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Scheduling doctors' appointments: optimal andempirically-based heuristic policiesLAWRENCE W. ROBINSON a & RACHEL R. CHEN aa Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853-6201, USAE-mail: LWR2@cornell.edu or RC72@cornell.eduPublished online: 29 Oct 2010.

To cite this article: LAWRENCE W. ROBINSON & RACHEL R. CHEN (2003) Scheduling doctors' appointments: optimal andempirically-based heuristic policies, IIE Transactions, 35:3, 295-307, DOI: 10.1080/07408170304367

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Scheduling doctors’ appointments:optimal and empirically-based heuristic policies

LAWRENCE W. ROBINSON* and RACHEL R. CHEN

Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853-6201, USAE-mail: LWR2@cornell.edu or RC72@cornell.edu

Received May 2001 and accepted February 2002

Consider the problem facing a doctor (or other service provider) who is setting patient appointment times in the presence ofrandom service times. He or she must balance the patients’ waiting times (if the appointments are scheduled too closely together)against the doctor’s idle time (if the appointments are spaced too far apart). Although this problem is fairly intractable, this paperuses the structure of the optimal solution as the basis for a simple closed-form heuristic for setting appointment times. Over a widetest bed of problems, this heuristic is shown to perform on average within 2% (and generally within 0.5%) of the optimal policy.

1. Introduction

Everyone who has visited their doctor (or a health clinic)has had plenty of time to observe that the scheduledappointment times can be only marginally related to thetime at which they actually see the doctor. Anecdotesabout patients billing the doctor for the value of theirwaiting time periodically surface. The problem of ap-pointment scheduling (setting appointment times thatbalance the patients’ waiting time against the doctor’s idletime) is thorny and largely unsolved. Even with the as-sumption of timely patient arrivals, this problem is stilltoo intractable to be solved optimally, apart from a fewspecial cases (e.g., two patients, phase-type service timedistributions). There is also a dearth of generally appli-cable heuristics which are accessible to practitioners. Al-most all of the published heuristics are tested only againstother heuristics, and not against the optimal policy. Mostdo not even recognize that the means of the service timescan be eliminated from the formulation, and so can yieldarbitrarily poor performance.In this paper we use the optimal appointment times as

the basis for a very simple closed-form heuristic, whichcan be quickly implemented using only a calculator. Ourmodel is quite general, and can accommodate arbitrarymean service times for every patient, as well as an arbi-trary common standard deviation for the set of patients.The standardized service time distribution is derived from

an empirical distribution of surgery times. We show thatthis heuristic generally averages within 0.5% of the costof the optimal policy, with worst-case performance usu-ally within 20% of the optimal. We also show that ourheuristic’s performance is fairly robust with regards todistributional misspecification.More formally, consider setting the appointment times

for a sequence of n customers who need to be seen by aservice provider over a period of time. Examples of suchsystems include patients and a medical practitioner, doc-tors and an operating room, clients and a consultingprofessional (lawyer or accountant), automobiles and aservice center, tractor-trailers and a receiving bay, legalcases and a courtroom, or even students and a professor’soffice hours. For convenience and clarity, we will use theparadigm of a daily sequence of patients seeing a doctorthroughout this paper, although the applicability of thisproblem is considerably broader. Because the patients’service times are uncertain, the problem of setting ap-pointment times is not a trivial one. Setting them too closetogether leads to excessive patient waiting times, which isof some slight concern, even for doctors. Setting them toofar apart, on the other hand, will cause excessive doctoridle times. The optimal appointment schedule mustsomehow balance the patients’ waiting time against thedoctor’s. Further, the balance will vary over the course ofthe day: falling behind in the morning might delay patientsthroughout the day, while falling behind near closing timemight not affect anyone besides the doctor.Thus the problem is to choose the appointment times

to minimize the expected weighted sum of the waiting*Corresponding author

0740-817X � 2003 ‘‘IIE’’

IIE Transactions (2003) 35, 295–307

Copyright� 2003 ‘‘IIE’’

0740-817X/03 $12.00+.00

DOI: 10.1080/07408170390175512

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time for the patients and the idle time of the doctor. Weassume that the decision-maker (generally, the doctor)considers patient i’s time to be worth some positivefraction ai of the doctor’s time. In other words, we havedefined the dimensionality of waiting costs such that thedoctor’s time has value 1.0. In this paper we consideredthe range 0:01 � ai � 1:0, although there could certainlybe situations where ai falls outside of this range. Forexample, the Congressional barber would certainly con-sider ai � 1, whereas financial constraints might forcestate-run medical clinics to consider ai < 0:01. But ingeneral, we felt that as the decision maker, the doctor (orother service provider) would generally view his or hertime to be at least as valuable as the patients’ time, so thatai � 1:0 is reasonable.The outline of this paper is as follows. In Section 2, we

review some of the most relevant research in this area. InSection 3, we lay out the assumptions behind our problemand formulate the model. We then transform the decisionvariables from appointment times to job allowances (thetime between appointments), and rescale the variables toeliminate the means of the service times from the prob-lem. Finally, we show how Monte Carlo integration canbe used to approximate this intractable problem as astochastic linear program. In Section 4 we develop a widetest bed of problems, and model the service time by fittingthe four-parameter generalized lambda distribution to theempirical distribution of Goldman et al. (1970). We usethe form of the optimal policy as the basis for an athe-oretic closed-form heuristic appointment policy. In Sec-tion 5 we calculate the increase in cost of the heuristicpolicy relative to the optimal policy, both in terms ofexpected value (within 2%, and often 0.5%) and worst-case performance (within 60%, and often 20%). Section 6describes our overall conclusions and outlines severalpromising directions of future research.

2. Literature review

The problem of controlling arrivals to queues has beenstudied since the 1950s, using queuing, dynamic pro-gramming, and simulation methodologies. The queuing-based papers by Mercer (1960), Jansson (1966), andSabria and Daganzo (1989) all study the steady-statebehavior of a single server queue with an infinite numberof identical customers and constant interarrival times.However, as pointed out by Bailey (1954), these steady-state queuing results are generally inappropriate for ap-pointment systems, which involve only finite number ofpatients, and do not achieve steady state.Brahimi and Worthington (1991) use discrete service

time distributions to formulate this problem as a Markovchain in discrete time, which allows them to compute thesystem state probabilities numerically. They use theseprobabilities to calculate various performance measures

for evaluating different appointment systems. They usethese measures to approximate the queue characteristicsunder continuous service time distributions.Because this problem is analytically intractable, a

number of authors have explored various heuristics forassigning patient appointment times. One commonscheduling rule is to break up the day into m blocks ofequal length, and then assign various numbers of patientsto arrive at the beginning of these blocks. Bailey (1952,1954) and Welch and Bailey (1952) propose ‘individualappointment scheduling rules’, under which the first kpatients are scheduled to arrive at the start of the day,with succeeding patients scheduled at intervals equal totheir expected service time l. Using manual simulation,they conclude that setting k ¼ 2 often performs well.More recently, Ho and Lau (1992) use simulation toevaluate the performance of a large number of job al-lowance heuristics.Soriano (1966) compares a number of commonly-used

heuristics for scheduling appointments in steady state andconcludes that a ‘two-at-a-time’ system, which schedulespairs of patients to arrive simultaneously, is generallybest. Blanco White and Pike (1964) allow for unpunctualarrivals, and use simulation to arrive at a similar con-clusion. Fries and Marathe (1981) generalize this modelto allow for blocks of variable sizes, and use dynamicprogramming to determine the optimal sizes of eachblock, assuming that the service times are exponentiallydistributed. Liao et al. (1993) also use dynamic pro-gramming. They assume that the service times are Erlangdistributed, and allow for dynamic scheduling over thecourse of the day as service times are revealed.The above studies assume that the service times are

identically distributed for all patients. Charnetski (1984)models heterogeneous patients, where the service time forpatient i has a mean of li and a variance of r2i . Heevaluates the performance of a heuristic which sets a joballowance of li þ zri for each patient, using simulation inhis search for the optimal value of the safety factor z.The shortcoming of all of these heuristics is that this

safety factor z is constant. (Ho and Lau (1992) do allowthe safety factor z to jump from one value to a second.)These heuristics will not generally be optimal; in particu-lar, Wang (1993) showed that the optimal job allowancetimes exhibit a ‘dome’ pattern, rather than being constant,when the service times are exponentially distributed.More recently, the literature has focused on the opti-

mal scheduling of a finite number of customers. Weiss(1990) shows that when there are only two patients, theproblem is analogous to a simple newsvendor problem.Pegden and Rosenshine (1990) consider the case in whichthe patient service times are exponentially distributed, andderive the optimal job allowances for two and three pa-tients. Wang (1993) generalizes this problem to an arbi-trary number of patients, and shows that the customerwaiting time will have a phase-type distribution, which

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allows him to find the optimal job allowances by solving aset of nonlinear equations. As was previously mentioned,he shows that for a fixed number of identical patients withexponentially-distributed service times, the optimal joballowances are ‘dome’ shaped, with higher job allowancesfor patients in the middle of the day. Wang (1997) deter-mines the optimal appointment times when the servicetime can be approximated by a phase-type probabilitydistribution. Vanden Bosch and Dietz (2001) proposemethods for scheduling appointments when customer ar-rivals are constrained to a discrete lattice of times.Robinson et al. (1996) solve the three-patient problem

with general service time distributions. However, theiranalytical approach for characterizing the optimal joballowances is intractable for more patients. They thenpropose a stochastic programming approach, whichcombines Monte Carlo integration with linear program-ming, which they use to solve the problem for up to 16identical patients with normally-distributed service times.Their results also show the dome pattern for patient al-lowance times. Denton and Gupta (2001) present a sto-chastic linear programming model for determining theoptimal appointment schedule when the patient servicetimes follow general discrete distributions. They also de-velop an algorithm that generates converging upper andlower bounds when the patient service times follow gen-eral continuous distributions.An important practical issue in all of these models is

how to determine the appropriate distribution to use inmodeling the stochastic patient service time. For reasonsof tractability, Jansson (1966), Soriano (1966), Pegdenand Rosenshine (1990), and Wang (1993) all assume thatthe service times are exponentially distributed. Robinsonet al. (1996) assume that the service times are normallydistributed. In an empirical study, O’Keefe (1985) findsthat patient service times often have a coefficient of vari-ation cv¼: r=l between 0.58 and 0.70, but demonstrate avariety of skewness and kurtosis, and so cannot always becharacterized by a single simple two-parameter probabil-ity distribution. Ho and Lau (1992) use the four-para-meter distribution developed by Schmeiser and Deutsch(1977), and observe that the ranking of the performance oftheir heuristic rules is not affected by the skewness andkurtosis of the patient service time distribution. Althoughthey conclude that only the first two moments of durationdistributions are important in determining job allowances,their conclusion is based on the relative performance of asmall set of heuristic rules, rather than on the optimalscheduling policy.Welch (1964), Goldman et al. (1970), and Brahimi and

Worthington (1991) all provide empirically-derived his-tograms of the patient service times. All three histogramsdisplay the same general form: unimodal and right-skewed. In our numerical study, we will model patientservice times as following the Generalized Lambda Dis-tribution (GLD) developed by Ramberg and Schmeiser

(1972, 1974) which matches the histogram of Goldmanet al. (1970). This histogram reflects 1000 observations, ascompared to 151 observations for Welch (1964), and 114observations for Brahimi and Worthington (1991).

3. Model assumptions and development

3.1. Model assumptions

The first and strongest assumption of our model is thatthe patients will arrive promptly at the established ap-pointment times. This is a standard assumption in thescheduling literature, and holds in many important ap-plications; e.g., hospital operating rooms and non-medi-cal professional services (lawyers and accountants).Admittedly, this assumption is less than completely rep-resentative of other real systems, particularly (in ourpersonal experience) of doctors’ offices. However, webelieve that patient lateness in these other settings islargely in response to past experiences of poor schedulingand long wait times. So if a better scheduling system weresuccessful in reducing expected waiting times, patientswould start to make more of an effort to arrive on time,as they currently do for other service providers, and thisassumption would become more realistic. As a particularexample, Schafer (1986) describes the steps he took in hispediatric practice both to ensure prompt arrivals and tomaintain his patient schedule. He reports that 99% of hispatients are seen at their scheduled appointment time.More generally, Hall (1991) describes a number of tech-niques for changing both the arrival and service processesin order to reduce delays.Note that in some situations it may be the case that

some patients arrive early, but do not mind waiting priorto their scheduled appointment times. Allowing for earlyarrivals with no waiting cost is very close to, but not quitethe same as, our assumption of prompt patient arrivals,since it would be possible for an early patient to startservice before his or her appointment time whenever thedoctor is available earlier.Finally, relaxing this assumption would raise a number

of thorny issues in addition to the game-theoretic patientarrival times discussed above. For example, with uncer-tain arrival times the sequence in which patients are seenby the doctor may differ from the sequence of their ap-pointment times. Additionally, variable arrival times arelikely to require a two-tier patient waiting cost, wherewaiting prior to the appointment time is perceived to beless onerous than waiting beyond it. These issues aresubstantial enough to warrant their own paper, which iscurrently under development.A second assumption is that the sequence of n patients

has already been specified. A reasonable rule-of-thumbfor sequencing might be to do so in order of increasingvariability of their service times, in order to minimize the

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negative consequences of uncertainty. However, the op-timality of this appealing heuristic has proved difficult toestablish. Wang (1993) showed this sequencing rule to beoptimal if the service times were exponentially distribut-ed. Robinson et al. (1996) showed that this rule is optimalfor n ¼ 3 patients facing a common distributional form(scaled by both mean and standard deviation). The per-formance of this (or any other) sequencing rule in moregeneral cases is still an open issue.Finally, we assume that the doctor is punctual, and has

no other responsibilities over the course of the day be-yond seeing these n patients. An alternative formulationmight consider additional doctors’ responsibilities to in-clude a set of ‘postponable tasks’ (e.g., returning phonecalls, arguing with HMO’s, etc.), which can occupy idletime and which must in any case be completed by the endof the day.

3.2. Model development

For each patient i, we assume that the value of his or hertime, relative to the doctor’s, is defined by some multipleai. Further, we assume that the service time for patient i,ni, is stochastic with a mean of li and a standard devia-tion of ri. We assume that these service times are mutu-ally independent. We can eliminate from considerationany patient i < n with a deterministic service time by‘collapsing’ him or her into the following patient in a pre-processing stage as:

liþ1 li þ liþ1; aiþ1 ai þ aiþ1:

Although these assumptions are sufficient for the fol-lowing problem formulation, we would like to identify atthis point the additional restrictions that will be imposedlater in the paper for the numerical testing. First, we willassume that while the mean service time li can differ fromone patient to another, the distribution of the service timeabout these differing li will be the same for the first n� 1patients. In particular, this implies that the first n� 1service times are homoskedastic, with a common stan-dard deviation of r. Note the service time of the finalpatient, nn, will not affect anyone’s waiting time, and socan follow an arbitrary distribution. Second, we will as-sume that the cost of waiting for the last n� 1 patientswill all be identical, at some common value a. Because thefirst patient will never wait, the value of his or her time(a1) can be arbitrary.For each patient i ¼ 1; . . . ; n, define

Ai ¼: appointment (arrival) time for patient i ðA1 � 0Þ;Si ¼: actual starting time of service for patient i;¼ max Ai; Si�1 þ ni�1f g i ¼ 2; . . . ; n;

S1 ¼ A1 ¼ 0:In other words, patient i will arrive at the appointment

time Ai and start service at time Si, which is the earliest

time at which both the patient is available (Ai) and thedoctor has finished the previous patient (Si�1 þ ni�1).Patient i will finish service at time Si þ ni. The waitingtime for patient i is simply Si � Ai. The doctor’s totalwaiting time is equal to the sum of the times betweenfinishing service for one patient and starting service forthe next,

Xn�1i¼1

Siþ1 � ðSi þ niÞ½ � ¼ Sn �Xn�1i¼1

ni;

since S1 ¼ 0: Equivalently, the doctor’s idle time can becalculated as the time the doctor starts examining thefinal patient (Sn), minus the total time that he or she hasspent examining the first n� 1 patients. The doctor’s costof waiting has been normalized to 1.0. Defining A ¼A2; . . . ;Anf g and n ¼ n1; . . . ; nn�1f g allows the problem tobe written as

ðP1Þ minA En Sn �Xn�1i¼1

ni

!þXni¼2

ai Si � Aið Þ" #( )

; ð1Þ

subject to

Si ¼ max Ai; Si�1 þ ni�1f g i ¼ 2; . . . ; n; ð2ÞS1 ¼ 0:

The formulation becomes much more tractable if weredefine the decision variables. Define Xi ¼: Aiþ1 � Ai tobe the job allowance time for patient i; i.e., it is the timeallotted between patients i and iþ 1. We can always re-cover Aj from X ¼ X1; . . . ;Xn�1f g as Aj ¼

Pj�1i¼1 Xi:

Also, by explicitly defining Wi ¼: Si � Ai to be thewaiting time for patient i, we can rewrite Equation (2) as

Si ¼ max Ai; Si�1 þ ni�1f g;Si � Ai ¼ max 0; Ai�1 þ Wi�1ð Þ þ ni�1 � Aif g;

Wi ¼ Wi�1 þ ni�1 � Xi�1ð Þþ: ð3ÞNoting that

Sn ¼ An þ Wn ¼Xn�1i¼1

Xi þ Wn

allows (P1) to be rewritten as

ðP2Þ minX En

Xn�1i¼1

Xi � nið Þ þ Wn þXni¼2

aiWi

" #( ); ð4Þ

subject to

Wi ¼ Wi�1 þ ni�1 � Xi�1ð Þþ i ¼ 2; . . . ; n;W1 ¼ 0:

At this point we invoke our assumption of homoske-dasticity in the service times (with r denoting this com-mon standard deviation), and rescale both the decisionvariables and random variables as:

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xi ¼: Xi � lið Þ=r;fi ¼: ni � lið Þ=r; andwi ¼: Wi=r;

this allows us to factor out the r from the model. Fornotational convenience, we redefine an an þ 1: Withx ¼: x1; . . . ; xn�1f g, we can substitute these redefinitionsinto (P2) to yield

ðP3Þ r �minxXn�1i¼1

xi þXni¼2

aiEf wi½ �( )

; ð5Þ

subject to

wi ¼ wi�1 þ fi�1 � xi�1ð Þþ i ¼ 2; . . . ; n;w1 ¼ 0;

since E fi½ � ¼ 0 by definition. Note that the objectivefunction is proportional to r, and that (P3) is indepen-dent of all of the following:

• The expected service times l1; . . . ; ln.• The common standard deviation of the service times,

r.• The probability distribution of the service time for thelast patient, nn.

• The value of time for the first patient, a1.

The formulation depends only on the transformedrandom variables ffig, which are i.i.d. with a mean ofzero and a standard deviation of one. In essence, we aredefining the unit of time such that r ¼ 1. The final servicetime nn will not affect the waiting times for either thedoctor or any other patients. Because the first patient willnever wait, his or her waiting cost a1 is irrelevant.

3.3. Solution methodology

Generally speaking, the definition of the waiting times asthe positive part of another function makes this problemfairly intractable to solve exactly, outside of a couple ofspecial cases. Weiss (1990) shows that for n ¼ 2, thisproblem can be formulated as a newsvendor model.Wang (1993) is able to find the optimal solution for ex-ponentially-distributed service times. More generally,Denton and Gupta (2001) ‘discretize’ the continuousservice time distributions and apply Bender’s decompo-sition to the resultant stochastic linear program to findoptimal and heuristic solutions for up to n ¼ 10 patients.An alternative solution methodology is Monte Carlo

integration, which replaces the true distribution of f withan approximate discrete distribution consisting of asample of K randomly generated points fk

� �. The well-

known advantage of Monte Carlo integration is that thegoodness of the resulting approximate solution dependson the size of the sample (K) and not on the dimension-ality of the underlying problem (n� 1); Hammersley andHandscomb (1964), Halton (1970), and Fishman (1996)

all provide good introductions to this methodology. Inthis case, the expected waiting time will be replaced by theaverage waiting time across the K sampled points.Dropping the extracted r term, the optimization problembecomes

ðP4Þ minxXn�1i¼1

xi þXni¼2

ai1

K

XKk¼1

wki

" #( ); ð6Þ

subject to

wki ¼ wk

i�1 þ fki�1 � xi�1� �þ

i ¼ 2; . . . ; n; k ¼ 1; . . . ;K;wk1 ¼ 0 k ¼ 1; . . . ;K;

ð7Þ

where wki is the waiting time for patient i under service

time realization fk.Robinson et al. (1996) show that this problem could be

formulated as a linear program, by replacing Equation (7)with the pair of inequalities

wki � wk

i�1 þ fki�1 � xi�1i ¼ 2; . . . ; n; k ¼ 1; . . . ;K; ð8Þ

wki � 0 i ¼ 2; . . . ; n; k ¼ 1; . . . ;K; ð9Þ

for k ¼ 1; . . . ;K. Because the optimization routine willdrive the wk

i terms to their lowest feasible level, each pairof constraints will correspond to Equation (7) above. Thelinear program is quite large, with Kðn� 1Þ constraints,Kðn� 1Þ non-negative decision variables, and n� 1 un-restricted-in-sign decision variables. However, the factthat it can be represented as a linear program means thatit is convex and quite well behaved; a greedy search inn� 1 dimensions will arrive at the optimal x. In thispaper we used a conjugate gradient search, based (forconvenience) on the left-hand partial derivatives

@wk‘

@xi¼ �a‘ if ‘ > i and wk

j > 0 8j 2 ½iþ 1; ‘�0 otherwise

� �:

ð10ÞIntuitively, the choice of xi will never affect the waitingtime of earlier patients. A later patient will only be affectedif he or she, and all intervening patients, had to wait; inthis situation a marginal increase in the job allowance xiwill decrease the waiting times for all these delayed pa-tients. Note that this partial derivative will be discontin-uous whenever xj�1 ¼ wk

j�1 þ fkj�1 for any j between iþ 1and ‘, in which case the right-hand partial derivativewould be zero. The right-hand partial derivative cannot bedetermined from the values of wk

j , but instead can be de-fined by replacing the condition wk

j > 0 in Equation (9)with the more-cumbersome xj�1 � wk

j�1þ fkj�1. Define ‘kðiÞ

to be the index of the last of these delayed patients

‘kðiÞ ¼: max ‘ > i j wkj > 0 8j 2 ½iþ 1; ‘�

n o: ð11Þ

Then the partial derivatives of the objective function canbe concisely represented as

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@

@xi¼ 1� 1

K

Xkjwk

iþ1>0

X‘kðiÞj¼iþ1

aj

!: ð12Þ

Define x� ¼ x�1; . . . ; x�n�1

� �to be the optimal solution to

(P4). In the discussion of the heuristic policies in thefollowing section, this policy will be referred to as the‘optimal’ policy, although the reader should keep in mindthat (P4) is only an approximation to the original (P3).But because the Monte Carlo integrations used in thisresearch were particularly thorough (employing Cornell’ssupercomputer to calculate x� as the average across 1000replication of (P4), each of which used K ¼ 10 000 ran-domly-generated vectors of service times), this distinctionwill be trivial.

4. Development of the heuristic policy

While Monte Carlo integration can certainly be used tosolve this problem, its high computational requirementreduces its applicability in many practical situations. Thepurpose of this section is to develop a simple and verytractable heuristic for determining the job allowances.Towards this end, we develop a test bed of the parametervalues of interest, investigate and characterize the form ofthe optimal policy over this test bed, derive a simplerapproximate form for this optimal policy, and then finallydevelop a tractable approximation for the policy pa-rameters within this approximate policy.

4.1. Problem test bed

Our first step in developing a heuristic policy was to de-fine an appropriate test bed of problems, and calculatethe optimal policy for each of those problems. We con-sidered n identical patients, for 10 different values of thenumber of patients n between three and 16, and 21 dif-ferent values for the (common) value of patient’s waitingtime a between 0.01 and 1.0; the full range of values isgiven in Table 1. We choose 16 as the test bed’s upperbound for n because this would allow for half-hourappointments over the course of an 8-hour day, or15-minute appointments if the morning and afternoonpatients were ‘decoupled’ by a lunch break halfwaythrough the day.The second step was to decide upon an appropriate

continuous probability distribution to model the servicetimes. A priori, we felt that in many service settings, andcertainly for doctor visits, the distribution of service timeswould be positively skewed. For lack of a clearly superioralternative, we decided to base our distribution on theempirical study of Goldman et al. (1970), who provide ahistogram for 1000 sample ratios of actual/forecastedsurgery times, which is shown in Fig. 1. Because themeans and (common) standard deviation do not affectour optimization (P4), we focus on the third and fourth

moments. This histogram shows a moderate degree ofskewness ða3 ¼ 0:646Þ, with Pearson kurtosis somewhatgreater than a normal distribution ða4 ¼ 3:678, versus3.0), as identified in Table 2. Although we could havesampled service times directly from the original histo-gram, this would have resulted in only 17 distinct ob-servations. We instead represented his findings throughthe four-parameter Generalized Lambda Distribution(GLD) developed by Ramberg and Schmeiser (1972,1974) and described in detail by Karian and Dudewicz(2000), which is both appealing in shape and quite ver-satile in matching empirical data. With f ðxjkÞ and F ðxjkÞdefining the p.d.f. and c.d.f. of the service time respec-tively, the GLD is defined by four parametersk ¼ k1; . . . ; k4f g in terms of the inverse c.d.f. as

x ¼ F �1ðpjkÞ ¼: k1 þ pk3 � 1� pð Þk4k2

; ð13Þ

Table 1. Test bed values

Number of patients (n) Value of patient’s time (a)

3 0.01 0.14 0.0125 0.1255 0.015 0.156 0.02 0.27 0.025 0.258 0.03 0.310 0.04 0.412 0.05 0.514 0.065 0.6516 0.08 0.8

1.0

Fig. 1. Standardized distribution of patient service times, afterGoldman et al. (1970).

Table 2. Skewness and kurtosis for various data sets

Data source a3 a4

Goldman et al. (1970) 0.646 3.678Welch (1964) 0.747 2.958Brahimi and Worthington (1991) 1.292 4.512Normal 0.000 3.000

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with

f ðxjkÞ ¼: 1 dF �1ðpjkÞdp

����p¼F ðxjkÞ

,

¼ k2

k3F ðxjkÞk3�1 þ k4 1� F ðxjkÞ½ �k4�1: ð14Þ

We wanted to choose k to maximize the likelihood ofobserving the empirical data of Goldman et al. (1970).Representing their histogram as a set f xi; pið Þg, wherepi = ProbfX ¼ xig, then with N ¼ 1000 observations,this problem can be represented as

maxk

Yi

f ðxijkÞpiN( )

: ð15Þ

For tractability, we will instead maximize the log ofthis likelihood function. Substituting in Equation (13)and factoring out the irrelevant N yields the equivalentproblem

mink

Xi

pi ln k3F ðxijkÞk3�1h(

þ k4 1� F ðxijkÞ½ �k4�1i� ln k2ð Þ

); ð16Þ

the optimal value of k is given in the first row of Table 3,after standardizing to l ¼ 0 and r ¼ 1. Figure 1 showsthat this fitted GLD provides an excellent match to thedata of Goldman et al. (1970). Although there is noreason to believe that this distribution is universally ap-plicable to all service settings, it is at least empiricallybased. The sensitivity of the heuristic’s performance tothe service time distribution will be examined in Section 5.

4.2. Approximating the form of the optimal policy

The next step of the heuristic development was to inves-tigate the form of the optimal policy, in the hope that asimpler approximate policy form might be able to ade-quately capture most of the structure of the optimalpolicy. Each of the 210 test problems was formulated as(P4) with K ¼ 10 000 Monte Carlo generated samples ofthe service times, and was solved using a conjugate gra-dient search 1000 times. Define x� ¼ x�1; . . . ; x

�n�1

� �to be

the average ‘optimal’ solution, over these 1000 replica-

tions. Again, to the degree that (P4) is only an approxi-mation of (P3), this solution will only be an approximationto the true optimal solution.Two representative cross sections of the 210 solutions

are given in Figs. 2 and 3. A number of general obser-vations can be made about the form of these optimalsolutions:

• The job allowances follow a ‘dome’ pattern for all nand a, with more time allotted to patients in the middleof the day. Wang (1993) found this same result forexponentially-distributed service times.

• The first job allowance, x�1, is always much lower thanthe other x�i ’s, and varies only slightly with n, for afixed.

• The final job allowance, x�n�1, is also somewhat lowerthan the other x�i ’s.

• The intermediate job allowances, x�2; . . . ; x�n�2, are all

about the same.• Unsurprisingly, all of the x�i ’s increase with a, for nfixed.

This suggests that an approximate policy which uses asingle job allowance for all intermediate patients mightperform almost as well as the optimal solution. The nextstep was to investigate, on an ad hoc basis for n ¼ 8, justhow complicated of a policy was required in order toadequately capture the structure of the optimal policy.The four simple policies that were examined are outlinedin Table 4. The zero-parameter model sets all job allow-ance times equal to zero. Two one-parameter modelswere examined. The first sets all n� 1 job allowances tothe same value, which is equivalent to the heuristic pro-posed by Charnetski (1984), although his is based ontruncated normal distributions for up to 20 patients perday. The second model solves for the first job allowance~xx1, and sets the remaining n� 2 job allowances equal tozero. This generalizes the heuristic proposed by Bailey(1952), who proposed that the first two patients be as-signed the same appointment time (A2 ¼ A1 ¼ 0, orx1 ¼ �l1=r), with the others offset by their expectedservice time (Aiþ1 ¼ Ai þ li, or xi ¼ 0, for i � 2). Notethat both one-parameter models are generalizations ofthe zero-parameter model, and so will always outperformit. The two-parameter model is a combination of the twoone-parameter models: It solves for the first job allow-ance ~xx1, and for single job allowance ~xx2 for each of theremaining n� 2 patients.

Table 3. Generalized Lambda Distribution parameters (l = 0, r = 1)

Data source k1 k2 k3 k4

Goldman et al. (1970) )0.504 073 0.122 036 0.041 722 0.113 048Welch (1964) )0.963 222 0.206 155 0.032 474 0.298 743Brahimi and Worthington (1991) )0.831 499 0.049 985 0.006 313 0.050 239Normal 0.000 000 0.197 451 0.134 912 0.134 912

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The same conjugate gradient search methodology wasagain used to solve these one- and two-parameter heu-ristic policies; the average percentage increase in costabove the optimal is shown in Fig. 4 for n ¼ 8, as afunction of a. It is clear that while both the zero- and one-

parameter policies can result in expected costs of morethan 20% above optimal, the two-parameter policy con-sistently performs within 1% of optimal. Similar resultswere found for other values of n. Because of the strongperformance of this two-parameter policy, more compli-cated policies were not investigated.

4.3. Approximating the parameters of the two-parameterheuristic policy

With the form of the heuristic policy approximated bytwo parameters (~xx1 for the first job allowance, and ~xx2 foreach of the remaining job allowances), the next step wasto find the optimal values for these two parameters,

Fig. 2. Representative optimal job allowance times (a ¼ 0:1).

Fig. 3. Representative optimal job allowance times (n ¼ 8).

Table 4. Heuristic policies examined

Number of parameters x1 x2 ¼ x3 ¼ � � � ¼ xn�1

0 0 01 ~xx1 ~xx11 ~xx1 02 ~xx1 ~xx2

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within this approximate policy, for the 210 test problems.Again, each of these test problems was solved 1000 times,each using K ¼ 10 000 Monte Carlo generated observa-tions of the service times. The final step was to constructtwo concise atheoretic functions of n and a which ap-proximate the optimal values of these two parametersacross the test bed. An afternoon of unstructured exper-imentation resulted in the following two closed-formatheoretic functions, which approximate the optimal joballowances for the heuristic policy.

x̂x1ðn; aÞ ¼: aþ b lnðaÞ; ð17Þx̂x2ðn; aÞ ¼: cþ ad � c

� �n�e þ 1ð Þ; ð18Þ

with parametersa ¼ 0:111 878;b ¼ 0:473 760;c ¼ 2:221 271;d ¼ 0:301 939;e ¼ 0:444 411:

ð19Þ

Note that x̂x1 depends on a but not n. The approximateand optimal parameter values are graphed in Figs. 5 and6 as functions of a, for representative values of n. Thesefigures show that these atheoretic functions do a rea-sonably good job of fitting the optimal parameter values.For both x̂x1 and x̂x2, the fit is the worst for n ¼ 3, especiallyfor extreme values of a. Table 5 summarizes the sequenceof approximations in both policy and parameter valueswhich were taken.

5. Numerical results

In this section we examine the performance of the heu-ristic policy x̂x, relative to the optimal policy x�. We pre-sent both average and worst-case results for the relative

increase in cost from using the heuristic policy in lieu ofthe optimal policy, and end this section with a brief in-vestigation into the sensitivity of the performance of theheuristic policy x̂x to distributional misspecification.Define V ðxjfÞ to be the weighted waiting time, given

(scaled) vectors of job allowance times x and realizedservice times f

V ðxjfÞ ¼:Xn�1i¼1

xi � fið Þ þXni¼2

aiwi; ð20Þwhere

wi ¼ wi�1 þ fi�1 � xi�1ð Þþ i ¼ 2; . . . ; n;w1 ¼ 0:

Then the expected relative increase in cost will be

Ef V ðx̂xjfÞ � V ðx�jfÞ½ �Ef V ðx�jfÞ½ � ; ð21Þ

Fig. 4. Simple heuristic performance: average percent increaseabove optimal cost (n ¼ 8).

Fig. 5. Atheoretic x̂x1 versus optimal ~xx1 parameter values withinthe heuristic policy.

Table 5. Summary of policies examined

Policy type Number ofparameters

Parametervalues

Job allowance notation

First Subsequent

Optimal n� 1 ‘optimal’ x�1 x�2; . . . ; x�n�1

Heuristic 2 ‘optimal’ ~xx1 ~xx2Heuristic 2 fitted x̂x1 x̂x2

Fig. 6. Atheoretic x̂x2 versus optimal ~xx2 parameter values withinthe heuristic policy.

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which are estimated for each of the 210 test cases bytaking the average over 10 000 000 generated realizationsof f. The results are graphed in Fig. 7 for five represen-tative values of n, for a between 0.01 and 1.0. The per-formance of the heuristic policy is uniformly excellent; itsaverage cost is always within 2% of the cost of the op-timal policy, and is generally within 0.5%.Another performance measure of interest is the worst-

case increase in relative cost: For which observation ofservice times f would we most regret using x̂x instead of x�?

maxf V ðx̂xjfÞ � V ðx�jfÞ½ �Ef V ðx�jfÞ½ � : ð22Þ

It is straightforward to show that the numerator ofEquation (21) will be maximized by f ¼ x�, for whichV ðx�jx�Þ ¼ 0; note that the denominator of (21) remainsEf V ðx�jfÞ½ �. These worst-case results are graphed in Fig. 8,

again for a between 0.01 and 1.0 and for five represen-tative values of n. The worst-case performance of theheuristic policy seems acceptable; it is never more than60% above the average optimal cost, and for n � 4 anda � 0:04 it is within 20% of the optimal cost.So in terms of both the average and the worst-case

performance, this simple closed-form atheoretic heuristicpolicy performs exceptionally well over the fairly broadtest bed of parameters (n � 16, 0:01 � a � 1:0). Thisstrong performance is somewhat surprising (albeit grati-fying), given that both the form of the policy, and thevalues used within that policy, are approximated.Up to now, all of the numerical results are based on the

patient service times following the GLD fitted to the dataset of Goldman et al. (1970) as shown in Fig. 1. Althoughthis distribution is intuitively appealing (being unimodaland positively skewed), it is of course not the only choice.A thorough investigation into the effects of distributionalmisspecification for this problem would warrant its ownpaper, but it is important to develop some rough idea asto the robustness of this heuristic with regards to thedistributional form of the service times. In order to dothis, we examined its performance under three differentservice time distributions: the GLD’s fitted to the datasets of Welch (1964) and of Brahimi and Worthington(1991), and the normal distribution. As with the data setof Goldman et al. (1970), the two GLD’s were constructedthrough maximum likelihood estimation, and were scaledto yield a mean of zero and a standard deviation of one.(The alternative of sampling directly from the originalhistogram was not chosen, as it would have resulted inonly 18 distinct observations for either data set.) Theoriginal histograms, and the fitted GLD’s, are given inFigs. 9 and 10. Summary statistics (skewness and kurtosis)for all four distributions are given in Table 2, while theGLD parameter values are listed in Table 3. For com-parative purposes, the distributions are graphed togetherin Fig. 11. It is clear that the data set of Brahimi andWorthington (1991) is considerably different than that ofGoldman et al. (1970), with twice the skewness (1.292versus 0.646) and decidedly more kurtosis (4.513 versus3.678).For testing purposes, we chose a single intermediate

number of patients (n ¼ 8), and found the optimal poli-cies x� for each of the three distributions for each of 21different values of a. Again, we used K ¼ 10 000 obser-vations and 1000 replications in calculating the optimalpolicy. We then used Equation (20) to calculate the av-erage increase in costs from using the heuristic policy x̂x

based on the data of Goldman et al. (1970), based on10 000 000 generated observations of service times. Theresults are shown in Fig. 12; note that the solid linerepresenting Goldman is the same as in Fig. 7, and isincluded as a baseline to indicate the average cost increaseof using a heuristic policy, without distributional mis-specification. Figure 12 shows that performance of the

I

Fig. 7. Average heuristic performance: percent increase aboveoptimal cost.

I

Fig. 8. Worst-case heuristic performance: percent increaseabove optimal cost.

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heuristic policy given by Equations (16) and (17) is onlyslightly worse for different service time distributions, andgenerally remains under 1.5%. An exception is for small

values of a, under the very skewed data of Brahimi andWorthington (1991), where the expected cost penaltyapproaches 7% above optimal. This is because small

Fig. 9. Standardized distribution of patient service times, after Welch (1964).

Fig. 10. Standardized distribution of patient service times, after Brahimi and Worthington (1991).

Fig. 11. Comparison of different standardized service time distributions.

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values of a will yield values of x�1 near the very bottom ofthe left tail of the service time distribution. Because theleft tail of Brahimi and Worthington is so much shorterthan those of the other three distributions, values of x1derived from these other distributions (including theheuristic x̂x1) will be fit poorly.

6. Conclusions and future research

The problem of setting appointment times is very im-portant in many professional services, ranging fromdoctors and dentists to auto repair, particularly as awider recognition is given to the importance of servicequality as a means of competitive advantage. This modelcan also be applied to broader problems of schedulingphysical resources over time, including the scheduling ofan operating room and the setting of departure times fora train or an aircraft, for example. Even in the simplifiedcase where the variability in service times dominates allother sources of uncertainty, the problem is quite in-tractable for more than two patients, requiring MonteCarlo integration or decomposition techniques.In this paper we develop a very simple closed-form

heuristic for setting job allowances (and appointmenttimes) for up to 16 homoskedastic and equally-important(in terms of the value of their waiting time) customers.We show that this heuristic performs very well: usuallyaveraging within 0.5% of the cost of the optimal policy,with worst-case performance usually within 20% of op-timal. Although the service time distribution used for thisheuristic was based on a single empirical study of surgerytimes, some limited testing shows that this heuristic per-forms quite well for other service times distributions,provided that their skewness is not so different as todrastically shift their lower tail.

There are a wide range of directions for possible futureresearch. It is easy to extend this model to account fordoctor tardiness (rather than idleness), or to includepostponable tasks by the doctor (e.g., calling pharmacies,arguing with HMO’s, etc.) which can be used to fill his orher idle time. Another generalization would allow formultiple (and interchangeable) doctors. Staying withinthe general model of this paper, other obvious extensionswould be to expand the test bed to evaluate a > 1,a < 0:01, or n > 16, or to consider heteroskedastic cus-tomers, or customers with different waiting costs ai.As this research stream develops, it will be important to

capture other uncertainties within the system. One suchexample would be the uncertainties of emergency andunscheduled appointments within the day. A first cut atthis inclusion might replace the distribution of the lengthof a patient’s service time with the multi-modal distri-bution of the length of the busy period of serving thatcustomer and any and all emergency patients who pre-empt that patient.Another important and significant source of uncer-

tainty is the patient arrival times. The simplest modelwould be to model patient arrival times by a given dis-tribution about his or her appointment time. A morerealistic extension would also incorporate the differencebetween lateness and tardiness from the patient’s per-spective, where waiting may be perceived as less oner-ous prior to the scheduled appointment time. Modelingstochastic arrivals would allow us to evaluate the com-monly-used ‘block schedules’, where multiple patients areassigned the same appointment time, and are seen by thedoctor in the order of their arrival.Microeconomic methodologies are being more fre-

quently applied to problems of operations management.For this problem, an important application would be touse game theory to model patient arrival times. For ex-ample, if patients believe that their doctor is more likelyto run behind schedule at the end of the day, they will notmake much of an effort to arrive promptly at their ap-pointment times. A formal model of this common be-havior is apt to be quite rich.

References

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Bailey, N.T.J. (1954) Queuing for medical care. Applied Statistics, 3,137–145.

Blanco White, M.J. and Pike, M.C. (1964) Appointment systems inout-patients’ clinics and the effect of patients’ unpunctuality.Medical Care, 2, 133–145.

Brahimi, M. and Worthington, D.J. (1991) Queuing models for out-patient appointment systems—a case study. Journal of the Oper-ational Research Society, 42, 733–746.

Fig. 12. Effects of distributional misspecification.

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Biographies

Lawrence W. Robinson is an Associate Professor at the JohnsonGraduate School of Management at Cornell University. He receivedhis M.B.A. and Ph.D. degrees from the Graduate School of Business atthe University of Chicago in 1986. His research focuses on problems ofoperating in an uncertain service environment; in particular, on de-veloping practical heuristic policies that perform well and can be easilycalculated. He is a member of INFORMS, ASQ, and Rotary.

Rachel R. Chen is currently a Ph.D. student at the Johnson GraduateSchool of Management at Cornell University. Her research focuses onsupply chain management and the scheduling of service operations. Shehas been a member of INFORMS since 1999.

Contributed by the Feature Applications and Technology ManagementDepartment

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