delays both upstream and downstream from the
PACU, as eliminating the laer can effectively eliminate at least some of the former. The
downstream delays are represented in solid black, corresponding to the same format as
points ‘D’ and ‘E’ in Figure 1, and these are delays in moving patients from the PACU to
either PACU2 or to a ward bed. Checkered black intervals represent upstream delays mov-
ing patients into the PACU (at point ‘C’ of Figure 1) coming from any one of the ORs. Brief
gaps between intervals correspond to the patient transit times and thereby separated time-
stamp gaps in the hospital data.
Figure 2. Patient flow examples.
Figure 1. Perioperative system flow.
The stage of perioperative flow of greatest concern to management (point ‘C’ in
Figure 1) lies on the most common path followed by patients immediately after surgery. If
this pathway is blocked, due to a PACU hold, a patient must remain under anesthesiologist
surveillance in the OR, preventing room turnover and delaying the start of the next surgery.
In these situations, the hospital incurs the opportunity costs of an idle and expensive room
and OR team. Additional costs are incurred if overtime is required at the end of the day to
make up for the delay or if surgeries have to be postponed. Surgeons lose some of their
limited OR block time to perform revenue-generating (and backlog-reducing) activities.
The health care system and patients awaiting treatment suffer, as fewer elective surgeries
performed in a given day, week, month, and year translate into larger queues and longer
wait-times.
Figure 2shows several examples of patient flow. The characters in the ‘Flow’ and
‘Delay’ columns correspond to critical points in the process as depicted in Figure 1, that
is, the path through which each example patient flowed and the location(s) at which they
encountered any delays. The patterns within the timelines correspond to the various
locations in Figure 1. We are concerned with delays both upstream and downstream from
the PACU, as eliminating the latter can effectively eliminate at least some of the former.
The downstream delays are represented in solid black, corresponding to the same format
as points ‘D’ and ‘E’ in Figure 1, and these are delays in moving patients from the PACU
to either PACU2 or to a ward bed. Checkered black intervals represent upstream delays
moving patients into the PACU (at point ‘C’ of Figure 1) coming from any one of the ORs.
Brief gaps between intervals correspond to the patient transit times and thereby separated
time-stamp gaps in the hospital data.
Analytics 2023, 2, FOR PEER REVIEW 5
make a conservative assumption that all surgeries will be followed by a PACU stay, as is
most often the case. Some patients may be diverted rarely on a given day, but these cases
will serve as a counterbalance to some of the added PACU load from unplanned emergent
and urgent cases that arise.
Admission/
Preparation
Day
Surgery
Recovery
(PACU 2)
OR 1 OR 2 …. OR R
Wards
PACU
C
B
A
E
FD
Recovery
(PACU 2)
Figure 1. Perioperative system flow.
The stage of perioperative flow of greatest concern to management (point ‘C’ in Fig-
ure 1) lies on the most common path followed by patients immediately after surgery. If
this pathway is blocked, due to a PACU hold, a patient must remain under anesthesiolo-
gist surveillance in the OR, preventing room turnover and delaying the start of the next
surgery. In these situations, the hospital incurs the opportunity costs of an idle and ex-
pensive room and OR team. Additional costs are incurred if overtime is required at the
end of the day to make up for the delay or if surgeries have to be postponed. Surgeons
lose some of their limited OR block time to perform revenue-generating (and backlog-
reducing) activities. The health care system and patients awaiting treatment suffer, as
fewer elective surgeries performed in a given day, week, month, and year translate into
larger queues and longer wait-times.
Figure 2 shows several examples of patient flow. The characters in the ‘Flow’ and
‘Delay’ columns correspond to critical points in the process as depicted in Figure 1, that
is, the path through which each example patient flowed and the location(s) at which they
encountered any delays. The paerns within the timelines correspond to the various loca-
tions in Figure 1. We are concerned with delays both upstream and downstream from the
PACU, as eliminating the laer can effectively eliminate at least some of the former. The
downstream delays are represented in solid black, corresponding to the same format as
points ‘D’ and ‘E’ in Figure 1, and these are delays in moving patients from the PACU to
either PACU2 or to a ward bed. Checkered black intervals represent upstream delays mov-
ing patients into the PACU (at point ‘C’ of Figure 1) coming from any one of the ORs. Brief
gaps between intervals correspond to the patient transit times and thereby separated time-
stamp gaps in the hospital data.
Figure 2. Patient flow examples.
Figure 2. Patient flow examples.
The hospital typically opens 15–17 ORs on weekdays, with 1–3 operating outside of the
core scheduled surgery period of 8:00–18:00. ORs can run overtime, but the management
of perioperative care has discretion to cancel a final OR procedure if it is deemed likely to
induce substantial overtime. The number of scheduled surgeries per day at our partner
Analytics 2023,2661
hospital typically falls between 50 and 65, with 8 to 12 urgent or emergent (unscheduled)
procedures added on most days.
The PACU is rarely empty as it often begins with an overnight patient trauma patient,
or perhaps one from the previous day still waiting for a ward bed as in Ex. 9 of Figure 2.
PACU occupancy rises quickly in the morning to roughly the number of open ORs as
their first procedures are completed. It continues to rise through the mid-afternoon before
declining rather slowly at the end of the day. This pattern stems from patient recoveries
being on average longer than the associated procedure times or the OR service rate being
greater than the PACU service rate; consequently, the PACU cannot release patients as
quickly as it receives them, even in the best case without any downstream delays. Finally,
as new PACU patient arrivals slow toward the end of the day, patients with long recoveries
keep the PACU load from dissipating quickly. Finally, the unavailability of a ward bed or
the inability to move a patient back to Day Surgery/PACU 2 will cause PACU occupancies
to remain above manageable preferred levels.
2. Model Development
We develop a scheduling model aimed at mitigating the problem of blocking at points
‘C’ and ‘E’ (in Figure 1) in this complex and multidimensional system. Blocking at point ‘D’
can be addressed with simpler heuristics, as presented in the Section 3.4 of the paper. We
begin through stating several assumptions, conventions, and simplifications that we use to
provide for a tractable, generalized model that can be refined to accommodate a variety of
additional constraints, as required.
2.1. Assumptions
First, we assume that procedure and recovery durations can be predicted with rea-
sonable accuracy and are available well enough in advance to inform the scheduling of
procedure sequences and start times. We note that [
22
] provide a compelling case for
machine learning techniques to improve such time estimates; however, to take advantage
of these methods, information systems and internal communication require more details
about patients and procedures than are currently available at the hospital. As one example,
we found that ASA scores (a measure of patient physical condition established by the
American Society of Anesthesiologists (ASA)) could substantially improve recovery time
estimates, but these are only determined just prior to surgery by the attending anesthetist
and so are not useful in OR scheduling.
Second, we disregard unscheduled (trauma/priority) cases as they are difficult to
anticipate and not known at the time of scheduling. We assume that managing scheduled
cases to minimize likelihood of PACU blockages will better accommodate unscheduled
cases as they arise. We also assume that all patients will recover in the PACU, despite this
not being required in roughly 10% of cases, and thus serving to counterbalance the effect of
not including unscheduled cases in our PACU load estimates.
Third, we assume that surgeons, OR teams, and patients are indifferent toward case
sequence and timing, so long as the procedure will be done in the designated OR during
the block surgery time allocated to the surgeon. However, we note that our model can be
adapted to address specific needs or preferences regarding case sequencing if necessary.
Fourth, our objective is minimizing the total completion time (flow time) of all surg-
eries scheduled for the day. This is an appropriate collective goal, although it may not be
individually the most preferred objective for every OR and surgeon involved.
Finally, we follow the convention that the hospital wishes to maintain scheduled start
times only at :00 and :30 of the hour, and that OR cleanup (not including closure and setup)
can generally be accomplished in 20 min or less.
2.2. Hospital Data
We obtained data regarding more than 27,000 surgical procedures performed at the
hospital from 1 April 2017 to 31 March 2018 and developed two regression models for
Analytics 2023,2662
predicting procedure and recovery durations based on information available at time of
schedule release (normally 2:00 p.m. on the afternoon of the day preceding surgery). We
reserved four weeks of this data for testing our models and to provide a proof of concept.
We later used the model repetitively over three weeks of surgical activity from 25 March to
12 April 2019 to pilot its use with actual daily schedules.
2.3. Procedure and Recovery Durations
We developed linear regression models to predict procedure and recovery times. In
both regression models (Figure 3), the independent variables included initial OR time as
booked by the surgeon (30 min increments), patient age, and surgical discipline (binary
indicator variables). After some surgical discipline, binary variables were removed from
the recovery model due to a lack of statistical significance. The model for procedure
durations has an R-squared of 0.83 whereas the model for recovery durations is much less
explanatory with an R-squared of 0.19. These two models provide a reasonable starting
point for anticipating arrivals of patients into the PACU based on their procedure start
times and to predict their length of stay in the PACU. We note that the standard error of
both predictions is in the realm of 40 min, reflecting the substantial variability among cases,
but this is a similar magnitude of error as reported elsewhere in the literature ([23,24]).
Analytics 2023, 2, FOR PEER REVIEW 7
Finally, we follow the convention that the hospital wishes to maintain scheduled start
times only at :00 and :30 of the hour, and that OR cleanup (not including closure and
setup) can generally be accomplished in 20 min or less.
2.2. Hospital Data
We obtained data regarding more than 27,000 surgical procedures performed at the
hospital from 1 April 2017 to 31 March 2018 and developed two regression models for
predicting procedure and recovery durations based on information available at time of
schedule release (normally 2:00 pm on the afternoon of the day preceding surgery). We
reserved four weeks of this data for testing our models and to provide a proof of concept.
We later used the model repetitively over three weeks of surgical activity from 25 March
to 12 April 2019 to pilot its use with actual daily schedules.
2.3. Procedure and Recovery Durations
We developed linear regression models to predict procedure and recovery times. In
both regression models (Figure 3), the independent variables included initial OR time as
booked by the surgeon (30 min increments), patient age, and surgical discipline (binary
indicator variables). After some surgical discipline, binary variables were removed from
the recovery model due to a lack of statistical significance. The model for procedure du-
rations has an R-squared of 0.83 whereas the model for recovery durations is much less
explanatory with an R-squared of 0.19. These two models provide a reasonable starting
point for anticipating arrivals of patients into the PACU based on their procedure start
times and to predict their length of stay in the PACU. We note that the standard error of
both predictions is in the realm of 40 min, reflecting the substantial variability among
cases, but this is a similar magnitude of error as reported elsewhere in the literature
([23,24]).
Figure 3. Linear regression models for procedure and recovery durations.
Except for emergent cases, surgical ORs cycle through four states while they are
open: setup, surgery, closure, and cleanup, where a patient’s stay in the OR coincides with
all phases but the cleanup. Although we had access to hospital data that would allow us
to distinguish between setup, surgery, and closure times, we treated them together as one
procedure time estimate, both in formulating and applying the linear models and to pre-
dict procedure times. One exception is that if a PACU delay was indicated at the end of
the procedure, we excluded closure time from the procedure duration and used it instead
as a measure of PACU delay, even though some of this time would have in fact been re-
quired to close the procedure. Where no such delays occurred, as happened in the major-
ity of ORs on most days, the sum of procedures times divided by total OR open time pro-
vides a direct measure of OR utilization.
Figure 3. Linear regression models for procedure and recovery durations.
Except for emergent cases, surgical ORs cycle through four states while they are open:
setup, surgery, closure, and cleanup, where a patient’s stay in the OR coincides with all
phases but the cleanup. Although we had access to hospital data that would allow us to
distinguish between setup, surgery, and closure times, we treated them together as one
procedure time estimate, both in formulating and applying the linear models and to predict
procedure times. One exception is that if a PACU delay was indicated at the end of the
procedure, we excluded closure time from the procedure duration and used it instead as a
measure of PACU delay, even though some of this time would have in fact been required to
close the procedure. Where no such delays occurred, as happened in the majority of ORs
on most days, the sum of procedures times divided by total OR open time provides a direct
measure of OR utilization.
2.4. Constraint Programming (CP) Model
We developed a CP model as detailed below using IBM’s ILOG CPLEX 12.9 CP
Optimizer. Variable inputs to the model include a set of planned procedures for a given
day with initial start times and booking durations as requested by the surgeon, along with
procedure and recovery durations estimated using the linear models above. In addition to
this procedure information, the model takes as input a set of OR block times which specify
the surgical discipline to which the OR is allocated, and the opening and closing times
within which the set of procedures must be scheduled.
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2.4.1. Preliminaries
Model definitions are presented under the following groupings, after which we pro-
vide a model formulation and a discussion:
1. Tuple Sets (Patients PRooms R);
2.
Deducible Parameters and Relationships (Occupancy Durations
Xpr
, Assignment
Matrix M);
3. Decision Variables;
a. leftmargin
Interval (Occupancies
Opr
, Patient PACU Occupancies
Up
, OR Occupied
Times Vr);
b. leftmargin Sequence (OR Loads Lr, Patient Paths Πp);
4. Interval Parameters (OR Open Windows Wr, Surgical Day Time T);
5. Setting Specific Parameters (Allowable OR Start Times Yr, OR Cleaning Times Yr);
6.
Intermediate Functions and Expressions (Patient PACU Pulse
pUt
, Patient Flow
Time Fp);
7. Objective Function.
Tuple sets provide for the simple extraction of model input data from the hospital’s
pre-schedule. These data are processed into deducible parameters related to patient as-
signments to and durations in specific ORs and, subsequently, in the PACU (if required.)
The ultimate decision variables fully specify a solution that is a feasible set of occupancy
intervals (start-to-end times) of all patients in their respective ORs as well as in the PACU,
as required. However, the time locations of these occupancy intervals are partially a func-
tion of complementary decisions regarding occupancy interval sequences. These interval
sequences are determined for both sets of procedures within each room and each patient’s
flow through the two-stage system. Additional parameters help to constrain the problem
according to local practices, and additional functions translate any solution candidate set
of interval decisions into metrics against which the objective is measured and optimized.
2.4.2. Some Properties
The interval decision variables and interval parameters defined below each possess
the following properties:
.start
an integer value lying with a specified range ‘.in ..’ (see below)
.end
an integer value lying in the same specified range ‘.in ..’:
.size
an integer value representing the difference
.optional
a Boolean value (default: False) if interval not required in a solution
.in..
a specified range constraining the above.
2.4.3. Tuple Sets
Ra set of discrete ORs (operating rooms)
Rthe union of the set Rwith the PACU (a special room): R=R∪PACU
rFor each r∈Rwe introduce a tuple of input data with the following
.id
key field r=1, 2, .., |R|
.room
OR # or PACU
.service
name of the surgical group (e.g., Ortho)
.open
(.start) of first prescheduled surgical procedure
.close
(.start + .booking) of last prescheduled surgical procedure
Pa set of patients on each of whom a surgical procedure is to be performed
pFor each p∈Pwe introduce a tuple of inputs with the following properties:
.patient
key field p=1, 2, .., |P|
.service
name of surgical group performing the service
.room
OR # or PACU;
.start
prescheduled start time (e.g., 480 min = 08:00);
.booking
a pre-scheduled booking window (e.g., 30, 60, 120 min);
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.procTime
the expected duration of the procedure;
.recTime
the expected duration of the patient’s recovery;
.patient
a real number lying in e−q:q∈[0, 1].
2.4.4. Deducible Parameters
Xpr an integer representing the time a patient pis expected to spend in room r
Xpr =
p.proctime, if r:r.room =p.room
p.rectime, if r:r.room =PACU
0, otherwise
, for all p∈P, for all r∈R
Ma matrix of dimensions |P|×|R|with binary values:
Mpr =1, if Xpr =Opr.size >0
0, if Xpr =Opr.size =0.
2.4.5. Decision Variables
Intervals:
Opr an interval decision variable such that Opr.size =Xpr , for all P, for all R
note: Mpr =0⇔Opr.optional =True
Up
an interval decision variable representing the time a patient
p
will occupy a bed
in PACU
Vr
an interval decision variable representing the time during which room
r
is expected
to be open, beginning at r.open and ending with the latest Opr.end in that room
Span interval decision variable represent a patient’s total length of stay
Sequences
Lra sequence decision variable on every Opr of type r,
such that Lr.start =min(Opr.start)and Lr.end =max(Opr.end)
Πpa sequence decision variable on Opr of type p,
such that Πp.start =min(Opr.start)and Πp.end =max(Opr .end)
We refer to Lras the load on room r, and Πpas the path of patient p.
2.4.6. Interval Parameters
Wran interval parameter representing the range [r.open..r.close], for all r∈R
2.4.7. Setting-Specific Parameters
T a fixed range from 480..1440 (08:00 a.m.–24:00 p.m.)
Yra step function which restricts possible start times of procedures
ka minimum allowable clean-up time gap between procedures
Krsets of pi,pij ,kspecifying minimum time between procedures i→jin r
Q a maximum allowable number of patients in the PACU at any time.
2.4.8. Intermediate Functions and Expressions
pUt=∑ppulse[Upr], the sum of patients in the PACU at points-in-time
for all t∈Tover minutes of the day.
Fp=Opr.end −Wr.open, flow time for each patient, from the opening of
for all p∈P,r:Mpr =1 their OR until the expected completion of their surgery
completion, given the procedure start time in a proposed schedule solution.
Br=min
p∈P:Xpr =1{startO f (Opr)}, the earliest procedure start time in a room
Cr=max
p∈P:Xpr =1{endO f (Opr )}, the latest procedure end time in a room.
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2.4.9. Objective Function
minimize ∑p∈PFp, the sum of patient flow times
Alternatively,
minimize ∑pFp×p
.
precedence
the sum of patient flow times weighted by their respective
precedence scores.
2.5. Formulation
min ∑
p∈P
Fp(1)
s.t.
for all p∈P
span(Sp,al l(r∈R,Opr )), (2)
lastΠp,Op|R|(3)
noOverlap(Πp)(4)
le ngthO f (Sp)=∑
r∈R
Opr.size (5)
sta rtO f (Up)=Op|R|.start (6)
for all r∈R,r6=|R|
noOverlap(Lr,Kr), (7)
Br≥r.open (8)
Cr≤1200 (i.e. , 20 : 00 pm)(9)
for all r∈R,r6=|R|, for all p∈P
f orbidStart(Ppr ,Yr), (10)
pUt≤Q, for allt(11)
In comparison with the two-part mixed integer program of [
22
] that has a similar
objective of minimizing the squared PACU load, the advantage of more compact and
intuitive CP formulations is clear, subject to some explanations for those unfamiliar with
the special types of CP constraints used above. Note that the first five constraints apply to
all patients, whereas the next three apply to all rooms excluding the PACU.
(1)
is the objective function minimizing total flow time.
(2)
specifies that decision intervals
Sp
, which represent each patient’s total length of stay
from the start of procedure to the end of recovery, must span their two occupancies in
the OR and PACU.
(3)
specifies that the last (2nd) occupancy a patient’s path must be in the PACU, numeri-
cally the |R|th member of R.
(4)
specifies no overlap between a patient’s occupancies in the OR and PACU.
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(5)
complements both Equations (2) and (4) specifying further that the size of
Sp
should
exactly match the sum of the patient’s occupancies. A solution must have no delay
between them.
(6)
specifies that the decision interval Upfor each patient begins with PACU occupancy.
(7)
specifies that intervals in sequences
Lr
cannot overlap and furthermore must be
separated by at least a minimum clean time (for two patients with occupancies
Opi,r→Opj,rin a sequence Lr.
(8)
specifies that room rcannot start before the room’s block time opening Br.
(9)
specifies the latest any procedure can be expected to end is 20:00 p.m. (minute 1200).
(10)
requires that patient procedure scheduled starts are only at allowable times in Yt.
(11)
specifies a peak number of patients in the PACU at any time cannot exceed an integer
amount
Q
. This is initially set to a high number to ensure that it will not be binding and
thus to discover the unbounded peak PACU expected from a solution with the best
objective function value.
Q
is lowered incrementally through successive iterations of
the model until its lowest possible value (with a feasible schedule solution) is achieved.
We explain this more next.
2.6. Iterative Solution Approach
We use our model iteratively to generate a recommended schedule. Our iterative
method works as follows.
1.
Solve the problem with an overly generous PACU capacity limit
QM
and the objective
of minimizing total patient flow time, which also translates into minimizing total
OR makespans.
2.
Determine the resulting expected peak PACU patient load,
max{pU}≤Q
. Call this
amount
QU
, being the threshold below which capacity has a negative effect, i.e., a
restriction on flow time optimization. Set
Q=QU
before proceeding to the next step.
3.
Invoke a PACU capacity constraint one less than the peak determined in the previous
step; that is, set
Q=Q−
1 and re-solve for the objective of minimizing total patient
flow time, subject to: max{pU}≤Q.
4.
If a feasible solution is not found in the most recent step (3), accept the feasible solution
found in the second-most recent step as the recommended schedule and stop. (In
effect, after making one too many progressively constrained solve attempts, return to
the last successful one.)
5.
Otherwise, a feasible solution was found in step 3, so repeat steps 3 and 4 until
reaching a stop.
2.7. Discussion of Model Features
One of the features of our model is the creation of a binary matrix representing whether
a patient requires service in each of the rooms
r
. For each patient row of the matrix, there
will be two columns with entry 1, one for a specific OR and the other for their PACU
stay. This allows us to define the set of occupancy intervals
Opr
for all patient and room
combinations, while setting the irrelevant ones as optional and of zero duration .size =0.
Another is a set of intervals that serve as parameters for the block time windows that,
despite being called a room in our model, may be one of multiple blocks within a single
OR during a surgery day, each with its own time window Wr.
We also introduce three active and interrelated interval decision variables for each
patient. One represents specifically their stay in PACU,
Up
. Two other intervals
Opr
repre-
sent the patient’s expected occupancies in two rooms, one of them being the same PACU
stay and therefore exactly overlapping
Up
. Through defining two equivalent, parallel, and
concurrent intervals, we can constrain them independently and/or in combination. For
example, a patient’s path includes two intervals beginning with their surgical procedure in
an OR, which must be followed immediately by their recovery in the PACU. The procedure
alone is part of an operating room’s sequence, whose intervals cannot overlap, whereas the
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recovery alone is part of the PACU load, whose intervals are allowed overlap, as there are
often numerous patients in the PACU concurrently.
The hospital schedules surgeries beginning at 08:00 a.m. (minute 480 of the day) on
every weekday except Wednesday, on which they begin at 09:00 a.m. The latest scheduled
OR booking ends at 20:00 p.m. (minute 1200 of the day), although 18:00 p.m. is when most
ORs are expected to close each day. Any schedule solution must have interval decision
variables
Opr
that lie within the specified OR time windows
Wr
. The upper bound of
T
,
midnight (minute 1440), is given to accommodate lengthy expected recoveries, especially if
arriving toward the end of the OR day (maximum 1200.) The extra length of
T
allows for
extended PACU occupancies but has no impact on the model’s objective of minimizing the
average OR makespan.
The decision expressions
Fp
represent flow time for each patient. That is the length of
time from the opening of the OR in which their surgery is performed until the expected
completion of their surgery, assuming specified start time for their procedure in a solution.
We note that it is a simple change in the CP model to calculate patient flow time using the
expected completion of their recovery. However, our choice has an important and more
direct effect on the objective function. The primary effect is to produce a schedule following
a shortest processing time (SPT) heuristic [
7
], all else being equal; that is, OR sequences
Lr
are initially scheduled in order of ascending duration within each room. However,
as we incrementally constrain the peak PACU load, some value in the objective function
is foregone, to allow for the shift of long surgery intervals forward and short surgeries
backward within the room sequences, in order to eliminate PACU peaks.
Practically speaking, we are not concerned whether patients finish sooner or later on
average in the day, as we understand that to them the day itself is/has been their only
serious concern (with exceptions discussed below.) A more important effect of the objective
function
min ∑pFp
is to minimize the total flow time of patients as it is important to them as
well as the hospital how many procedures can be performed in a given time, or conversely,
in how little time a number of procedures can be performed, to reduce the incidence of
overtime and perhaps create opportunities for additional surgeries.
To accommodate a requirement that procedures be scheduled to start only at specific
times, as is currently standard practice at the hospital, we created the step function
Yr
. We
first chose a start frequency of 30 min (which could be changed as desired) and a range of
start time epochs from 0 to 23 (such that 30 times 23 + 1 = 12 h from the start of the day.) We
then generated an ordered set of tuples
{h0.480i,h100.481i,h0.510i,h100.511i, . . .}
forming
the basis of a step function
Yr
which alternates from 0 (off) to 100 percent or 1 (on) every
30 min
throughout the day. These tuples can be interpreted as 0 before
08:00 a.m.
, then 1
until 08:01 a.m., then 0 until 8:30 a.m., and so on. Constraint (9) prevents any OR occupancy
Opr
,
r6=|R|
from beginning whenever the step function has a value of 0, therefore allowing
it to begin only at one time (minute) every 30 min.
To account for minimum changeover times between procedures in each OR, that is, to
allow for cleaning, we first generated a set of triplets composed of two patient identifiers:
effectively, the procedures ending and beginning in the OR changeover, followed by a
constant minimum OR clean-up time
k=
5. We chose this value despite the expected clean-
up time requirement being between 10 and 20 min. We rationalize this in combination with
the step functions
Yr
that allow scheduled starts only at 30 min intervals. If one assumes
a random uniform distribution of procedure end times over the minutes of any hour,
imposing a minimum five-minute gap will result in a schedule gap of anywhere between
5 and 34 min
before any subsequent scheduled start (at :00 or :30). Our assumption is that
surplus and deficit cleanup times will offset each other in many cases or, in the worst case,
cause only minor delays with no greater impact than already anticipated, stochastically,
due to surgeries whose durations turn out to be unpredictably short or long. However, we
recognize in the latter case that short scheduled clean-up times could compound problems
from surgery time overruns.
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We noted previously an alternate form for the objective function
min ∑p(Fp.precedence)
.
This followed an initial review of our model with hospital perioperative management; they
requested that some additional factors be incorporated in the model. The first related to the
problem of ward beds often becoming available only late in the day, such that it is preferable
for SDA patient procedures to be scheduled later, especially for services whose wards are
most notorious in this regard. Secondly, since there are often delays in the preparation
of very early ODS and SDA patients, as when they arrive late in the morning, preference
should be given for inpatient surgeries to be scheduled in the early part of the day as they
can be prepared quite early, being already in the hospital. The management team also
suggested that younger patients should have surgical priority early in the day for two
reasons. The first is that young pediatric patients are less tolerant of long flow times from
arrival to completion, and the second is that recovery times tend to vary in proportion with
age, so that younger PACU patients first means quicker bed turnover in the earlier part of
the day for the PACU, which is helpful for mitigating patient load surges during the later
peak hours.
Based on these inputs, we added the final property (.precedence) to the patient tuple
described above, and we developed a simple method for translating patient factors into
a single ordinal variable, regardless of whether these factors themselves may be binary,
categorical, interval, or ratio in nature. For type of flow (ODS, SDA, IP), we begin through
assigning the values (0,
−
1, 1), respectively, because we would like to advance IP procedures
to earlier in the day and defer SDA procedures later. However, for patient age, we count
infants as 1 and patients 100 or older as -1; then, for any patient, we use (1
−
age/50) as a
score in the range between (
−
1, 1). Referring to the resulting flow and age scores as
f
and
a
, we determine an aggregate precedence score using the expression
lnef+a
, which then
varies between [
−
2, 2], although it could easily be normalized and/or weighted differently
for each underlying factor.
We note that a time interval for physical movements from the ORs to the PACU was
not included in our model, as these are generally accomplished within a couple of minutes
at the hospital, thanks to the physical design of the perioperative suite.
3. Results
3.1. Timeline Comparision
Figure 4depicts the results of the iterative-CP-model-optimized solution in terms of
expected patient loads in the PACU. It shows the average PACU patient loads by minute
of the day across all sample days, expected if following the day’s pre-schedule versus
expected if following the CP-optimized schedules on each day.
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Analytics 2023, 2, FOR PEER REVIEW 14
Figure 4. Pre-schedule vs. model—forecast averages: patient load by time of day.
We modeled the expected peak PACU patient load given the established schedule for
each day, using our estimated procedure and recovery durations, and assuming on-time
starts. A surprising result was that in one-third of the days analyzed, the expected PACU
peak from the ‘Pre-schedule’ was higher than what was expected from our unconstrained
‘CP optimized’ schedule. On only 40% of the days did the minimized patient flow time of
the unconstrained CP-optimized schedule impose an increase in the expected peak PACU
load beyond what it was already with the pre-schedule. Yet in those cases, the CP model
could reduce expected patient flow time by an average of 15%. Overall, our model
achieved an average 11% reduction in patient flow time at PACU load peaks that were
equal to or less than those expected by following the pre-schedule, similar to the particular
case depicted on the right side of Figure 5.
Figure 5. Efficient frontier: average patient flow time vs. peak PACU patient load.
Figure 5 displays two scaerplots of average patient flow times versus peak PACU
patient loads as predicted by our model. We show averages rather than the total flow time
objective to normalize days with different numbers of patients. A comparison is made
between pre-schedules as established by hospital surgeons and CP-optimized schedules
as proposed by our model. On the right side of Figure 5 is an example of a single day for
which the lowest average patient flow time of 260 min is expected from the unconstrained,
‘CP Optimized’ schedule (versus 310 min with the ‘Pre-Schedule’.) However, that sched-
ule is expected to result in a peak PACU load of 18 patients at some point in the day. In
Figure 4. Pre-schedule vs. model—forecast averages: patient load by time of day.
We modeled the expected peak PACU patient load given the established schedule for
each day, using our estimated procedure and recovery durations, and assuming on-time
starts. A surprising result was that in one-third of the days analyzed, the expected PACU
peak from the ‘Pre-schedule’ was higher than what was expected from our unconstrained
‘CP optimized’ schedule. On only 40% of the days did the minimized patient flow time of
the unconstrained CP-optimized schedule impose an increase in the expected peak PACU
load beyond what it was already with the pre-schedule. Yet in those cases, the CP model
could reduce expected patient flow time by an average of 15%. Overall, our model achieved
an average 11% reduction in patient flow time at PACU load peaks that were equal to
or less than those expected by following the pre-schedule, similar to the particular case
depicted on the right side of Figure 5.
Analytics 2023, 2, FOR PEER REVIEW 14
Figure 4. Pre-schedule vs. model—forecast averages: patient load by time of day.
We modeled the expected peak PACU patient load given the established schedule for
each day, using our estimated procedure and recovery durations, and assuming on-time
starts. A surprising result was that in one-third of the days analyzed, the expected PACU
peak from the ‘Pre-schedule’ was higher than what was expected from our unconstrained
‘CP optimized’ schedule. On only 40% of the days did the minimized patient flow time of
the unconstrained CP-optimized schedule impose an increase in the expected peak PACU
load beyond what it was already with the pre-schedule. Yet in those cases, the CP model
could reduce expected patient flow time by an average of 15%. Overall, our model
achieved an average 11% reduction in patient flow time at PACU load peaks that were
equal to or less than those expected by following the pre-schedule, similar to the particular
case depicted on the right side of Figure 5.
Figure 5. Efficient frontier: average patient flow time vs. peak PACU patient load.
Figure 5 displays two scaerplots of average patient flow times versus peak PACU
patient loads as predicted by our model. We show averages rather than the total flow time
objective to normalize days with different numbers of patients. A comparison is made
between pre-schedules as established by hospital surgeons and CP-optimized schedules
as proposed by our model. On the right side of Figure 5 is an example of a single day for
which the lowest average patient flow time of 260 min is expected from the unconstrained,
‘CP Optimized’ schedule (versus 310 min with the ‘Pre-Schedule’.) However, that sched-
ule is expected to result in a peak PACU load of 18 patients at some point in the day. In
Figure 5. Efficient frontier: average patient flow time vs. peak PACU patient load.
Figure 5displays two scatterplots of average patient flow times versus peak PACU
patient loads as predicted by our model. We show averages rather than the total flow time
objective to normalize days with different numbers of patients. A comparison is made
between pre-schedules as established by hospital surgeons and CP-optimized schedules
as proposed by our model. On the right side of Figure 5is an example of a single day for
which the lowest average patient flow time of 260 min is expected from the unconstrained,
‘CP Optimized’ schedule (versus 310 min with the ‘Pre-Schedule’.) However, that schedule
is expected to result in a peak PACU load of 18 patients at some point in the day. In contrast,
the pre-schedule is expected to result in an average patient flow time nearly 20% higher
(worse), but with a lesser (better) peak load of 14 patients in the PACU. Solid circles in
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Figure 5represent a series of intermediate schedule solutions that constitute the frontier
of lowest possible patient flow times and PACU peak loads. We obtained these through
successively lowering
K
in the
pU <K
constraint (10), as described in the Section 2above.
We see in Figure 5that a load equal to the pre-schedule expectation (14 in this example)
is achieved with little increase in the average patient flow time (only 10 min higher than
for the unconstrained optimized solution.) Furthermore, a one-third reduction in peak
PACU patient load is apparently achievable with a resulting average patient flow time that
is equal to that expected with the pre-schedule. A much greater (~45%) reduction in PACU
peak load appears to be achievable with only a modest increase in patient flow time. The
left side of Figure 5shows a similar trend over our sample of testing days.
We conducted an ex post analysis using actual procedure and recovery durations in
both the pre- and CP-optimized schedules. Figure 6below shows two paired charts of
PACU loads for a typical surgery day. The left side refers to the pre-schedule, and the right
side refers to the CP-optimized schedule. On both charts, a solid line depicts the expected
count of PACU patients as predicted by our model. The dashed line on the left depicts
actual events (considering only scheduled patients) in the PACU that day. The dashed line
on the right is slightly different, as we can only simulate what would have happened if the
CP-optimized schedule had been adopted—using actual durations for both procedure and
recovery, but with different timing.
Analytics 2023, 2, FOR PEER REVIEW 15
contrast, the pre-schedule is expected to result in an average patient flow time nearly 20%
higher (worse), but with a lesser (beer) peak load of 14 patients in the PACU. Solid circles
in Figure 5 represent a series of intermediate schedule solutions that constitute the frontier
of lowest possible patient flow times and PACU peak loads. We obtained these through
successively lowering 𝐾 in the 𝑝𝑈<𝐾 constraint (10), as described in the Model Devel-
opment section above.
We see in Figure 5 that a load equal to the pre-schedule expectation (14 in this exam-
ple) is achieved with lile increase in the average patient flow time (only 10 min higher
than for the unconstrained optimized solution.) Furthermore, a one-third reduction in
peak PACU patient load is apparently achievable with a resulting average patient flow
time that is equal to that expected with the pre-schedule. A much greater (~45%) reduction
in PACU peak load appears to be achievable with only a modest increase in patient flow
time. The left side of Figure 5 shows a similar trend over our sample of testing days.
We conducted an ex post analysis using actual procedure and recovery durations in
both the pre- and CP-optimized schedules. Figure 6 below shows two paired charts of
PACU loads for a typical surgery day. The left side refers to the pre-schedule, and the
right side refers to the CP-optimized schedule. On both charts, a solid line depicts the
expected count of PACU patients as predicted by our model. The dashed line on the left
depicts actual events (considering only scheduled patients) in the PACU that day. The
dashed line on the right is slightly different, as we can only simulate what would have
happened if the CP-optimized schedule had been adopted—using actual durations for
both procedure and recovery, but with different timing.
Figure 6. Efficient frontier: average patient flow time vs. peak PACU patient load.
3.2. Flow and Occupancy Visualization Model
Unique durations were previously observable in the patient flow timelines of Figure
2. However, the nine discrete examples of flow delay shown in that figure represent fewer
than half of the possible permutations when also considering actual versus scheduled pro-
cedure start times. (These features are included in the visualization model provided to the
hospital, but not in Figures 2 or 7, due to the complexity of depicting these additional
elements in gray-scale.) As one example, we found several instances of OR delays in mov-
ing patients into the PACU that were arguably a result of the procedure starting earlier
than scheduled, since these delays would not have occurred and would not have been
recorded had the surgery started at the scheduled (later) time. We mention the laer point
to emphasize that the assignment of responsibility for delays is not always straightfor-
ward, and that best outcomes are achieved when parties commit to the specific timing of
their activities [25].
The aim of the patient flow visualization is to provide a systematic view of the pro-
cess in spite of the uniqueness of a great many cases across many ORs. Effective manage-
ment requires a clear understanding of both ‘the forest and the trees’, and appropriate
aggregated levels between these extremes. The top of Figure 7 shows timelines for 16
Figure 6. Efficient frontier: average patient flow time vs. peak PACU patient load.
3.2. Flow and Occupancy Visualization Model
Unique durations were previously observable in the patient flow timelines of Figure 2.
However, the nine discrete examples of flow delay shown in that figure represent fewer
than half of the possible permutations when also considering actual versus scheduled
procedure start times. (These features are included in the visualization model provided
to the hospital, but not in Figure 2or Figure 7, due to the complexity of depicting these
additional elements in gray-scale.) As one example, we found several instances of OR
delays in moving patients into the PACU that were arguably a result of the procedure
starting earlier than scheduled, since these delays would not have occurred and would
not have been recorded had the surgery started at the scheduled (later) time. We mention
the latter point to emphasize that the assignment of responsibility for delays is not always
straightforward, and that best outcomes are achieved when parties commit to the specific
timing of their activities [25].
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Analytics 2023, 2, FOR PEER REVIEW 16
patients passing through the first six PACU bays, shown at the boom. They represent
roughly a quarter of scheduled procedures on a sample day. Through cross-referencing
the two views, it is straightforward to trace delays in the PACU to a patient’s destination
ward (or PACU2) and to match OR delays with concurrent activity in the PACU.
Patient View
—
with Originating OR and Destination PACU Bay
Bay View
—
with Patient Occupancies and Originating OR
Figure 7. Patient flow visualization: patient-OR view and PACU-bay view.
The final six cases shown in the top of Figure 7 are examples of ODS patients who
did not pass through the PACU, but rather had only minor recovery in PACU2. It hap-
pened on this sample day that OR 16′s caseload was a series of similar orthopedic proce-
dures (a RASTOR room as described earlier), all performed under a regional anesthetic
rather than a general one and therefore not requiring PACU recovery but PACU2 only.
This example reinforces the value of characterizing ORs by their case loads to understand
the impact of these loads on the PACU, i.e., the central stage of the system.
Whereas some OR blocks contain homogenous, predictable procedure lengths and
flow paerns, others involve procedures that vary greatly in duration and patient type.
These provide more opportunities to affect a different PACU load via re-sequencing cases,
whether using something like our CP optimization model or heuristics such as the longest
or shortest durations ‘first’ and/or SDA patients and oldest ones ‘last’. The incremental
PACU impacts may vary markedly depending on the portion of patients receiving a re-
gional versus general anesthetic (although, when both are practical, the choice is subject
to patient agreement, and that is not always predictable.)
3.3. Other PACU Considerations
Although some perioperative units have explored the practice of pre-assigning
PACU bays [26] prior to patient arrivals, this was not under consideration by the hospital
Figure 7. Patient flow visualization: patient-OR view and PACU-bay view.
The aim of the patient flow visualization is to provide a systematic view of the process
in spite of the uniqueness of a great many cases across many ORs. Effective management
requires a clear understanding of both ‘the forest and the trees’, and appropriate aggregated
levels between these extremes. The top of Figure 7shows timelines for 16 patients passing
through the first six PACU bays, shown at the bottom. They represent roughly a quarter
of scheduled procedures on a sample day. Through cross-referencing the two views, it is
straightforward to trace delays in the PACU to a patient’s destination ward (or PACU2)
and to match OR delays with concurrent activity in the PACU.
The final six cases shown in the top of Figure 7are examples of ODS patients who did
not pass through the PACU, but rather had only minor recovery in PACU2. It happened
on this sample day that OR 16
0
s caseload was a series of similar orthopedic procedures (a
RASTOR room as described earlier), all performed under a regional anesthetic rather than
a general one and therefore not requiring PACU recovery but PACU2 only. This example
reinforces the value of characterizing ORs by their case loads to understand the impact of
these loads on the PACU, i.e., the central stage of the system.
Whereas some OR blocks contain homogenous, predictable procedure lengths and
flow patterns, others involve procedures that vary greatly in duration and patient type.
These provide more opportunities to affect a different PACU load via re-sequencing cases,
whether using something like our CP optimization model or heuristics such as the longest or
shortest durations ‘first’ and/or SDA patients and oldest ones ‘last’. The incremental PACU
impacts may vary markedly depending on the portion of patients receiving a regional
versus general anesthetic (although, when both are practical, the choice is subject to patient
agreement, and that is not always predictable.)
3.3. Other PACU Considerations
Although some perioperative units have explored the practice of pre-assigning PACU
bays [
26
] prior to patient arrivals, this was not under consideration by the hospital at the
time of our study. Management felt strongly that bay assignments were best allocated
Analytics 2023,2672
dynamically based on actual arrivals, especially considering the unpredictability of either
arrival times or recovery lengths. Moreover, certain rules applicable to PACU nursing
operations would need to be incorporated in any assignment algorithm. PACU bays were
opened in pairs, with a single nurse assigned to both bays, for example, 07 and 08. A nurse
should not receive two new patients within a span of 30 min. Also, a nurse should not have
two SDA or IP patients at once, although one of either in addition to an ODS patient was
acceptable (given that the latter were typically less complicated patients). Some pairs of
PACU bays were reserved for pediatric patients, with very young patients requiring a fully
dedicated nurse, sometimes with an additional nurse’s assistance.
Nevertheless, the PACU ‘Bay View’ of the patient flow visualization reveals important
facts about the efficiency with which bays are utilized. For example, in the bottom of
Figure 7, the last patient assigned to PACU bay 05 and the last two patients in bay 06 could
have all been accommodated in bay 01, thereby allowing the third pair of bays (05 and 06)
to be closed seven hours earlier through keeping the first pair of bays (01 and 02) open only
two hours later. The four instances of extended wait for ward beds in PACU bays 02 and
04 should also warrant investigation, as the early vascular surgery SDA patient in OR 18
might better have been scheduled later in the day.
3.4. Discussion
The perioperative process includes a high degree of uncertainty and substantial in-
herent variety; consequently, there is no magic solution to ensure the smoothest and most
efficient patient flow, particularly through the centralized PACU stage. However, despite
the uncertainties, we believe much of the case variety that directly impacts timing and
flow can be captured in modeling and optimization techniques that are reasonably ac-
cessible to practitioners and use operational data that is already being used in heuristic
scheduling approaches.
Our choice of patient flow time as the performance metric (objective) for our CP opti-
mization model may be challenged on the notion that patients who are waiting months for
surgery might not care about how long a procedure takes on the day of surgery. However,
this is a commonly applied metric in static scheduling problems when it is assumed, as
in our case, that all jobs arrive simultaneously at the beginning of the scheduling period.
Furthermore, through choosing the minimization of the total patient flow time as the objec-
tive, we are at the same time maximizing global OR utilization, i.e., through completing
the given set of procedures within the least overall OR time.
Whether individual surgeons who control their own OR schedules can be convinced
of the benefits of a more collective scheduling approach remains to be seen. Autonomy
is a difficult thing to forego without a clear understanding of, and or certainty about, the
personal impact of allowing someone else to decide the order in which patients should be
treated. Surgeons have individual preferences, such as whether two similar procedures
should be sequenced contiguously or not, and whether longer procedures are better per-
formed earlier or later in the day. As a modeling platform, CP lends itself to the addition of
constraints incrementally and with relative ease, as these specific surgeon needs arise.
The models we developed enable, within roughly one hour, the translation of schedules
(in PDF, portable document format) into data input files for the CP model, the running
of the model, and the output of a proposed alternate CP-optimized schedule for review,
including, for each procedure, the direction and length of time of any proposed change in
scheduled start time. In some cases, only a handful of changes in a few ORs are enough
to reduce a projected peak in the PACU load by one or two patients. It was left to the
perioperative management team to negotiate sequence changes with surgeons where they
found them most compelling. A dominant factor in these decisions was whether a sequence
change supported the precedence of patient types (young people will be treated earlier,
and IP patients precede SDA patients) without blocking upstream patients from entering
the PACU. It is for this reason that we provided the option of incorporating a precedence
weighting factor to individual patient flow times in the alternate objective function, which
Analytics 2023,2673
we note can be easily tuned according to the relative importance of age versus flow, and
other factors can also be added. The aim was to find a balance between the CP model
recommendations based mainly on procedure times with qualitative factors of importance
in patient sequencing.
The rapid translation of a pre-schedule to a CP-optimized schedule enabled man-
agement to spot improvement opportunities more quickly (moreover, to justify them as
having been proposed objectively), as opposed to managers having to scan lists of five
dozen procedures to identify these opportunities. Similarly, on the day following surgery,
as actual time-stamp data became available for extraction, summary reports including
patient flow visualizations like Figure 7were constructed. These reports facilitated the
timely and efficient reflection on the events of the previous day and allowed managers to
identify and communicate where and when problems occurred, decisions that turned out
to be positively or negatively impactful, and to assess the accuracy with which estimates
had been made. The patient flow visualizations provided to the hospital include identifiers
within the timelines that depict the model’s predictions of procedure times and durations
(and where absent, indicating unscheduled cases whose impact could be assessed.) In
many cases, the time predictions proved to be reasonably accurate, even when shifted in
time due to other events. A rationale was sought in cases where time predictions proved to
be inaccurate, including questions regarding how the predictive models could be improved.
We note that aside from the CP model, which was developed in and for use with IBM ILOG
CP Optimizer, other model components were developed in Excel VBA (Visual Basic for
Applications), which was readily available to hospital management and staff and thereby
facilitated implementation.
3.5. Opportunities for Further Research
We are not aware of any previous attempts to model the problem of assigning PACU
bays while respecting constraints such as one nurse per two PACU bays, described in
the previous section. We believe that these factors could be incorporated in an extended
CP model, although doing so would amplify complexity without necessarily bolstering
confidence in the accuracy of recovery time predictions. If such a model were to be dynamic
and applicable in or near ‘real-time’ using updated data to plan PACU bay assignments
forward in the day (subject to change), that could certainly enable higher occupation and
patient flow, and thus the possibility of running multiple RASTOR rooms on some days,
and thereby address the surgery backlog and wait-time problems in the system.
One of the challenges in developing solutions to perioperative patient flow problems
is they are often setting-specific. Our partner hospital chooses to schedule procedure start
times at :00 and :30, whereas another one can work in five-minute intervals. The hospital
handles a lot of orthopedic, obstetric-gynecological, general, and pediatric procedures,
whereas its other location handles a higher portion of neurology and cardiology patients.
The compatibility of such different surgery block cases warrants exploration, including an
assessment of the value of simple heuristics to the benefit of the system as a whole.
In this regard, our CP model can enable the simple translation of any hospital’s OR
daily block schedule into a time-series prediction of PACU load across the day (a) as a basis
for more sophisticated analysis of the PACU stage itself or (b) supporting any multi-OR
scheduling study via assessing its downstream effect on the PACU.
Further research should aim to quantify the impacts of uncertain procedure durations
and stochastic disruptions [
27
] in determining an optimal plan. Various CP solvers can
be compared for performance, and decomposition methods [
28
,
29
] can be developed and
compared. Open-source CP options should be investigated to lower financial barriers for
hospitals seeking to implement and tune these scheduling technologies.
4. Conclusions
In this paper, we have extended the view of the perioperative process to distinguish
between patients according to a variety of paths they follow, as in-patients, one-day surg-
Analytics 2023,2674
eries, and same-day admissions. We have also introduced a constraint program to develop
coordinated OR schedules aimed at minimizing peak patient loads in the PACU, to better
ensure that OR delays will not be incurred due to the PACU reaching full capacity and
being unable to accept new patients from the ORs.
The model required a mechanism to predict procedure and recovery times, and these
are open to further refinement through improved data collection and methods such as
machine learning. We have developed and described a model for patient flow visualization
which can help perioperative care managers and surgeons to quickly locate problems in
the time and stage of the process, to better understand the interactive effects of schedule
sequence decisions, to propose additional practical factors that can be incorporated in the
optimization model, and to provide a simple and common foundation for the ongoing
review and refinement of OR scheduling practices.
Our model has contributed to organizational learning and improved communication
and cooperation among the many parties involved in scheduling the ORs at the hospital.
In a letter supporting our research, the management team wrote: “The Ivey research team
has provided us important insight to the possibilities and significant potential benefits
of adopting a more methodical and coordinated approach to daily OR scheduling. We
look forward to the next phase of development in what we believe can be a valuable and
implementable tool in streamlining the costs and timeliness of surgical operations, not
only within ‘our hospital’ but throughout the health care system, if successful. In the
meantime, we strongly support not only the research team’s continued work on our behalf,
but also the sharing of their methods and conclusions (ongoing as they are) with members
of operational research community focused on similar health care challenges.”
Author Contributions:
Conceptualization, J.S.F.L. and M.A.B.; methodology, J.S.F.L., M.A.B. and
P.C.B.; software, J.S.F.L.; validation, J.S.F.L. and M.A.B.; formal analysis, J.S.F.L., M.A.B. and P.C.B.;
data curation, J.S.F.L.; writing—original draft preparation, J.S.F.L.; writing—review and editing,
J.S.F.L., M.A.B. and P.C.B.; visualization, J.S.F.L.; supervision, M.A.B. and P.C.B.; project administra-
tion, M.A.B. and P.C.B. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement:
Ethical review and approval were not required for this study
due to the absence of any medical or individually identifiable data of any kind in the study.
Informed Consent Statement: Not applicable.
Data Availability Statement:
Upon request, data de-identified to a level suitable for public release
may be provided after approval from the hospital.
Acknowledgments:
The authors would like to thank the hospital for the support and collaboration
in this research project.
Conflicts of Interest: The authors declare no conflict of interest.
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