improving operating room schedules

Health Care Manag SciDOI 10.1007/s10729-015-9318-2

Improving operating room schedules

Fei Li · Diwakar Gupta · Sandra Potthoff

Received: 13 October 2014 / Accepted: 13 January 2015© Springer Science+Business Media New York 2015

Abstract Operating rooms (ORs) in US hospitals are costlyto staff, generate about 70 % of a hospital’s revenues, andoperate at a staffed-capacity utilization of 60-70 %. Manyhospitals allocate blocks of OR time to individual or groupsof surgeons as guaranteed allocation, who book surgeriesone at a time in their blocks. The booking procedure fre-quently results in unused time between surgeries. Realizingthat this presents an opportunity to improve OR utilization,hospitals manually reschedule surgery start times one ortwo days before each day of surgical operations. The pur-pose of rescheduling is to decrease OR staffing costs, whichare determined by the number of concurrently staffed ORs.We formulate the rescheduling problem as a variant of thebin-packing problem with interrelated items, which are thesurgeries performed by the same surgeon. We develop alower bound (LB) construction algorithm and prove that theLB is at least (2/3) of the optimal staffing cost. A key featureof our approach is that we allow hospitals to have two shiftlengths. Our analytical results form the basis of a branch-and-bound algorithm, which we test on data obtained from

Electronic supplementary material The online version of thisarticle (doi:10.1007/s10729-015-9318-2) contains supplementarymaterial, which is available to authorized users.

F. Li · D. Gupta (�)Industrial & Systems Engineering Department,University of Minnesota, Minneapolis, MN, USAe-mail: [email protected]

F. Lie-mail: [email protected]

S. PotthoffUniversity of Minnesota School of Public Health,420 Delaware Street SE, Minneapolis, MN 55455 USAe-mail: [email protected]

three hospitals. Experiments show that rescheduling savessignificant staffing costs.

Keywords Operating rooms · Surgery scheduling · Healthcare operations management · Math programming

1 Introduction

Operating rooms (ORs) in US hospitals generate about 70 %of revenues and 20-40 % of operating costs while operatingat a staffed capacity utilization of 60-70 % [1]. ORs are alsoresponsible for a significant proportion of hospital admis-sions [2] and a recent estimate puts the cost of a staffed ORat approximately $15-20 per minute ([3]). Therefore, hos-pitals spend considerable administrative resources to ensurethat OR time is used efficiently. A typical scenario in manyhospitals is that each day the OR management team looksat the two-day-ahead surgical schedule, and tries to manu-ally revise case start times to reduce the number of operatingrooms that would need to run concurrently. This reducesstaffing costs. At this point in time, there is already a surgi-cal schedule in place with planned start times and plannedsurgical case lengths. The latter are provided by schedulingsoftware used by hospitals, with some adjustments based ondiscussions with the surgeons at the time of booking proce-dures. The management team treats the surgical case lengthsas fixed, changing only the case start times. The purpose ofthe model we develop is to aid in this daily schedule revisionprocess.

By rearranging the scheduled procedure start times oneto two days in advance, it may be possible to lower costsby reducing the number of staff needed, or by reducingthe amount of staff overtime needed. Alternatively, the re-optimized schedule may free up OR time that can be used to

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F. Li et al.

fit additional procedures into the upcoming day’s schedule,thereby increasing case volume and revenue.

The OR rescheduling problem mentioned above is a vari-ant of the bin-packing problem with bins being the staffedORs and items or jobs being the surgeries. The goal is tominimize the weighted sum of bins used (i.e. cost of staffedORs), where the weight of a bin is proportional to its size.Two features of the OR rescheduling problem make it dif-ferent from problem formulations studied in the literature.These are (1) surgeries performed by the same surgeon mustnot overlap, and (2) hospitals may employ staff with dif-ferent shift lengths. The bin-packing and therefore the ORrescheduling problems are NP hard. Therefore, we establisha lower bound on the cost of staffing ORs that is guaran-teed to be at least (2/3) of the optimal staffing cost for anysubset of surgeries. The lower bound is used in a branch-and-bound algorithm developed to solve the reschedul-ing problem. Upon testing our approach on data obtainedfrom three hospitals, we identify significant opportunitiesfor reducing OR staffing costs. We also analyze result-ing OR schedules to study how rescheduling would affectsurgeons’ work days, delays in surgery start times, andovertime.

OR management practices vary from one hospital toanother. We present the ensuing institutional background asbroadly representative of common practices at US hospitalsthat allocate periodically occurring (e.g. weekly, biweeklyor monthly) blocks of OR time to individual or groupsof surgeons as guaranteed allocation. Surgeons holdingblocks may book surgeries in their blocks up until theauto-release date. On the auto-release date, which mayoccur between 0 to 14 days in advance of the day ofsurgery, any unused block time reverts back to the hos-pital. This OR time may be used either by surgeons whodo not have assigned blocks or by those whose demandexceeds their block times, or for urgent and emergentcases. Non-block surgeons’ cases are typically booked on afirst-come-first-served basis. Different hospitals may followdifferent approaches to deal with urgent cases. Some sched-ule urgent blocks with zero auto-release dates, whereassome others reserve dedicated ORs for urgent and emer-gent cases. All hospitals also try to “fit” urgent casesinto available open times between scheduled non-urgentcases. Finally, any remaining urgent cases are scheduledas add-on cases at the end of shifts, incurring overtimecharges.

Because staffed OR utilization is low even after hospi-tals’ manual attempts to reorganize surgery schedules (seeSection 3 for details), we focus in this paper on develop-ing an algorithm that would allow hospitals to accommodatethe same number of surgeries with fewer staffed ORs byreworking the case start times. Such rearrangements areoften feasible because blocks typically have 2 to 5 day

auto-release dates and surgeons are willing, within rea-son, to accept some changes to the start times of theircases. Moreover, patients are typically asked to arrive sev-eral hours before the start of their surgeries to preventdelays due to peri-operative activities, which facilitatesrescheduling.

Our algorithm for improving OR schedules significantlyreduces the number of open times between scheduled non-urgent cases. Therefore, upon implementing this approach,hospitals would need to schedule dedicated ORs for han-dling urgent cases. The management team would need tocommit to having these ORs staffed before knowing thetrue urgent and emergent demand. We evaluate the perfor-mance of our algorithm both with and without accountingfor urgent cases. In each scenario, our approach results insignificant total cost savings, including overtime charges.Emerging evidence in biomedical literature supports ourapproach by finding that dedicated urgent/emergent ORsalso help improve health outcomes (e.g. [7]).

Our analytical framework involves three steps. In stepone, we classify connected sequences of surgeries that wecall “chains.” We also classify doctors into different cate-gories based on the properties of chains formed by theirsurgeries. Surgeon classification is used in step two to assignsurgeries to ORs in a particular sequence, which not onlyproduces a lower bound but also helps us in step three torecover a feasible solution that is no more than (3/2) of thelower bound.

Because our algorithm may increase surgeons’ idle orunused time, we test the quality of our solution by apply-ing our algorithm to data from three hospitals. This leadsto several insights. First, rescheduling works as expectedby flattening the peak number of concurrently staffed ORsand scheduling surgeries uniformly throughout the day. Wealso find that efficiency is greater when a hospital hasthe flexibility to schedule some long shifts because thatleads to more efficient packing of surgical cases. Second,efficiency gains come at the expense of increased staff over-time, surgeon idle time, and the total amount of time thatsurgeons spend at the hospital. We quantify the impact ofrescheduling on surgeons and staff and find that savingsfrom efficiency gains are high, suggesting that hospitalsmay be able to obtain surgeons’ cooperation through anappropriate gain sharing plan.

The remainder of this paper is organized as follows. Wereview literature in Section 2, summarize data from threehospitals in Section 3, formulate a data-supported model inSection 4, and present key analytical results in Section 5,which form the basis of our branch-and-bound algorithmin Section 6. Numerical experiments that utilize real dataare presented in Section 7, and extensions and concludingremarks are discussed in Section 8. In the interest of brevity,some details are provided in an Online Supplement.


2 Literature review

There are three bodies of literature that are related to ourwork. These are OR/surgery scheduling, bin packing, andresource-constrained scheduling. We position our work nextin relation to each of these bodies of literature.

Surveys of OR scheduling literature are provided in sev-eral recent papers; see, for example, [4, 8–10], and [12].OR capacity planning problems fall into six broad cate-gories: (i) determining the number of ORs and the equip-ment/capability of each OR, (ii) determining staffing needsand corresponding shift lengths, (iii) assigning blocks ofOR time to surgeon groups or individual surgeons, (iv)putting in place booking rules for the use of OR time andthe release of exclusive blocks, (v) rescheduling, and (vi)coping with day-of-surgery variations. Planning problemsin each category arise with different frequency and there-fore relevant models need to consider different levels ofgranularity and time scales. For instance, the problem ofdetermining the number of ORs and equipment may berevisited once every few years and relevant models mayconsider aggregate demand over a quarter or a year. It maybe appropriate for such models to assume that surgeriesare packed in a fluid fashion. In contrast, when choosingplanned start times of surgeries, resulting schedules must fitsurgeries into available shift lengths.

Surgery scheduling problems, i.e. problems mentionedin items (iv) and (v) above, can be categorized in severaldifferent ways. For example, by the assumed booking pro-tocol (online or offline), by procedure lengths (constant orrandom), and by urgency status (emergent/urgent or non-urgent) ([12]). Online scheduling occurs when surgeriesare booked one at a time. Offline means all requests forsurgeries that need to be performed on a particular dayare known before determining scheduled procedure lengthsand the sequence in which surgeries will be performed. Inmany US hospitals, surgeries are booked in an online fash-ion, booking clerks assume that estimated case lengths areconstant, and non-urgent cases are booked first, followedby urgent and emergent cases. Case length estimates maydepend on a whole host of factors including the surgerytypes, patients’ characteristics, and track records of sur-geons performing the surgeries. When surgeons holdingblocks book non-urgent cases, they determine the sequencein which surgeries will be performed. In contrast to whatis common in practice, the Operations Management litera-ture focuses primarily on the problem of determining thescheduled duration and the sequence of surgeries assumingsurgeries are booked offline. Most papers in the schedul-ing literature consider only one urgency type, i.e. theyfocus either entirely on non-urgent cases, or entirely onurgent/emergent cases. Articles that consider both types,e.g. [6], do not model discrete surgery durations. That is,

they assume that surgeries can be scheduled in a fluidfashion.

Uncertain actual surgery duration is an important consid-eration in surgery planning and scheduling. Consequently,many papers focus on the problem of estimating timeallowances for different surgical procedures. Two vari-ants of such models exist. All models assume an offlinescheduling environment. In the first case, surgery durationsare random but their distributions are assumed known orit is assumed that actual durations can be sampled froman existing database of surgery durations. In the secondcase, surgery durations are unknown. Examples of mod-els of the former type can be found in [14–16], and [17],whereas an example of the latter can be found in [18].In these models, overlap-avoidance constraints are absent,which is a key feature of the analysis presented in thispaper. The absence of overlap-avoidance constraints is jus-tified by considering either only one OR, or assumingthat surgeons work in the same OR on a single day. Incontrast, our data show that surgeons routinely operate inseveral rooms on a given day. Also, our objective is notto determine optimal time allowances for different sur-gical procedures. Instead, we focus on creating surgeryschedules that require fewer staff shifts upon assuming thathospitals’ estimates of case lengths are not affected bysequencing of cases. This assumption is commonly made bypractitioners. We present supporting evidence from data inSection 3.

The above-mentioned problem types, the importance ofconsidering uncertainty, scheduling constraints, and possi-ble concern for smoothing downstream resources (e.g. hos-pital beds) give rise to many variants of the surgical schedul-ing problem. Because structured reviews exist that discusseach problem class, we do not describe these probleminstances in detail, except to point out that none of the exist-ing models addresses the problem of rescheduling surgeriesto minimize staffing costs. There are a few papers, however,that consider the difficulty of scheduling surgeries whensurgeons perform multiple surgeries on the same day andoverlap avoidance is an important consideration. We discussthese papers below.

Fei et al. [19] model the constraint that scheduled surg-eries performed by the same surgeon must not overlap. Theauthors formulate the daily surgery scheduling problem asa two-stage hybrid flow-shop problem with the objectiveof minimizing the cost induced both by the ORs and therecovery rooms. A hybrid genetic algorithm is proposed tosolve this model. The paper does not provide either boundsor performance guarantees, which are key elements of ourapproach. Hsu et al. [20] consider the surgery allocationproblem in an ambulatory surgical center. The authors for-mulate the sequencing step as a variant of the two-stageno-wait flow shop scheduling problem. A tabu search based

F. Li et al.

heuristic is used to find a near-optimal solution. Once again,neither bounds nor performance guarantees are established.

Turning next to the bin-packing and machine schedul-ing literatures, we find several problem formulations thathave elements in common with the problem studied in thispaper. Our problem is closer to bin-packing as opposedto machine-scheduling (see [21] for a survey of machine-scheduling literature) because our goal is to find the min-imum number of ORs (bins) needed to fit all procedures.In contrast, in machine scheduling problems, the numberof machines is known and the goal is to minimize themakespan, i.e. to complete all work at the earliest pos-sible time. However, our problem has some features ofmachine scheduling because we do have the constraint thatprocedures done by the same surgeon cannot overlap.

Coffman et al. [22] provide a review of the online andoffline approximation algorithms for bin packing. Our set-ting is offline and our problem is significantly differentfrom the standard bin-packing problem because we incor-porate overlap-avoidance constraints. Upon placing suchconstraints, existing offline bin-packing algorithms may noteven find a feasible solution if applied to our problembecause same-surgeon jobs may overlap. Overlap avoidanceprovides a natural segue into a discussion of papers on binpacking with conflicts ([23]). Given a set of items, the goalin such problem formulations is to find a partition of itemssuch that items that are predefined to be in conflict cannotbe placed in the same bin. Conflicts in this setting are “hori-zontal” – i.e. they need to be avoided when certain items areplaced in the same bin. In contrast, in our problem, the con-flicts are “vertical” – i.e. overlap in time must be avoidedacross all bins for jobs performed by the same surgeon.

Many papers in the bin-packing literature present onlinealgorithms, i.e., heuristically pack items one at a time(see [22], and [24]). The online bin-packing literature alsoincludes problems with variable-sized bins (see [25], and[26]), which is relevant because we consider different shiftlengths. In this literature, it is not common for papers tofocus on developing lower bounds on the number of binsneeded. This is the case in part because online algorithms donot rely on a branch-and-bound type approach and the needto develop lower bounds does not arise. In contrast, we pro-vide lower bounds for the relevant problem formulation andshow that it is no less than (2/3) times the cost of a feasiblesolution.

Key papers on resource-constrained scheduling are [27,28], and [29], which define the problem as that of minimiz-ing the makespan of a set of unit-length independent jobsthat cannot be scheduled before their start times on iden-tical processors. Each job needs a certain amount of eachresource from a set of available resources. All resourcesare available throughout the planning horizon, but the avail-able quantity of each resource is bounded. Srivastav and

Stangier [29] provide a (1 + ε)-approximation algorithmfor such problems. Even with the unit job-length assump-tion, the approximation algorithm studied in [29] does notapply to our setting for two reasons. The first reason is thatthese models assume that the number of machines (ORs inour model) is fixed, and minimize the finish time of jobs(i.e. makespan). A guaranteed bound of the scheduling prob-lem does not result in a guaranteed bound for our problemof minimizing the number of ORs. Second, we considera model with two shift lengths, which makes our prob-lem significantly different from the resource-constrainedscheduling problem.

Another related stream of work concerns resource-constrained project scheduling in which each activity hasa potentially different processing time and the goal is tominimize the makespan; see the survey in [30]. Papersin this literature usually develop branch-and-bound algo-rithms. The lower bounds on the makespan are calculatedby solving a relaxed problem, e.g. by relaxing the resourceconstraints ([31]), or the precedence constraints and allow-ing preemption ([32]). The key difference relative to ourapproach is that the lower bound in these papers is formakespan, which does not translate into a lower bound forthe number of bins needed.

3 Motivating examples

We obtained surgical scheduling data from three hospitals,which included scheduled surgery start times, scheduledprocedure lengths, surgeon codes, names of surgical ser-vices and surgical groups, dates and times when surgerieswere booked, OR numbers, actual surgery start times anddurations, and assigned staff codes. Note that a particularsurgical service, which is also sometimes called a surgicaldepartment, could have multiple surgical groups with blockassignments. Our data did not contain patient, surgeon orstaff identifying information. Only one hospital kept recordsof cancelations and only for those cases that were canceledon the day of surgery. We obtained block schedules and autorelease time information separately because these data arenot stored in computerized scheduling records. Table 1 sum-marizes these data. Non-urgent refers to deferrable surgeriesthat are booked at least two days in advance of the day whenthey are performed. Non-urgent cases are booked primar-ily on non-holiday weekdays. All cases include non-urgentcases, cases scheduled on weekends, and those scheduledwithin 2 days of each surgery day, i.e. urgent and emergentcases. We use a 2-day threshold because the auto-releasedate is at least 2 days for the vast majority of surgical groupsin all hospitals.

Hospital 1 had the largest number of ORs, whereas Hos-pital 2 had the smallest. Hospital 3 had the most number of


Table 1 Basic Data Summary

Hospital 1 Hospital 2 Hospital 3

All Non-urgent All Non-urgent All Non-urgent

Operation Days in Data 364 257 252 213 530 379

No. of ORs 18 10 14

Percent of OR time blocked 60 % 100 % 84 %

Shift lengths 8 or 12 hr 8 hr 10 hr

Scheduled Surgeries 10,191 7,483 10,866 9,446 12,394 8,875

Cancelations 222 167 N/A N/A

Surgical Services 14 3 17

Surgeons 209 187 106 102 82 82

N/A means data were not availableCancellations for Hospital 1 refer to those cases that were cancelled on the surgery day

surgical services, followed by Hospital 1, and then Hospi-tal 2. Before the auto-release date, the blocked OR time was60 % of OR capacity in Hospital 1, 100 % in Hospital 2 and84 % in Hospital 3. Hospital 1 used two shift lengths – 8hours and 12 hours, whereas the other two hospitals used asingle shift length. However, the shift lengths were differ-ent in Hospitals 2 and 3. The presence of two shift lengthsprovides greater flexibility in scheduling cases, but it alsocomplicates the corresponding optimization problem. Thethree hospitals differed a great deal in the mix of surgeriesperformed and the volume of each major surgery type – seeAppendix A in the Online Supplement for details.

We plot the range of procedure durations (complexity)and volume (percent of total number of procedures) inFigure A-1 of the Online Supplement. This figure is dividedinto three parts because two of the three hospitals supportsurgical services with very small volumes but highly vari-able surgical procedure durations. The horizontal axis showsmean procedure duration and the 95 % confidence intervalof scheduled procedure durations of each surgery type. Thevertical axis shows the volume of each surgery type in termsof percent of all procedures performed. Not surprisingly,complexity decreases as volume increases.

Next, in Table 2 we report the current performance statis-tics of the three hospitals. Note that all three hospitals usesome form of manual rescheduling and the results shown inTable 2 are obtained after such efforts. Data show that his-torical utilization was highest in Hospital 1 (60-65 % range)and lowest in Hospital 3 (48-52 % range). We explain howwe calculate utilization in the Online Supplement, AppendixB. On any given day, at least a third of the surgeons per-form multiple surgeries. If we count surgeon-days (eachsurgeon performing cases on a particular day counts as asingle surgeon-day), then within the vast majority of sur-geon days, surgeons perform multiple cases (87-99 % in thethird to last row of Table 2). Recall that multiple cases give

rise to the key difficulty in rescheduling because cases thatbelong to the same doctor must not overlap. In practice, ifa surgeon has a helper, he or she may overlap proceduresscheduled in different ORs. However, usually this creates ashort overlap (in our data, usually 10 minutes or less), whichwe ignore in our rescheduling procedure. Although manysurgeons perform multiple surgeries on their OR day, thenumber of surgeries performed by any particular surgeon ondays on which he or she performs multiple cases, is oftensmall. The average number of cases per MD per day liesbetween 2.76 and 3.76 for non-urgent cases. We utilize thisfact in constructing our approach for efficiently solving therescheduling problem. Data also reveal that the number ofsurgeons who operate in multiple rooms on any given dayranges from 1 to 8 among the three hospitals.

All three hospitals in our data use a similar process forestimating surgical case lengths. The scheduling softwareis used to suggest an initial estimate based on a movingaverage of historical surgical case lengths by surgeon andprocedure code. The booking clerk then confirms this timewith the surgeon’s office. Surgeons may make adjustmentsto the planned case lengths suggested by the computersoftware based on patients’ characteristics. We find thatbetween 68 to 70 percent of surgeries at the three hospitalsfinish within the allotted time. Among surgeries that takelonger than scheduled, the amount of extra time needed ison average between 22 and 41 minutes. All three hospitalsschedule slightly more time on average than the actual caselength. This is not surprising because the cost of delays,which result in patient inconvenience, surgeon idleness, andstaff overtime, is high.

Because rescheduling changes the sequence in whichsurgeries are performed, it is important to test whetherit will be appropriate to continue to use original plannedcase lengths. In order to do so, we fitted each hospital’sdata to separate generalized linear mixed-effect models.

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Table 2 Performance Statistics


All Non-urgent All Non-urgent All Non-urgent

AVG Number of ORs used 10.7, 3.2* 11.9, 1.6* 9.0 8.6 8.6 7.9

SD ORs used 2.5 1.5* 1.9 1.1* 1.5 1.3 1.5 1.2

AVG Utilization (%) 61.56 65.24 58.76 54.06 47.69 49.34

SD Utilization (%) 17.36 8.77 16.50 9.24 40.65 28.93

AVG MDs/day 17.29 18.23 10.73 11.04 8.64 8.65

SD MDs/day 9.44 4.10 4.89 2.92 4.53 2.23

AVG MDs With > 1 Case/day 6.06 6.19 7.90 8.35 5.33 5.43

SD MDs With > 1 Case/day 4.04 2.39 4.06 2.52 3.27 1.98

MD-days With > 1 Case (%) 88 98 87 99 93 99

AVG Cases/MD/day† 2.75 2.76 5.10 4.98 3.76 3.72

SD Cases/MD/day† 1.38 1.39 4.15 4.11 2.21 2.26

AVG=Average, SD=Standard Deviation, MD = Surgeon.* Shifts marked with an asterisk are long shifts, † Only MDs with > 1 case/day were counted

The dependent variable in each model was the actual caselength. The independent variables belonged to two groups.The surgeon ID was the random effect (intercept). Note,this makes it a factor variable. The fixed effects were theplanned case lengths (which were a proxy for procedurecomplexity and patient characteristics), the sequence num-ber of a surgery in its assigned OR, the difference betweenactual and planned surgery start times, and an indicator thatwas set to 1 if the difference in start times could be attributedto the surgeon (e.g. when the surgeon arrived late) and 0 oth-erwise. Results of this analysis are shown in Table A-1 inthe Online Supplement, Appendix A. Below we summarizekey findings.

In Hospitals 2 and 3, we found that the surgeon ran-dom effect was strong and that after controlling for thesurgeon effect, only planned duration had a significant fixedeffect. In particular, this implies that actual surgery dura-tions in Hospitals 2 and 3 are not correlated with surgerysequence numbers and early or late start relative to plannedstart time. Hospital 1 was different in the sense that thesurgeon effect was not strong, and planned surgery times,surgery sequence number, and difference between actualand planned start times were all significant. However, thecoefficients of surgery sequence number and difference instart time were small, indicating that their practical impacton actual case lengths was small. We use this analysisto support our assumption in Section 4 that planned caselengths may be left unchanged when rescheduling.

A different way of explaining the results of our statis-tical analysis is that among variables that could explain

variability in actual case lengths, given data limitations,surgery sequence number was not significant. The variabil-ity in actual procedure times was largely explained by thefixed effects of planned case lengths and random effects ofsurgeon ID. The latter is a factor variable. Because thereis no measure of goodness-of-fit for mixed-effects models,we also ran a hypothetical fixed-effects regression model inwhich actual duration was regressed on independent vari-ables of planned duration and surgeon ID (the latter beinga factor variable). The model was significant with adjusted-R2 of 94 %, 87 % and 89 % for the three hospitals,respectively. The mixed-effects model is more appropriatefor identifying significant factors because cases performedby a particular surgeon form a cohort of observations withindependent but separate variability in each cohort.

4 Notation and model formulation

We model a hospital with multiple ORs staffed by anesthesi-ologists, nurses and health technicians for either αT (calledshort shift) or T minutes (called long shift), where α ≤ 1.Note that α = 1 means that there is only one shift type.In this section, we formulate the rescheduling problem fora particular day, which we call the tagged day. The num-ber of surgeries (jobs) to be scheduled on the tagged dayis known. A job j is characterized by its physician indexμ(j), duration dj and originally scheduled start time s0

j . Foreach surgeon indexed i, J (i) denotes the set of jobs that areperformed by that surgeon, and J is the set of all jobs. We


make the following assumptions to develop a parsimoniousmodel.

Assumption 1 Time is discrete and 1 minute is a unit oftime.

Assumption 2 All ORs are interchangeable and there areno equipment constraints.

Assumption 3 There are enough staff for both long andshort shifts and enough ORs to accommodate all proce-dures.

Assumption 4 The relationship between scheduled andactual surgery durations remains unchanged when surgeriesare rescheduled.

Assumption 1 is justified by the fact that scheduledsurgery durations are measured in whole minutes. Assump-tions 2 – 4 are made for mathematical tractability. We dis-cuss extensions of our model that allow us to relax Assump-tion 2 in Section 8. The availability of ORs in Assumption3 is typically not an issue in problems of practical inter-est because there is a feasible solution that schedules allsurgeries into available rooms. Assumption 3 is thereforeequivalent to the assumption that the OR manager canchoose any number of long shifts when rescheduling, whichmay not be possible in practice. Therefore, we allow con-straints on the availability of long shifts when implementingour branch-and-bound algorithm (see Section 6). Finally,Assumption 4 is consistent with practice and justified by theanalysis presented at the end of Section 3.

Our goal in rescheduling is to choose a set of new starttimes, denoted sj , that minimize the staffing cost. Beforerescheduling, we pre-process data to remove (1) all jobs thathave dj > T , and (2) all those surgeons’ cases whose case-length sum (denoted by d�) exceeds T . In both instancessuch cases are scheduled into rooms with single long shiftsand the minimum necessary overtime, which is trivially anoptimal strategy.

A key decision variable in our formulation is yj,t , whichis 1 if job j is rescheduled to start at time t , and 0 other-wise. In particular, if yj,t = 1, then sj = t is the new starttime of surgery j . To prevent overlap among surgeries per-formed by the same surgeon, we introduce binary variablespjk , which equal 1 if jobs j and k are performed by the samesurgeon and j is scheduled before k, and 0 otherwise. Whenμ(j) = μ(k), pjk + pkj = 1 must hold because either jobj is performed before job k, or its opposite occurs. Becauseeach job that is active (being performed) at time t must bescheduled in a separate OR, the minimum number of staffedORs required at time t equals ht = ∑

j

∑τ :τ≤t≤τ+dj

yj,τ ,the number of active jobs, where the inner sum identifies if

a job j is active at time t and the outer sum counts all activejobs. An arbitrary job j is active at time t if it started attime τ and t occurs no later than dj after τ . Problem param-eters and decision variables are summarized in Table 3 forconvenience.

4.1 Model formulation

With the above notation in hand, we formulate the ORrescheduling problem as the following integer program.

z∗ = min αn1 + n2 (1)

Subject to:∑

t

tyj,t + djpjk − Tpkj

≤∑

t

tyk,t , ∀j, k such that μ(j) = μ(k) (2)

pjk + pkj = 1, ∀j, k such that μ(j) = μ(k) (3)

n1 + n2 ≥ ht , t = 1, · · · , αT (4)

n2 ≥ ht , t = αT + 1, · · · , T (5)

ht ≥∑

j

∑

τ :τ≤t≤τ+dj

yj,τ , t = 1, · · · , T (6)

∑

t

yj,t = 1, ∀j (7)

pjk ∈ {0, 1}, ∀j, k such that μ(j) = μ(k) (8)

yj,t ∈ {0, 1}, ∀j, t (9)

The objective Eq. 1 minimizes staffing cost, i.e. the totalnumber of staffed ORs after weighting the shorter stafflengths by a factor α. Constraints Eq. 2 can be explainedas follows. Suppose jobs j and k belong to the same sur-geon. Then, either pjk or pkj must equal 1 (from ConstraintEq. 3). Suppose pjk = 1. Then, Constraint Eq. 2 ensuresthat sj + dj ≤ sk because sj = ∑

t tyj,t and sk = ∑t tyk,t .

Conversely, if pkj = 1, then Constraint Eq. 2 reduces tosj − T ≤ sk , which is trivially true because sj ≤ T andstart times are non-negative. Constraints Eq. 2 thus enforcea non-overlapping ordering of job start times if they belongto the same surgeon. Constraints Eq. 6 count the numberof active jobs at each time t and Constraints Eqs. 4 and 5ensure that the number of ORs needed is at least equal to themaximum of ht across all t . Constraints Eq. 7 are needed toensure that each job is assigned a start time. Finally, Con-straints Eqs. 8 and 9 require that pjk and yj,t must be binaryvariables.

The OR rescheduling problem Eqs. 1 – 9 is NP hardbecause upon ignoring constraints Eqs. 2 and 3 and settingα = 1, we obtain the well known bin-packing problem.Therefore, we focus in this paper on developing a lowerbound with a performance guarantee, which is utilized in abranch-and-bound algorithm.

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Table 3 Notation Used inModel Formulation Parameters

αT , T = shift lengths, α ≤ 1

t = time index, t ∈ {1, · · · , T }m = number of jobs (surgeries) scheduled on the tagged day

J = job index set, J = {1, · · · , m}dj = planned case length of job j

s0j = originally scheduled start time of job j

μ(j) = index of the surgeon who performs job j

J (i) = set of jobs that are performed by surgeon i

d�(i) = ∑j∈J (i) dj = the sum of case lengths of surgeon i

Decision variables

ni = number of type-i shifts used after rescheduling, i = 1, 2

yj,t = 1 if job j starts at time t , 0 otherwise

sj = new start time of job j

pjk = 1 if jobs j and k are performed by the same doctor and j is

scheduled before k, 0 otherwise

ht = the minimum number of ORs that need to be staffed at time t

z∗(J ) = minimum cost of serving jobs in set J

Other notation

L = lower bound cost

z = cost associated with a feasible solution, L ≤ z∗ ≤ z

5 Analytical approach: Lower and upper bounds

Our approach consists of three steps. In the first step,we develop a classification of surgeon types based onwhether or not their cases can be divided into two connectedsequences such that each sequence is no longer than half ofa shift length. We do not differentiate between those sur-geons who have block assignments and those who do not.Taking advantage of the surgeon classification, we constructa staffing cost lower bound L in the second step. Finally,in the third step, we develop a procedure for recovering afeasible schedule z from the lower bound construction suchthat z ≤ (3/2)L, which immediately leads to the conclu-sion that the constructed lower bound is at least (2/3) ofthe optimal solution. Put differently, we use the argumentthat L

z∗ ≥ Lz

≥ 2/3. In this section, we present details ofour analysis with one shift type. When there are two shifttypes, the proof of the lower bound’s performance guar-antee requires three cases to be considered separately: (i)α ≤ 1/2, (ii) 1/2 < α < 2/3, and (iii) α ≥ 2/3.In each case, we define surgeon types, then develop a LBconstruction algorithm, and finally an approach to convertthe LB into a feasible solution that is at most (3/2) of thelower bound. Note that both the lower bound and the fea-sible solution costs are calculated in terms of the weightedequivalent number of ORs used. Details of the analysis with

two shift types are presented in the Online Supplement,Appendix E.

5.1 Step 1: Surgeon types

Suppose Y is a set of q ≥ 1 jobs with indices {j1, · · · , jq},then a chain of jobs in Y satisfies the property that sjk

+djk

= sjk+1 , where sj1 is arbitrary. In other words, any arbi-trary connected sequence of jobs is called a chain. Note thatY could be either all jobs of a particular surgeon, or a subsetof his or her jobs, and that sjq + djq < T .

Definition 1 A chain of jobs in Y is called an O-chain withrespect to shift length T if upon splitting the chain in themiddle, i.e. at a point t = (sjq + djq + sj1)/2, one of thefollowing two properties holds (1) either no job is cut intotwo pieces, or (2) if a job is cut, then upon taking the jobthat is cut and assigning it to either the first or the secondpiece of the chain, both sides of the chain are no longer than(T /2) in at least one of the two assignments.

From the above definition, it should be clear that if an O-chain is split at a point that is not the midpoint of the chain,and if the job that is cut (if any) is combined with either oneof the two pieces of the chain, then at least one of these twopieces (after combining the cut job) must be no more than


Table 4 Additional NotationChain = a connected sequence jobs

P2||Cmax(J ) = the two-machine makespan-minimization problem with job set J

C2(J ) = the optimal value of P2||Cmax(J )

A-type surgeon = surgeon i whose jobs satisfy the property: C2(J (i)) > T/2

O-type surgeon = surgeon i whose jobs satisfy the property: C2(J (i)) ≤ T/2

Sk = the index set of k-type surgeons, where k ∈ {A, O}nk = number of k-type surgeons, where k ∈ {A, O}

(T /2). This is an important property of O-chains that weuse later in this paper.

Definition 2 P2||Cmax(Y ) refers to a two-machine mini-mum makespan problem ([33]) for job set Y . In the mini-mum makespan problem formulation, there are no overlapavoidance constraints, such as constraints Eqs. 2 and 3 in theOR rescheduling problem. The optimization problem canbe written as C2(Y ) = min Cmax, subject to

∑j xij dj ≤

Cmax, i = 1, 2,∑

i xij = 1, ∀j ∈ Y , xij ∈ {0, 1}, ∀i, j .The decision variable xij equals 1 if job j is assigned tomachine i, and 0 otherwise.

We use P2||Cmax(Y ) to identify those surgeon typeswhose jobs can be arranged in an O-chain. Note thatP2||Cmax(Y ) is also NP hard ([34]). However, in the ORrescheduling context, we find that surgeons who performmultiple surgeries on a particular day perform a relativelysmall number of surgeries (typically in single digits, seeTable 2 in Section 3) and pseudo-polynomial algorithmsexist for solving such problems (for example, via a dynamicprogramming algorithm for an equivalent knapsack prob-lem with size (d�/2) – see [35]). Therefore, in the intendedapplication of our approach, the P2||Cmax problem thatarises is easy to solve.

Definition 3 Consider an arbitrary surgeon indexed i withjob set J (i). This surgeon is referred to as an A-type if andonly if C2(J (i)) > T/2. Similarly, a surgeon is O-type ifand only if C2(J (i)) ≤ T/2.

Clearly, a surgeon may be either A-type or O-type, butnot both. The importance of this surgeon classification isthat if a surgeon is A-type, then there does not exist anO-chain of his or her jobs. Conversely if a surgeon is O-type, then there must exist at least one O-chain of his orher jobs. We prove this preliminary result in Lemma 1, butbefore doing so, we summarize the additional notation usedin this Section in Table 4. In this table, we introduce nota-tion Sk to denote the index set of k-type surgeons and nk todenote the number of k-type surgeons, where k ∈ {A, O}.

A proof of Lemma 1 is provided in the Online Supplement,Appendix C.

Lemma 1 Given a surgeon i with job index set J (i), thefollowing statements are true.

1. If surgeon i is A-type, then there does not exist an O-chain of jobs in J (i).

2. If surgeon i is O-type, then there exists at least one O-chain of jobs in J (i).

5.2 Step 2: Lower bound construction

A key step in the construction of the lower bound involvesarranging surgeons’ jobs in a chain and filling them in avail-able empty spaces of previously activated operating roomsin a fluid fashion. We refer to this step as fluid filling. Essen-tially, this means that we use all open time in a staffed roombefore choosing to staff more rooms and do not worry aboutthe fact that this procedure may cause a particular surgeon’schain to be split, i.e. placed in more than one room. Splittingmay cause a conflict, which means at least one job is placedin multiple rooms and/or some same-surgeon jobs overlap.We focus in Section 5.4 on eliminating all splits, and thuseliminating all conflicts. In the LB construction algorithm,shown in a graphical form in Fig. 1, we ignore splits.

5.3 LB algorithm

Step 1: Arrange A-type surgeons’ jobs in arbitrary chainsand place them in separate rooms, using nA

rooms. Each room may have some unused time.The remaining surgeons are all O-type surgeons.

Step 2: Arrange O-type surgeons’ jobs in arbitrary O-chains and assign these chains one at a timeto available rooms in a fluid fashion. Use extrarooms as needed if spaces left in nA rooms are notenough to fit jobs of all O-type surgeons.

A count of the number of shifts needed is L :=nA +

⌈∑j /∈SA

dj −(nAT −∑j∈SA

dj )

T

⌉+, where the notation �·�

F. Li et al.

Fig. 1 LB Construction Example (splits are shown by dotted lines)

denotes the integer ceiling of its argument. In order to provethat L is a valid lower bound, we first prove that at least onejob of an A-type surgeon must cross (T /2) in any feasibleassignment of his or her jobs. This is a crucial step becauseit immediately implies that each A-type surgeon requires atleast one room.

Lemma 2 If a surgeon is A-type, then in any feasible solu-tion, one of the surgeon’s jobs crosses (T /2). That is, theremust exist one job jk such that sjk

< T/2 < sjk+ djk

.

Proof We prove the result by contradiction. Suppose thereis no sjk

such that sjk< T/2 < sjk

+ djk. Then, (T /2)

divides the surgeon’s jobs into two non-overlapping parts.Each of these parts need not be scheduled in a single room.In one part, each job starts and ends before (T /2) and inthe other part, each job starts and ends after (T /2). Thisimplies that C2(J (i)) ≤ T/2 and contradicts the definitionof A-type surgeons. Hence proved.

L is a valid lower bound because at least nA rooms areneeded for A-type surgeons and O-type surgeons’ jobs areassigned in a fluid manner. Lemma 3 presents this resultwithout a formal proof.

Lemma 3 L = nA +⌈∑

j /∈SAdj −(nAT −∑

j∈SAdj )

T

⌉+is a

valid lower bound.

5.4 Step 3: Feasible solution construction

Next, we obtain a feasible solution from L that uses nomore than (1/2)L more ORs. Our main result is presentedin Theorem 1 below.

Theorem 1 L is a (2/3)-lower bound. Specifically, thereexists a feasible solution z such that L ≥ (2/3)z ≥ (2/3)z∗for every instance of the OR rescheduling problem.

Proof The LB algorithm causes at most (L − 1) O-typesurgeons’ chains to be split (see Fig. 1 for an example).We show next that splits can be removed by consideringthe following three cases. In these arguments, μ denotesan arbitrary O-type surgeon whose O-chain is split by thefluid-filling routine.

1. |J (μ)| = 1, i.e. surgeon μ has only one job, labeled k.Because C2(J (μ)) ≤ T/2, it follows that job k can bescheduled in a new room and it will occupy no morethan (T /2) of that room’s time. That is, we can elimi-nate the assignment conflict of one room by adding atmost (1/2) more room. This is shown graphically inFig. 2.

2. |J (μ)| > 1 and surgeon μ’s O-chain is split at leasttwice (i.e. occupies time in at least three different ORs).In this case, we take all jobs in J (μ) and schedule themin a new room. This resolves potential scheduling con-flict of at least two rooms, each of which would havecontained pieces of O-chain of the same surgeon. Thus,for each room whose assignment conflict is resolved,this step adds at most (1/2) extra room. An exampleshowing the LB and feasible solution construction whena surgeon’s O-chain is split twice is shown in Fig. 3.

3. |J (μ)| > 1 and surgeon μ’s O-chain is split only once.If the split does not cause a job to be cut, then conflictmay arise because surgeon μ’s jobs may overlap. Suchconflict can be avoided relatively easily by schedulingthe two pieces of surgeon μ’s jobs at opposite ends of

Fig. 2 Feasible Solution Construction when |J (μ)| = 1


Fig. 3 Feasible Solution Construction When Surgeon μ’s Chain is Split Twice

the two rooms. Next, we consider the case in whichsplitting causes a job to be cut.

Using Definition 1 and the discussion that followsthis definition, we can argue that upon taking the splitjob and combining it with one of the two pieces of theO-chain, at least one piece must be no more than (T /2).We remove the piece that is less than (T /2) and assignit to a new room, utilizing at most (1/2) extra room toresolve the conflict – see example in Fig. 4. Moreover,we schedule this surgeon’s jobs at the two ends of theORs that contain his or her jobs to avoid overlap.

In all cases discussed above, the task of turning the surgeryschedule of a room into a feasible schedule adds at most halfextra room. That is, we require at most � 1

2 (L−1)� additionalroom to obtain a feasible assignment, which establishes ourclaim. �

The above procedure gives us a (3/2)-approximationalgorithm. We use this approach to generate the initial feasi-ble solution in our implementation of the branch-and-boundalgorithm.

6 The branch and bound (B&B) algorithm

We modify the standard branch-and-bound algorithm toaccount for two shift types. A pseudo code for this algo-

rithm is provided in the Online Supplement, Appendix D.We first obtain an upper bound on the number of long shiftsthat can be used. A theoretical bound is the number of sur-geons who operate on any given day. However, because wealready have a feasible solution, a practical bound is pro-vided by either (1) the number of shifts that the existingschedule uses, or (2) the maximum number of long shiftsavailable in a particular hospital. For a fixed upper limit, n,of the number of long shifts, we run the branch-and-boundprocedure at most (n + 1) times: once for each iterationindex i, where i goes from 1 to (n + 1), and in the i-thiteration, we fix the first (i − 1) shifts to be long shifts. Fur-thermore, during execution of the algorithm, if we find thatin the k-th run there is a feasible solution in which the first(k − 1) long shifts accommodate all work, then we do notneed to consider additional iterations. Note that which shiftindices are assigned to long shifts and which are assignedto short shifts is not relevant because the B&B algorithmexhausts all possible assignments with (i − 1) long shifts initeration i.

Each iteration finds an optimal assignment with a fixednumber of long shifts. Within each iteration, we use back-tracking to undo a recently assigned job and place it at theend of the queue of available jobs. Backtracking is per-formed when (1) there is no feasible assignment of thecurrent job either in existing rooms or in a new roombecause of same-surgeon overlap, or (2) the sum of the cur-

Fig. 4 Feasible Solution Construction When Surgeon μ’s Chain is Split Exactly Once

F. Li et al.

rent partial assignment’s cost and the lower bound on thecost of assigning the remaining jobs is not strictly smallerthan the current best feasible solution. Note that we cal-culate a lower bound cost for the remaining jobs at eachjob assignment epoch. In this way, backtracking searchesall possible assignments for a fixed number of long shiftswithin each iteration.

The algorithm terminates at each iteration if either (1) theglobal lower bound calculated at the beginning of the iter-ation is achieved by a feasible solution, or (2) all possibleassignments are exhausted, or (3) the maximum number ofsteps is reached. We set the maximum number of steps equalto 100,000. Our implementation of the algorithm requiredaverage, standard deviation and maximum run times of(158, 231, 686), (23, 111, 865), and (4, 27, 243) seconds forthe three hospitals data when run on a PC with 2.40 GHzprocessor and 4 GB of RAM.

7 Numerical experiments and insights

We implemented our approach on data from the three hospi-tals and tabulated two types of impacts: (1) on staffing costs,and (2) on surgeons. These experiments reveal the essentialtradeoffs for hospitals considering gainsharing with physi-cian groups to realize staffing cost reductions. We beginwith the results related to efficiency (hospital perspective),which are presented in Table 5.

In Table 5, we first calculate efficiency gains from non-urgent cases only. Later, we consider the combined effectof both urgent and non-urgent cases. Because the datacontained instances in which urgent cases were “fitted”in open time between non-urgent cases, and this was notpossible after rescheduling (which created a more tightlypacked schedule), we included the cost of staffing dedi-cated rooms for urgent and emergent cases in the second

Table 5 OR Efficiency Metrics


Non-Urgent Only Planned Realized Planned Realized Planned Realized

Utilization (before) 65.24 60.41 54.06 50.68 49.34 45.66

95 % C.I. [64.35,66.13] [59.53,61.29] [53.05,55.06] [49.72,51.64] [48.39,50.30] [44.79,46.55]

Utilization (after) 88.22 80.83 86.78 75.61 83.55 69.15

95 % C.I. [87.66,88.80] [80.18,81.48] [85.74,87.88] [74.35,76.88] [82.59,84.52] [76.67,78.83]

# of staffed ORs (before) 11.9, 1.6* (14.3) 8.3 7.7

95 % C.I. [11.5,12.1], [1.5,1.8]* ([14.0,14.3]) [8.2,8.6] [7.6,7.8]

# of staffed ORs (after) 4.0, 4.6* (10.9) 5.7 4.5

95 % C.I. [3.5,4.3], [4.3,4.9]* ([10.5,11.2]) [5.5,5.8] [4.3,4.6]

Daily Overtime (before) 281 305 36

95 % C.I. [254,309] [276,334] [29,44]

Daily Overtime (after) 133 321 280

95 % C.I. [110,155] [288,354] [254,307]

$ Savings/day 29,970 18,360 23,310

95 % C.I. [28,544, 31,391] [17,563, 18,135] [22,021, 24,637]

With Urgent Cases

# of staffed ORs (before) 10.7, 3.2* (15.5) 9.0 8.7

95 % C.I. [10.4,11.0], [3.0,3.3]* ([15.1,15.8]) [8.8,9.2] [8.4,8.8]

# of staffed ORs (after) 6.0, 4.6* (12.9) 6.7 5.5

95 % C.I. [5.8,6.2], [4.5,4.7]* ([12.6,13.2]) [6.5,6.9] [5.4,5.6]

Daily Overtime (before) 281 + 170 = 451 305 + 60 = 365 36 + 70 = 106

95 % C.I. [432, 469] [340, 381] [99, 113]Daily Overtime (after) 133 + 380 = 513 321 + 90 = 411 280 + 200 = 480

95 % C.I. [490, 532] [392, 432] [456, 502]$ Savings/day 17,325 15,525 20,385

95 % C.I. [16,316, 18,147] [14,615, 18,147] [19,411, 21,309]

$ Savings/day are based on $15/minute of regular OR time and $22.5/minute of overtime∗Entries marked with an asterisk show ORs with long shiftsNumbers in parentheses show equivalent number of 8-hour shift


part of our analysis. In Table 5, “before” refers to statisticsbased on data obtained from the hospitals and “after” refersto similar statistics obtained after applying our reschedul-ing algorithm. After each metric, we also report the 95 %confidence intervals. Note that all three hospitals exhibitsubstantial decrease in staffed OR requirements for non-urgent cases and concomitant gains in utilization. Plannedutilization gains range from 23 to 34 percent with Hospitals2 and 3 showing above 30 percent gains. The realized uti-lization is calculated using actual case lengths as opposedto planned case lengths. In these calculations, we includeddelays that were caused in the original schedule by the sur-geon arriving late. However, delays that were caused by thesequence of surgeries were recalculated based on the newsequence. We also kept day-of-surgery cancelations intactwhen calculating the effect of rescheduling.

The realized utilization gains are smaller. The differencecomes from the fact that the new schedule uses significantlyfewer rooms. Therefore, idleness introduced in the revisedschedule by late surgeon-arrival, sequence-related delays,and surgeries completing earlier than planned, occupy amuch greater percent of the total staffed time. The numberof staffed ORs are calculated using the planned case lengthsand we count the amount of overtime that would be neededto accommodate non-urgent cases in the original and revisedschedules. Savings are counted only when an OR is notstaffed for the entire day and overtime costs are subtractedfrom such savings. The numbers we report are average dailysavings.

The last five rows of Table 5 show the impact of consider-ing urgent and emergent cases. Using a simple local search,we find the fixed number of dedicated ORs that would min-imize the cost of scheduling urgent and emergent casesfor each hospital. Hospitals would commit to staffing theserooms and their staffing costs would be incurred regardlessof realized urgent/emergent demand. The optimal numberof dedicated rooms was 2 (8-hour shifts) for Hospital, 1(8-hour shift) for Hospital 2 and 1 (10-hour shift) for Hos-pital 3. Urgent cases are scheduled as compactly as possiblein the order of arrival. Cases that cannot be accommo-dated in dedicated rooms are scheduled as add-on cases atthe end of shift and incur overtime charges. Note that pro-jected savings decline, but remain substantial nonetheless.Notwithstanding potential cost savings, dedicated ORs forurgent/emergent cases may also improve health outcomesbecause urgent cases no longer have to wait until a suitableopening in the existing schedule of non-urgent cases (see[7]).

For Hospital 1, we observe a decline in the number of8-hour ORs needed after accounting for urgent cases in theoriginal schedule. This happens because of the manner inwhich 8- and 12-hour shifts are ascertained – see detailsin Appendix B in the Online Supplement. Specifically, we

determine if an OR was open for 12 hours by tallying thework scheduled from the 8th hour to the 12th hour. Duringthis 4-hour period, if at least 2 hours of work is scheduled,then the OR is considered to be open for 12 hours; otherwiseit is considered to be open for 8 hours, and work scheduledbeyond the 8-th hour is considered overtime. Upon addingurgent cases, some shifts that were earlier deemed 8-hourshifts were converted to 12-hour shifts causing a decline in8-hour ORs.

Table 5 shows that the effect on the use of overtime fornon-urgent cases is quite different in the three hospitals.Overtime use decreases in Hospital 1, remains about thesame in Hospital 2, and increases in Hospital 3. This canbe explained based on structural differences among thesehospitals. Hospital 1 has the ability to use long shifts. There-fore, by planning to staff more ORs with long shifts, as ouralgorithm recommends, it can reduce the use of overtimewhile at the same time reducing the requirement to staff alarge number of concurrent rooms. Hospital 2 does not havethis flexibility and its use of overtime increases as one wouldexpect. Hospital 3 has long open times between surgeries inthe original schedule. This results in an unusually low over-time usage in the original schedule. Such open times areeliminated by our algorithm, resulting in overtime use thatis similar to that in other hospitals.

One of the key structural differences among the threehospitals is the relative size of open intervals between sched-uled cases in the data. We find that Hospital 3 tends toleave larger intervals open. This difference explains, tosome extent, the differences in the realized performance ofthe rescheduling algorithm. We illustrate the differences byplotting the proportion of total idle time (in the plannedschedule) that is accounted for by a certain count of openintervals, after these intervals are arranged from the longestto the shortest – see Fig. 5, which shows the distribu-tion of open intervals. We find that Hospital 1 requires a

Fig. 5 Distribution of Open Intervals in Planned Schedule

F. Li et al.

stochastically larger number of intervals to achieve the sameproportion of idle time within its schedule. Put differently, ittends to leave open small intervals of unused time betweenprocedures. In contrast, Hospital 3 leaves larger chunks ofopen time and Hospital 2 lies somewhere between thesetwo.

How are the number of staffed ORs affected by therescheduling procedure? Table 5 suggests that surgeries arepacked more efficiently and that Hospital 1 will need morelong shifts. In order to provide greater insight into howgreater efficiency is realized, we plot the profile of numberof ORs in use in each 15-minute interval of the day from7:30 AM till 7:30 PM in Fig. 6. What we show here arethe average number of ORs in use based on original (dash-dotted lines) and rescheduled start times (solid lines) andactual surgery durations. We also show 95 % confidence

intervals because the actual usage of ORs changes from oneday to the next.

A common trend across all hospitals is the flattening ofOR-use requirements. Rescheduling creates more uniformutilization of ORs throughout the day, which allows the hos-pital to staff fewer ORs concurrently and achieve greaterutilization. We also see that Hospital 1 benefits from plan-ning to open more long rooms than it currently does. Incases such as these, the hospital administration may needto work with nursing coordinators to identify new staffingmodels that enable additional long-shift rooms. Later in thissection, we also evaluate the potential benefit to Hospitals 2and 3 from using both long and short shifts.

We turn our attention next to the effect of rescheduling onsurgeons. We developed several surgeon-sensitive metrics tocompare before and after rescheduling results. These results

Fig. 6 Number of ORs In Use by Time of Day. Dash-dotted lines show original schedules and solid lines show revised schedules


are summarized in Table 6 for non-urgent cases. Becauseurgent cases would be dealt with differently after reschedul-ing, we did not include urgent cases in these calculations.OR managers may be concerned that a denser packing ofsurgical procedures may lead to greater surgeon delays. Wecalculate three types of delays. Type 1 delays occur whenthe affected doctor is delayed by a late finish of a precedingprocedure performed by a different surgeon, Type 2 delaysoccur when the affected doctor is delayed but he or sheperformed the previous case in a different OR, and Type 3delays occur when the affected doctor is running late foran earlier procedure performed by the same surgeon in thesame room. Clearly, Type 1 delays are more serious fromdoctors’ perspectives than Type 2 or Type 3 delays. Calcu-lations reported in Table 6 show that doctors operating inHospital 1 and 2 can expect greater frequency of Type 1delays, but the average number of minutes delayed will besmaller. This happens because our algorithm schedules pro-cedures performed by doctors with multiple cases and longdurations in the same room. For reasons explained in Fig. 5,Hospital 3 is different. Doctors in that hospital will experi-ence relatively more Type-1 delays because in the originalschedule, they experience very small delays on accountof having large chunks of unused times between sched-

uled procedures. The differences between mean delaysare statistically significant (p-values ≈ 0 in all threecases).

The effect on Type 2 and Type 3 delays are quite differ-ent. Generally, both the frequency and mean delays increaseupon rescheduling. Similarly, the total time that a doctorspends performing surgeries (from the start time of theirfirst case to the end time of their last case), which we callspread, increases as a result of rescheduling. Upon per-forming statistical tests, we found the mean differences tobe statistically significant between before and after meanspreads across all hospitals. We also find that reschedulingwill cause a significant proportion of doctors to either reportearlier or later than the first case in their original sched-ule. These statistics are reported under the row headings“Early Report Time” and “Late Report Time”. The statis-tics show the percent of surgeons’ cases affected and theaverage number of minutes by which cases may be resched-uled earlier or later than their original schedule, if they aremoved. As a consequence of increased spread time, sur-geons have more idle time in between surgical procedures.For an arbitrary surgeon, the average increase in idle timewas 32 minutes, 74 minutes, and 58 minutes in the three hos-pitals. We found that the average daily increase in idleness

Table 6 Impact on Surgeons (non-urgent cases only)


Before After Before After Before After

Type-1 Delay

Count (%) 557 (8.7 %) 1352 (21.2 %) 300 (3.7 %) 3023 (37.7 %) 85 (1.0 %) 1470 (17.25 %)

Avg (min) 95 80 88 78 53 124

Type-2 Delay

Count (%) 931 (14.6 %) 709 (11.1 %) 395 (4.9 %) 2216 (27.6 %) 178 (2.1 %) 616 (7.2 %)

Avg (min) 45 220 62 193 46 244

Type-3 Delay

Count (%) 676 (10.6 %) 442 (6.9 %) 2411 (30.1 %) 731 (9.1 %) 1586(18.6 %) 1716 (20.1 %)

Avg (min) 36 32 51 84 79 97

Spread

Avg (min) 248 280 250 324 279 337

SD 149 173 151 198 157 232

Early Report Time (planned)

Count (%) 1465 (37.0 %) 1386 (60.1 %) 765(23.7 %)

Avg (min) 174 114 165

Late Report Time (planned)

Count (%) 1688 (45.7 %) 753 (32.6 %) 1316(40.8 %)

Avg (min) 207 117 198

Avg = average, SD = standard deviationType-1: Same room different MD; Type-2: Same MD different room; Type-3: Same room same MD

F. Li et al.

across all surgeons amounted to 249 minutes in Hospital 1(with SD = 577 minutes), 272 minutes in Hospital 2 (SD= 570 minutes) and 376 minutes in Hospital 3 (SD = 343minutes).

Doctors often decide the sequence in which they prefer toperform surgeries on their OR day. Rescheduling may pro-duce an undesirable sequence. However, if a doctor’s casesare scheduled consecutively, it will be possible for that doc-tor to rearrange the sequence of his or her surgeries withoutaffecting the overall schedule. When a doctor has multiplelong cases, our algorithm favors placing that doctors’ jobsin the same room. In order to calculate the flexibility thata hospital will have to re-sequence surgeries according to adoctor’s wishes after running our algorithm, we calculatedthe percent of total surgery durations that occur in connectedsequences. Connected means that the procedures are doneby the same doctor and are placed consecutively in the sameroom. We found that in Hospital 1 and 2, 53.3 and 81.2 per-cent of surgery durations occurred in connected sequencesin the original schedule. In contrast, after running our algo-rithm, these percentages were 59.9 and 86.9, respectively.Therefore, for these two hospitals, there will be flexibilityto re-sequence surgeries if desired. Hospital 3 is once againdifferent. In that hospital, 84.4 percent of surgery durationswere in connected sequences in the data, whereas our algo-rithm produces a schedule in which 65.4 percent of thesurgery durations occur in connected sequences. We believethat these differences relate to the way in which Hospital 3scheduled cases and the case-mix of doctors who performsurgeries at that hospital.

We used our approach next to gain deeper understandingof the relative benefit of having two shift types – one longand one short, and of not placing all cases of a surgeon inone room. Such concerns and questions have arisen repeat-edly in our conversations with OR managers. In the interestof brevity, this material is presented in Appendix F of theOnline Supplement.

The potential savings from rescheduling need to beweighed against the impact on surgeons. Across the threehospitals, daily savings are sufficiently high that we believeit is not inconceivable that doctors will find it attractiveto allow more flexible scheduling of their cases. The threehospitals may have available up to $70, $57, and $54 perphysician idle minute, respectively, to share with physiciansor alternatively incur as additional cost of having salariedphysicians idle. The exact details of gain-sharing plan needto be worked out separately in each situation because hos-pitals are likely to have a mix of independent and employedphysicians. It is also possible to place additional constraintson the degree to which case start times may be changed.That will reduce the extent of savings, but may lead togreater doctor participation. We discuss extensions of ourwork in the next section.

8 Extensions and concluding remarks

Practitioner considerations may lead to alternate formula-tions and further extensions of our work. Surgeons may wishto have all of their cases scheduled within a short time win-dow, i.e. without too many breaks in between so they canutilize their time more effectively. A surgeon may also wishto have all of his or her cases scheduled either in the AM orthe PM block if the total duration is no more than 4 hours.We refer to such constraints as spread constraints. Ourbranch-and-bound algorithm can deal with such constraints,and our lower bounds will be valid, but its worst-case per-formance will be reduced to (1/2) from (2/3). The key toobtaining a bound with guaranteed performance is that eachtime a chain is split, we place the chain (which includes allof a surgeon’s jobs) into a new empty room. This way, werecover feasibility by using at most twice as many roomsas in the lower bound. Investigation of better ways of con-structing lower bounds and feasible solutions are topics forfuture research.

Some hospitals have specialized equipment in somerooms, but not all rooms, which gives rise to a constraint thatcertain cases can be scheduled only in some rooms. If therooms with specialized equipment are not used for routinecases, then the problem of rescheduling cases can be dividedinto two separate problems and solved using our method-ology. However, when rooms with specialized equipmentare also routinely used for cases that do not require suchequipment, the problem of rescheduling cases remains achallenge. Similarly, some hospitals have limited copies ofmovable equipment that they wheel from one OR to another.In this case, it would be necessary to make sure that thenumber of concurrently scheduled cases that require a par-ticular piece of equipment do not exceed the number ofavailable pieces of that equipment, creating an additionalnon-overlapping constraint. Such constraints are also diffi-cult to deal with. In both scenarios, our branch-and-boundalgorithm and lower bounds will remain valid, but the worst-case performance guarantee will not apply. We believe suchproblem settings provide important areas for future work.

Consistent with common practice, we assume that at thetime when surgeries are rescheduled, the hospital does notconsider using strategic overtime. In some instances, it maybe more economical to use a small amount of overtime andavoid staffing a room for the entire shift length. Reschedul-ing with the use of strategic overtime is a hard problem,which requires a great deal of information about work rulesand availability of scheduled overtime. One of the primaryreason why hospitals do not consider strategic overtimeis that rescheduling is done at least two days before thesurgery date. Many more surgeries will be booked after therescheduling is done, which may use open time in staffedrooms and also lead to the use of overtime anyway. That is,


there is a potential that the empty space in an OR that isnot well utilized will be required for other surgeries that arescheduled late. If we allow strategic overtime, our theoret-ical bounds may not remain intact. We believe consideringextensions of our model with strategic overtime is anotherarea for future research.

The analysis presented in this paper leads to severalmanagerial insights. First, it shows that significant improve-ments in OR utilization are possible. Hospitals that areable to obtain cooperation from their surgeons can increasecase volumes with the same number of ORs and lowerstaffing costs, or open up block time for additional sur-geons. Second, our analysis identifies patterns of surgicalcase durations that should be placed in a single OR and thosethat may be spread across multiple rooms. These patternscan be explained to OR schedulers and may lead to betterinitial schedules. Third, the analysis shows that the use ofan appropriate number of long shifts is beneficial. In partic-ular, Hospital 1 in our data sample used two shift lengths.Upon rescheduling, we found that Hospital 1 realized thegreatest efficiency gains, which is likely due to the fact thatour algorithm selected an optimal number of long shifts. Atake away for hospital executives is to determine the opti-mal mix of short and long shifts, and to incentivize staff towork long shifts.

Acknowledgments The authors are grateful to two anonymous ref-erees for their constructive comments on earlier versions of this paper.This material is based upon work supported (in part) while DiwakarGupta was serving at the National Science Foundation as programdirector for the Service Enterprise Systems and Manufacturing Enter-prise Systems programs. Any opinions, findings, and conclusions orrecommendations expressed in this material are those of the authorsand do not necessarily reflect the views of the National ScienceFoundation.

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