Effect of two-level provider capacities on the performanceof open access clinics

Xiuli Qu & Jing Shi

Received: 19 December 2007 /Accepted: 29 August 2008 /Published online: 23 September 2008# Springer Science + Business Media, LLC 2008

Abstract The successful implementation of open accessscheduling requires the match of daily healthcare providercapacity and patient demand at the high level of totalcapacity and the low levels of individual capacities fordifferent types of appointments. In this paper, we introduce12 scheduling rules for determining the two-level providercapacities and compare them in terms of four performancemetrics: the probabilities of granting requests for fixed andopen appointments, and the expectation and the variance ofthe number of patients consulted. Our analytical resultsshow that adjusting low level provider capacities canreduce the difference between the two probabilities. Whenthe ratios of low level capacities to the high level providercapacity are fixed, the two probabilities increase with theincrease in the high level capacity. Meanwhile, ournumerical results demonstrate that the expectation and thevariance of the number of patients consulted increase withthe increase in the high level capacity. The results provideinsights in determining optimal two-level provider capaci-ties to match daily patient demand. Potential approaches tooptimality are also proposed based on the results.

Keywords Open access . Appointment scheduling rules .

Provider capacity . Performance metrics . Outpatient clinic

1 Introduction

Health expenditures have grown more rapidly than theeconomy in all developed countries, which accounted for8–11% of the GDPs in Canada, France, Germany, Japanand UK, and 16% of GDP in US [1]. Despite recordhealthcare spending, many patients have trouble accessingboth routine check-up and acute condition appointmentswith primary care providers. Increasing the accessibility tostate-of-the-art healthcare will require improvements inpatient scheduling. While acute conditions are scheduledwith short notice, traditional appointment schedulingsystems schedule a patient’s next visit months in advanceat the end of his/her current visit which has led to longwaiting lists for both routine check-up and acute careappointments in outpatient clinics [2, 3]. Furthermore,valuable resources are wasted with the higher no-showrates seen from historical comparisons of short-noticeversus long lead-time appointment scheduling. Long wait-ing also undermines the timely delivery of healthcare andoften breaks the continuity of care [4, 5].

To improve healthcare accessibility and reduce patientno-shows, open access scheduling was introduced in theearly 1990s. Its key principle is to enable patients to seetheir own care providers on the day they want to be seen forany problem [6, 7]. It should be noted that most openaccess clinics still reserve appointment slots for long lead-time appointments due to patient preference and clinicalnecessities, such as medical treatment preparations. How-ever, the majority of the appointment slots will be reservedfor short-notice appointments. The short-notice appoint-ments, called open appointments, are scheduled within ashort time period (e.g. 24 h). The long lead-time appoint-ments, called fixed appointments, can be scheduled weeksin advance. In open access scheduling systems, clinicadministrators limit the number of fixed appointments to

Health Care Manag Sci (2009) 12:99–114DOI 10.1007/s10729-008-9083-6

X. Qu (*)Department of Industrial and Systems Engineering,North Carolina A&T State University,1601 East Market Street,Greensboro, NC 27411, USAe-mail: xqu@ncat.edu

J. ShiDepartment of Industrial and Manufacturing Engineering,North Dakota State University,1410 14th Avenue N.,Fargo, ND 58105, USA

be scheduled in a clinic session. When the number of fixedappointments scheduled reaches the limit, patients whorequest fixed appointments in the session have to choosefixed appointments in other sessions or call again later foran open appointment in the session.

Over the past 10 years, successful implementation ofopen access scheduling has been reported in various typesof healthcare practices [6–11]. The potential benefitsinclude the reductions of appointment lead time and patientno-shows, and the improvements of the accessibility tohealthcare, continuity of care, operational efficiency andprovider productivity [12–14]. As a result, replacingtraditional appointment scheduling with open access sched-uling has become a nationwide trend for outpatient clinicsin the United States.

For open access scheduling, clinic administrators need tospecify provider capacities at two levels in a clinic session,which is a period for providers to see patients consecutive-ly. The high level capacity is the total provider capacity,represented by the total number of appointments to bescheduled. The low level capacities are the providercapacities for fixed and open appointments, representedby the numbers of fixed and open appointments to bescheduled, respectively. Matching daily provider capacityto patient demand at both levels is critical for the successfulimplementation of open access scheduling [5, 9]. Intraditional appointment scheduling systems, the schedulingrule is usually provider-based. In other words, the clinicadministrators make decisions on the number of appoint-ments to be scheduled per session based on the number ofproviders and their availability. This approach may cause awaste of resource if there is not sufficient demand, and thedenial of access to care for some patients if the demandexceeds the capacity. As the healthcare industry is movingtowards patient-centred care, the clinic scheduling rules arerequired to determine the total provider capacity based onpatient demand. This new approach can alleviate thepotential problems caused by the conventional approach.In this approach, the clinic administrators will adjust thelength of sessions and the number of providers in eachsession according to the estimate of patient visits.

Although many capacity management models havebeen proposed for other service industries, such as airlines,hotels, and delivery services [15–17], only a few studiesfocus on allocating capacity to different types of medicalservices [18, 19]. Green et al. [18] propose service priorityrules to schedule outpatients, inpatients and emergencypatients in a magnetic resonance imaging (MRI) center.Unlike outpatient appointment scheduling, MRI examina-tion scheduling manages a lower percentage of patient no-shows, and possesses a higher degree of flexibility toincrease capacity. For example, an emergency MRI

examination can be scheduled at 11 p.m. while an urgentoutpatient visit would not be scheduled outside normalclinic hours. Furthermore, same-day MRI examinations areonly available for inpatients and emergency patients whileopen access clinics provide open appointments for bothurgent patients and routine check-up patients. Becauseof unique challenges in open access clinics, such as ahigher percentage of patient no-shows, less flexiblecapacity, and open appointments available for all patients,the service priority rules proposed in the study are notapplicable to allocating provider capacity to open versusfixed appointments.

Green and Savin [19] use the newsvendor-type analysisto determine the proper panel size for each provider tomaximize the expected daily profit which consists of theprofit made from seeing patients during regular hours andthe loss caused by seeing patients during overtime.However, the approach is still provider-based because itcontrols the patient panel size to match patient demand withthe high level provider capacity. In addition, allocating highlevel provider capacity to open versus fixed appointmentdemand is not considered. For low level provider capaci-ties, Qu et al. [20] present a closed-form analyticalapproach to determine the optimal percentage of openappointments that maximizes the expected number ofpatients consulted. By considering the volatility caused bythe stochastic nature of patient demand and no-shows, thework is extended to a mean-variance model to maximizethe expectation and minimize the variance of the number ofpatients consulted [21]. However, in both studies [20, 21],the high level provider capacity is assumed to be givenbased on the number of appointment slots in each clinicsession and their overbooking policy. Further research iscalled for on how to simultaneously determine the highlevel and low level provider capacities to match patientdemand, and how the two level provider capacities affectthe performance of open access scheduling. The primarygoal of this study is to address this research needs.

In this paper, we propose a series of methods todetermine the provider capacities at both levels andcompare the performances of the proposed schedulingrules. The rest of the paper is organized as follows. InSection 2, we propose 12 scheduling rules to match dailyprovider capacities to patient demands in open accessclinics. Then we discuss performance metrics to evaluatethe match in Section 3. In Section 4, we analyticallyanalyze the effect of the scheduling rules on the clinicperformance, which is followed by the performancecomparison of these scheduling rules in a real clinic caseand the extended cases in Section 5. According to theanalytical and numerical results in Sections 4 and 5, wediscuss how to determine optimal high level and low level

100 Health Care Manag Sci (2009) 12:99–114

provider capacities in Section 6. Finally, conclusions aredrawn in Section 7.

2 Open access scheduling rules to determine two-levelprovider capacities

2.1 Notation

Before introducing the methods to determine the two-levelprovider capacities and the quantitative metrics to evaluatethe match of provider capacity and patient demand, wesummarize the symbol notation used in Sections 2 and 3 inTable 1.

2.2 Methods to determine two-level provider capacities

In an open access clinic, to match daily providercapacities to patient demands, clinic administrators firstneed to determine the total number of appointments thatcan be scheduled in each clinic session, namely, high levelprovider capacity. We compare three methods to deter-mine the total number of appointments to be scheduled(N). One method is N ¼ E Dfð Þ þ E Doð Þd e, where �d emeans to round up to the closest integer. In this method,only the average demands for fixed and open appoint-ments are considered to determine N. This is the

traditional approach used by outpatient clinics to deter-mine provider capacity [22]. The second and thirdmethods determine N by considering the no-show ratesand the average demands for fixed and open appointments.The no-show rate is defined as the proportion of the totalscheduled appointments that are missed. The secondmethod is N ¼ 1� gfð ÞE Dfð Þ þ 1� goð ÞE Doð Þd e. This isvery reasonable because patient no-shows actually reducethe provider capacity needed to match patient demand. Onthe other hand, the third method increases N to reduce theadverse effects of patient no-shows, which is N ¼E Dfð Þ= 1� gfð Þþd E Doð Þ= 1� goð Þe. This idea has beenadopted by many outpatient clinics to counter thedetrimental effect of high patient no-shows [23–25].

After determining the high level provider capacity, clinicadministrators still need to determine how many fixed andopen appointments to schedule in a given session, namely,low level provider capacities. This is equivalent todetermining the percentage of open appointments to bescheduled in a clinic session (w). We propose four methodsfor determining this percentage by: (1) setting N1=N (i.e.w=0); (2) w ¼ E Doð Þ= E Dfð Þ þ E Doð Þ½ �; (3) maximizingthe expected number of patients consulted, E Mð Þ; (4)minimizing the variance of the number of patients con-sulted, Var Mð Þ. The first method indicates that noappointment slots are reserved for open appointments,which is the exact practice of traditional appointmentscheduling. The second method requires the percentage ofopen appointments to be the proportion of requests for openappointments. For the remaining two methods, quantitativeapproaches have been developed to determine the percent-age of open appointments given high level providercapacity [20, 21].

By combining the three methods determining the totalnumber of appointments (N) and the four methodsdetermining the percentage of open appointments (w), wegenerate 12 open access scheduling rules (OAS rules)summarized in Table 2. To evaluate the performances of theproposed OAS rules, we assume that the OAS rules do notchange patient demand. Also we assume that patient no-shows are independent of each other, and no-show ratesincrease with the increase in appointment lead time [3, 9]. Itis assumed that in an open access clinic, the no-show ratesof fixed and open appointments are known, and the demanddistribution is also known.

3 Performance metrics to evaluate the match of dailyprovider capacity and patient demand

The probability of granting appointment requests influencespatient satisfaction and measures the accessibility to care,

Table 1 Notation

Symbol Description

Cov Sf ;Soð Þ Covariance of Sf and SoDf Demand for fixed appointments in a clinic sessionDo Demand for open appointments in a clinic sessionE(●) Expectation of a random variableM Number of patients consulted in a clinic sessionN Total number of appointments to be scheduled in a

clinic session, i.e. high level provider capacityN1 Limit of fixed appointments to be scheduled in a

clinic sessionp(i, j) Joint probability mass function of demands Df

and Do

pf(i) Probability mass function of demand Df

qf Probability of granting fixed appointment requestsqo Probability of granting open appointment requestsγf No-show rate of fixed appointmentsγo No-show rate of open appointmentsSf Number of fixed appointments scheduled in a clinic

sessionSo Number of open appointments scheduled in a clinic

sessionVar(●) Variance of a random variablew Percentage of open appointments to be scheduled

in a clinic session

Health Care Manag Sci (2009) 12:99–114 101

which is directly affected by the degree of match betweendaily provider capacity and patient demand. Insufficientprovider capacity leads to a low probability of grantingappointment requests, while a probability close to 100%may imply excess provider capacity. Furthermore, asignificant difference between the probabilities of grantingrequests for fixed and open appointments in a sessionreflects the mismatch of low level provider capacities to thedemands for fixed and open appointments. Therefore, thefirst two performance metrics to compare the 12 proposedOAS rules are the probability of granting fixed appointmentrequests and the probability of granting open appointmentrequests. The second two performance metrics are theexpectation and the variance of the number of patientsconsulted in a session. The reasons are that the averagenumber of patients consulted affects the revenue and cost ofa clinic, and that a high variability in the number of patientsconsulted results in provider overtimes and long patientwaiting times in clinic sessions. Therefore, the 12 schedul-ing rules proposed in Section 2 are compared in terms offour performance metrics: the probability of granting fixedappointment requests (qf), the probability of granting openappointment requests (qo), the expected number of patientsconsulted, E Mð Þ, and the variance of the number ofpatients consulted, Var Mð Þ.

3.1 Probabilities of granting requests for fixed and openappointments

For a proposed OAS rule, the probability of granting fixedappointment requests equals the proportion of all requestsfor fixed appointments that are granted. Thus

qf ¼ E Sfð Þ=E Dfð Þ; ð1Þ

where the expectation of Sf can be calculated by

E Sfð Þ ¼ E min Df ;N1ð Þ½ � ¼ N1 �XN1

i¼0

N1 � ið Þpf ið Þ

¼ E Dfð Þ �X1i¼N1

i� N1ð Þpf ið Þ;

ð2Þ

where N1 ¼ N � wNb c ( �b c means to round down to theclosest integer).

Similarly, the probability of granting open appointmentrequests equals the proportion of all requests for openappointments that are granted. Thus

qo ¼ E Soð Þ=E Doð Þ; ð3Þwhere the expectation of So can be calculated by

E Soð Þ ¼ E min Do;N �min Df ;N1ð Þð Þ½ �¼ N � N1ð Þ þPN1

i¼0N1 � ið Þpf ið Þ

�PN1

i¼0

PN�i

j¼0N � i� jð Þp i; jð Þ

� P1i¼N1þ1

PN�N1

j¼0N � N1 � jð Þp i; jð Þ

¼ E Doð Þ �PN1

i¼0

P1j¼N�iþ1

j� N þ ið Þp i; jð Þ

� P1i¼N1þ1

P1j¼N�N1þ1

j� N þ N1ð Þp i; jð Þ:

ð4Þ

3.2 Expectation and variance of the number of patientsconsulted

According to the results in [20], we know that for aproposed OAS rule, the expected number of patientsconsulted, E Mð Þ, isE Mð Þ ¼ 1� gfð ÞE Sfð Þ þ 1� goð ÞE Soð Þ; ð5Þ

Table 2 Proposed open access scheduling rules (OAS rules)

OAS rule Method to determine the total number of appointments (N)a Method to determine the percentage of openappointments (w)b

R1 N ¼ E Dfð Þ þ E Doð Þd e w=0R2 N ¼ E Dfð Þ þ E Doð Þd e w ¼ E Doð Þ= E Dfð Þ þ E Doð Þ½ �R3 N ¼ E Dfð Þ þ E Doð Þd e Maximizing E Mð ÞR4 N ¼ E Dfð Þ þ E Doð Þd e Minimizing Var Mð ÞR5 N ¼ 1� gfð ÞE Dfð Þ þ 1� goð ÞE Doð Þd e w=0R6 N ¼ 1� gfð ÞE Dfð Þ þ 1� goð ÞE Doð Þd e w ¼ E Doð Þ= E Dfð Þ þ E Doð Þ½ �R7 N ¼ 1� gfð ÞE Dfð Þ þ 1� goð ÞE Doð Þd e Maximizing E Mð ÞR8 N ¼ 1� gfð ÞE Dfð Þ þ 1� goð ÞE Doð Þd e Minimizing Var Mð ÞR9 N ¼ E Dfð Þ= 1� gfð Þ þ E Doð Þ= 1� goð Þd e w=0R10 N ¼ E Dfð Þ= 1� gfð Þ þ E Doð Þ= 1� goð Þd e w ¼ E Doð Þ= E Dfð Þ þ E Doð Þ½ �R11 N ¼ E Dfð Þ= 1� gfð Þ þ E Doð Þ= 1� goð Þd e Maximizing E Mð ÞR12 N ¼ E Dfð Þ= 1� gfð Þ þ E Doð Þ= 1� goð Þd e Minimizing Var Mð Þa �d e means to round up to the closest integer, E Dfð Þ and E Doð Þ denote the average demands for fixed and open appointments, respectively, andγf and γo denote the no-show rates of fixed and open appointments, respectively.bE Mð Þ and Var Mð Þ denote the expectation and the variance of the number of patients consulted, respectively.

102 Health Care Manag Sci (2009) 12:99–114

where E Sf

� �and E Soð Þ can be determined using Eqs. 2

and 4.Similarly, the variance of the number of patients

consulted, Var Mð Þ, can be determined by

Var Mð Þ ¼ 1� gfð Þ2Var Sfð Þ þ gf 1� gfð ÞE Sfð Þþ 1� goð Þ2Var Soð Þ þ go 1� goð ÞE Soð Þþ2 1� gfð Þ 1� goð ÞCov Sf ;Soð Þ;

ð6Þ

where the variances and covariance of Sf and So can becalculated by

Var Sfð Þ ¼ Var min Df ;N1ð Þ½ � ¼XN1

i¼0

N1 � ið Þ2pf ið Þ

�XN1

i¼0

N1 � ið Þpf ið Þ" #2

;

ð7Þ

Var Soð Þ ¼ Var min Do;N �min Df ;N1ð Þð Þ½ �

¼XN1

i¼0

N1 � ið Þ2pf ið Þ þXN1

i¼0

XN�i

j¼0

N � N1 � jð Þ2� N1 � ið Þ2h i

p i; jð Þ þX1

i¼N1þ1

XN�N1

j¼0

N � N1 � jð Þ2p i; jð Þ

�XN1

i¼0

N1 � ið Þpf ið Þ �XN1

i¼0

XN�i

j¼0

N � i� jð Þp i; jð Þ �X1

i¼N1þ1

XN�N1

j¼0

N � N1 � jð Þp i; jð Þ" #2

;

ð8Þ

and Cov Sf ;Soð Þ ¼ Cov min Df ;N1ð Þ;min Do;N �min Df ;N1ð Þð Þ½ �

¼ �XN1

i¼0

N1 � ið Þ2pf ið Þ þXN1

i¼0

XN�i

j¼0

N1 � ið Þ N � i� jð Þp i; jð Þ

þXN1

i¼0

N1 � ið Þpf ið ÞXN1

i¼0

N1 � ið Þpf ið Þ �XN1

i¼0

XN�i

j¼0

N � i� jð Þp i; jð Þ �X1

i¼N1þ1

XN�N1

j¼0

N � N1 � jð Þp i; jð Þ" #

:

ð9Þ

4 Analytical results

In the previous section, we discussed four performancemetrics to evaluate the match of daily provider capacity andpatient demand. In this section, we discuss the effects of thepercentage of open appointments (w) and the total numberof appointments to be scheduled (N) based on the analyticalresults. Due to the complexity of the effects of w and N onthe expectation and the variance of the number of patientsconsulted [21], we only analytically examine the effects ofw and N on the probability of granting fixed appointmentrequests (qf) and the probability of granting open appoint-ment requests (qo).

Proposition 1 qf is a non-increasing function of w giventhe total number of appointments to be scheduled (N) andthe demand distribution.

Proof Let wI and wII denote the percentages of openappointments to be scheduled using two OAS rules,

respectively, and qIf and qIIf denote the probabilities ofgranting fixed appointment requests using the two rules,respectively. Thus, the numbers of fixed appointmentsallowed to be scheduled using the two rules are N I

1 ¼N � wINb c and N II

1 ¼ N � wIINb c. According to Eq. 2, forthe two OAS rules, the expected numbers of fixed appoint-ments scheduled are

E Sf N I1

� �� � ¼ E min Df ;NI1

� �� � ¼ E Dfð Þ �X1i¼N I

1

i� N I1

� �pf ið Þ;

ð10Þ

and

E Sf N II1

� �� � ¼ E min Df ;NII1

� �� � ¼ E Dfð Þ �X1i¼N II

1

i� N II1

� �pf ið Þ;

ð11Þ

Health Care Manag Sci (2009) 12:99–114 103

where Sf N I1

� �and Sf N II

1

� �denote the numbers of fixed

appointments scheduled in a clinic session using the twoOAS rules, respectively, given the demand distribution.

Without loss of generality, we assume wI<wII. Thus, wehave N I

1 � N II1 given N. According to Eqs. 10 and 11, we

know E Sf N I1

� �� � � E Sf N II1

� �� �because

E Sf N I1

� �� �� E Sf N II1

� �� � ¼ XN I1

i¼N II1

i� N II1

� �pf ið Þ

þX1

i¼N I1þ1

N I1 � N II

1

� �pf ið Þ � 0 for N I

1 � N II1 :

ð12Þ

According to Eq. 1, when E Sf N I1

� �� � � E Sf N II1

� �� �, we

have qIf � qIIf . Therefore, qf is a non-increasing function ofw given N and the demand distribution. ■

Proposition 2 qo is a non-decreasing function of w giventhe total number of appointments to be scheduled (N) andthe demand distribution.

Proof Let wI and wII denote the percentages of openappointments to be scheduled using two OAS rules,respectively, and qIo and qIIo denote the probabilities ofgranting open appointment requests using the two rules,respectively. Thus, the numbers of fixed appointmentsallowed to be scheduled using the two rules are N I

1 ¼

N � wINb c and N II1 ¼ N � wIINb c. According to Eq. 4, for

the two OAS rules, the expected numbers of open appoint-ments scheduled are

E So N I1

� �� � ¼ E Doð Þ �XN I

1

i¼0

X1j¼N�iþ1

j� N þ ið Þp i; jð Þ

�X1

i¼N I1þ1

X1j¼N�N I

1þ1

j� N þ N I1

� �p i; jð Þ;

ð13Þ

and

E So N II1

� �� � ¼ E Doð Þ �XN II

1

i¼0

X1j¼N�iþ1

j� N þ ið Þp i; jð Þ

�X1

i¼N II1 þ1

X1j¼N�N II

1 þ1

j� N þ N II1

� �p i; jð Þ;

ð14Þ

where So N I1

� �and So N II

1

� �denote the numbers of open

appointments scheduled in a clinic session using the twoOAS rules, respectively, given the demand distribution.

Without loss of generality, we assume wI<wII. Thus, wehave N I

1 � N II1 given N. According to Eqs. 13 and 14, we

know E So N I1

� �� � � E So N II1

� �� �because

E So N I1

� �� �� E So N II1

� �� � ¼ � PN I1

i¼N II1 þ1

PN�N II1

j¼N�iþ1j� N þ ið Þp i; jð Þ � PN I

1

i¼N II1 þ1

P1j¼N�N II

1 þ1

i� N II1

� �p i; jð Þ

�X1

i¼N I1þ1

XN�N II1

j¼N�N I1þ1

j� N þ N I1

� �p i; jð Þ �

X1i¼N I

1þ1

X1j¼N�N II

1 þ1

N I1 � N II

1

� �p i; jð Þ � 0 for N I

1 � N II1 :

According to Eq. 3, when E So N I1

� �� � � E So N II1

� �� �, we

have qIo � qIIo . Therefore, qo is a non-decreasing function ofw given N and the demand distribution. ■

Proposition 3 qf and qo are non-decreasing functions of Ngiven the percentage of open appointments to be scheduled(w) and the demand distribution.

Proof According to Inequality (Eq. 12), E Sfð Þ is a non-decreasing function of N1 given the demand distribution.

Given w, since N1 is a non-decreasing function of N, E Sfð Þis a non-decreasing function of N. Therefore, according toEq. 1, qf is a non-decreasing function of N given w and thedemand distribution.

Next, we prove that qo is a non-decreasing function of Ngiven w and the demand distribution. Let NI and NII denotethe total numbers of appointments to be scheduled usingtwo OAS rules, respectively, and N I

1 and N II1 denote the

numbers of fixed appointments allowed to be scheduledusing the two rules, respectively. Thus N I

1 ¼ N I � wN Ib c

104 Health Care Manag Sci (2009) 12:99–114

and N II1 ¼ N II � wN IIb c. Since So is a function of N1 and

N, E So N1;Nð Þ½ � is used to explicitly denote the expectationof So for a OAS rule by which at most N appointments canscheduled and at most N1 fixed appointments can bescheduled in a clinic session.

Without loss of generality, we assume NI>NII. Since w isgiven, we know N I

1 � N II1 . Thus, according to Eq. 4, we

obtain

E So N I1;N

I� �� �� E So N I

1;NII þ N I

1 � N II1

� �� � ¼ XN I1

i¼0

X1j¼N I�iþ1

N I � N II � N I1 þ N II

1

� �p i; jð Þ

þXN I

1

i¼0

XN I�i

j¼N IIþN I1�N II

1 �iþ1

j� N II � N I1 þ N II

1 þ i� �

p i; jð ÞþX1

i¼N I1þ1

X1j¼N I�N I

1þ1

N I � N I1 � N II þ N II

1

� �p i; jð Þ

þX1

i¼N I1þ1

XN I�N I1

j¼N II�N II1 þ1

j� N II þ N II1

� �p i; jð Þ � 0;

ð15Þ

and

E So N II1 þ 1;N II þ 1

� �� �� E So N II1 ;N

II� �� �

¼XN II

1

i¼0

X1j¼N II�iþ1

p i; jð Þ � 0:

ð16Þ

Since E So NI1 ;N

I� �� �� E So N II

1 ;NII

� �� � ¼ E So N I1;N

I� �� �� E So N I

1;NII þ N I

1 � N II1

� �� �� �þ E So N I

1;NII þ N I

1 � N II1

� �� �� E So N II1 ;N

II� �� �� �

¼ E So N I1;N

I� �� �� E So N I

1;NII þ N I

1 � N II1

� �� �� �þ

XN I1�N II

1 �1

l¼0

E So N II1 þ l þ 1;N II þ l þ 1

� �� �� E So N II1 þ l;N II þ l

� �� �� �;

we know E So N I1;N

I� �� � � E So N II

1 ;NII

� �� �for NI>NII

according to Inequalities (Eq. 15) and (Eq. 16). That meansthat given w, E Soð Þ is a non-decreasing function of N.Therefore, according to Eq. 3, qo is a non-decreasingfunction of N given w and the demand distribution. ■

Given the high level provider capacity and the demanddistribution, Propositions 1 and 2 reveal that as thepercentage of open appointments to be scheduled (w)increases, the probability of granting fixed appointmentrequests (qf) decreases while the probability of grantingopen appointment requests (qo) increases. As a result, foreach group with identical high level provider capacity (e.g.

Rules R1–R4, Rules R5–R8, or Rules R9–R12), theprobability of granting fixed appointment requests using arule with w=0 is not less than those using the other threerules in the group, and the probability of granting openappointment requests using a rule with w=0 is not greaterthan those using the other three rules in the group. Forexample, for the group of Rules R5–R8, the probabilityof granting fixed appointment requests using Rule R5with w=0 is not less than those using Rules R6–R8, andthe probability of granting open appointment requests usingRule R5 is not greater than those using Rules R6–R8.

Proposition 3 demonstrates that the probabilities ofgranting requests for fixed and open appointments increase

Health Care Manag Sci (2009) 12:99–114 105

with the increase in the high level provider capacity givenw and the demand distribution. Therefore, the probabilitiesof granting requests for fixed and open appointments usingRule R9 with w=0 are not less than those using R1 withw=0, which is not less than those using Rule R5 with w=0,because the high level provider capacity in Rule 9 is higherthan that in Rule R1, which is higher than that in Rule R5.Similarly, among Rules R2, R6 and R10 which use thesame percentage of open appointments to be scheduled,Rule R10 produces the highest probabilities of grantingrequests for fixed and open appointments, and Rule R6results in the lowest probabilities of granting requests forfixed and open appointments.

Propositions 1–3 also provide insights for clinic admin-istrators on how to achieve the desired probabilities ofgranting requests for fixed and open appointments. Whenboth probabilities of granting requests for fixed and openappointments are lower than the desired values, the totalnumber of appointments that can be scheduled should beincreased to improve the accessibility to care. If only theprobability of granting fixed appointment requests is lowerthan the desired value, clinic administrators should reducethe percentage of open appointments. On the other hand, ifonly the probability of granting open appointment requestsis lower than the desired value, clinic administrators shouldincrease the percentage of open appointments.

5 Numerical results

In the previous section, we analyze the effects of thepercentage of open appointments (w) and the total numberof appointments to be scheduled (N) on the probabilities ofgranting requests for fixed and open appointments. Sincethere are no closed-forms for the percentage of openappointments in Rules R3, R4, R7, R8, R11 and R12, wecompare the 12 proposed OAS rules through a real cliniccase and the cases extended from the real case.

5.1 Base case and extended cases

The data used to compare the proposed OAS rules arederived from patient non-attendance and appointmentscheduling studies for an academic clinic in Indianapolis,

Indiana [26–28]. In the clinic, there were about 8.4 full-time equivalent (FTE) faculty physicians and about 50residents, who formed two groups to work in two separatedsites. Correspondingly, each group has about 5 FTEphysicians. According to its historical appointment data,143,991 appointments were made in the clinic during 2003and 2004, of which 38,182 appointments were missed.Thus, for each provider group, patient demand was about70 per session and the no-show rate was 26.5%. One yearlater, the clinic implemented an open access schedulingsystem, in which only patients who needed a follow-upvisit within 14 days, medical treatment preparations or pre-arranged transportation to the clinic could request fixedappointments. As a result, the demands for fixed and openappointments in the clinic were almost separated accordingto the types of patient visits. Thus, independent Poissondistributions are used to approximate the distribution ofdemands for fixed and open appointments. The clinicadministrator estimated that about 25% of total requestsfor appointments were eligible for fixed appointments, andthat the no-show rate of open appointments was about 7%.These data are used as a base case, which is summarized inTable 3.

We extend the base case to investigate the impacts of theno-show rates and the demand distribution for fixed andopen appointments on the performances of the proposedOAS rules. In literature, the no-show rate for long lead-timeappointments could reach as high as 50–55% [3]. On theother hand, the no-show rate of open appointments isgenerally much lower than that of long lead-time appoint-ments. For example, the no-show rate resulting from anopen access pilot project for primary care conducted inNorth Carolina decreased from 16% to 11% [11]. Theaverage no-show rate decreased from 31% to 16% afterimplementing open access scheduling in a primary careclinic [9]. Therefore, we consider two extended levels of16% and 50% for the no-show rate of fixed appointments,and one extended level of 10% for the no-show rate of openappointments.

In practice, many US clinics form provider groups withtwo to five providers in each group, but some small clinicsonly have one regular provider. Usually, a provider groupwith more providers has a higher level of total patientdemand. To investigate the effect of total patient demand

Table 3 Clinic characteristics in the base case and extended cases

Factor Base case (real clinic case) Levels in extended cases

No-show rates of fixed appointments (γf) 26.5% 16%, 50%No-show rates of open appointments (γo) 7% 10%Total average demand for fixed and open appointments, (E Dfð Þ þ E Doð Þ) 70 per clinic session 14, 42 per clinic sessionPercentage of demand for fixed appointments, (E Dfð Þ= E Dfð Þ þ E Doð Þ½ �) 25% 50%Correlation coefficient of demands for fixed and open appointments Independent 0.4

106 Health Care Manag Sci (2009) 12:99–114

level on the performances of the proposed OAS rules, weconsider two other levels of total patient demand inextended cases. The two levels are 14 and 42 per session,corresponding to one FTE physician and three FTEphysicians, respectively, according to the patient/physicianratio for the real case. Due to the difference of open accessscheduling policies implemented in clinics, a higher orlower percentage of requests for fixed appointments may beobserved. In addition, in some open access clinics, patientsare given more freedom to choose fixed or open appoint-ments for their visits, which results in the correlationbetween the demands for fixed and open appointments.Therefore, in the extended cases, we also examine theeffects of the percentage of requests for fixed appointmentsand the positive correlated demand distribution for fixedand open appointments. Table 3 summarizes the levels ofthe no-show rates and the demand distributions in 71extended cases. Next, we compare the performances of the12 OAS rules in the base case, and investigate the impactsof the no-show rates and the demand distribution on therule performances in both the base case and the extendedcases.

5.2 Performances of the proposed OAS rules in the basecase

Table 4 summarizes the performances of the proposed OASrules in the base case in terms of four performance metrics:the probabilities of granting requests for fixed and openappointments, and the expectation and the variance of thenumber of patients consulted. By comparing the perform-ances of three groups of Rules R1–R4, Rules R5–R8, and

Rules R9–R12, each of which has an identical high levelprovider capacity, it can be observed that the variance of thenumber of patients consulted increases as the high levelprovider capacity increases. Except Rule R12, the expectednumber of patients consulted also increases as the highlevel provider capacity increases. These findings hold truefor all extended cases as to be shown in later sections. Thisimplies that a compromise between the expectation and thevariance of the number of patients consulted has to be madewhen determining the high level provider capacity.

However, the high level provider capacity does not affectthe difference between the probabilities of granting requestsfor fixed and open appointments. According to Propositions1 and 2, an appropriate percentage of open appointments(i.e. low level provider capacities) can reduce the differencebetween the probabilities of granting requests for fixed andopen appointments. The results in Table 4 show that for thegroup of Rules R1–R4 with high level provider capacityequal to the total average demand for both fixed and openappointments, Rule R3 generates the smallest differencebetween the two probabilities. In addition, although the fourOAS rules have similar performance in terms of theexpected number of patients consulted, Rule R2 has slightlyworse performance than Rules R1, R3 and R4 in terms ofthe variance of the number of patients consulted.

For the group of Rules R5–R8 with high level providercapacity less than the total average demand, Rules R6 andR7 use the same value for the percentage of open appoint-ments, and hence have the same performance. Rules R6 andR7 outperform Rules R5 and R8 in terms of the differencebetween the probabilities of granting requests for fixed andopen appointments and the expected number of patients

Table 4 Performances of the proposed OAS rules in the base case

OASrule

Decisions using an OAS rule Performance metrics

Total number of appointmentsto be scheduled (N)

Percentage of openappointments (w) (%)

Probability of granting requests forappointments

Number of patientsconsulted

Fixed appointments(qf) (%)

Open appointments(qo) (%)

ExpectationE Mð Þ

VarianceVar Mð Þ

R1 70 0 100 93.65 58.59 23.90R2 70 75.71 89.06 95.89 58.27 26.83R3 70 67.14 98.76 94.00 58.60 24.06R4 70 54.29 99.99 93.65 58.59 23.90R5 62 0 100 83.40 53.58 9.92R6 62 75.81 81.90 88.69 53.84 11.38R7 62 75.81 81.90 88.69 53.84 11.38R8 62 66.13 97.12 84.33 53.67 9.82R9 81 0 100 99.23 61.31 51.90R10 81 75.31 95.80 99.52 60.91 50.93R11 81 58.02 100 99.23 61.31 51.90R12 81 100 0 100 48.82 48.81

Health Care Manag Sci (2009) 12:99–114 107

consulted. However, they are worse than Rules R5 and R8in terms of the variance of the number of patients consulted.

For the group of Rules R9–R12 with high level providercapacity greater than the total average demand, theperformances of Rules R9 and R11 are the same in termsof all the four metrics. Rules R9 and R11 outperform RulesR10 and R12 in terms of reducing the difference betweenthe two probabilities and the expected number of patientsconsulted, but they are outpaced by Rules R10 and R12 interms of the variance of the number of patients consulted.

From above, Rules R3, R7 and R11 are obviously thepreferred rules. Since they use the same method todetermine the percentage of open appointments, thismethod appears to be a good choice for determining thelow level provider capacities in the base case. Next, wecompare the performances of the proposed OAS rules inextended cases, and investigate the effects of no-show ratesand patient demands.

5.3 Impact of no-show rates on the expectationand the variance of the number of patients consulted

To investigate the impact of the no-show rates of fixed andopen appointments on the performances of the proposedOAS rules, we first compare them in terms of theexpectation and the variance of the number of patientsconsulted. Table 5 summarizes the expected numbers ofpatients consulted using the proposed OAS rules in sixcases, which have the real clinic demands for fixed andopen appointments, but different no-show rates of fixed andopen appointments. Table 6 summarizes the variances ofthe number of patients consulted using the OAS rules in thesix cases. One of the six cases is the base case, which ishighlighted in Tables 5 and 6. The results in Tables 5 and 6support the observation, mentioned in Section 5.2, that theexpectation and the variance of the number of patientsconsulted increase with the increase in the high levelprovider capacity.

The results in Table 5 demonstrate that for each OASrule except Rule R12, the expected number of patientsconsulted decreases with the increase in the no-show rate offixed or open appointments. Meanwhile, the results inTable 6 show that for each OAS rule except Rule R12, thevariance of the number of patients consulted increases withthe increase in the no-show rate of fixed or open appoint-ments. In other words, for each of Rules R1–R11, as the no-show rate of fixed or open appointments decreases, theperformance of the OAS rule improves in terms of theexpectation and the variance of the number of patientsconsulted.

5.4 Impact of patient demands on the expectationand the variance of the number of patients consulted

The impact of demands for fixed and open appointments onthe performances of the proposed OAS rules is firstexamined by comparing the expectations and the variabilityof the number of patients consulted using the OAS rules in12 cases summarized in Tables 7 and 8. The 12 cases use theno-show rates of fixed and open appointments from the realclinic, but different demand distributions for fixed and openappointments except for the base case, which is highlightedin Tables 7 and 8. Three levels of total patient demand areconsidered in the 12 cases. Without taking the level of totalpatient demand into consideration, the comparison based onthe expected number of patients consulted and its variabil-ity could be biased. Thus, to ensure a fair comparisonbetween the cases with different total patient demands, wecompare the OAS rules in terms of the ratio of the expectednumber of patients consulted to total patient demand,E Mð Þ= E Dfð Þ þ E Doð Þ½ �, and the ratio of the standarddeviation of the number of patients consulted to totalpatient demand,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar Mð Þp �

E Dfð Þ þ E Doð Þ½ �. Tables 7and 8 summarize the ratios of E Mð Þ= E Dfð Þ þ E Doð Þ½ �and the ratios of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar Mð Þp �

E Dfð Þ þ E Doð Þ½ �, respective-ly, when using the proposed OAS rules in the 12 cases. The

Table 5 Expected numbers of patients consulted using the proposed OAS rules in six cases with the real clinic demands for fixed and openappointments

Casea Expected number of patients consulted, E Mð Þ

γf (%) γo (%) R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12

16 7 60.42 59.91 60.43 60.42 56.90 56.75 56.92 56.91 62.79 62.20 62.79 61.9310 58.95 58.40 58.95 58.95 54.11 53.94 54.13 54.11 61.49 61.04 61.49 47.25

26.5 7b 58.59 58.27 58.60 58.59 53.58 53.84 53.84 53.67 61.31 60.91 61.31 48.8210 57.11 56.76 57.12 57.11 51.51 51.75 51.75 51.58 59.89 59.44 59.89 47.25

50 7 54.47 54.61 54.63 54.49 46.18 47.67 48.91 47.18 57.56 57.46 57.56 48.8210 53.00 53.10 53.13 53.02 43.28 44.74 46.43 44.74 55.99 55.89 55.99 47.25

a γf and γo denote the no-show rates of fixed and open appointments, respectively.b This case is the base case.

108 Health Care Manag Sci (2009) 12:99–114

results in Tables 7 and 8 also support the observation,mentioned in Section 5.2, that the expectation and thevariance of the number of patients consulted increase withthe increase in the high level provider capacity.

The results in Table 7 demonstrate that for each OASrule except Rule R12, the ratio of E Mð Þ= E Dfð Þþ½ E Doð Þ�decreases with the increase in the percentage of demand forfixed appointments and/or the correlation coefficients ofdemands for fixed and open appointments. Meanwhile,Table 8 shows that for each OAS rule, the ratio offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var Mð Þp �E Dfð Þ þ E Doð Þ½ � increases with the decrease

in the total patient demand and/or the increase in thecorrelation coefficient of demands for fixed and openappointments. Therefore, the increase in the correlationbetween demands for fixed and open appointments hasnegative impact on the performances of all proposed OAS

rules in terms of the expectation and the variance of thenumber of patients consulted.

5.5 Impact of no-show rates on the probabilities of grantingrequests for fixed and open appointments

In this section, we examine the impact of the no-show ratesof fixed and open appointments on the probabilities ofgranting requests for fixed and open appointments whenusing the 12 OAS rules in the base case and 71 extendedcases. Figure 1 illustrates the pairs of the two probabilitiesusing the OAS rules in all 72 cases, which are groupedbased on the no-show rates. It can be seen that for RulesR1, R5 and R9, the probability of granting fixed appoint-ment requests is greater than the probability of grantingopen appointment requests for all 72 cases. This means that

Table 6 Variances of the number of patients consulted using the proposed OAS rules in six cases with the real clinic demands for fixed and openappointments

Casea Variance of the number of patients consulted, Var Mð Þ

γf (%) γo (%) R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12

16 7 23.97 26.37 23.98 23.97 11.28 13.06 11.38 11.27 46.84 45.96 46.84 45.8810 24.37 26.43 24.37 24.37 9.79 11.43 9.88 9.79 50.48 48.79 50.48 47.23

26.5 7b 23.90 26.83 24.06 23.90 9.92 11.38 11.38 9.82 51.90 50.93 51.90 48.8110 24.24 26.88 24.32 24.24 9.95 10.94 10.94 9.85 53.82 51.99 53.82 47.25

50 7 23.73 26.97 25.90 23.69 10.58 8.50 18.17 8.32 56.94 56.27 56.94 48.8210 23.95 27.00 25.25 23.91 10.75 8.33 16.11 8.33 55.68 54.92 55.68 47.25

a γf and γo denote the no-show rates of fixed and open appointments, respectively.b This case is the base case.

Table 7 Ratios of the expected number of patients consulted to total patient demand when using the proposed OAS rules in 12 cases with the realno-show rates of fixed and open appointments

Case Ratio of the expected number of patients consulted to total patient demand, E Mð Þ= E Dfð Þ þ E Doð Þ½ �

Totaldemand

FixedApp.a (%)

Corr.coef.b

R1(%)

R2(%)

R3(%)

R4(%)

R5(%)

R6(%)

R7(%)

R8(%)

R9(%)

R10(%)

R11(%)

R12(%)

70 25 0c 83.7 83.2 83.7 83.7 76.5 76.9 76.9 76.7 87.6 87.0 87.6 69.70.4 83.0 82.9 83.1 83.0 76.1 76.7 76.7 76.2 87.2 86.9 87.2 69.7

50 0 78.8 78.6 78.9 78.8 68.2 69.4 69.4 69.0 83.1 82.9 83.1 46.50.4 78.0 78.2 78.3 78.0 67.8 69.3 69.3 68.6 82.9 82.8 82.9 46.5

42 25 0 82.4 81.8 82.4 82.4 77.0 77.0 77.2 77.1 87.0 86.2 87.0 85.70.4 81.5 81.4 81.6 81.5 76.3 76.7 76.7 76.3 86.4 86.0 86.4 69.7

50 0 77.5 77.2 77.7 77.5 66.8 67.9 68.0 67.6 82.8 82.4 82.8 46.50.4 76.5 76.7 76.8 76.5 66.2 67.7 67.7 67.1 82.4 82.2 82.4 46.5

14 25 0 78.3 77.0 78.3 78.3 74.7 74.1 74.8 74.8 84.9 83.4 84.9 84.90.4 76.7 76.4 76.9 76.7 73.2 73.5 73.7 73.2 83.6 82.9 83.7 78.4

50 0 73.4 72.8 73.6 73.5 65.6 66.0 66.4 66.1 81.2 80.3 81.2 46.50.4 71.6 71.9 72.2 71.6 64.1 65.3 65.3 64.6 80.0 79.7 80.0 46.5

a The parameters in this column are the percentage of demand for fixed appointments in the 12 cases.b The parameters in this column are the correlation coefficients of demands for fixed and open appointments in the 12 cases.c This case is the base case.

Health Care Manag Sci (2009) 12:99–114 109

in the traditional outpatient scheduling systems, it is moredifficult for patients to obtain a short-notice appointment.Figure 1 shows that Rules R2, R3, R6, R7, R10 and R11result in a smaller difference between the probabilities ofgranting requests for fixed and open appointments thanRules R1, R4, R5, R8, R9 and R12 in most cases. It shouldbe also noticed that only Rule R12 produces significantdifference in some cases. Therefore, Rules R1, R4, R5, R8,R9 and R12 are not good choices to ensure equalitybetween the probabilities of granting requests for fixed andopen appointments.

Figure 1 also demonstrates that for Rule R3 and RulesR5–R8, the two probabilities are sensitive to the no showrates. For Rule R3, the probability of granting openappointment requests is greater than the probability ofgranting fixed appointment requests in the cases with thehighest level of the no-show rate of fixed appointments, butis slightly less than or equal to the probability of grantingfixed appointment requests in the cases with a lower no-show rate of fixed appointments. For Rule R5, theprobabilities of granting open appointment requests in thecases with the highest level of the no-show rate of fixedappointments are less than those in the remaining cases. ForRule R6, the two probabilities decrease as the no-showrates increase. For Rules R7 and R8, the probability ofgranting fixed appointment requests decreases as the no-show rate of fixed appointments increases. However, for

Rules R1, R2, R4, and R9–R12, the no-show rates havelittle impact on the two probabilities.

5.6 Impact of patient demand on the probabilitiesof granting requests for fixed and open appointments

Similar to Fig. 1, Fig. 2 also illustrates the pairs of the twoprobabilities using the OAS rules in all 72 cases. However,the 72 cases are grouped based on the demand distributionfor fixed and open appointments instead of the no-showrates. It can be seen that for all rules except Rule R12, thecorrelation between demands for fixed and open appoint-ments has little impact on the two probabilities. Comparedwith Fig. 1, it can be noted that for Rules R1–R11, the totalaverage demand and the percentage of demand for fixedappointments have less significant impact on the twoprobabilities than the no-show rates.

6 Discussion on determining optimal two-level providercapacities

The numerical results in Section 5 reveal that for Rules R1–R11, the expectation and the variance of the number ofpatients consulted increase as the high level providercapacity increases. Meanwhile, except Rule R12, eachgroup of OAS rules with identical high level provider

Table 8 Ratios of the standard deviation of the number of patients consulted to total patient demand when using the proposed OAS rules in 12cases with the real no-show rates of fixed and open appointments

Case Ratio of the standard deviation of the number of patients consulted to total patient demand,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar Mð Þp �

E Dfð Þ þ E Doð Þ½ �

Totaldemand

FixedApp.a (%)

Corr.coef.b

R1(%)

R2(%)

R3(%)

R4(%)

R5(%)

R6(%)

R7(%)

R8(%)

R9(%)

R10(%)

R11(%)

R12(%)

70 25 0c 6.98 7.40 7.01 6.98 4.50 4.82 4.82 4.48 10.3 10.2 10.3 10.00.4 7.60 7.83 7.67 7.60 4.90 5.07 5.20 4.88 11.2 11.0 11.2 10.0

50 0 6.96 7.25 7.01 6.96 4.60 4.64 4.64 4.39 10.6 10.4 10.6 8.150.4 7.66 7.88 7.80 7.66 4.81 4.83 4.95 4.66 11.9 11.8 11.9 8.15

42 25 0 8.96 9.50 8.98 8.96 6.64 7.22 6.80 6.63 12.7 12.7 12.7 12.70.4 9.73 10.0 9.85 9.73 7.28 7.60 7.45 7.28 13.7 13.6 13.7 12.9

50 0 8.94 9.29 8.99 8.94 6.10 6.32 6.11 5.90 13.3 13.0 13.3 10.50.4 9.81 10.1 9.96 9.81 6.49 6.62 6.62 6.35 14.7 14.6 14.7 10.5

14 25 0 15.2 16.2 15.2 15.2 13.3 14.2 13.4 13.3 20.5 20.6 20.5 20.50.4 16.4 16.9 16.6 16.4 14.4 14.9 14.6 14.4 21.9 21.8 21.9 21.5

50 0 15.2 15.6 15.3 15.2 12.1 12.6 12.2 12.0 21.2 20.9 21.2 18.20.4 16.5 16.9 16.8 16.5 13.1 13.4 13.4 13.1 23.1 23.1 23.2 18.2

a The parameters in this column are the percentage of demand for fixed appointments in the 12 cases.b The parameters in this column are the correlation coefficients of demands for fixed and open appointments in the 12 cases.c This case is the base case.

110 Health Care Manag Sci (2009) 12:99–114

capacity have very close performance in terms of theexpected number of patients consulted. However, it isobvious that when the high level provider capacity is muchgreater than patient demand, the expectation and the varianceof the number of patients consulted do not further changewith the increase in the high level provider capacity. Basedon these observations, for an open access clinic in which theaverage number of patients consulted needs to be maximizedwhile an upper bound on the variability in the number ofpatients consulted is required, an optimal high level providercapacity should be the largest value that guarantees the limitfor the variability be satisfied. On the other hand, if the limitof variability is set too loose, the obtained optimal high levelprovider capacity could be significantly higher than thepatient demand, and this is not desired.

On the other hand, for an open access clinic in which thereis a lower bound requirement for the average number of

patients consulted and the variability in the number ofpatients consulted needs to be minimized, an optimal highlevel provider should be the smallest value that ensures thelower bound be satisfied. In addition, if there are lowerbound requirements for the probabilities of granting requestsfor fixed and open appointments in an open access clinic, therequirements can be converted to a lower bound for the highlevel provider capacity according to Proposition 3.

After determining an optimal high level providercapacity, an optimal percentage of open appointments tobe scheduled in a clinic session (i.e. the low level providercapacities) could be determined based on the chosenoptimization criterion (i.e. one or a combination ofperformance metrics). If the objective is to maximize theexpected number of patients consulted, E Mð Þ, or tominimize the variance of the number of patients consulted,Var Mð Þ, the studies [20, 21] present mathematical models

Pro

ba

bil

ity

of

gra

nti

ng

op

en

ap

po

intm

en

t req

uests

Probability of granting fixed appointment requests

w = 0 w minimizing Var ( )f o

w = E ( )/[E ( )+E ( )]f

w maximizing E ( )

N =

E

()+

E(

)

fo

||

Rule R4

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R3

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R2

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R1

20%

40%

60%

80%

100%

% 20% 40% 60% 80% 100%

(26.5%, 7%)

(16%, 7%)

(50%, 7%)

(26.5%,10%)

(16%, 10%)

(50%, 10%)

( , )f

o

N =

(1–

)E

(

)+(1

–

)E(

)

of

fo

||

Rule R8

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R7

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R6

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R5

0% 20% 40% 60% 80% 100%

N =

E

( )/

(1–

)+

E(

)/

(1–

)o

ff

o|

|

Rule R12

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R11

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R10

0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R9

0%

20%

40%

60%

80%

10

0% 20% 40% 60% 80% 100%

100%

0%

20%

40%

60%

80%

100%

0%

100% 0% 100% 0% 100% 0%

Fig. 1 Impact of no-show rates on the probabilities of granting requests for fixed and open appointments. Note: γf and γo denote the no-showrates of fixed and open appointments, respectively

Health Care Manag Sci (2009) 12:99–114 111

and quantitative procedures to determine an optimalpercentage of open appointments for a given high levelprovider capacity. If the objective is to minimize thedifference between the probabilities of granting requestsfor fixed and open appointments, Propositions 1 and 2provide an approach to finding an optimal percentage ofopen appointments to be scheduled. According to Proposi-tions 1 and 2, as the percentage of open appointments to bescheduled increases, the probability of granting fixedappointment requests decreases while the probability ofgranting open appointment requests increases. Therefore,the search could start from any percentage of openappointments. Then if the probability of granting fixedappointment requests is greater than the probability ofgranting open appointment requests, a larger percentageshould be used for the next try. On the contrary, if theprobability of granting fixed appointment requests is lessthan the probability of granting open appointment

requests, a smaller percentage should be for the next try.The search should be continued until the minimum (oracceptable) difference between the two probabilities isfound. According to the numerical results in Sections 5.5and 5.6, the percentage of open appointments equal toE Doð Þ= E Dfð Þ þ E Doð Þ½ � or by maximizing E Mð Þ shouldbe a good start point for the search.

7 Conclusions

Matching healthcare provider capacity and patient demandis critical to outpatient appointment scheduling. For openaccess clinics, it becomes a challenging issue in that thematch is required at both the high level of total providercapacity and the low level of individual capacities for fixedand open appointments. In this paper, we proposed threemethods to determine the high level capacity (i.e. the total

Rule R4 0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R3 0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R2 0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R1

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

(70, 25%) (70, 25%)(70, 50%) (70, 50%)(42, 25%) (42, 25%)(42, 50%) (42, 50%)(14, 25%) (14, 25%)(14, 50%) (14, 50%)

Independent = 0.4a

Rule R8 0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R7 0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R6 0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R5

20%

40%

60%

80%

0% 20% 40% 60% 80% 100%

Rule R12 0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R11 0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R10 0%

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Rule R90%

20%

40%

60%

80%

0% 20% 40% 60% 80% 100%100% 100% 100%

0%100%

0%100%

Pro

babi

lity

of g

rant

ing

open

app

oint

men

t req

uest

sρ

N =

E(

)+E

( )

fo

||

N =

(1−

)E

( )

+(1

− )

E(

) of

fo

N =

E(

)/(1

− )+

E(

)/(1

− ) o

ff

o

Probability of granting fixed appointment requests

0%0% 0%

w = 0 w minimizing Var ( )ow = E ( )/[E ( )+E ( )] w maximizing E ( )f f

Fig. 2 Impact of patient demands on the probabilities of grantingrequests for fixed and open appointments. a In the legend, ρ denotesthe correlation coefficient of demands for fixed and open appoint-

ments. In a pair of parentheses, the first parameter is the total averagedemand for both fixed and open appointments in a case, and thesecond parameter is the percentage of demand for fixed appointments

112 Health Care Manag Sci (2009) 12:99–114

number of appointments to be scheduled), and fourmethods to determine the low level capacities, which canbe represented by the percentage of open appointments.The combination of methods at two levels generated 12proposed OAS rules. Their performances were investigatedand compared in terms of the probabilities of grantingrequests for fixed and open appointments, and the expec-tation and the variance of the number of patients consulted.The findings are summarized as follows,

& The analytical results showed that when the percentageof open appointments is fixed, the probabilities ofgranting requests for fixed and open appointmentsincrease with the increase in the total number ofappointments to be scheduled. Meanwhile, the numer-ical results demonstrated that the expectation and thevariance of the number of patients consulted increasewith the increase in the total number of appointments tobe scheduled.

& The analytical results showed that adjusting the per-centage of open appointments could reduce the differ-ence between the two probabilities of granting requestsfor fixed and open appointments. This is supported bythe numerical cases, which showed that, among themethods of determining the percentage of open appoint-ments, two methods outperform the remaining in termsof reducing the difference between the probabilities ofgranting requests for fixed and open appointments.

& The numerical results also demonstrated that for allproposed OAS rules except Rule R12, the increase inthe no-show rate of fixed or open appointments has anegative impact on the expectation and the variance ofthe number of patients consulted. In addition, theincrease in the correlation between demands for fixedand open appointments has negative impact on theexpectation and the variance of the number of patientsconsulted, but it hardly affects the probabilities ofgranting requests for fixed and open appointments.

Our analytical and numerical results provide insights indetermining optimal two-level provider capacities to matchdaily provider capacity to patient demand. Although wediscussed the potential approaches to determining optimaltwo-level provider capacities based on these results, nomathematical models were presented to address the opti-mality issues. It is believed that for future research,optimization models are needed to determine optimal two-level provider capacities.

Acknowledgement The authors are grateful to the reviewers for thethoughtful suggestions to improve the quality of this paper. Theauthors would like to thank the administrators in the IndianaUniversity Medical Group—Primary Care clinics for providingnecessary help.

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