sequential clinical scheduling with service criteria sequential clinica… · sequential clinical...

European Journal of Operational Research 214 (2011) 780–795

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Innovative Applications of O.R.

Sequential clinical scheduling with service criteria

Ayten Turkcan a,⇑, Bo Zeng b, Kumar Muthuraman c, Mark Lawley d

a Northeastern University, Department of Mechanical and Industrial Engineering, Boston, MA 02115, USAb University of South Florida, Department of Industrial and Management Systems Engineering, Tampa, FL 33620, USAc University of Texas, McCombs School of Business, Austin, TX 78712, USAd Purdue University, Weldon School of Biomedical Engineering, West Lafayette, IN 47907, USA

a r t i c l e i n f o

Article history:Received 10 December 2009Accepted 17 May 2011Available online 23 May 2011

Keywords:Multiple criteria clinical schedulingMoment-based constraintsOverbookingPareto optimal set

0377-2217/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.ejor.2011.05.023

⇑ Corresponding author. Tel.: +1 617 373 2032.E-mail addresses: [email protected] (A. Turkcan),

[email protected] (K. Muthudu (M. Lawley).

a b s t r a c t

This study investigates sequential appointment scheduling with service criteria. It uses a constraint-based approach with service criteria bounded in a constraint set in contrast to the more typical weightedlinear objective function. Properties are derived and a sequential scheduling algorithm is developed. Fair-ness properties of generated schedules are considered in detail, where fairness is the uniformity of per-formance across patients. New unfairness measures are proposed and used to capture the inequity amongpatients assigned to different slots. Other criteria such as expectation and variance of patient waitingtime, queue length, and overtime are also considered. The fairness/revenue tradeoff is investigated asis the flexibility of the constraint-based approach in handling unavailable time periods.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

The majority of US patient care is delivered through outpatientclinics (US Census Bureau, 2002). Access to these clinics is providedthrough appointment scheduling. Many studies indicate that well-designed appointment scheduling can increase access to care,smooth clinic operations, increase patient satisfaction, improvequality of care, and reduce overall costs (AHRQ, 2007; Savin,2006). Gupta and Denton (2008) discuss the characteristics ofappointment scheduling in primary care and specialty clinics.Appointment scheduling is a sequential process where the demandfor appointments is realized online. When a patient calls for anappointment, the patient must be worked into a pre-existing (per-haps partially constructed) schedule without changing theappointment times of other patients. This must be done and the se-lected appointment time given to the patient before the call termi-nates. There are many complicating factors that affect theperformance of appointment scheduling methods such as patientno-show, variability in arrival and service times, and physicianand patient preferences.

Ho and Lau (1992) show that, among three factors (no-show,service time variability, and number of patients per clinic session),no-show is the major factor that affects clinic performance. Cayirliand Veral (2003) mention that no-show rates range from 5% to30%. There are many studies in medical and health informatics lit-erature that identify the predictors of no-shows using statistical

ll rights reserved.

[email protected] (B. Zeng),raman), [email protected]

methods (Goldman et al., 1982; Gruzd et al., 1986; Cashmanet al., 2004; Lee et al., 2005; Daggy et al., 2010). Several patient(demographic, behavioral, psychological), provider (demographics,specialty), clinical (diagnosis, severity), environmental (access toclinic, weather, day, family), and appointment-related (lead time,day and time of the scheduled appointment) factors are identifiedas predictors of no-show (Deyo and Inui, 1980). No-show patientstend to be younger, unmarried, uninsured, with psychosocial prob-lems and a history of no-showing (Daggy et al., 2010). Other factorsthat affect no-shows are the distance to the medical center, leadtime between the actual appointment and the time appointmentwas made, season, number of previous visits, diagnosis, and, thetime between two appointments (Daggy et al., 2010). Even thoughthe existing studies show that the no-shows can be predicted, thepredictors of no-show may change according to the patient popu-lation and clinic environment.

Clinics try to reduce no-shows with appointment reminders orby using open access scheduling. Under open access, a large por-tion of the appointment schedule is not booked until a day ortwo in advance. The aim is providing appointments to sick patientswhen they need, rather than having to wait several days or weeksfor the appointment time to arrive. There are studies in medical lit-erature that show the effect of open access on reducing patient no-shows and improving access and quality of care (Murray et al.,2003; Bundy et al., 2005; Parente et al., 2005; O’Connor et al.,2006). However, not all clinics have a positive experience. Neutralor negative effects include no impact on patient satisfaction(Parente et al., 2005; Mehrotra et al., 2008), no change in patientno-show rates (Mehrotra et al. 2008, Bennett and Baxley, 2009),no increase in continuity of care (Salisbury et al., 2007a,b),

http://dx.doi.org/10.1016/j.ejor.2011.05.023

mailto:[email protected]





http://dx.doi.org/10.1016/j.ejor.2011.05.023

http://www.sciencedirect.com/science/journal/03772217

http://www.elsevier.com/locate/ejor

A. Turkcan et al. / European Journal of Operational Research 214 (2011) 780–795 781

difficulties in teaching clinics (DeLaurentis et al. 2006), and failureto achieve implementation (Murray et al., 2003; DeLaurentis et al.2006).

In contrast to implementation studies in medical literature to re-duce no-show rates, operations research literature focuses onmathematical models and simulation studies for optimally over-booking to reduce the impact of no-shows on clinic operationsrather than reducing the no-show rates. Researchers typically de-velop a linear objective function that includes elements of revenuefor seeing patients combined with costs associated with patientwaiting, clinic overtime, and/or physician idle time (Cayirli and Ver-al, 2003). Service times are typically assumed to be either determin-istic or exponential, and most studies assume a single no-showprobability for all patients. Kim and Giachetti (2006) determinethe number of patients to book in a given session to maximize clinicprofit, modeled as the weighted sum of revenue, cost of overtime,and a penalty for patients who leave without being seen. They as-sume deterministic service times and a single no-show probability.(Laganga and Lawrence, 2007a,b) balance the utility of serving pa-tients with the costs of waiting time and overtime. They assumedeterministic service times and a single no-show probability anduse simulation to conclude that their approach works best withhigh no-show probability and low service time variances. Kaandorpand Koole (2007) assume exponential service times and attempt tominimize a linear combination of patient waiting time, physicianidle time, and overtime. To compute the schedule, they develop alocal search algorithm, which seeks incremental improvement byshifting patients between adjacent slots (note that this assumesthe set of patients to be scheduled is known beforehand and thuscannot be classified as sequential). They establish that this ap-proach yields a global optimum when the objective is multi-modu-lar (which is true when there is only one no-show probability).Hassin and Mendel (2008) consider the problem of scheduling a gi-ven number of patients with homogenous no-show. The objective isto minimize the cost of customer waiting and server availability.Their model indicates that appointments should be close togetherat the beginning and end of a schedule. Green and Savin (2008) bal-ance physician capacity and patient panel size. They assume thatthe patient no-show rate is a non-decreasing function of theappointment backlog (queue length). They show that queue lengthand the probability to get same day appointment depend on the pa-tient panel size for deterministic service times.

Most of the studies considering no-show assume that allpatients have the same no-show rate. However, patient no-showbehavior can be estimated at the individual patient level. Manystudies investigate the relationship between no-show behaviorand patient and provider characteristics, environmental factors,health status, lead time, etc. In practice, we observed that the clin-ics currently do not have these advanced scheduling techniques.The main reason is that most of the medical clinics are not awareof the potential benefits of using an advanced scheduling method,and therefore, do not request these capabilities from schedulingsoftware vendors. Another reason is that the implementation ofno-show prediction models requires statistical analyses of the cur-rent population characteristics. Each clinic might need a differentno-show prediction model. Even though the no-show predictionmodels can be developed in a short time, the data and the expertisemay not readily be available. Our research group have been devel-oping advanced appointment scheduling methods with heteroge-neous no-show probabilities to show the potential benefits ofusing no-show prediction models and advanced scheduling meth-ods on clinic performance. Muthuraman and Lawley (2008) is thefirst study that explicitly considers heterogeneous patients, i.e.patients with different no-show rates, in a sequential schedulingenvironment. They develop a recursive computation method anda sequential scheduling algorithm based on a myopic policy (one

that does not consider any information about the unknown futurecall-in sequence) to maximize the weighted average of expectedrevenue, patient slot overflow (a surrogate for waiting time), andovertime. The work is extended by Chakraborty et al. (2010) forgeneral service time distributions, thus relaxing the exponentialservice time assumption. Zeng et al. (2010) shows that theobjective is not multi-modular for heterogeneous no-show (imply-ing that finding an optimal schedule for a given set of patients iscomputationally hard), and develops non-myopic algorithms forthe sequential scheduling problem, illustrating the value of fore-casting the residual call-in sequence for a partially constructedschedule.

In contrast to the work discussed above, this research considerssequential scheduling with heterogeneous no-shows in the pres-ence of service constraints. The objective is to maximize revenuefrom seeing patients subject to constraints on patient waiting time,number of patients in system, and so forth. In our experience, it ismuch easier for clinic managers to specify bounds such as ‘‘the aver-age patient waiting time must be no more than 10 minutes’’ than it isto specify an actual cost. We also treat overtime as a constraintrather than including it in the objective. Again, in our experience,it is easier for clinic managers to specify a constraint on overtimethan to come up with an actual overtime cost. Further, when over-time is included in the objective, the ratio between revenue per pa-tient and overtime cost per patient becomes very important. Indeed,if overtime cost is less than revenue, there is potential for schedulingalgorithms to become unstable since the marginal revenue obtainedby adding one additional patient is always greater than the marginalovertime cost (see Muthuraman and Lawley, 2008).

We are particularly interested in schedule fairness measures.Cayirli and Veral (2003) define fairness as the uniformity of perfor-mance across patients. In their review paper, they suggest that fu-ture studies should consider multiple performance measuresincluding fairness to evaluate appointment systems. In the litera-ture, there are few studies that consider fairness while evaluatingthe quality of the appointment scheduling rules. Bailey (1952) con-siders the mean waiting time according to the sequence of patientsin the schedule. Yang et al. (1998) consider the variance of waitingtimes, Cox et al. (1985) consider the variance of queue lengths, andCayirli et al. (2008) consider the standard deviation of waitingtimes as the fairness measures. These papers use simulation to cal-culate fairness measures. In this study, we propose new unfairnessmeasures to evaluate schedule uniformity for patients assigned todifferent slots. We derive analytical expressions for these mea-sures, which are functions of expected waiting time and the num-ber of patients in the system.

The main contributions of this paper are as follows:

1. To the best of our knowledge, this is the first study that consid-ers fairness as the primary objective.

2. The first and second moments of waiting time, queue length,and number of patients in the system at each slot are analyti-cally derived for the first time in the literature.

3. The first moments, which are derived based on the assumptionof exponential service times, are shown to be upper bounds forgeneral service time distribution.

4. The sequential scheduling problem with service constraints isshown to have different properties than the problem with aweighted linear combination of objectives. With a weighted lin-ear objective, the assignment of a calling patient to the scheduleis independent of the patient no-show probability, and thus it isalways better for expected profit to schedule low no-showpatients. In contrast, when service constraints are in place, add-ing a patient with low no-show probability may violate con-straints, whereas a patient with higher no-show probabilitymight still be added to the schedule.

782 A. Turkcan et al. / European Journal of Operational Research 214 (2011) 780–795

5. The constraint-based approach can easily handle other con-straints such as provider unavailable times, as illustrated inthe paper.

The paper is structured as follows. Section 2 defines the sequen-tial scheduling problem with service constraints along with itsunderlying assumptions. A stochastic programming model is pro-posed with the objective of minimizing unfairness and maximizingrevenue. Service criteria such as the expectation and variance ofslot-conditioned waiting time, queue length, and number of pa-tients in the system are bounded in the constraint set. The proper-ties of the service criteria are derived. Section 3 presents asequential scheduling algorithm to solve the problem. Section 4presents a variety of computational studies, and Section 5 showshow the proposed approach can handle unavailable provider times.Finally, Section 6 discusses future work and provides some con-cluding remarks.

2. Problem definition

In this study, we assume a single server, single stage system.This means that patients are scheduled only for the examination,other stages such as registration and laboratory tests are not con-sidered. The queue for each provider is different, which is a com-mon practice for continuity of care with the primary physician.The service times of patients are assumed to be exponentially dis-tributed. We note that although this assumption (which is com-monly made by scheduling researchers) may not hold in practice,it allows us to develop and test results that give important insightsinto practical scheduling operations. Also, we show that our firstmoment derivations provide bounds in general service time set-tings. In other work, we have developed sequential scheduling re-sults for general service time distributions (Chakraborty et al.,2010), but we have not yet concisely expressed the performancecriteria. Table 1 provides the notation that will be used throughoutthe paper.

We consider heterogeneous patients, that is, patients with dif-ferent no-show rates. We assume that patients are punctual, thatis, if they arrive for their appointment, they arrive on time. The

Table 1Notation.

I number of slotsi slot index, i 2 {1, . . . , I}J number of patient typesj patient type index, j 2 {1, . . . , J}ti length of slot iXi number of patients arriving for slot iYi number of overflow patients from slot i to slot i + 1Li number of possible service completions in slot iQLi Number of patients in the queue at the beginning of slot i, after any

arrivalsNSi Number of patients in the system at the beginning of slot i, after any

arrivalsWi Waiting time of patients assigned to slot ir revenue per patient1 � pj no-show probability of patient of type jS an appointment schedule (2ZI�J)Sij number of patients of type j scheduled for slot i in schedule SRik probability that k patients will overflow from slot i to slot i + 1Qim probability that m patients will arrive to slot if(x) Probability density function of service timel Expected service time for each patientbeo Upper bound on expected overtimebew Upper bound on expected waiting timebeq Upper bound on expected queue lengthben Upper bound on expected number of patients in the systembvw Upper bound on variance of waiting timebvn Upper bound on variance of number of patients in the system

number of patients that can be scheduled per clinic session is notfixed, because the number may change due to patient mix with dif-ferent no-show rates. The slot lengths and usage are fixed in ad-vance according to provider preferences and patient mix (adult,pediatrics, new, established, acute, chronic, etc.). For example,the provider might specify 15 minutes slots during some sessions,20 minutes slots for others, reserve some slots for new patients, orblock out certain slots. In our experience, determination of slotlengths is rarely done analytically or optimally, and thus this isan excellent problem for research, which we address in Chakr-aborty et al. (2009). It is not, however, the focus of this paper,and thus we take the slot length for each slot (ti) as given.

In a fixed slot appointment system, the clinic session is dividedinto I slots of length ti. There are J types of patients with differentno-show rates. Appointment requests are not known in advance,but occur online at any time before the actual appointment sessionstart time. When a patient (say the (n + 1)st) calls for an appoint-ment in a given session, he/she should be scheduled withoutchanging the appointment times of the previous n patients in ses-sion schedule, S. A schedule S is a matrix of size I � J, where Sij rep-resents the number of scheduled patients of type j in slot i. Dij isalso a matrix of size I � J where dij = 1 and di0 j0 ¼ 0 for any i0 – i orj0 – j. If patient n + 1 of type j is assigned to slot i, the new scheduleis represented as S + Dij.

The expected number of arrivals might be less than the numberof scheduled patients because of patient no-shows. Let Xi be a ran-dom variable which denotes the number of arrivals to slot i and Yi

be the number of patients waiting for the completion of service atthe end of slot i. If Li is a random variable that shows the number ofpossible service completions assuming infinite demand, then theactual number of service completions is min (Li,Yi�1 + Xi). The over-flow from slot i to slot i + 1 is calculated as Yi�1 + Xi �min(Li,Yi�1 + Xi) = max (Yi�1 + Xi � Li,0). Fig. 1 shows the relationshipamong the variables in a fixed slot appointment system.

The arrival (Q) and overflow (R) probability matrices are definedas follows. Let P be the set of all non-negative, integer, J-vectorsp = hp1,p2, . . . ,pJi such that

PJj¼1pj ¼ m and pj 6 Sij for all j. Qim is

the probability that m patients arrive in slot i for a given scheduleS and is calculated by conditioning on the event that pj patientsarrive for their appointments.

Qim ¼ PrfXi ¼ mg ¼Xp2P

PrfXi ¼ mjðp1; . . . ;pJÞg

¼Xp2P

Prfðp1; . . . ;pJÞg ¼Xp2P

Yj

Sij!

pj!ðSij � pjÞ!p

pj

j ð1� pjÞðSij�pjÞ:

Rik is the probability that k patients overflow from slot i into sloti + 1 for a given schedule S.

Rik ¼ PrfYi ¼ kg ¼ PrfmaxðXi þ Yi�1 � Li;0Þ ¼ kg

¼ PrfXi þ Yi�1 � Li ¼ kg k > 0PrfXi þ Yi�1 � Li 6 0g k ¼ 0

�

¼

Pm

P~k

Prfmþ ~k� k ¼ LigQ imRi�1;~k k > 0

Pm

P~k

Prfmþ ~k 6 LigQ imRi�1;~k k ¼ 0

8><>:

¼

Pm

P~k

fLiðmþ ~k� kÞQ imRi�1;~k k > 0;

Pm

P~k

ð1� FLiðmþ ~kÞÞQ imRi�1;~k k ¼ 0;

8><>:

where FLiðlÞ ¼ PrfLi < lg and fLi

ðlÞ ¼ PrfLi ¼ lg are calculated usingthe distribution of service times. For additional detail, see Muthur-aman and Lawley (2008).

i−1 i ... Ii+1

Xi

Yi−1

...1Y

i

min (Li , X

i + Y

i−1 )

Fig. 1. The fixed slot model.


2.1. Stochastic programming model with service criteria

We now formulate a stochastic programming model that usesrevenue and fairness as objectives and service expressions as con-straints. We will first define new unfairness measures and thenpresent and discuss the stochastic programming model. (Deriva-tions for the quantities appearing in the constraints are providedin the Appendix A).

The first proposed measure is the minimization of differencebetween the number of patients in the system at the beginningof each slot (UNS).

UNS ¼ maxi¼f1;...;Ig

E½NSi� � mini0¼f1;...;Ig

E½NSi0 �: ð1Þ

The second unfairness measure is the minimization of differencebetween the expected waiting time for patients arriving at each slot(UW).

UW ¼ maxi¼f1;...;Ig

E½Wi� � mini0¼f1;...;Ig

E½Wi0 �: ð2Þ

Both unfairness measures are non-regular. Thus, they may alter-nately increase and decrease as more patients are scheduled, andthere is no natural criteria to stop scheduling more patients in orderto maximize expected profit and minimize unfairness. However, weuse other criteria such as overtime (E[YI]l), expected waiting time(E[Wi]) and expected number of patients in the queue/system atthe beginning of each slot (E[QLi], E[NSi]) to determine when to stop.We put these criteria into the constraint set and use upper boundsto control each criteria. Variance of waiting times (Var[Wi]) andnumber of patients in the system/queue at the beginning of eachslot (Var[NSi]) are the other unfairness criteria that we consider inthe constraint set. We propose the following stochastic program-ming model:

min UNS ¼ maxi¼f1;...Ig

E½NSi� � mini0¼f1;...Ig

E½NSi0 �

min UW ¼ maxi¼f1;...Ig

E½Wi� � mini0¼f1;...Ig

E½Wi0 �

max E½revenue� ¼X

i

rE½Xi�

st E½YI�l ¼X

Ik

kRIkl 6 beo; ð1Þ

E½Wi� ¼ E½Yi�1�ð1� Q i0Þ þE½Xi� � 1þ Qi0

2

� �l 6 bew

; ð2Þ

E½QLi� ¼ E½Yi�1� þ E½Xi� � 1þ Ri�1;0Q i;0 6 beq; ð3Þ

E½NSi� ¼ E½Yi�1� þ E½Xi� 6 ben; ð4Þ

Var½Wi� ¼ E½Yi�1�ð1� Q i0Þ þE½Xi� � 1þ Qi0

2

� �r2

þ E Y2i�1

h ið1� Qi0Þ þ E½Yi�1�ðE½Xi� � 1þ Qi0Þ

�

þ2E X2

i

h i� 3E½Xi� þ 1� Q i0

6

1Al2

� E½Yi�1�ð1� Q i0Þ þE½Xi� � 1þ Q i0

2

� �2

l26 bvw

; ð5Þ

Var½NSi� ¼ Var½QLi� ¼ Var½Yi�1� þ Var½Xi� 6 bvn: ð6Þ

The first constraint guarantees that the expected overtime does notexceed the upper bound beo. In the second, third, and fourth con-straints, the expected waiting time and number of patients in thequeue/system are bounded by upper bounds bew, beq, and ben,respectively. In the last two constraints, the variance of waitingtime and the variance of number of patients in the queue/systemare bounded by bvw and bvn, respectively.

Constraints (2)–(4) are functions of E[Xi], E[Yi�1], Qi0 and Ri�1,0.Some of the constraints may become redundant according to thechoice of the upper bounds. For example, if ben = bew/l, constraint(1) becomes redundant because E½Wi�=l ¼ ðE½Yi�1�ð1� Q i0Þ þ E½Xi ��1þQi0

2 Þ < E½Yi�1� þ E½Xi� ¼ E½NSi� 6 ben. Constraint (3)becomes redundant when ben = beq, since E[QLi] =E[Yi�1] + E[Xi] � 1 + Ri�1,0Qi,0 6 E[Yi�1] + E[Xi] = E[NSi] 6 ben. In thatcase, using only one constraint is sufficient to achieve the desiredeffect.

We note that the criteria considered in this study are calculatedfor each slot, not for the overall schedule. This is not a disadvan-tage, since the main focus of the paper is providing uniform sched-ules for patients assigned to different slots. However, to illustratehow composite schedule criteria can be computed from slot-basedcriteria, we provide a simple example. Table 2 illustrates expectedwaiting times for a schedule with 4 patients and 3 slots, i.e.S = (2,1,1). The no-show probability is 0.2 for all patients, the slotlength is one, and the service times are exponentially distributedwith a mean one slot length.

Table 2 shows the expected waiting time for all possible realiza-tions. For a given realization (X1,X2,X3), the expected waiting timeof patients arriving to slot i (E[Wij(X1,X2,X3)]) is calculated as(E[Yi�1] + (Xi � 1)/2) (which is Eq. (2) for l = 1, Xi > 0 and known).The total expected waiting time for the realization (X1,X2,X3) is

E½WjðX1;X2;X3Þ� ¼X

i

Xi � E½WijðX1;X2;X3Þ�,X

i

Xi;

The expected waiting time at each slot across all realizations is cal-culated as

E½Wi� ¼X

8ðX1 ;X2 ;X3ÞE½WijðX1;X2;X3Þ� � PrfðX1;X2;X3Þg;

where Pr{(X1,X2,X3)} is the probability of the realization.The overall composite measure for expected waiting time for

schedule S is calculated as

E½W� ¼X

8ðX1 ;X2 ;X3ÞE½WjðX1;X2;X3Þ� � PrfðX1;X2;X3Þg:

For example, when the realization is (2,0,1), E[W1j(2,0,1)] =(0 + (2 � 1)/2), (E[W2j(2,0,1)]) = 0 (since there is no arrival at slot2), and (E[W3j(2,0,1)]) = E[Y2]. The overall expected waiting timefor realization (2,0,1) is E[Wj (2,0,1)] = (2 � E[W1j(2,0,1)] +1 � E[W3j(2,0,1)])/3 = (2 � (0 + (2 � 1)/2) + 1 � E[Y2])/3 = 0.514. Theprobability of realizing (2,0,1) is 0.2 � (0.8)3 = 0.1024.

Applying these computations to all schedule realizations,we find E[W1] = 0.5 � (0.4096) + 0.5 � (0.1024) + 0.5 � (0.1024) +0.5 � (0.0256) + 0 � (0.0256) + � � � + 0 � (0.0016) = 0.320. Similarly,E[W2] = 0.659 and E[W3] = 0.714. The overall expected waiting time

Table 2Expected waiting time for all possible realizations (X1,X2,X3) of schedule S = (2,1,1).

(X1,X2,X3) Pr{(X1,X2,X3)} E[W1j(X1, X2,X3)] E[W2j(X1,X2,X3)] E[W3j(X1,X2,X3)] E[Wj(X1,X2,X3)]

(2,1,1) 0.4096 0.500 1.104 1.248 0.838(2,1,0) 0.1024 0.500 1.104 0 0.701(2,0, 1) 0.1024 0.500 0 0.541 0.514(2,0, 0) 0.0256 0.500 0 0 0.500(1,1,1) 0.2048 0 0.368 0.639 0.336(1,1,0) 0.0512 0 0.368 0 0.184(1,0, 1) 0.0512 0 0 0.135 0.068(1,0, 0) 0.0128 0 0 0 0(0,1,1) 0.0256 0 0 0.368 0.184(0,1,0) 0.0064 0 0 0 0(0,0,1) 0.0064 0 0 0 0(0,0,0) 0.0016 0 0 0 0


for the schedule (2,1,1) is E½W� ¼P8ðX1 ;X2 ;X3ÞE½WjðX1;X2;X3Þ�

PrfðX1;X2;X3Þg ¼ 0:567.As an aside, note that if we use the slot waiting time formula

given by (2), ((E[Yi�1](1 � Qi0) + (E[Xi] � 1 + Qi0)/2)l), we have tocalculate Qi0, E[Yi�1] and E[Xi]. According to the arrival and overflowmatrices, Q10 = 0.04, Q20 = 0.2, Q30 = 0.2, E[Y1] = 0.824, E[Y2] = 0.892,E[X1] = 1.6, E[X2] = 0.8 and E[X3] = 0.8. The expected waiting timesare calculated as E[W1] = (1.6 � 1 + 0.04)/2 = 0.320, E[W2] = 0.824(1 � 0.2) + (0.8 � 1 + 0.2)/2 = 0.659, and E[W3] = 0.892(1 � 0.2) +(0.8 � 1 + 0.2)/2 = 0.714, which is identical to that given above.

2.2. Criteria properties

We use the memoryless property of exponential distribution toderive the performance criteria incorporating the number ofpatients overflowing from slot i to slot i + 1, Yi. If the service timeshave a general distribution, the expected overflow calculated basedon the exponential service time assumption will be an over-estima-tion of the overflow value for the general distribution, which isshown by Property 1. Property 1 is an important result which showsthat a feasible solution found for the scheduling problem with theupper bounds and exponential service time distribution is also fea-sible for the problem with general service time distribution.

Property 1. Given a schedule S, the expected overtime (E[YI]l) andthe expected service time for patients overflowing from slot i � 1to slot i (E[Yi�1]l), which are calculated for the exponential servicetime distribution, are upper bounds for other service time distri-butions assuming identical means.

The proofs of all properties are provided in Appendix B. Zenget al. (2010) considered a weighted linear combination of expectedrevenue, overflow and overtime costs and showed that it was al-ways better to include patients of low no-show probabilities intoan existing schedule. This rule is not valid when a constraint-basedapproach is used, as is shown by Property 2.

Property 2. Let S1 = S + Di1 + Di4 and S2 = S + Di2 + Di3 be twoschedules with n patients, where the types of two patientsassigned to slot i differ. Schedule S1 has patients of types 1 and 4assigned to i and schedule S2 has patients of types 2 and 3 assignedto i, where p1 + p4 = p2 + p3 and p1 > p2 P p3 > p4.

(a) Both schedules have the same E[Xi].(b) Schedule S1 has smaller E[Yi] than schedule S2.(c) If E[Yi�1] < 1/2 for schedule S, schedule S1 gives smaller E[Wi]

than schedule S2.

Property 2 shows that a schedule with a combination of highand low no-show rate patients (schedule S1) gives lower E[Yi] com-pared to a schedule with medium no-show rates (schedule S2). Forschedule S2, the probability that there will be at least one arrival to

slot i is lower than that of schedule S1. Therefore, if the expectedoverflow from previous slot is high (E[Yi�1] > 1/2) in schedule S,schedule S2 gives a lower expected waiting time than scheduleS1. The result of Zeng et al. (2010), which shows that it is alwaysbetter to include patients of low no-show probabilities into anexisting schedule, is not valid for the clinical scheduling problemwith constraints.

Property 3 shows that the expected overflow (E[Yi]), expectedwaiting time at each slot (E[Wi]), expected queue length (E[QLi]),and the expected number of patients in the system (E[NSi]) at thebeginning of each slot are increasing functions of pj.

Property 3. For each slot, the expected overflow, expected waitingtime, expected queue length and expected number of patients inthe system increase as pj increases.

As a result of Property 3, if a patient of type j cannot be assignedto slot i due to constraint violation, none of the patients with high-er pj values can be assigned to that slot. Muthuraman and Lawley(2008) proposed a myopic scheduling algorithm with the objectiveof maximizing the profit, which is a function of expected revenueand expected overflow cost. In their algorithm, the decision to ac-cept (or reject) a patient of type j is independent of pj (as shown byZeng et al. (2010)). When a constraint-based approach is used, thestopping condition depends on pj since all constraints should besatisfied. Property 4 provides the condition to stop scheduling pa-tients to a session.

Property 4. For a given schedule S, let pci be the critical probability

such that no patients with higher pj values can be assigned to slot i.If a patient of type j has higher pj than all pc

i values pj > maxipci

� �,

then the patient cannot be assigned to any slot. If minjpj > maxipci ,

no patients can be added to the schedule S without violating atleast one constraint.

When more than one slot satisfies all the constraints, theappointment slot should be chosen carefully so that the total num-ber of patients scheduled for a session is maximized. This can bedone if the marginal increase in the service criteria is minimized.Property 5 shows that scheduling patients to earlier slots decreasesthe expected overtime, and hence increases the number of patientsthat can be scheduled.

Property 5. The expected increase in overflow at the end of theday is smaller if a patient is assigned to slot i1 instead of slot i2,where i1 < i2.

The proposed stochastic programming model should be solvedeach time a patient calls in for an appointment. However, the sche-dule for the existing patients cannot be changed. In order to max-imize expected revenue, the new patient should be assigned to oneof the appointment slots satisfying all constraints. Even though theexpected revenue increases, the unfairness measures may increaseor decrease. If only the unfairness objective is considered, the pro-


posed model may not schedule the new patient to the currentschedule, but may schedule another patient with the same no-show probability at a later time when the schedule changes. In or-der to avoid this, we proposed a sequential scheduling algorithm,which schedules the patient when a slot satisfying all constraintsis found. The algorithm does not accept or reject the patient basedon the current decrease or increase in unfairness. However, sincethe ultimate goal is finding a ‘‘fair schedule’’, the patient is not as-signed to the first slot satisfying all constraints. The assignment ismade according to the workload at each slot with the objective ofminimizing the difference between slots. Properties 3–5 are usedto reduce the computation times. In the following section, the pro-posed sequential scheduling algorithm is explained in detail.

3. Proposed sequential scheduling algorithm

We propose a sequential scheduling algorithm with the objec-tive of minimizing the unfairness measures UNS and UW. The aimis to find a fair schedule by assigning similar amount of expectedworkload to each slot. The basic steps of the proposed algorithmare as follows:

Step 1 Initialize n = 0, S = ;, pci ¼ 1, and all upper bounds (beo,bew, -

beq,ben,bvw, bvn).Step 2 Wait for the next patient call before the start time of first

slot. Let the patient be of type j. Set i⁄ to �1.Step 3 If pj > maxipc

i , patient cannot be assigned, go to Step 5.Otherwise, sort the slots with pc

i > pj in non-decreasingorder of max{0,E[NSi]l � ti}. In case of ties, the slot witha smaller index is chosen first. Let SI be the set of orderedslots.

Step 4 For all i 2 SI
4.1 Temporarily assign (n + 1)st patient to slot i.4.2 Calculate E½YI�; E½Wi0 �; E½QLi0 �; E½NSi0 �;Var½Wi0 � and
Var½QLi0 � ¼ Var½NSi0 � for all i0.4.3 If all constraints with an upper bound are satisfied,then i⁄ = i. Go to Step 5. Else if at least one of the con-straints among (1)–(4) are not satisfied, updatepc

i ¼ pj � � (due to Property 3). Go to Step 4.4.4.4 If all slots in SI are evaluated, go to Step 5. Otherwise,

go to Step 4.1 for the next slot in the sorted list SI.
Step 5 If i⁄ = �1, patient cannot be added to schedule S, go to Step
6. Otherwise, assign the patient to slot i⁄, increase n by 1and update the schedule as S ¼ Sþ Di� j

Step 6 If minjpj 6maxipci , go to Step 2 to wait for the next patient

call. Otherwise, terminate since no more patients can beadded to schedule S (due to Property 4).

The scheduling algorithm starts with an empty schedule of a gi-ven session. Since any patient can be added to an empty schedule,the critical probability pc

i is set to one in Step 1. In Step 2, thescheduler waits until a patient calls for an appointment. Whenthe patient calls, the no-show probability is estimated and the pa-tient is categorized in one of the predetermined patient groups. Letthe patient be of type j with no-show probability of 1 � pj. In Step3, pj is compared with the critical probabilities at each slot ðpc

i Þ. If pj

is greater than all pci values, then the patient cannot be assigned to

any of the slots (Property 4). Otherwise, the slots are sorted accord-ing to the expected overtime required to serve all the patients inthe system at the beginning of the slot (max{0,E[NSi]l � ti}). Ifthe expected time to serve all patients at the beginning of the slotis less than the slot length, the slot will have a higher priority whilethe patient is being scheduled. When there are multiple slots withthe same priority, the earlier slot is chosen since it gives smaller in-crease in expected overtime (Property 5). The main reason for sort-

ing the slots according to the expected workload is finding a ‘‘fair’’schedule with similar workload at each slot. Once the slots aresorted, the patient is assigned to the first slot in the sorted list tem-porarily (Step 4.1). In Step 4.2, the expectation and variance ofwaiting time, queue length and number of patients in the systemare calculated. In Step 4.3, constraint feasibility is checked for allslots. If all constraints are satisfied, the patient is assigned to thatslot i, i.e. i⁄ = i. If at least one of the constraints among (1)–(4) arenot satisfied, which means the patient of type j cannot be assignedto slot i, no patients with higher pj0 values can be assigned to slot i(Property 3). In order to ensure that, the critical probability of slot i,pc

i , is updated as pj � �, where � is a very small number such thatpj � � > pk, "pk < pj. Please note that the critical probability is up-dated only when constraints (1)–(4) are violated. Property 3 cannotbe generalized for constraints (5) and (6), because the variance isnot an increasing function of pj. For example, the variance of arriv-als to a slot (Var[Xi]) is minimum when pj = 0. As pj increases, Var[-Xi] increases until pj becomes 0.5. After that point, Var[Xi] decreasesas pj gets closer to 1. If the patient cannot be assigned to slot i dueto constraint violation, Steps 4.1–4.5 are repeated for the next slotin the sorted list SI. In Step 5, if a slot satisfying all the constraints isfound, the patient is assigned to that slot. Otherwise, the patient isnot scheduled for that day. Another day is searched for an availableslot using the same algorithm. In Step 6, the stopping condition ischecked. If minjpj > maxipc

i , no more patients can be added to theschedule and the algorithm terminates for the given session.Otherwise, the scheduler waits for next patient call.

4. Computational study

The computational study is in four parts. In the first part, ouraim is to show trade-off among multiple criteria. The proposedmodel is solved with different upper bound values to find thetrade-off between expected revenue and unfairness and to showthe effect of different bounds on the service criteria, which are inthe constraint set. In the second part, we show the effect of differ-ent appointment slot lengths on expected revenue and unfairness.In the third part, we show the effect of using general service timesinstead of exponential service times. The schedules are generatedwith the assumption of exponential service times and then theyare simulated with Gamma and Lognormal service time distribu-tions with the same mean and different variances. In the last part,we show the effect of using different number of no-show classes oncomputation times and performance measures.

4.1. Trade-off among multiple criteria

We consider a 4 hours session with 16 slots (each slot is 15 min-utes). The service times are assumed to be exponentially distrib-uted with mean l = 1, which corresponds to one slot (expectedservice time is 15 minutes). Three types of patients with no-showprobabilities of 0.1, 0.3, and 0.5 are considered. When the patientcalls in, the no-show probability of the patient is estimated andthe patient is categorized into one of the three patient groupswhere p1 = 0.9, p2 = 0.7 and p3 = 0.5. We consider the objective ofminimizing UNS and UW, and maximizing expected revenue. Theupper bounds considered in the first part of the computationalstudy are 1 and 2 slots for expected overtime and expected waitingtime, 2 and 3 patients for expected number of patients in the sys-tem, 2 and 4 for the variance of waiting time and variance of num-ber of patients in the system (beo = 1 or 2, bew = 1 or 2, ben = 2 or 3,bvw = 2 or 4, and bvn = 2 or 4). Since all the criteria in the constraintset are functions of Rik and Qim, only one upper bound is fixed for agiven experiment and the other bounds are set to infinity, resultingin 8 different problem settings. One hundred different call-in

0 2 4 6 8 10 12 14 16 180.5

1

1.5

2

2.5

3

3.5

Slot i

Exp

ecte

d nu

mbe

r of

pat

ient

s in

the

syst

em, E

[NS

i]

beo=1 bew=1 ben=2 bvw=2 bvn=2


0 2 4 6 8 10 12 14 160.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Slot i

Exp

ecte

d w

aitin

g tim

e, E

[Wi]


beo=2 bew=2 ben=3 bvw=4 bvn=3a b

Fig. 2. (a) Expected number of patients in the system at the beginning of each slot (E[NSi]), (b) Expected waiting time at each slot (E[Wi]).


sequences are randomly generated for each problem setting.Figs. 2-5 plot the average of the 100 replications.

Fig. 2. (a) shows the expected number of patients in the systemat the beginning of each slot (E[NSi]). As the upper bounds increase,the expected number of patients at the beginning of each slotincreases. The expected number of patients in the system at thebeginning of slot 17 shows the expected number of patients whoare served during overtime hours. When upper bounds are tight,the number of patients served during overtime hours is between1–1.25. As the bounds increase, the number of patients increaseto 1.8–2.2. Fig. 2. (b) shows the expected waiting time at each slot(E[Wi]). The expected waiting time at each slot shows the samepattern as E[NSi].

Fig. 3 shows the expected number of arrivals at each slot. Theexpected number of arrivals in the first two slots is higher thanthe ones in later slots. When the upper bounds increase, the ex-pected number of arrivals at each slot increases, resulting in im-proved expected profit.

0 2 4 6 8 10 12 14 160.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Slot i

Exp

ect

ed

nu

mb

er

of

arr

iva

ls,

E[X

i]



Fig. 3. Expected number of arrivals at each slot (E[Xi]).

Fig. 4. (a) shows the trade-off between the expected revenueand unfairness measure UNS. The expected revenue increases asthe upper bounds increase. However, the unfairness also increases.The connected points are non-dominated solutions. When upperbounds are tight, the solutions found for the problems with the ex-pected overtime, waiting time, and number of patients in the sys-tem form the non-dominated solution set. When the boundsincrease, all schedules, except the schedule generated for theproblem with upper bounds on variance of waiting time, formthe non-dominated solution set. Fig. 4. (b) shows the trade-off be-tween expected revenue and UW. We can see that putting a tightupper bound on variance of number of patients in the system doesnot generate a non-dominated solution.

Fig. 5 shows the variance of the number of patients in the sys-tem and waiting time. When bounds are tight, the variance in thefirst three slots increases rapidly and then stabilizes for theremaining slots. As the upper bounds increase, the variance in-creases as the slot number increases. Thus, patients scheduled tolater slots experience more variable waiting times.

When we analyze individual schedules, we can see that, most ofthe time, only one patient is scheduled to each slot when thebounds are tight. As the bounds increase, the slots are overbooked.The following example shows how the schedules change for differ-ent upper bounds. The call-in sequence for the example is (1, 1, 1,3, 3, 2, 2, 1, 3, 3, 2, 1, 3, 2, 3, 2, 1, 2, 1, 1, 2, 3, 3, 2, 2, 1, 1, 1, 3, 3, 1),where the first call-in patient is of type 1, the second call-in patientis type 1, and so on. Fig. 6 shows a Gantt chart of the schedules fordifferent upper bounds on expected overtime. When the upperbound on expected overtime is 1 (corresponding to 15 minutes),the first 17 patients are scheduled. The schedule is S = [2,0,0;1,0,0; 0,0,2; 0,0,0; 0,1,0; 0,1,0; 1,0,0; 1,0,0; 0,0,1; 0,0,1; 0,1,0;1,0,0; 0,0,1; 0,1,0; 0,0,1; 0,1,0], where two patients of type 1(p1 = 0.9) are scheduled to arrive in the first slot, one patient of type1 is scheduled to arrive in the second slot, two patients of type 3(p3 = 0.5) are scheduled to arrive at the third slots, and so on. Asthe upper bound on expected overtime increases, the schedulednumber of patients also increases. When the upper bound is 2,the first 20 patients are scheduled. When upper bound is 3, the first24 patients are scheduled. The expected number of arrivals are11.9, 14.4 and 17.0, and unfairness measures (UW) are 0.9267,1.2543 and 2.4914 for the upper bounds 1, 2 and 3, respectively.When the upper bound is reduced from 3 to 1, the unfairness

0.8 0.9 1 1.1 1.2 1.3 1.4 1.511

11.5

12

12.5

13

13.5

14

14.5

15

15.5

Unfairness, UNS

Expe

cted

num

ber o

f pat

ient

arri

vals

beo

=1

beo

=2

bew

=1

bew

=2

ben

=2

ben

=3

bvw

=2

bvw

=6

bvn

=2

bvn

=4

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.511

11.5

12

12.5

13

13.5

14

14.5

15

15.5

Unfairness, UW

Expe

cted

num

ber o

f pat

ient

arri

vals

beo

=1

beo

=2

bew

=1

bew

=2

ben

=2

ben

=3

bvw

=2

bvw

=6

bvn

=2

bvn

=4

a b

Fig. 4. (a) Trade-off between expected revenue and unfairness measure UNS, (b) Trade-off between expected revenue and unfairness measure UW.

0 2 4 6 8 10 12 14 16 180

1

2

3

4

5

6

Slot i

Var

ianc

e of

num

ber

of p

atie

nts

in th

e sy

stem

, Var

[NS

i]



0 2 4 6 8 10 12 14 160

1

2

3

4

5

6

7

Slot i

Var

ianc

e of

wai

ting

time,

Var

[Wi]


beo=2 bew=2 ben=3 bvw=4 bvn=3a b

Fig. 5. (a) Variance of number of patients in the system at the beginning of each slot (Var[NSi]), (b) Variance of waiting time at each slot (Var[Wi]).


improves by 1.5647 (2.4914–0.9267), which corresponds to 23.5minutes decrease in the difference between maximum and mini-mum expected waiting times. However, the expected number ofarrivals decreases by 5.1 patients. If the expected revenue per pa-tient is $150, the expected revenue loss is $765. The revenue lossper one minute of improvement in unfairness is $32.6 ($765/23.5minutes).

To summarize, we make the following observations based onour computational results:

� When upper bounds are tight, the proposed algorithm sched-ules less patients leading to less expected profit. As the upperbounds increase, more patients are scheduled. Overbookingincreases the expected profit.� When the bounds are tight, the expectation and variance of

waiting time and number of patients in the system become sta-ble after the third slot. When the upper bounds are high, the

variance of waiting time and number of patients in the systemincreases as the slot index increases. This leads to more extremepatient waiting times for the patients scheduled to later slots.� The service criteria at each slot show similar behavior when the

expectation and variance of waiting time and number ofpatients in the system at each slot are bounded. When expectedovertime is bounded, the proposed algorithm schedules morepatients to earlier slots and less patients to later slots.� Non-dominated solutions that maximize expected profit and

minimize unfairness can be generated by changing the upperbounds on constraints to provide alternatives to the decisionmaker. As the upper bounds increase, more patients are sched-uled and higher expected profit is achieved.� When the bounds are tight, the proposed algorithm gives lower

unfairness values (UNS,UW). Higher upper bounds increase theunfairness between slots. We believe that the main reason forhigh unfairness is the high variance of exponential service

8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00

1

2

3

1,1

1,1

1,1

1

1

2,1

3,3

3,3

3,3

2

2

2

2

2,2

2

2

2,2

1

1

1

1

1

1

3

3

3,3

3

3,1

3,1

2

2

2

1

1

1

3

3

3

2

2

2

3

3,1

3,1

2

2

2

Time

Upp

er b

ound

on

over

time

UW = 0.9267UNS = 0.7760E[X] = 11.9N=17

UW = 1.2543UNS = 1.1098E[X] = 14.4N=20

UW = 2.4914UNS = 2.4120E[X] = 17.0N = 24

Fig. 6. Gantt chart of the schedules for a call-in sequence of (1,1,1,3,3,2,2,1,3,3,2,1,3,2,3,2,1,2,1,1,2,3,3,2,2,1,1,1,3,3,1), where p1 = 0.9, p2 = 0.7 and p3 = 0.5.

0.5 1 1.5 2 2.5 313.5

14

14.5

15

15.5

16

16.5

17

17.5

Unfairness, UNS

Exp

ecte

d nu

mbe

r of

pat

ient

arr

ival

s, E

[X]

Slot length = 30 minutesSlot length = 15 minutesSlot length = 5 minutes

0 10 20 30 40 500.5

1

1.5

2

2.5

Slot i

Ave

rage

num

ber

of a

rriv

als

(E[X

i])


a b

Fig. 7. (a) Trade-off curve between unfairness (UNS) and expected profit for slot lengths 30, 15 and 5 minutes, (b) Average number of arrivals at each slot.

5 10 15 20 25 3013.5

14

14.5

15

15.5

16

16.5

17

17.5

Unfairness, UW

Exp

ecte

d nu

mbe

r of

pat

ient

arr

ival

s, E

[X]


5 10 15 20 25 3013.5

14

14.5

15

15.5

16

16.5

17

17.5

Unfairness, UW

Exp

ecte

d nu

mbe

r of

pat

ient

arr

ival

s, E

[X]


a b

Fig. 8. Trade-off curve between unfairness (UW) and expected profit for slot lengths 30, 15 and 5 minutes, considering (a) all slots, and (b) slots with at least one scheduledpatient.



times, which significantly increases the expected number ofpatients and expected waiting times in later slots due to moreoverbooking in earlier slots. Since the unfairness measures areirregular performance measures, which do not monotonicallyincrease or decrease as more patients are scheduled, this obser-vation is not generalizable.

4.2. Effect of slot length

In the first part of our study, we assume that the slot lengths areequal to the expected service time. However, the slot length andthe number of slots in each session would affect the schedulingdecisions. In the second part of the computational study, our aimis to show the effect of slot length on multiple moment-basedcriteria. We consider 30, 15, and 5 minutes slots in a 4 hourssession. We assume that the service times are exponentiallydistributed with mean 15 minutes. Three types of patients withno-show probabilities of 0.1, 0.3, and 0.5 are considered. The upperbound for expected waiting time at each slot is set as ubew = 30

0 2 4 6 8 10 12 14 165

10

15

20

25

30

Slot i

Aver

age

wai

ting

time

at e

ach

slot

(in

min

utes

)

Exponential σ=15Gamma σ=10Gamma σ=5Lognormal σ=15Lognormal σ=10Lognormal σ=5

a

Fig. 10. (a) Average waiting time of patients assigned to each slot for different service timshow probabilities for all patients are under-estimated or over-estimated.

0 5 10 15 20 25 30 35 40 45 505

10

15

20

25

30

Slot i

Ave

rage

wai

ting

time

at e

ach

slot

(m

inut

es)


Fig. 9. Expected waiting time at each slot and expected numb

minutes. All other upper bounds are set to infinity. The primaryobjective is minimization of the unfairness measure UNS, that con-siders the expected number of patients in the system at the begin-ning of each slot. One hundred different call-in sequences aregenerated for all slot length alternatives.

In Fig. 7 (a), the trade-off between the expected number ofarrivals and unfairness (UNS) can be seen. When slot length is 5minutes (which is shorter than the average service time), the num-ber of patients that can be scheduled increases. However, unfair-ness also increases due to the slots with no patients. Unfairnessis minimized when 15 minutes appointment slot lengths are used.We believe it is easier to generate a schedule with a more balancedworkload throughout the session when slot lengths are equal toaverage service time. Fig. 7 (b) shows the expected number ofarrivals (E[Xi]) at each slot. The average number of arrivals at eachslot gets smaller as the slot length decreases.

In Fig. 8 (a), the trade-off between the expected number ofarrivals and the second unfairness measure (UW) can be seen. Theunfairness increases as the slot length decreases. This is due to

0 2 4 6 8 10 12 14 160

5

10

15

20

25

30

35

40

45

Slot i

Aver

age

wai

ting

time

at e

ach

slot

(in

min

utes

)

pj + 0.05pj + 0.1pj + 0.2pjpj − 0.05pj − 0.1pj − 0.2

b

e distributions, (b) Average waiting time of patients assigned to each slot where no-

0 10 20 30 40 501.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

Slot i

Ave

rage

num

ber

of p

atie

nts

in th

e sy

stem

at th

e be

ginn

ing

of e

ach

slot

(E

[NS

i])Slot length = 30 minutesSlot length = 15 minutesSlot length = 5 minutes

er of patients in the system at the beginning of each slot.

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

Patient call−in sequence number

Cpu

tim

e (in

sec

onds

)

C=100C=10C=3

0 2 4 6 8 10 12 14 165

10

15

20

25

30

35

Slot i

Aver

age

wai

ting

time

at e

ach

slot

(in

min

utes

)

C=100C=10C=3a b

Fig. 11. (a) Cpu times according to patients call-in sequence (average of 100 call-in sequences), (b) Average waiting time of patients assigned to each slot for different no-show classes and exponential service time.

Table 3No-show classes and computation times to generate appointment schedules with theproposed algorithm.

Number of no-show classes, C

Probability of arrival to anappointment [p1 p2 . . .pC]

Computation times perpatient (in seconds)

3 [0.9 0.7 0.5] 0.67210 [0.95 0.85 0.75 0.65 0.55 0.45

0.35 0.25 0.15 0.05]0.804

100 [0.995 0.985 0.975 . . . . . . . . .

0.025 0.015 0.005]0.981


empty slots, higher number of scheduled patients in earlier slots,and high variability in service times. When there are empty slots,the unfairness measure shows the maximum expected waitingtime rather than the difference of expected waiting times for thepatients assigned to different slots. Since the number of emptyslots will be higher when slot lengths smaller than service timesare used, the unfairness measure will be more than the actual dif-ference between expected waiting times. In Fig. 8 (b), the unfair-ness measure is calculated for the slots with at least onescheduled patient. The actual difference between the expectedwaiting times of the patients scheduled to different slots has de-creased for slot length of five minutes. The maximum unfairnessfor all slot lengths is 24 minutes. The unfairness has less variabilityamong different call-in sequences for smaller slot lengths.

Fig. 9 shows the average waiting times at each slot for the slotswith at least one scheduled patient and the average number of pa-tients in the system at the beginning of each slot. The average wait-ing time and the number of patients in the system at each slot islower for 5 minutes appointment slots.

Based on our computational results, we make the followingobservations:

� The expected waiting times at each slot and number of patientsin the system at the beginning of each slot are minimized andthe profit is maximized when smaller slot lengths are used.� The average UW increases as the slot length decreases. This is

due to empty slots, higher number of scheduled patients in ear-lier slots, and high variability in service times.� The average UNS is lower when slot length is equal to average

service time.

4.3. Sensitivity of the results with respect to general service times andno-show probability estimations

To show the performance of the schedules suggested from theexponential based model in a more realistic system with non-exponential service times, we performed a simulation study. 100call-in sequences with 40 randomly selected patients are gener-ated. An initial schedule is generated for each call-in sequencebased on the exponential service time assumption with an upperbound of 30 minutes (2 slots) on expected waiting at each slot,i.e. E[Wi] 6 2 slots. The schedules are simulated 2000 times for dif-ferent service time distributions: (i) Exponential (15) with l = 15,

r = 15, (ii) Gamma (2.25,6.67) with l = 15, r = 10, (iii) Gamma(9,1.67) with l = 15, r = 5, (iv) Lognormal (2.3615,0.8326) withl = 15, r = 15, (v) Lognormal (2.5242,0.6064) with l = 15, r = 10,(vi) Lognormal (2.6554,0.3246) with l = 15, r = 15.

Fig. 10. (a) shows the average waiting time at each slot for dif-ferent service time distributions. As the variance of service timesdecreases, the average waiting time decreases. For example, theaverage waiting time for all patients is 23.8 minutes when the ser-vice times are exponential, reduces to 17.4 minutes when servicetimes are Gamma with standard deviation of 10, and further re-duces to 12.5 minutes when service times are Gamma with stan-dard deviation of 5. Lognormal distribution gives similar resultsas the Gamma distribution. Computational results show that theexponential service assumption provides an upper bound fornon-exponential service times. However, the expected waitingtimes at later slots reduces significantly as the variance of servicetimes decreases.

The no-show predictions are sensitive to the amount of data apatient has. The no-show probabilities for long-time establishedpatients are more reliable (with narrower confidence intervals).The no-show probabilities for new patients are less reliable withwider confidence intervals. Fig. 10. (b) shows the average waitingtime for extreme cases where all no-show probabilities are over-estimated or under-estimated. When no-show probabilities areover-estimated (pj � 0.05,pj � 0.1,pj � 0.2), the expected waitingtimes decrease due to less number of actual arrivals. Whenno-show probabilities are under-estimated (pj + 0.05,pj + 0.1,pj + 0.2), the waiting times increase due to more arrivals. For thisextreme case, where all no-show probabilities are assumed to bebiased, the change in waiting time is significant. In a real setting,the amount of increase/decrease will depend on the percentage

late start

i...... u i +1u

i −1u

unavailabletime period

...... ......

Xi + 1u X u

i + 1Yi − 1u 0

1 Planned schedule

Realized schedule

I......

Fig. 13. The unavailable period where the patients in the system at the beginning of unavailable period are served, but the patients scheduled after the unavailable periodstart to be served late.

assigned for slot i to

1 Ii...... ......

Xi

u i +1u

i −1u

i

Zi, i −1u Z u

i, i=

unavailabletime period

overflow of patients

slot iu

Fig. 12. The unavailable period where the patients in the system at the beginning of unavailable period have to wait until the physician becomes available.


of patients who have more no-show probability estimation error.For example, if only 10% of the patients are relatively new patientswith less data and wider confidence intervals, the difference wouldbe less.

According to these computational results, we can conclude thatthe performance measures are sensitive to the variance of servicetimes and the no-show prediction errors. As the variance of servicetimes decrease, the expected waiting times at each slot decrease.As the no-show estimation error increases, the performance mea-sures increase or decrease significantly. As a future research, newscheduling methods that consider the variance of both servicetimes and no-show probabilities can be proposed.

4.4. Computation times

The computation times of the scheduling algorithm depend onthe number of slots and the number of no-show classes used toclassify patients according to the estimated no-show probabilities.To show the effect of number of no-show classes on computationtimes, three settings are used (shown in Table 3). When a patientcalls, the no-show probability is estimated according to the no-show prediction models and the patient is put into the closestno-show class. For example, if the no-show probability is 0.65,the patient will be assigned to no-show class 2 (with p2 = 0.7) insetting 1, no-show class 4 in setting 2 (p4 = 0.65), and no-showclass 35 in setting 3 (p35 = 0.655). The proposed algorithm usesthe no-show probability of the corresponding no-show class ratherthan individualized estimate. To show the sensitivity of the resultsand computation times to the number of no-show classes, we gen-erated 100 call-in sequences. A call-in sequence shows the pre-dicted no-show values of each patient Then, each patient isassigned to the closest no-show type. For example, if the call-in se-quence is [0.65 0.33 0.23 0.85 0.15 . . . ], the no-show probabilitieswould be [0.7 0.5 0.5 0.9 0.5 . . . ] when three no-show classes areused. Three schedules (with 3, 10, and 100 no-show classes) aregenerated for each call-in sequence.

Fig. 11. (a) shows the average computation times according topatient’s call-in sequence. When the schedule is not full (the call-in sequence of the patient 616, where 16 is the number of slots),the computation times to find an available slot for the call-in pa-tient is less than half second. As the schedule becomes full, thecomputation times increase because the algorithm has to checkall slots for a possible assignment. The maximum computationtime is approximately six seconds for all no-show classes. Whenwe look at the expected waiting times at each slot (Fig. 11. (b)),we can see a significant difference between different no-show clas-ses. As the number of no-show classes increase from 3 to 10 and100, better solutions with less expected waiting times are found.The average unfairness measures (UW) are 23.9, 25.03, and 25.52for no-show classes 100, 10, and 3, respectively.

The results show that the computation times are very lowregardless of the number of no-show classes used. However, theperformance measures become better as more no-show classesare used. That means, using individualized no-show probabilitiesgives better schedules. This result also shows the importance ofconsidering heterogenous no-show probabilities in appointmentscheduling.

5. Unavailable time periods

The constraint-based approach can be used to handle severalother constraints such as unavailable time periods. Unavailabletime periods occur due to providers’ other clinical and administra-tive duties such as making rounds in the hospital, performing sur-geries, having clinic sessions in different clinics, being on-call,attending meetings, teaching, etc. In this section, we considerunavailable time periods to show how the proposed approachcan handle these other constraints.

We consider two cases of provider unavailability. In the firstcase, when the provider becomes unavailable, patients in the sys-tem have to wait until the provider becomes available. This casemight occur when the physician has to attend another scheduledevent such as meetings, classes, surgery, etc. It might also occur


when the physician is on-call and has to leave the clinic for emer-gency patients. In the second case, the provider serves all patientswho are waiting at the beginning of unavailable period and thenbecomes unavailable for a fixed period of time. Patients who arescheduled to arrive during the unavailable period will have to waituntil the provider becomes available. This case occurs when theprovider has to be in different places before and after the unavail-able time period. For example, if the provider arrives to the clinicafter making rounds in the hospital or performing surgeries, thepatients have to wait until the provider comes to the clinic. If theprovider has his/her morning and afternoon sessions in differentclinics, the provider sees all the patients in one clinic and then goesto the other clinic. The patients in the second clinic have to waituntil the provider arrives. In the following subsections, we willshow how the expectation of waiting time, queue length and num-ber of patients in the system at each slot is calculated for the pa-tients affected by the unavailable time periods.

Case 1: The patients in the system at the beginning of unavailableperiod have to wait until the physician becomes availableFig. 12 is a pictorial representation of the first case. In this study,the moment-based criteria are calculated at each slot for thepatients assigned to that slot. If there is an unavailable periodduring the day (iu), then the criteria at each slot i < iu shouldbe re-calculated.The arrival matrix for the unavailable period should be Q iu ;0 ¼ 1and Q iu ;m ¼ 0 for m > 0. This will be guaranteed by putting anadditional constraint to the model, i.e. E[Xi] = 0. The overflowmatrix for the unavailable period should be Riu ;k ¼ Riu�1;k forall k. This can be guaranteed by putting an additional constrainton service rate in the unavailable slot, i.e., PrfLiu ¼ 0g ¼ 1. Theexpectation and variance of number of patients in the queue/system are functions of E[Xi], E[Yi�1], Ri�1,0, Qi0, Var[Xi] andVar[Yi�1] and can be calculated based on the arrival and over-flow matrices, which are updated after scheduling each patient.In order to calculate the expected waiting time at each slot, weneed to define a new variable ðZi;iu�1Þ to denote the number ofpatients who are scheduled to arrive at slot i and are still inthe system at the end of slot iu � 1. These patients have to waituntil the provider becomes available at the beginning of slotiu + 1. The probability that there are z patients, who are sched-uled to arrive at slot i and are still in the system at the end ofslot iu � 1, is calculated as follows:

PrfZi;iu�1 ¼ zg

¼

Pk

PmPz

Ri�1;kQ imPrfLi;iu�1 ¼ mþ k� zg z > 0;Pk

PmPz

Ri�1;kQ imPrfLi;iu�1 P mþ kg z ¼ 0:

8><>:

The calculation of PrfZi;iu�1 ¼ zg is very similar to the calculation ofRik = Pr{Yi = k}. When Rik is calculated, overflow from slot i to sloti + 1 is considered for all patients in the system at the beginning of sloti. When PrfZi;iu�1 ¼ zg is calculated, overflow from slot i to slot iu � 1 isconsidered only for the patients scheduled to arrive at slot i. The ex-pected waiting time at each slot i (i 6 iu � 1) is calculated as follows:

E½Wi� ¼ ðE½Yi�1�ð1� Q i0Þ þE½Xi� � 1þ Q i0

2þ E½Zi; i

u � 1�tiuÞl;

where tiu is the length of the unavailable time period. There will notbe any change in the basic steps of the proposed algorithm, becausethe arrival and overflow matrices are already calculated at eachstep, where the patient is temporarily assigned to a slot. However,there will be additional calculations to find PrfZi;iu�1 ¼ zg andE½Zi;iu�1� for all slots i < iu.

Case 2: The patients in the system at the beginning of unavailableperiod are served, but start time for the patients scheduled after theunavailable period might be lateA pictorial representation of the second case can be seen inFig. 13. In this case, the moment-based criteria for the slotsbefore the unavailable time period are not affected. However,there may be additional patient waiting due to providerunavailability at each slot after the unavailable time period(i > iu).

The arrival time of the provider depends on the total time re-quired to see the patients overflowing from slot iu � 1 to slot iu

and the duration of unavailable time period. The duration of pro-vider lateness is equal to l � Yiu�1. If we just ignore the unavailableperiod, and calculate the arrival and overflow matrices for theavailable periods assuming that the unavailable period length iszero, then we can calculate expectation and variance of waitingtime and number of patients in the queue/system at each slotwithout making any changes in our calculations. The only perfor-mance criterion that will change is the expected overtime, whichwill be calculated as E½YI�lþ tiu , where tiu is the length of unavail-able time period. The second case do not add any additional com-plexity to the execution of the proposed algorithm.

In both cases, there is no change in the basic steps of the pro-posed constraint-based approach. However, new constraints mightbe added to the model and additional calculations might be neces-sary to calculate the moment-based criteria.

6. Conclusion

In this study, we considered the clinical scheduling problemwith stochastic service times and patient no-shows. The main con-tribution of the study is using unfairness and congestion measuresas primary objectives. We proposed new unfairness measures in or-der to find uniform schedules for the patients assigned to differentslots. The proposed measures are the minimization of the differencebetween maximum and minimum expected waiting times at eachslot and the number of patients in the system at the beginning ofeach slot. In order to calculate these measures, we derive analyticalexpressions of the criteria for each slot. The derived expressions areexact expressions for the exponential service time distribution andupper bounds for the general distributions.

We also considered several other moment-based criteria suchas expectation and variance of waiting times and number of pa-tients in the system/queue. The variance of waiting time and num-ber of patients in the system/queue can also be considered asfairness measures. Instead of minimizing a weighted linear combi-nation of all the criteria, we use a constraint-based approach. Webelieve that it is easier for the decision maker to give an upperbound on these criteria rather than assigning a cost value. We de-rived the properties of the criteria and showed that the no-showrate of the patients is actually important in accepting or rejectingthe patient for a given session when a constraint-based approachis used. We also showed the flexibility of constraint-based ap-proach in handling unavailable time periods. The basic steps ofthe proposed algorithm do not change. There would be additionalconstraints and calculations to calculate the moment-based crite-ria and find a slot satisfying all constraints.

We proposed a sequential scheduling algorithm in order to min-imize the unfairness and maximize the revenue. We performed acomputational study in order to show the trade-off between multi-ple criteria. According to the results, unfairness can be minimizedwhen the upper bounds are tight. However, this reduces the num-ber of patients scheduled, which might have adverse effects onunscheduled patients’ health due to reduced access to care. As


the upper bounds increase, the expected revenue increasesbecause of overbooking. The unfairness measures, especially thevariance of waiting times and expected number of patients in thesystem, become worse for the patients assigned to later slots. Inour computational study, we also showed the effect of slot lengthson the performance measures. As the slot length decreases, theexpected revenue increases, and the expected waiting time andnumber of patients in the system decreases. The unfairnessincreases due to empty slots, where no patient can be scheduled.

Future research directions include considering other constraintssuch as service level constraints, unexpected provider unavailabil-ity (due to emergency) and restrictions on the assignment of pa-tient types (such as new and return patients) with differentservice time requirements to slots. The multiple patient typescan have different revenue values and constraints in the model.Another challenging future research direction might be findingthe worst case performance bounds for the objectives based onbounds on the constraints.

Acknowledgement

This work was supported by grants from the National ScienceFoundation (0729463) and the Regenstrief Center for HealthcareEngineering at Purdue.

Appendix A. Derivations of service criteria

A.1. Expected waiting time of the patients assigned to slot i

Expected waiting time of a patient assigned to slot i (E[Wi]) iscalculated by conditioning on the number of patients in the systemwho will be served before that patient, di.

E½Wi� ¼ E½E½Wijdi��¼ E½di�l ðservice times are i:i:d: with mean lÞ:

E[di] is calculated by conditioning on the number of patients in thesystem at the beginning of slot i.

E½di� ¼ E½E½dijYi�1;Xi�� ¼X

k

Xm

Ri�1;kQ im1m

Xm�1

s¼0

ðkþ sÞ

¼X

k

Xm

Ri�1;kQimðkþ ðm� 1Þ=2Þ

¼ E½Yi�1�ð1� Q i0Þ þE½Xi� � 1þ Q i0

2:

Then the expected waiting time becomes

E½Wi� ¼ E½Yi�1�ð1� Q i0Þ þE½Xi� � 1þ Qi0

2

� �l:

A.2. Expected queue length at the beginning of slot i

The queue length at the beginning of a slot (QLi) is the sum ofoverflow from previous slot (Yi�1) and the number of arrivals forthe current slot (Xi) minus 1. The expected queue length at thebeginning of slot i is calculated as:

E½QLi� ¼ E½E½QLijYi�1;Xi��

¼XkP1

XmP0

Ri�1;kQ imðkþm� 1Þ þXmP1

Ri�1;0Q imðm� 1Þ

¼ E½Yi�1� þ ðE½Xi� � 1Þð1� Ri�1;0Þ þ Ri�1;0ðE½Xi� � 1þ Q i0Þ¼ E½Yi�1� þ E½Xi� � 1þ Ri�1;0Q i0:

A.3. Expected number of patients in the system at the beginning of sloti

The number of patients in the system at the beginning of a slot(NSi) is the sum of overflow from previous slot (Yi�1) and the num-ber of arrivals for the current slot (Xi). The expected number of pa-tients in the system at the beginning of slot i is:

E½NSi� ¼ E½Yi�1� þ E½Xi� ðsince Yi�1 and Xi are independentÞ:

A.4. Variance of waiting time for the patients assigned to slot i

The variance of waiting time of the patients assigned to slot i(Var[Wi]) is calculated as

Var½Wi� ¼ E½Var½Wijdi�� þ Var½E½Wijdi�� ðconditional varianceÞ;

where di is the number of patients in the system who will be servedbefore a patient.

Var½Wijdi� ¼ dir2 ðservice times are i:i:d:with variance r2Þ;E½Wijdi� ¼ dilðservice times are i:i:d: with mean lÞ:

The variance of waiting time (Var[Wi]) becomes

Var½Wi� ¼ E½Var½Wijdi�� þ Var½E½Wijdi�� ¼ E½di�r2 þ Var½di�l2:

Var½di� ¼ E d2i

� �� ðE½di�Þ2 and E d2

i

� �is calculated as:

E d2i

� �¼ E E d2

i

Yi�1;Xi� ��

¼X

k

Xm

Ri�1;kQ im1m

Xm�1

s¼0

ðkþ sÞ2

¼X

k

Xm

Ri�1;kQ im k2 þ kðm� 1Þ þ ðm� 1Þð2m� 1Þ6

� �

¼ E Y2i�1

h ið1� Qi0Þ þ E½Yi�1�ðE½Xi� � 1þ Q i0Þ

þ2E X2

i

h i� 3E½Xi� þ 1� Qi0

6:

The variance of waiting time for patients assigned to slot i is

Var½Wi� ¼ E½di�r2 þ Var½di�l2

¼ E½Yi�1�ð1� Qi0Þ þE½Xi� � 1þ Q i0

2

� �r2

þ E Y2i�1

h ið1� Q i0Þ þ E½Yi�1�ðE½Xi� � 1þ Q i0Þ

�

þ2E X2

i

h i� 3E½Xi� þ 1� Qi0

6Þl2

� E½Yi�1�ð1� Qi0Þ þE½Xi� � 1þ Q i0

2

� �2

l2:

A.5. Variance of queue length and number of patients in the system atthe beginning of slot i

The variance of queue length and the variance of number of pa-tients in the system at the beginning of slot i are:

Var½QLi� ¼ Var½Yi�1� þ Var½Xi�;Var½NSi� ¼ Var½Yi�1� þ Var½Xi�:

Appendix B. Proof of properties

Proof of Property 1. To calculate the expected overtime and theexpected service time for the patients overflowing from slot i � 1to slot i, we use the memoryless property of exponential distribu-tion. One of the overflowing patients will be with the provider at


the beginning of the slot. For the exponential distribution, the timespent in the previous slot will not have any effect on the expectedservice time in the current slot due to the memoryless property. Ifthe service time distribution is not exponential, the time spent inthe previous slot must be considered. For example, if the servicetime has a general distribution with mean l for all patients, thenthe expected service time of the overflowing patients will becalculated as follows:

XkP1

E trsi�1

� �þX

1<k06k

E tsi�1;k0

h i0@

1APrfYi�1 ¼ kg

¼XkP1

E trsi�1

� �þ ðk� 1Þl

�Ri�1;k

¼XkP1

E trsi�1

� �Ri�1;k þ

XkP1

klRi�1;k �XkP1

lRi�1;k

¼ E trsi�1

� �XkP1

Ri�1;k þ lXkP1

kRi�1;k � lXkP1

Ri�1;k

¼ E trsi�1

� �ð1� Ri�1;0Þ þ lE½Yi�1� � lð1� Ri�1;0Þ

¼ E trsi�1

� �� l

�ð1� Ri�1;0Þ þ lE½Yi�1�;

where E trsi�1

� �is the expected remaining service time of the patient

being served at the end of slot i � 1 and E tsi�1;k0

h iis the expected

service time for all other patients overflowing from slot i � 1 to i.Because E trs

i�1

� �6 l, we have

lE½Yi�1� þ E trsi�1

� �� l

�ð1� Ri�1;0Þ 6 lE½Yi�1�;

implying that total expected service time for overflowing patientsunder general service time is not greater than that under exponen-tial. h

Proof of Property 2. Let Xi be the expected number of arrivals toslot i in schedule S, and, Yi, YS1

i , and YS2i be the expected overflow

from slot i � 1 to slot i in schedules S, S1 and S2, respectively.

(a) The expected number of arrivals to slot i is E[Xi] + p1 + p4 forschedule S1 = S + Di1 + Di4, and is E[Xi] + p2 + p3 for scheduleS2 = S + Di2 + Di3, which are equal to each other.

(b) The expected increase of overflow from slot i in scheduleS1 = S + Di1 + Di4 is:

E YS1i � Yi

h i¼ 2p1p4Pr YS1

i � Yi ¼ 2A14

n oþ p1p4Pr YS1

i � Yi ¼ 1A14

n oþ p1ð1� p4ÞPr YS1

i � Yi ¼ 1A1

n oþ ð1� p1Þp4Pr YS1

i � Yi ¼ 1A4

n o;

where Pr YS1i � Yi ¼ 2

A14

n ois the probability that the arrival

of patients of types 1 and 4 leads to two more patients over-flowing from slot i in schedule S1. This occurs when the num-ber of patients served is less than or equal to the number ofpatients in slot i in schedule S. Thus,

Pr YS1i � Yi ¼ 2

A14

n o¼ PrfLi 6 Xi þ Yi�1g:

Similarly, Pr YS1i � Yi ¼ 1

A1gðPr YS1i � Yi ¼ 1

A4

n on �is the

probability that the arrival of patient of type 1 (4) lead toone more patient overflowing from slot i in schedule S1,which is equal to Pr{Li 6 Xi + Yi�1}. So, we have

E YS1i � Yi

h i¼ 2p1p4PrfLi 6 Xi þ Yi�1g

þ p1p4Pr YS1i � Yi ¼ 1

A14

n oþ p1ð1� p4ÞPrfLi 6 Xi þ Yi�1gþ ð1� p1Þp4PrfLi 6 Xi þ Yi�1g¼ ð2p1p4 þ p1ð1� p4Þþ ð1� p1Þp4ÞPrfLi 6 Xi þ Yi�1g


A14

n o¼ ðp1 þ p4ÞPrfLi 6 Xi þ Yi�1g


A14

n o:

Similarly, the expected increase of overflow from slot i inschedule S2 = S + Di2 + Di3 is:

E YS2i � Yi

h i¼ ðp2 þ p3ÞPrfLi 6 Xi þ Yi�1gþ

p2p3Pr YS2i � Yi ¼ 1

A23

n o:

Since p1 + p4 = p2 + p3 and p1p4 6 p2p3, E YS1i � Yi

h i6

E YS2i � Yi

h i. Thus, schedule S1 gives smaller expected in-

crease in overflow than schedule S2, implying E YS1i

h i6

E YS2i

h i.

(c) By Eq. (2), the expected waiting time of schedule S1 iscalculated as:

E WS1i

h i¼ð2E½Yi�1��1Þð1�Qi0ð1�p1Þð1�p4ÞÞþE½Xi�þp1þp4

2l:

The expected waiting time of schedule S2 is:

E WS2i

h i¼ð2E½Yi�1��1Þð1�Qi0ð1�p2Þð1�p3ÞÞþE½Xi�þp2þp3

2l:

Since p1 + p4 = p2 + p3 and (1 � p1)(1 � p4) 6 (1 � p2)(1 � p3),E½WS1

i � > E½WS2i �when 2E[Yi�1] � 1 > 0.) If E[Yi�1] < 1/2, then

E WS1i

h i< E WS2

i

h i. h

Proof of Property 3. Let S be the initial schedule and nth patient oftype j is assigned to slot i⁄ to form the schedule S1.

(a) The expected overflow from slot i (i P i⁄) in schedule S1 iscalculated as follows:

E YS1i

h i¼ E½Yi� þ E YS1

i � Yi

h i

¼ E½Yi� þ pj

Yi�6l6i

PrfLl 6 Xl þ Yl�1g( )

;

when a patient is scheduled for slot i⁄, the expected overflowfrom slot i (i P i⁄) increases as pj increases.

(b) The expected waiting time at slot i⁄ in schedule S1 is calcu-lated as follows:

E WS1i�

h i¼

2E Yi��1½ � � 1ð Þ 1� Q i�0ð1� pjÞ �

þ E Xi�½ � þ pj

2l

¼ E Wi�½ � þ2E Yi��1½ �Q i�0 � Q i�0 þ 1ð Þpj

2l:

The expected waiting time at slot i⁄ increases as pj increases.

For slots i > i⁄, the expected waiting time E WS1i

h i� �increases

because E YS1i�1

h iincreases (due to Property 3.a).


(c) The expected queue length at the beginning of slot i⁄ in sche-dule S1 is calculated as follows:

E QLS1i�

h i¼ E½Yi��1� þ E½Xi� � þ pj � 1þ Ri��1;0Qi�0ð1� pjÞÞ

¼ E½QLi� � þ pjð1� Ri��1;0Q i�0Þ:

Since ð1� Ri��1;0Qi�0ÞP 0; E QLS1i�

h iincreases as pj increases.

For slots i > i⁄, the expected queue length at the beginning

of slot i is E YS1i�1

h iþ E½Xi� � 1þ RS1

i�1;0Qi0. As pj increases,

E YS1i�1

h iincreases (due to Property 3.a) and RS1

i�1;0 decreases.

Therefore, E½QLS1i � increases.

(d) The expected number of patients in the system at the begin-ning of slot i⁄ in schedule S1 is calculated as follows:

E NSS1i�

h i¼ E½Yi��1� þ E½Xi� � þ pj ¼ E½NSi� � þ pj:

E NSS1i�

h iincreases as pj increases. For slots i > i⁄, the expected

number of patients in the system at the beginning of slot i in-creases, because E YS1

i�1

h iincreases (Property 3.a) h

Proof of Property 5. Let S be the initial schedule. If patient of typej is assigned to slot i1, schedule S1 is formed. If the patient isassigned to slot i2, schedule S2 is formed. We define Aj as the arri-val of patient j. The expected overflow at the end of the day forschedule S1 is calculated as:

E YS1I

h i¼ E½YI� þ E YS1

I � YI

h i¼ E½YI� þ pjPr YS1

I � YI ¼ 1Aj

n o

¼ E½YI� þ pj

Yi16i6I

PrfLi 6 Xi þ Yi�1g( )

:

When the patient is assigned to slot i2, the expected overflow at theend of the day is:

E YS2I

h i¼ E½YI� þ pj

Yi26i6I

PrfLi 6 Xi þ Yi�1g( )

:

Since i1 < i2,Q

i16i6IPrfLi 6 Xi þ Yi�1gn o

6Q

i26i6IPrfLi 6 Xi

nþYi�1gg,

and E YS1I

h i6 E YS2

I

h i. Thus, the expected increase in overflow at the

end of the day is smaller if a patient is assigned to slot i1 instead ofslot i2, where i1 < i2 h

References

AHRQ. National Quality Health Report, 2007. http://www.ahrq.gov/qual/nhqr07/nhqr07.pdf.

Bailey, N., 1952. A study of queues and appointment systems in hospital outpatientdepartments with special reference to waiting times. Journal of the RoyalStatistical Society 14, 185–199.

Bundy, D., Randolph, G., Murray, M., Anderson, J., Margolis, P., 2005. Open access inprimary care: results of a North Carolina pilot project. Pediatrics 116 (1), 82.

Cashman, S.B., Savageau, J.A., Lemay, C.A., Ferguson, W., 2004. Patient health statusand appointment keeping in an urban community health center. Journal ofHealth Care Poor Underserved 15, 474–488.

Cayirli, T., Veral, E., 2003. Outpatient scheduling in healthcare: A review ofliterature. Production and Operations Management 12 (4), 519–549.

Cayirli, T., Veral, E., Rosen, H., 2008. Assessment of patient classification inappointment system design. Production and Operations Management 17 (3),338–353.

U.S. Census Bureau, Ambulatory health care services, 2002.Chakraborty, S., Muthuraman, K., Lawley, M.A., 2009. Sequential clinical scheduling

under patient no-show in absence of fixed appointment slots. Working paper,Purdue University,

Chakraborty, S., Lawley, M.A., Muthuraman, K., 2010. Sequential clinical schedulingwith patient no-shows and general service time distributions. IIE Transactions42 (5), 354–366.

Cox, T.F., Birchall, J.F., Wong, H., 1985. Optimizing the queuing system for anear, nose and throat outpatient clinic. Journal of Applied Statistics 12, 113–126.

Daggy, J., Lawley, M.A., Willis, D., Thayer, D., Suelzer, C., DeLaurentis, P.C.,Turkcan, A., Chakraborty, S., Sands, L.P., 2010. The effect of accurate no-showmodeling on clinical scheduling outcomes. Health Informatics Journal 16 (4),246–259.

Goldman, L., Freidin, R., Cook, E.F., Eigner, J., Grich, P., 1982. A multivariate approachto the prediction of no-show behavior in a primary care center. Archives ofInternal Medicine 142 (3), 563–567.

Green, L.V., Savin, S., 2008. Reducing delays for medical appointments: A queueingapproach. Operations Research 56 (6), 1526–1538.

Gruzd, D.C., Shear, C.L., Rodney, W., 1986. Determinants of no-show appointmentbehavior: the utility of multivariate analysis. Family Medicine 18, 217–220.

Gupta, D., Denton, B., 2008. Appointment scheduling in healthcare: Challenges andopportunities. IIE Transactions 40, 800–819.

Hassin, R., Mendel, S., 2008. Scheduling arrivals to queues: A single-server modelwith no-shows. Management Science 54 (3), 565–572.

Ho, C., Lau, H., 1992. Minimizing total cost in scheduling outpatient appointments.Management Science 38 (12), 1750–1764.

Kaandorp, G., Koole, G., 2007. Optimal outpatient appointment scheduling. HealthCare Management Science 10, 217–229.

Kim, S., Giachetti, R., 2006. A stochastic mathematical appointment overbookingmodel for healthcare providers to improve profits. IEEE Transactions onSystems, Man, and Cybernetics-Part A: Systems and Humans 36, 1211–1219.

Laganga, L., Lawrence, S., 2007a. Clinic overbooking to improve patient access andprovider productivity. Decision Sciences 38, 251–276.

Laganga, L., Lawrence, S., 2007b. An appointment overbooking model to improveclient access and provider productivity, Technical report, University of Coloradoat Boulder.

Lee, V.J., Earnest, A., Chen, M.I., Krishnan, B., 2005. Predictors of failed attendances ina multi- specialty outpatient center using electronic databases. BMC HealthServices Research 5 (51).

Murray, M., Bodenheimer, T., Rittenhouse, D., 2003. Improving timely access toprimary care:case studies of the advanced access model. The Journal of theAmerican Medical Association 289, 1042–1046.

Muthuraman, K., Lawley, M.A., 2008. A stochastic overbooking model for outpatientclinical scheduling with no-shows. IIE Transactions 40 (9), 820–837.

OConnor, M.E., Matthews, B.S., Gao, D., 2006. Effect of open access scheduling onmissed appointments, immunizations, and continuity of care for infant well-child care visits. Arch Pediatr Adolesc Med 160, 889–893.

Parente, D., Pinto, M., Barber, J., 2005. A pre-post comparison of service operationalefficiency and patient satisfaction under open access scheduling. Health CareManagement Review 30 (3), 220.

Salisbury, C., Montgomery, A., Simons, L., Sampson, F., Edwards, S., Baxter, H.,Goodall, S., Smith, H., Lattimer, V., Pickin, D., 2007a. Impact of Advanced Accesson access, workload, and continuity: Controlled before-and-after andsimulated-patient study. The British Journal of General Practice 57 (541), 608.

Salisbury, C., Goodall, S., Montgomery, A., Pickin, D., Edwards, S., Sampson, F.,Simons, L., Lattimer, V., 2007b. Does Advanced Access improve access toprimary health care? Questionnaire survey of patients. The British Journal ofGeneral Practice 57 (541), 615.

Savin, S., 2006. Managing patient appointments in primary care. In: Hall, RandolpW. (Ed.), Patient Flow: Reducing Delay in Healthcare Delivery. Springer, NewYork.

Yang, K.K., Lu, M.L., Quek, S.A., 1998. A new appointment rule for a single-server,multiple customer service system. Naval Research Logistics 45, 313–326.

Zeng, B., Turkcan, A., Lin, J., Lawley, M.A., 2010. Clinic scheduling models withoverbooking for patients with heterogeneous no-show probabilities. Annals ofOperations Research 178 (1), 121–144.

http://www.ahrq.gov/qual/nhqr07/nhqr07.pdf

http://www.ahrq.gov/qual/nhqr07/nhqr07.pdf

sequential clinical scheduling with service criteria sequential clinica… · sequential clinical...

Documents