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Strategic allocation of limited operating room (OR) capacity to surgeons is crucial for the coordination of surgical work flow, including planning of consultation and surgery days, and staff assignment to perioperative teams. However, it is a challenging problem in practice, since the capacity allocation needs to be cyclic for schedule predictability and surgical team coordination, and also needs to satisfy surgeons’ preferences. It is further complicated by the practice of surgeons sharing ORs. In this study, we propose a mathematical optimization model to coordinate capacity allocation among surgeons in order to improve the utilization of surgical capacity. We introduce the concept of capacity allocation patterns to account for schedule cyclicity and surgeons’ preferences. Further, we develop a data-driven approach to coordinate OR sharing among surgeons based on their historical OR usage. The proposed methodology is applied to a case study with data from a surgical division at Mayo Clinic. Compared with the state-of-the-practice, the proposed approach shows a substantial potential in reducing the maximum number of ORs allocated daily to the division with little overtime. With a solution time of less than 0.5 s, the proposed methodology can be readily used as a decision support tool in surgical practice.
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Pattern-based Strategic Surgical Capacity Allocation I
Miao Baia, Kalyan S. Pasupathya, Mustafa Y. Sira,
aDepartment of Health Sciences Research, Mayo Clinic, 200 1st Street SW, Rochester,
Minnesota 55905
Abstract
Strategic allocation of limited operating room (OR) capacity to surgeons is cru-
cial for the coordination of surgical work flow, including planning of consultation
and surgery days, and staff assignment to perioperative teams. However, it is a
challenging problem in practice, since the capacity allocation needs to be cyclic
for schedule predictability and surgical team coordination, and also needs to sat-
isfy surgeons’ preferences. It is further complicated by the practice of surgeons
sharing ORs.
In this study, we propose a mathematical optimization model to coordinate
capacity allocation among surgeons in order to improve the utilization of surgical
capacity. We introduce the concept of capacity allocation patterns to account for
schedule cyclicity and surgeons’ preferences. Further, we develop a data-driven
approach to coordinate OR sharing among surgeons based on their historical
OR usage.
The proposed methodology is applied to a case study with data from a
surgical division at Mayo Clinic. Compared with the state-of-the-practice, the
proposed approach shows a substantial potential in reducing the maximum num-
ber of ORs allocated daily to the division with little overtime. With a solution
time of less than 0.5 second, the proposed methodology can be readily used as
a decision support tool in surgical practice.
Keywords: Surgery Capacity Allocation; Pattern-based; Data-driven;
Mathematical Optimization
IThis study meets exemption criteria established by the Mayo Clinic Institutional Review
Board.
Corresponding author
Email address: sir.mustafa@mayo.edu (Mustafa Y. Sir)
Preprint submitted to Journal of Biomedical Informatics March 29, 2019
1. Introduction
Operating rooms (ORs) are among the most capital-intensive resources in
a hospital [1]. As a consequence of a large aging population and payment-
related regulations, hospital administrators are under growing pressure to seek
more sophisticated OR management approaches to reduce costs and improve5
performance [2, 3].
Strategic surgical capacity allocation is a critical OR management problem,
which involves determining the number of ORs to staff each day and allocating
OR times to surgical groups or individual surgeons [4]. The following consider-
ations complicate the problem:10
Surgical capacity allocation should be cyclic (e.g., it has a biweekly re-
peating pattern) and with high predictability [5], which is essential to the
coordination of surgical workflow. With an acyclic allocation, a surgeon
could perform surgeries on varying weekdays and consequently have dif-
ferent weekly consultation hours. This could also result in inconsistent15
surgical and medical teams assigned to a surgeon, which may adversely
impact surgical duration and patient safety [6–8].
It is common practice for two or more surgeons share an OR (e.g., morning
and afternoon sessions). This further complicates the surgical capacity
allocation, in that mismatch between surgeons’ usage of a shared OR20
could result in under- or over-utilization of surgical capacity. For instance,
if surgeons who share an OR all have high surgical demand, a significant
OR overtime could be incurred.
Surgical demand is highly seasonal [9–11] requiring OR managers to pe-
riodically revise capacity allocation to adopt fluctuations in the surgical25
demand and balance workload among surgeons. Therefore, the surgical
capacity allocation method should be computationally tractable and data-
driven.
In this study, we propose a mathematical optimization model to coordinate
capacity allocation patterns among surgeons in order to improve the utilization30
of surgical capacity. We introduce the concept of capacity allocation patterns to
account for schedule cyclicity and surgeons’ preferences. Further, we develop a
data-driven approach to coordinate OR sharing among surgeons based on their
historical OR usage. The proposed methodology is applied to a case study with
data from a surgical division at Mayo Clinic.35
2
The rest of the paper is organized as follows. Background of surgical capacity
allocation and relevant literature are discussed in Section 2. Our methodology
and its application to a surgical division at Mayo Clinic are presented in Section
3. We discuss our findings in Section 5 based on numerical results in Section 4.
Section 6 concludes this study.40
2. Background and Relevant Literature
Strategic surgical capacity allocation arises from the widely-implemented
block scheduling system where surgeons are assigned to blocks of uninterrupted
OR time [4]. With a planning horizon of several months to a year, surgical
capacity allocation is periodically re-evaluated when surgical capacity or demand45
changes [4, 12]. The models for making strategic capacity allocation decisions
in the literature typically do not involve tactical and operational decisions, such
as when to release unused block time, how to coordinate surgeons’ OR time on
a day and how to schedule specific surgeries (c.f. [13–15]) are not considered.
These decisions will also be considered as out of scope in our study.50
Strategic decisions in surgical capacity allocation include the total amount
of OR time allocated for each surgeon and the exact days on which each surgeon
is allocated OR time. Some previous studies only consider the former. For ex-
ample, Gupta [4] and Dexter et al. [16] studied the distribution of newly-added
surgical capacity among surgeons to maximize the overall financial benefits.55
Beli¨en and Demeulemeester [17] and Hosseini and Taaffe [18] optimized the
allocation of OR time to minimize bed shortage and to maximize surgical ca-
pacity utilization, respectively. Our study offers a more practical approach by
integrating both decisions.
Surgical capacity allocation should be cyclic and should account for surgeons’60
preferences as much as possible to achieve a desired level of work-life balance [5].
These requirements have been mostly addressed by the construction of a master
surgical schedule (MSS) in previous studies [5, 11, 19–21]. MSS is a short-term
(e.g. 2-week) surgical capacity allocation plan that is repeatedly implemented to
preserve allocation cyclicity. MSS construction is a special case of the strategic65
capacity allocation problem we study in this paper. While all patterns share
the same cycle length in MSS construction, our methodology accommodates
distinct cycle length for different surgeons, which improves allocation flexibility.
Furthermore, we explicitly consider surgeons’ preferences, which may not be
easily modeled and have not been addressed in MSS studies.70
3
Many surgical practices let surgeons share an OR on a given day in order to
ensure efficient use of OR capacity. Some studies on surgical capacity allocation
have addressed this practice: Denton and Miller [22] developed a model to
coordinate blocks of surgeries among ORs on a specific day where OR sharing
is allowed. Differently from our study, they studied decisions that are made75
after surgeons have booked their surgeries according to the strategic capacity
allocation. Day et al. [23] proposed to designate some OR time to be shared by a
selected set of surgeons. However, as the authors discuss, the proposed strategy
may create schedules in which surgeons OR time is divided into multiple shorter
intervals, which are undesirable in practice. Moreover, cyclicity is not addressed80
in their proposed method.
In summary, we investigate strategic surgical capacity allocation when ac-
counting for cyclicity, surgeons’ preferences and the practice of OR sharing. Our
methodological framework can help OR operations managers to determine an
surgical capacity allocation with high predictability, which improves OR uti-85
lization and patient safety [6–8] and facilitates more consistent scheduling of
perioperative staff [5].
3. Material and Methods
Our methodology relies on the concept of surgical capacity allocation pat-
terns, which enable us to preserve the cyclicity of a surgical schedule and to90
satisfy surgeons’ preferences. An allocation pattern is defined as a set of pre-
determined surgery days in the planning horizon that are selected by a surgeon
after negotiation with administrators.
Surgical capacity allocation patterns can handle very specific preferences
and different cycle length. For instance, a surgeon may request to work either95
only on Mondays and Tuesdays or only on Wednesdays and Thursdays. This
request can be modeled as two allocation patterns as shown in Figures 1 and 2.
One of these allocation patterns would eventually be selected for this surgeon
in the subsequent capacity allocation optimization. In addition, patterns could
have different cycle time, which further increases the flexibility of capacity al-100
location. For instance, Figures 1 and 2 are allocation patterns with biweekly
and triweekly cycles, respectively. This is particularly helpful in practice where
surgical demands of different surgeons demonstrate distinct cycle lengths. Note
that surgeons do not have to work on every day in the assigned allocation pat-
tern. For example, if a surgeon is assigned to the pattern in Table 1, he/she may105
4
MTuWThF
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Figure 1: A biweekly pattern
MTuWThF
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Figure 2: A triweekly pattern
not have any OR time assigned on some Mondays or Tuesdays if he/she does
have surgical demand. Assigning a surgeon to an allocation pattern represents a
strategic decision, which can be adjusted as part of daily operational decisions.
Surgical capacity allocation pattern is inspired by the shift sequences in the
nurse rostering problem in which a nurse roster is constructed by assigning110
nursing teams to different preselected shift sequences [24–27]. In contrast, in
our study, the consideration of each surgeon’s preference along with the practice
of allowing surgeons share an OR significantly complicates the problem.
The rest of the section is organized as follows. We present a data-driven ap-
proach to coordinate OR sharing among surgeons in Section 3.1. In Section 3.2,115
we propose a bi-objective integer programming model to optimize the surgical
capacity allocation. The proposed model is customized for a surgical division
at Mayo Clinic in Section 3.3.
3.1. Data-driven Coordination of OR Sharing
Because of the variability in surgical demand and the uncertainty in surgical120
duration, OR sharing may result in costly overtime. Intuitively, one way to
address this problem is to match surgeons with high shared-OR utilization with
those with low utilization. For example, suppose that surgeon A, B and C
typically use 3.5, 4.5, and 4.5 hours of OR time, respectively, when they share
an OR with others in the past. It could be better to pair surgeon A with B125
or A with C, rather than to pair surgeon B with C which may have a higher
chance of overtime. Following this idea, we propose a data-driven surgeon-
pairing approach based on historical surgical scheduling data to limit overtime
in OR sharing.
In this study, OR sharing is limited to at most two surgeons sharing an OR130
on given day and each surgeon works in a single uninterrupted block of time.
This is preferred in practice for two reasons. First, a change between surgeons
may incur a certain amount of setup time and cost [17]. Second, the surgeon
5
who works earlier may run late leading to delays in and cancellations of the
subsequent surgeries. Therefore instead of assigning multiple surgeons in an135
OR, it is preferable to assign an OR to a surgeon for the whole day (referred to
as full-day block) or to split the OR time between two surgeons (referred to as
shared-OR blocks) [17, 20].
We define some parameters to explain the proposed data-driven approach.
Sis the set of surgeons; indexed by s.140
κis a predetermined regular OR operational hours (e.g. 10 hours from
7am to 5pm). Any extra surgery time beyond κis considered as overtime.
Tsis a random variable representing the hours surgeon sSuses his/her
assigned shared-OR blocks.
T(s1,s2)is a random variable representing the total hours used by surgeon145
s1Sand s2Swhen they share an OR.
Ω(Ts) is the set of all realizations of Tsobserved in historical data; Tω
s
corresponds to an observation.
βis a selected threshold on the probability that total surgery hours in two
consecutive shared-OR blocks in a particular OR does not exceed κ.150
γis a predetermined acceptable amount of expected overtime in OR.
We measure whether a pair of surgeons can share ORs based on two criteria.
First, surgeons can share ORs only when the probability of overtime Pr(Ts1+
Ts2κ) is below the selected threshold 1 β. The empirical cumulative
distribution function b
Fof (Ts1+Ts2) can be derived based on the historical155
usage of shared-OR blocks, Ω(Ts1) and Ω(Ts2). An estimate Pr(Ts1+Ts2κ)
can be derived based on b
Fas defined in Equation (1).Note that this estimation
is based on the assumption that Ts1and Ts2are independent, which is true in
our case study in Section 3.3.
Ω(T(s1,s2)) = (Tω1
s1, T ω2
s2)
Tω1
s1Ω(Ts1), T ω2
s2Ω(Ts2)
b
FΩ(T(s1,s2))(t) = P(Tω1
s1,T ω2
s2)Ω(T(s1,s2))1(Tω1
s1+Tω2
s2t)
Ω(T(s1,s2))
Pr(Ts1+Ts2κ) = b
FΩ(T(s1,s2))(κ) (1)
6
where 1() is an indicator function taking the value of 1 when the statement in160
parenthesis is true; 0 otherwise.
Second, we ensure that the expected overtime E(max(Ts1+Ts2κ, 0)) < γ.
The expected overtime can be calculated as follows:
E(max(Ts1+Ts2κ, 0)) = P(Ts1,Ts2)Ω(T(s1,s2))max(Ts1+Ts2κ, 0)
Ω(T(s1,s2))
(2)
Note that self-pairing {(s, s),sS}is also included in the set of feasible
pairs. This could correspond to an idle shared-OR block if overall surgical165
workload is less than the allocated OR capacity. This could also occur if a
surgeon frequently runs late in his/her shared-OR block and no other surgeon
is a compatible match.
Based on the two criteria, a set of compatible surgeon pairs can be defined
as
A={(s, s)|∀sS}[(s1, s2)|Pr(Ts1+Ts2κ)β,
E(max(Ts1+Ts2κ, 0)) < γ, s1, s2S:s16=s2(3)
The choice of βand γdepends on a hospital’s tolerance on overtime. In the
extreme case, β= 100% (γ= 0) indicates that two surgeons can share an OR170
only if overtime occurs with 0 probability (if the expected overtime is 0). Note
that these could result in no OR sharing if shared-OR blocks frequently run late
in the past. In Section 4.1, we illustrate the impact of βand γvalue in our case
study.
Note that compatible surgeon pairs are later used as inputs to our surgical175
capacity allocation model. They are selected beforehand based on criteria of
interests to specific surgical practice. We chose two criteria related to overtime
because excessive overtime negatively impacts patient safety and also staff well-
beings [28, 29]. Based on our discussion with practitioners, both the frequency
and the amount of overtime were thresholded.180
3.2. Strategic Surgical Capacity Allocation Optimization
We formulate an integer programming model that simultaneously minimizes
the maximum number of ORs allocated daily to a surgical division and the total
number of ORs used in a surgical planning period. Three sets of decisions are
7
made: 1) how to assign surgical capacity allocation patterns to surgeons; 2) how185
to coordinate surgeons’ OR blocks; and 3) how many ORs to open on each day.
3.2.1. Sets and Parameters
Sis the set of surgeons; indexed by s.
Psis the set of capacity allocation patterns that surgeon sScan be
assigned to; indexed by p.190
Dis the set of surgery days in the planning horizon, i.e., the implementa-
tion period of the surgical capacity allocation until the next re-allocation;
indexed with d.
Ais the set of surgeon pairs who can share an OR in a day, A={(s1, s2)|s1s2;s1, s2S}.
Note that self-pairing is also included in Awhere a surgeon uses a sec-195
ondary OR for the whole day.
Asis the set of surgeon pairs that surgeon sbelongs to, where As=
{(s1, s2)A|s {s1, s2}}
NF D
s, N HD
sare the number of full-day blocks and shared-OR blocks that
are currently assigned to surgeon sS200
cpd = 1 if allocation pattern pPcovers day dD; 0 otherwise
αis the weight on the total number of ORs used in the planning horizon,
0α1
NOR is the surgical capacity, i.e., the total number of available ORs
Note that our proposed methodology does not determine the number of full-205
day (NF D
s) or the number of half-day blocks (NHD
s) that are assigned to each
surgeon in the planning horizon. Instead, NF D
sand NHD
sare assumed to have
been determined by surgical suite administrators according to the specifics of
their surgical practice.
3.2.2. Decision Variables210
xsp:xsp = 1 if surgeon sis assigned to allocation pattern p; 0 otherwise
ysd:ysd = 1 if surgeon sis assigned a full-day block on day dD; 0
otherwise
8
zss0d:zss0d= 1 if surgeons sand s0((s, s0)A) share an OR on day
dD; 0 otherwise215
wdis the number of ORs allocated to the surgical division on day dD.
wis the maximum number of ORs allocated to the surgical division on
any day dD.
3.2.3. Optimization Model
min αX
dD
wd+ (1 α)w(4)
s.t. X
pPs
xsp = 1 sS(5)
X
pPs
cpdxsp ysd sS, d D(6)
X
pPs
cpdxsp X
(s,s0)As
zss0dsS, d D(7)
X
dD
ysd =NF D
ssS(8)
X
dD
X
(s,s0)As
zss0d=NHD
ssS(9)
X
sS
ysd +X
(s,s0)A
zss0dwddD(10)
wdwdD(11)
wNOR (12)
ysd +X
(s,s0)As
zss0d1sS, d D(13)
The objective function in (4) optimizes the utilization of surgical capacity in220
two ways. First, it minimizes the total number of ORs used over the planning
horizon, which could release redundant capacity for other usage. Second, it
minimizes the maximum number of ORs allocated daily to the surgical division
to maintain the number of allocated OR to a steady level. This enables OR
managers to balance surgical load over the planning horizon and thus balance225
the workload of medical teams. Note that if the second term in the objective
is excluded, an extremely-unbalanced capacity allocation (e.g., 14 open ORs
on the first day and 0 ORs on the second) could be deemed optimal. Ideally,
9
a capacity allocation should achieve the minimum in both components of the
objective function. However, in case that these components are conflicting, a230
weight parameter α[0,1] is included to represent the relative importance of
each component.
Constraint set (5) ensures that a surgeon is assigned to exactly one allocation
pattern. Constraint sets (6) and (7) require that surgeons perform surgeries,
either in full-day blocks or shared-OR blocks, only on the days covered by235
their assigned allocation patterns. Note that a surgeon can be assigned to an
allocation pattern that includes day d, but he/she may be not assigned any
blocks on day das discussed earlier in Section 3.
Constraint sets (8) and (9) ensures that the number of full-day and shared-
OR blocks assigned to each surgeon remain unchanged from the current practice.240
These constraints are of great significance in practice. Block assignment, includ-
ing the number of blocks and size of blocks allocated to surgeons, is made based
on demand in the past and on factors such as surgeons’ seniority and financial
contribution. It is approved by the surgical leadership after many iterations of
negotiations and revisions [20] and is implemented until a re-evaluation [4, 12].245
It is generally challenging to change the block assignment that has been ap-
proved [18]. Therefore we coordinate the allocation of surgeons’ blocks without
modifying the number or the size of assigned blocks.
Constraint set (10) calculates the number of ORs allocated on each day and
Constraint set (11) computes the maximum number of ORs allocated daily to250
the surgical division in the planning horizon. Constraint set (12) ensures that
OR usage does not exceed the number of ORs that are assigned to this surgical
division. Constraint set (13) ensures that a surgeon can only get either a full-day
or a shared-OR block on the same day.
Pattern construction for each surgeon can be automated by taking some basic255
inputs. Inputs include: 1) cyclicity of the allocation (see Section 3 and Figures 1
and 2. 2) days when a surgeon prefer to work. Note that cyclicity specifies how
often each pattern repeats itself in the planning horizon (e.g., biweekly patterns
repeat every two weeks in an 8-week planning horizon). Cyclicity could be made
the same for the whole surgical division or could be distinct across surgeons.260
For example, a surgeon prefers a biweekly pattern in which he/she works
1) only on the Thursday of the first week and on the Wednesday of the second
week 2) or only on the Friday of the first week and the Thursday of the second
week. His/her pattern in an 8-week planning horizon includes Figure 3 and 4.
As discussed, surgeons do not have to work on every surgery day in the assigned265
10
MTuWThF
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Figure 3: Pattern 1
MTuWThF
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Figure 4: Pattern 2
allocation pattern. If a surgeon is assigned to the pattern in Figure 3, he/she
may not have any OR time assigned on some Thursdays or Fridays if he/she
does not have surgical demand.
Our model could be directly applied to capacity management in a single sur-
gical division, or across multiple surgical divisions if surgical capacity is pooled270
and shared, since it is essentially equivalent to a large “centralized” division.
If surgical capacity is not shared among surgical divisions, they are typically
managed in a decentralized manner: each division is assigned a fixed number of
ORs and manages their own capacity. Constraint 12 in the optimization model
ensures that the number of ORs that are used on any day does not exceed the275
capacity assigned to the surgical division NOR.
Note that one can integrate OR sharing coordination in Section 3.1 into
the capacity allocation model in Section 3.2. As presented in Appendix D,
the resulting optimization model is a large-scale two-stage mixed integer pro-
gramming (MIP) model with a big-M formulation, which is difficult to solve.
Our proposed methodology is however responsive with a solution time of less
than 0.5 seconds, which can be iteratively revised and rerun as a decision
support tool in practice. Solving the integrated optimization model is an area
of future study.
3.3. Application to a Surgical Division at Mayo Clinic
We conduct a case study with data from a surgical division at Mayo Clinic
to evaluate the performance of our methodological framework.280
3.3.1. Background and Current Practice
As one of the largest surgical practices in the world, Mayo Clinic holds the
value that “the needs of the patient come first” [30] and designs a unique “blue-
11
Mon Tue Wed Thu Fri
Week 1
Week 2
Figure 5: “Blue-orange” system at Mayo Clinic
orange” system, illustrated in Figure 5, to provide timely access to surgery
patients. In this “blue-orange” system, a “blue” surgeon performs surgeries on285
“blue” days and hold clinic consultations on “orange” days, and vice versa. This
system dates back to the Mayo brothers who performed surgeries on alternate
days so that one of them was always available for clinic consultation [31]. Still
in use today, this system allows surgeons to balance their time between clinic
consultations and surgeries.290
On a surgery day, a surgeon is assigned a dedicated OR and possibly a
secondary dedicated OR if the case load is over the capacity of a single OR.
Two parallel ORs enable a surgeon to perform a surgery in one OR while the
other OR is being set up or cleaned. Although it is not the standard practice
in all medical institutions, parallel ORs have been adopted by multiple large295
hospitals to reduce surgeons’ waiting time [32–34].
3.3.2. Operational Issues Faced by the Surgical Division and Proposed Changes
The surgical capacity allocation policy at Mayo Clinic grants surgeons suffi-
cient surgical capacity and also a high level of autonomy. However, this policy is
suboptimal to the organization because of the consequent inefficient OR usage.300
First, a dedicated secondary OR is assigned no matter how much OR time is
needed in addition to the primary OR, which results in low OR utilization.We
find that the average utilization of the secondary OR time is around 40% for the
studied surgical division. 96.3% of the secondary OR usage (across 8 surgeons
over 4125 calendar days) involves only 1 or 2 surgeries. Second, the current305
surgical capacity allocation policy creates a significant variation in the number
of daily allocated ORs, as illustrated in Figure 6. This variability results in a
large OR blueprint, where each specialty is assigned a large number of reserved
ORs, and it leads to large fluctuations in daily workload assigned to the surgical
teams and the perioperative staff.310
For more efficient surgical capacity allocation, we propose two modifications
to the current policy. First, we suggest two surgeons to share a secondary OR by
assigning each surgeon a shared-OR block in the secondary OR. The data-driven
12
0
1
2
3
4
5
6
7
8
1 6 11 16 21 26 31 36
Number of Allocated ORs
Day
Figure 6: Number of allocated ORs in 38 consecutive surgery days
method proposed in Section 3.1 is used to identify compatible surgeon pairs who
can share secondary ORs. Second, we propose to optimize the surgical capacity315
allocation by considering both primary and secondary OR assignments. The
mathematical optimization methodology proposed in Section 3.2, with minor
modifications described in Appendix A, is used to determine the surgical ca-
pacity allocation. The modified optimization model adheres to the traditional
Mayo Clinic policy of that allocating a dedicated primary OR to each surgeon320
on their surgery days. We discuss the coordination of surgeons’ primary and
secondary OR usage in Section 5.
3.3.3. Data
We acquired the historical OR usage data over 4125 calendar days from the
surgical division, which had |S|= 8 surgeons and had access to NOR = 11 ORs325
during this period. The total number of surgeons in the surgical division varies
from 10 to 14 in the studied time range. This set of 8 surgeons perform surgeries
every year within the studied time range, and their surgeries contribute 54.7% to
82.9% of the yearly caseload. The data included information about all surgeries
performed by the division, including the date of surgery, surgeon, location (i.e.,330
the specific OR where the surgery was performed), and duration of surgery from
the time when a patient was wheeled in until he/she was wheeled out of the OR.
The planning horizon was set as 8 weeks with |Dc|= 56 calendar days as
being used by the studied surgical division; i.e., surgical capacity allocation is
planned for the next 8 weeks. Within 8-week time, there are in total |D|= 40335
surgery days. Note that weekend surgeries are rare in this surgical division and 6
13
public holidays that are observed at Mayo Clinic were ignored for simplification.
However both can be accommodated by adjusting the value of |D|if weekend
surgeries are planned or a public holiday is within the planning horizon. The
length of the regular work hours was defined as κ= 10 hours. In this case study,340
we assume that surgeons strictly follow the “blue-orange” system in Section 3.3.1
and they will be assigned to either a “blue” or an “orange” pattern without
additional personal preferences.
4. Numerical Results
In this section, we report the results of several numerical experiments using345
real-life surgical data described in Section 3.3.3. The proposed methodologies
are implemented in C++ and the optimization model is solved by commer-
cial solver CPLEX 12.7.1. Numerical experiments are conducted on a PC with
an E3-1270 CPU and 64GB memory. We first study the properties and per-
formance of secondary OR sharing (Section 4.1) followed by an analysis of the350
improvements achieved by the optimal surgical capacity allocation (Section 4.2).
4.1. Secondary OR Sharing
We used the data-driven method described in Section 3.1 to identify com-
patible surgeon pairs associated with low overtime probability and expected
overtime. Different values of βand γwere tested to illustrate their impact on355
the overall performance. Note that surgeons’ usage of secondary ORs in the
past is mutually independent, which satisfies our assumption in Section 3.1.
Secondary OR usage data of each surgeon in n= 4125 days was randomly
partitioned into training and test dataset with an 80%20% ratio. We identified
compatible surgeon pairs using data in the training dataset (80% of the data).360
It is to “simulate” the process in which administrators could identify compatible
surgeon pairs in practice. Compatible surgeon pairs identified were tested in the
test dataset (20% of the data). By case-resampling bootstrapping, secondary
OR usage data in 8 consecutive weeks was randomly sampled from the test
dataset and was used to evaluate the resulting overtime in ORs.365
This cross-validation process was repeated 100 times so that the overall per-
formance could be evaluated. Note that this section focuses on evaluations
of pairing between different surgeons and hence self-pairing cases where a sec-
ondary OR is assigned to a single surgeon are not reported in the analysis.
14
γ(minutes)
0 15 30 45 60
β
100% 0 0 0 0 0
95% 0 0 0 0 0
90% 0 1.99 1.99 1.99 1.98
85% 0 3.77 4.14 4.08 4.09
80% 0 3.94 10.24 10.17 10.03
75% 0 3.89 17.59 17.96 17.95
70% 0 3.92 17.46 18.30 18.33
65% 0 3.93 17.59 21.63 25.45
60% 0 4.05 17.46 21.72 28
Table 1: Average number of compatible surgeon pairs (excluding self-pairings)
Table 1 lists the average number of compatible surgeon pairs (denoted by370
ntrain
pair ) derived from training datasets. As expected, less strict requirements
on overtime, i.e., lowering the requirement on overtime probability (lower β) or
increasing the tolerance on expected overtime (higher γ), leads to more flexibility
in surgeon pairing and thus increases ntrain
pair . As can be seen in Figure 7, βand
γjointly determines ntrain
pair in a nonlinear fashion. For example, when γ= 15375
minutes, reducing βfrom 85% to 75% does not impose a significant impact on
ntrain
pair , whereas the impact of the same change in βis significant when γ= 30.
After the identification of surgeon pairs using the training data, we evaluated
these evaluated these pairs in terms of mean overtime (denoted by γtest) and
mean percentage of zero overtime (denoted by βtest) using Monte Carlo sampling380
of the test dataset. The confidence intervals for βtest and γtest are presented
in Table 2 and Table 3, respectively. As can be seen, the identified compatible
surgeon pairs perform consistently well in the test dataset by staying within the
tolerance limits γand β, i.e., γtest < γ and βtest > β. In addition, similar to
the behavior in ntrain
pair ,βtest and γtest are jointly dependent on both βand γ.385
4.2. Surgical Capacity Allocation Optimization
In this section, we evaluated the surgical capacity allocation methodology
described in Section 3.2 using historical data and predict the potential improve-
ments over the current policy.
The set of historical data in Section 3.1 was randomly partitioned into train-390
ing and test dataset with an 80%20% ratio to carry out numerical experiments.
We used 80% of n= 4125 days to “simulate” the process in which surgical suite
administrators determine the number of full-day (NFD
s) and half-day blocks
15
0 15 30 45 60
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0
5
10
15
20
25
gamma (minutes)
beta
Figure 7: Average number of compatible surgeon pairs (excluding self-pairings)
γ(minutes)
0 15 30 45 60
β
100% NA NA NA NA NA
95% NA NA NA NA NA
90% NA [90.9%,91.6%] [91.1%,91.7%] [91.2%,91.8%] [91.0%,91.8%]
85% NA [88.5%,89.3%] [88.3%,89.2%] [88.6%,89.3%] [88.7%,89.4%]
80% NA [88.2%,89.1%] [84.4%,85.1%] [84.5%,85.1%] [85.0%,85.6%]
75% NA [88.5%,89.3%] [81.3%,81.9%] [81.6%,82.2%] [81.6%,82.1%]
70% NA [88.4%,89.2%] [81.9%,82.4%] [81.2%,81.8%] [81.5%,82.0%]
65% NA [88.0%,89.0%] [81.6%,82.1%] [79.3%,79.9%] [77.2%,77.9%]
60% NA [88.0%,88.9%] [81.8%,82.3%] [79.3%,79.9%] [76.2%,76.9%]
Table 2: The 95% confidence interval for the mean percentage of zero overtime (βtest) when
compatible surgeon pairs share secondary ORs (excluding self-pairings) calculated based on
Monte Carlo sampling of the test dataset
16
γ(minutes)
0 15 30 45 60
β
100% NA NA NA NA NA
95% NA NA NA NA NA
90% NA [7.8,8.7] [7.6,8.5] [7.4,8.3] [7.6,8.5]
85% NA [10.8,11.9] [10.9,12.1] [10.9,11.9] [10.4,11.4]
80% NA [11.3,12.5] [16.2,17.1] [16.2,17.2] [15.4,16.3]
75% NA [10.9,12.0] [21.0,21.9] [20.5,21.3] [20.5,21.3]
70% NA [11.0,12.2] [20.2,21.0] [21.0,21.9] [20.7,21.5]
65% NA [11.4,12.7] [20.6,21.4] [24.4,25.4] [27.9,29.0]
60% NA [11.5,12.7] [20.4,21.1] [24.3,25.3] [29.1,30.3]
Table 3: The 95% confidence interval for the expected overtime (γtest) when compatible
surgeon pairs share secondary ORs (excluding self-pairings) calculated based on Monte Carlo
sampling of the test dataset.
(NHD
s) needed by each surgeon in the 8-week planning horizon. In our experi-
ments, we used the average number of full-day and half-day blocks in different395
8-week time periods in the training set, as shown in (14). However, administra-
tors can do it differently in practice, either based on their experience or based
on prediction.
NF D
s=ceil(|Dc| × NFD
s,hist
|Dc,hist|)
NHD
s=ceil(|Dc| × NHD
s,hist
|Dc,hist|) (14)
where NF D
s,hist and NHD
s,hist are the total number of primary ORs and secondary
ORs assigned to surgeon s,|Dc,hist|is the number of calendar days in the dataset400
and |Dc|is the number of calendar days in the planning horizon.
Note that in a relevant study [35], a methodological framework was developed
to test different sizes of historical training data for strategic OR management
decisions. The size of our training set is larger than those recommended values in
[35]. How to determine the number of full-day or half-day blocks is considered405
out of the scope of this study, but it is an important future research. It is
particularly helpful to examine whether strategic decisions on surgical capacity
allocation can be made based on a limited amount of historical data.
The estimated values of NFD
sand NHD
swere then fed into our allocation
optimization model as parameters to determine the optimal allocation. The410
17
derived allocation was tested in the remaining 20% of n= 4125 days (the test
dataset). By case-resampling bootstrapping, usage data in 8 consecutive weeks
was randomly sampled from the test dataset and was used to evaluate how the
derived allocation satisfies the caseload demand in the sampled 8 weeks.
This cross-validation process was repeated 500 times to measure the overall415
performance. The evaluation was based on two criteria. Using the training
dataset, we first computed the maximum number of ORs allocated daily to
the division (wc) and the total number of ORs used over the planning horizon
(PdDwc
d) under the current policy and compared them with those (wand
PdDw
d, respectively) under the optimal allocation. Second, we calculated the420
amount of overtime (reported as OT ) in the shared secondary ORs using the
test dataset.
Determining of the weight parameter αin these types of multiple objective
functions is a challenge that is frequently posed to researchers. However, in
our case study, we found that the choice of αdid not impact wand PdDw
d
425
under the optimal surgical capacity allocation with (β, γ ) in Table 1 . We
further explore this unique property in Appendix B. When this property holds,
the healthcare administrators can implement the proposed methodology with
any αand still ensure that the maximum number of ORs allocated daily to the
division and the total number of ORs used over the planning horizon are both430
minimized. In all of the experiments below, we calculated an optimal capacity
allocation using an arbitrarily selected α= 0.5.
We present results obtained with different (β, γ) in Table 4. When β=
100% or γ= 0, no secondary OR is shared (see Table 1). After capacity
allocation optimization, the maximum number of ORs allocated daily to the435
division is reduced by 46.1%, which implies a significant reduction in daily OR
usage variability. In the meanwhile, the total number of ORs used over the
planning horizon only increases by 4.1%.
When secondary OR sharing is allowed (i.e., β < 100%, γ > 0), the total
number of ORs used over the planning horizon (PdDwd) under the optimal440
allocation could improve over the current allocation policy. The improvement is
as much as 9.9% when (β, γ) = (70%,60). This behavior is expected since less
strict requirements on overtime creates additional eligible surgeon pairs npair in
sharing secondary ORs as discussed in Section 4.1.
Intuitively, the reduction in the total number of opened ORs comes only445
from the sharing of secondary ORs. The number of full-day OR blocks (NF D
s)
assigned to each surgeon, each of which is associated with an open OR, is treated
18
(wc,PdDwc
d) (β, γ ) (w,PdDw
d)
95% confidence interval
for total OT in the
planning horizon (minutes)
(7.4, 131.7)
β= 100% or γ= 0 (4.0, 137.1) (0.0, 0.0)
(90%,15) (4.0, 133.5) (43.0, 47.4)
(90%,30) (4.0, 133.5) (43.0, 47.4)
(90%,45) (4.0, 133.5) (43.0, 47.4)
(90%,60) (4.0, 133.5) (43.0, 47.4)
(80%,15) (4.0, 130.8) (82.5, 89.1)
(80%,30) (3.9, 125.5) (136.1, 144.8)
(80%,45) (3.9, 125.5) (136.1, 144.8)
(80%,60) (3.9, 125.5) (136.1, 144.8)
(70%,15) (4.0, 130.8) (82.5, 89.1)
(70%,30) (3.9, 122.5) (163.9, 172.9)
(70%,45) (3.5, 118.7) (228.3, 239.9)
(70%,60) (3.5, 118.7) (228.3, 239.9)
Table 4: Comparison of the maximum number of ORs allocated daily to the division and the
total number of ORs used over the planning horizon under the current v.s. optimal capacity
allocation policies.
as a parameter and is not changed after our allocation optimization (see Section
3.2.3).
Moreover, when secondary OR sharing is allowed, the improvement on the450
maximum number of ORs allocated to the surgical division is maintained around
at least 46%. These improvements can be achieved at the expense of incurring
as few as 43 minutes of overtime in the whole 8-week planning horizon.
As discussed in Section 3.1, when setting overtime tolerance (β, γ), one needs
to consider the tradeoff between the gain of improving OR usage and the loss due455
to overtime on a surgery day. For example in Table 4, when β= 80%, changing
γfrom 15 to 30 reduces PdDw
dby 5.3, while as γfurther increases from 30
to 60, PdDw
dremains the same. This allows healthcare administrators to
improve OR usage without incurring excessive overtime.
In general, our allocation optimization model can be solved within 0.5 sec-460
onds in the case study. Additional results are included in Appendix C to study
the relationship between solution time and different parameters including |S|,
|Ps|,|A|and NOR.
19
5. Discussion
The effectiveness of the proposed methodological framework has been demon-465
strated in numerical results. Our proposed methodological framework has ad-
dressed three challenges faced by healthcare administrators when making strate-
gic surgical capacity allocation decisions:
Surgical capacity allocation patterns address the cyclicity of the capacity
allocation and preferences of surgeons.470
The proposed data-driven approach coordinates surgeons’ OR sharing in
practice.
Our bi-objective integer programming model coordinates capacity pat-
terns among surgeons to optimize the surgical capacity allocation. It is
data-driven and is responsive with a solution time of less than 0.5 second,475
which can be iteratively revised and rerun as a decision support tool in
practice.
In the application in a surgical division at Mayo Clinic, our methodology is
insensitive to the choice of the weight parameter αin the objective function.
This property allows the healthcare administrators to implement the proposed480
methodology with any αand still ensure that the maximum number of ORs al-
located daily to the division and the total number of ORs used over the planning
horizon are both minimized. In addition, healthcare administrators are able to
explicitly control overtime, maximum number of daily OR allocation and total
OR usage by setting parameters in our methodology.485
Another benefit of our proposed methodology is the capability of reserv-
ing the stability of OR allocation plan. In our optimization model, allocation
patterns are used as inputs and therefore can be constructed beforehand to ac-
count for stability. For example, if a surgeon is assigned to work only on Mon-
days,Wednesdays and Fridays in the first 8 weeks, when constructing his/her490
allocation patterns for the next 8 weeks, we can limit his/her work days only to
these 3 weekdays (e.g., the set of pattern Psincludes “Mondays+Wednesday”,
“Mondays+Fridays”, “Mondays+Wednesdays+Fridays”, etc). If it is preferred
to maintain exactly the same weekdays for surgeries, we can limit Psto only
contain “Mondays+Wednesdays+Fridays” for the next 8 weeks. When being495
implemented, the OR assignment optimization model only need to be rerun,
when there are changes on surgeons’ estimated caseload (NF D
sand NHD
s) or
20
Figure 8: Example of Secondary OR Sharing
surgeons’ preferences (in terms of their possible patterns in Ps). Our model can
help surgeons keep their current work schedule by constructing pattern set Ps
as described.500
The proposed method addresses the strategic surgical capacity allocation
problem and thus the daily coordination of surgeons’ primary and secondary
OR usage is out of scope of this study. Nonetheless, we briefly explain how
such coordination can be achieved under the optimal capacity allocation. First,
note that a secondary OR is assigned only when the primary OR of a surgeon505
is fully booked. Therefore, surgeons have access to their primary ORs for the
whole day when they are also allowed to share a secondary OR with another
surgeon. It is thus possible to split the secondary OR time into two shared-
OR blocks and each surgeon works only in one block as illustrated in Figure 8.
Second, surgeons’ historical OR usage is measured from the first patient’s entry510
time to the last patient’s exit time, as discussed in Section 3.3.3. Historical
OR usage thus includes the idle time of an OR when a surgeon is physically
working in another parallel OR. Therefore, surgeons’ time in their primary ORs
and secondary shared-OR blocks under the optimal capacity allocation can be
coordinated in a similar way as the coordination of their primary and secondary515
OR usage in the current practice.
Note that there are two possible limitations of our numerical experiment.
First, we implicitly assume that surgeons’ OR usage in the future remains the
same as in the past. When identifying compatible surgeon pairs in Section
4.1, historical OR usage is used to “predict” surgeons’ future usage of OR.
When estimating NF D
sand NHD
sin Section 4.2, future OR usage is “pre-
dicted” to be the average of the past usage. For more accurate performance
evaluation of our methodology, prediction and time series analysis should be
carried out. Second, our numerical results apply to the studied surgical di-
21
vision at Mayo Clinic. Although OR utilization in our studied practice is
similar to some national and state-level statistics reported in previous litera-
ture [36, 37], specific benefits of implementing the proposed methodology may
depend on conditions of medical institutions (e.g., surgeons’ caseload and OR
utilization). That being said, our numerical experiment provides an estimate
of the value of implementing the proposed methodology and thus shows its
substantial potential in improving surgical capacity utilization.
6. Conclusions
We study a challenging but critical problem in healthcare operations man-520
agement: strategic surgical capacity allocation. We introduce the concept of
surgical capacity allocation patterns to ensure that capacity allocation is cycli-
cal and to satisfy surgeons’ preferences to the extent possible. We also propose
a data-driven approach to coordinate OR sharing among surgeons. These com-
ponents are incorporated into a mathematical optimization framework to assign525
surgical allocation patterns to surgeons in order to simultaneously minimize the
maximum number of ORs allocated daily to a surgical division and the total
number of ORs used over the planning horizon.
The proposed methodological framework is applied in a case study with data
from a surgical division at Mayo Clinic. The proposed data-driven approach can530
effectively identify surgeons who can share ORs without incurring an excessive
amount of overtime. Compared with the current practice, our optimization
model shows a substantial potential in reducing the number of ORs allocated
daily to the surgical division with little overtime. Moreover, with a solution
time of less than 0.5 second, the proposed methodology can be readily adopted535
by most surgical practices .
This proposed method allocates surgical capacity at a strategic level and
provides a more predictable work schedule for surgeons, which is not the case
under current policy (see Figure 9). We use surgeons’ average OR usage in
our surgical capacity allocation and focus on strategic planning instead of day-540
to-day allocation adjustments. Specifically, we do not consider operational de-
cisions of releasing unused ORs or assigning additional ORs on the day before
surgery. Therefore, directly applying the proposed approach to operational-level
OR management potentially leads to a mismatch between demand and capacity
due to highly unpredictable OR demand (e.g., see highly variable OR allocation545
22
0
5
10
15
010 20 30 40 50 60 70 80 90 100
Number of ORs
Different 8-week Periods
Primary ORs
Secondary ORs
Figure 9: Number of ORs allocated to a surgeon in 100 consecutive 8-week periods
to a surgeon in Figure 9). This issue can be alleviated by incorporating more ac-
curate OR demand predictions, which can be achieved by investigating temporal
effects in OR demand and including practitioners’ inputs. In addition, stochas-
tic optimization or scenario-based mixed integer programming (MIP) could be
used to better account for the varying OR demand. However, the resulting op-550
timization model has a two-stage MIP nature, a big-M formulation and a large
size (due to scenarios), which makes it difficult to solve. How to devise a com-
putationally efficient solution algorithm is still needed for its implementation in
practice. These directions will be explored in a future research.
Acknowledgments555
This work is funded in part by the Mayo Clinic Robert D. and Patricia E.
Kern Center for the Science of Health Care Delivery.
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Appendix A. Modification to the Mathematical Optimization Model
To accommodate the case study, Constraint (A.1) substitutes for Constraint
(13) in the original model in Section 3. This adjustment makes sure that a665
surgeon gets a shared-OR block in the secondary OR only on the day when
he/she is assigned a primary OR, as implemented in the Mayo Clinic surgical
practice.
Surgical capacity allocation model for the case study
min Objective (4)
s.t. Constraints (5) (12)
ysd X
(s,s0)As
zss0dsS, d D
(A.1)
Appendix B. Impact of weight parameter α
We show that the choice of αdoes not impact the total number of ORs used670
over the planning horizon and the maximum number of ORs allocated daily to
the surgical division (PdDw
d, w) in the optimal solution with (β, γ) in Table
1 in the case study.
We understand this behavior could be data dependent, but the proposed
analysis can be applied to similar problems to reduce their search space. Note675
that for some subset of the data in the computational test in Section 4.2, we
verify with the same method that w= 3 holds for different values of α.
Our analysis is conducted in three steps
1. Optimal solutions are derived for problems with α= 1, that is, with
a single objective of minimizing the total number of ORs used over the680
planning horizon (PdDwd). We show that w= 4 in these optimal
solutions.
27
(w,PdDw
d)
γ(minutes)
0 15 30 45 60
β
100% (4,149) (4,149) (4,149) (4,149) (4,149)
95% (4,149) (4,149) (4,149) (4,149) (4,149)
90% (4,149) (4,146) (4,146) (4,146) (4,146)
85% (4,149) (4,140) (4,140) (4,140) (4,140)
80% (4,149) (4,140) (4,136) (4,136) (4,136)
75% (4,149) (4,140) (4,133) (4,133) (4,133)
70% (4,149) (4,140) (4,133) (4,133) (4,133)
65% (4,149) (4,140) (4,133) (4,125) (4,123)
60% (4,149) (4,140) (4,133) (4,125) (4,123)
Table B.5: OR usage in the optimal allocation with α= 1 and different overtime tolerances
2. We prove that the optimal value wdecreases or remains the same as α
decreases.
3. By solving an extreme case, we show that the minimum possible win685
the studied problems is 4. Therefore wholds for any α[0,1] in our
problems.
The detail of the analysis is as follows.
1. We solve the optimization model with α= 1 and with values of (β, γ ). The
results are displayed in Table B.5. The optimal value wis consistently 4690
in all tested cases in presented cases.
2. We show that as αdecreases, the optimal value of the maximum number
of ORs allocated daily to the surgical division, wremains the same or
decreases. Let 0 α1< α21 and we denote the corresponding optimal
solution as (wα1,PdDwα1
d) and (wα2,PdDwα2
d). Since these solu-695
tions are optimal with regard to the corresponding αvalues, therefore we
have
α1X
dD
wα1
d+ (1 α1)wα1α1X
dD
wα2
d+ (1 α1)wα2
α2X
dD
wα2
d+ (1 α2)wα2α2X
dD
wα1
d+ (1 α2)wα1
28
Rearranging these two inequalities, we can deduce that
α1(X
dD
wα1
dX
dD
wα2
d)(1 α1)(wα2wα1)
α2(X
dD
wα2
dX
dD
wα1
d)(1 α2)(wα1wα2) (B.1)
Therefore we get
wα2wα1
α1
1α1
(X
dD
wα1
dX
dD
wα2
d)
α1
1α1
1α2
α2
(wα2wα1)
(wα2wα1)α2α1
α2(1 α1)0
wα2wα10
Therefore wα1wα2,0α1< α21.
3. We show the minimum possible win the studied problems is 4. We show
this by solving an extreme case Qext where all pairs of surgeons are allowed700
to share secondary ORs, i.e., (β, γ) = (0,+) and α= 0. It is not hard to
see that every feasible solution to the optimization problem (Qα=0) with
α= 0 and any (β, γ) is also feasible to Qext but not vice versa. Therefore
the optimal objective value of Qα=0 is no smaller than the optimal value
of Qext. The optimal solution of Qext in our case study is computed to705
be (w,PdDw
d) = (4,127). Therefore w4 holds for any allocation
problem with α= 0 and any (β, γ ) in our case study.
Since wincreases as αincreases, w4 for all problems Qwith α[0,1]
and any (β, γ). For cases in Table B.5, since w= 4 holds at α= 1, w= 4
holds for all α[0,1].710
Since wis fixed for (β, γ ) in Table B.5, based on inequalities (B.1), it
can be seen that PdDwα1
d=PdDwα2
d. Therefore the choice of α
does not affect the total number of ORs used over the planning horizon
(PdDw
d) or the maximum number of ORs allocated daily to the surgical
division (w) in the optimal allocation with any (β, γ ) in Table 1.715
Intuitively, two metrics in the objective, namely the total number of open
ORs in the planning horizon and the maximal number of open ORs on
any day, are somewhat related. In the ideal case, both reach minimum
29
if the same number of ORs are open everyday. In our case study, we
assume that surgeons follow “blue-orange” patterns without additional720
personal preferences. This allows significant flexibility to move surgical
blocks around and gives a better chance to reach minimum in both metrics.
Note that when α= 1, multiple optimal solutions exist, all of which reach
the minimum in the total number of open ORs. Some of optimal solutions
cause significant imbalance in the number of daily open ORs. However,725
one of the optimal solutions reaches minimum in the maximal number of
daily open ORs. Therefore to find this solution that reaches minimum in
both metrics, α= 1 instead of α= 1 is used in the optimization model
where is a very small number.
Appendix C. Solution Time730
The size of the optimization model in Section 3.2.3 could be affected by the
number of surgeons |S|, each surgeon’s number of capacity allocation patterns
|Ps|, the number of compatible surgeon pairs |A|and the number of ORs NOR .
In this section, we investigate whether solution time of the optimization model
is correspondingly changed.735
First, by adjusting overtime thresholds (β, γ ) in identifying compatible sur-
gical pairs, we adjusted the size |A|in the optimization model. From Table
C.6, we can see that solution time does not have a noticeable relationship with
|A|. However, in the worst case we have tested, the optimization model can be
solved within 0.5 seconds.740
Second, we studied the impact of each surgeon’s number of capacity al-
location patterns |Ps|on solution time. We fixed the overtime threshold as
(β= 90%, γ = 45) in the identification of compatible surgeon pairs and used
|S|= 8 surgeons in NOR = 11 ORs as in the original dataset. In current prac-
tice, each surgeon is assigned to either a “blue” or an “orange” pattern (|Ps|= 2)745
in the “blue-orange” system in Section 3.3.1. In the constructed hypothetical
problem, each surgeon is only available for one of the “blue” or “orange” pattern
(|Ps|= 1). In Table C.7, as |Ps|increases, solution time increases. However,
this increase rate is not linear, which may be related to structure changes in the
integer programming model with additional variables zss0d.750
Third, we studied the impact on solution time of changing the number of
surgeons |S|and the number of ORs NOR. We fixed the overtime threshold
as (β= 90%, γ = 45) in the identification of compatible surgeon pairs so that
30
(β, γ )
Average number of compatible surgeon
pairs (including self-pairings)
95% confidence in-
terval for solution
time
β= 100% or γ= 0 0 [0.067,0.068]
(90%,60) 9.98 [0.063,0.064]
(90%,15) 9.99 [0.063,0.064]
(90%,30) 9.99 [0.062,0.063]
(90%,45) 9.99 [0.063,0.064]
(70%,15) 11.92 [0.081,0.082]
(80%,15) 11.94 [0.082,0.083]
(80%,60) 18.03 [0.434,0.482]
(80%,45) 18.17 [0.433,0.479]
(80%,30) 18.24 [0.433,0.480]
(70%,30) 25.46 [0.173,0.175]
(70%,45) 26.30 [0.275,0.281]
(70%,60) 26.33 [0.274,0.280]
Table C.6: Solution time v.s. the number of compatible surgical pairs
Number of Patterns per surgeon 95% confidence interval
for solution time
1 [0.011,0.011]
2 [0.063,0.064]
Table C.7: Solution time v.s. the number of allocation patterns
31
Number of ORs Number of surgeons 95% confidence interval
for solution time
4 6 [0.057,0.059]
8 11 [0.063,0.064]
16 22 [1.012,1.245]
Table C.8: Solution time v.s. the number of ORs and the number of surgeons
they will not affect solution time. Note that in surgical practice, a larger |S|is
generally associated with an increased NOR . Therefore we experimented with755
different |S|and NOR and kept the ratio of |S|/NOR around 8
11 as in the original
dataset.
We tested cases with |S|= 4, 8 and 16 surgeons in the surgical division. We
arbitrarily pick 4 surgeons from the original dataset to construct a test problem
with |S|= 4. The test problem with 8 surgeons consists of all 8 surgeons in760
the dataset. The test problem with |S|= 16 is constructed by including 8
additional hypothetical surgeons who are replicates of 8 surgeons in the original
data set. Note the corresponding NO R are 6,11 and 22, respectively. As shown
in Table C.8, solution time increase as |S|and NOR increase. Note that when
|S|increases, the number of compatible surgeon pairs also increases, which may765
contribute to the increase in solution time.
In general, solution time of the proposed integer programming model for
capacity allocation cannot be simply estimated based on the size of the problem.
However, this model can be solved within 0.5 seconds for the case study in a
surgical division at Mayo Clinic. Due to it responsiveness, this approach can be770
used as a decision support tool in surgical practice.
Appendix D. Integrated Optimization Model for Capacity Allocation
and OR Sharing Coordination
One can integrate OR sharing coordination in Section 3.1 into the capac-
ity allocation model in Section 3.2. In the integrated optimization model,
threshold on overtime is formulated as constraints. The expected overtime is
minimized as part of the objective function, given that the weight is somehow
determined. To model these requirements, the following changes are made to
the current optimization model in Section 3.2.
Redefine zss0d:zss0d= 1 if surgeons sand s0(s, s0S:ss0) share
32
an OR on day dD; 0 otherwise.
Ω: scenario set that includes historical usage of half-day blocks.
dω
s: usage of half-day blocks (in hours) of surgeon sSin scenario
ωΩ.
M: a large enough number (for big-M approach).
uss0: binary variables; uss0= 1 if surgeon sand s0are a compatible pair
of surgeons; 0 otherwise.
rω
ss0: continuous variables to denote the resulting overtime if surgeon s
and s0hypothetically share an OR in scenario ωΩ.
qω
ss0: binary variables; qω
ss0= 1 if there is resulting overtime if surgeon s
and s0hypothetically share an OR in scenario ωΩ; 0 otherwise.
oss0d: continuous variables to compute the expected overtime if two
surgeons are assigned to the same OR in day d.
Additional term in objective: ρPs,s0S,dD:ss0oss0d, where ρis the
penalty for overtime.
Additional constraint 1: dω
s+dω
s0κrω
ss0,s, s0S, ω : ss0
where κis a predetermined regular OR operational hours (e.g. 10 hours
from 7am to 5pm). This is to compute the amount of overtime.
Additional constraint 2: dω
s+dω
s0κMqω
ss0,s, s0S, ω : ss0.
This is to determine the frequency of overtime.
Additional constraint 3: 1
||Pωrω
ss0γM(1 uss0), s, s0
S:ss0where γis a predetermined acceptable amount of expected
overtime in OR. This is to determine whether the pair of surgeons meets
the requirement on the expected amount of overtime.
Additional constraint 4: 1
||Pωqω
ss0βM(1 uss0), s, s0
S:ss0where βis a selected threshold on the probability that total
surgery hours in two consecutive shared-OR blocks in a particular OR
does not exceed κ. This is to determine whether the pair of surgeons
meets the requirement on the frequency of overtime.
775
33
Additional constraint 5: uss0zss0d,s, s0S, d D:ss0. This
makes sure that only compatible pairs would be assigned to the same
OR.
Additional constraint 6: oss0d1
||Pωrω
ss0(1 zss0d)M,s, s0
S, d D:ss0. This makes sure that expected overtime is penalized
only when two surgeons are assigned to the same OR.
The resulting optimization problem is a large-scale two-stage mixed integer
programming (MIP) model with a big-M formulation. First stage includes
decisions on block assignment and pairing of surgeons. Second stage evaluates
whether surgeon pairs are compatible and computes the expected overtime
based on scenarios of half-day block usage. The two-stage MIP nature, the
big-M formulation and also the size of the problem (due to scenarios) make
the resulting optimization problem difficult to solve. Solving this model is an
area of future study.
34
... 7 Prior studies in the literature have used modeling techniques to devise ways of maximizing the use of surgical block time. [8][9][10][11][12] These studies consider surgeon and patient preference, cost, and other factors in scheduling of elective surgery. 8,10,13,14 To our knowledge, a group scheduling system devised to optimize OR block time utilization and facilitate access to surgery for patients with cancer has not been previously described. ...
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Surgery planning and scheduling systems play a critical role in the efficient matching of supply and demand for surgery. Competing performance measures and a number of complicating factors make surgery planning and scheduling challenging. Operating room (OR) managers must consider uncertainty in the duration of surgery and other critical activities, the arrival of unexpected urgent add-on patients, and cancellations on the day of surgery while efficiently managing a variety of human and technical resources. Surgery schedules must be designed to balance competing performance criteria such as patient waiting time as well as idle time and overtime for both the OR team and other personnel. This article presents an introduction to surgery scheduling, a literature review, a small example, and a description of some of the important open challenges.Keywords:add-on cases;block-booking;open-booking;queueing models;simulation;optimization
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