Comparison of Traditional and Open-Access Appointment Scheduling for Exponentially Distributed Service Time

Transcription

1 Journa of Heathcare Engineering Vo. 6 No. 3 Page Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service Chongjun Yan, PhD; Jiafu Tang *, PhD; Bowen Jiang, PhD and Richard Y.K. Fung 3, PhD Coege of Management Science & Engineering, Dongbei University of Finance and Economic, Daian, China. Department of Systems Engineering, Northeastern University, Shenyang, China. c Department of System Engineering & Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowoon, Hong Kong, China. Submitted June 4. Accepted for pubication Apri. ABSTRACT This paper compares the performance measures of traditiona appointment scheduing (AS) with those of an open-access appointment scheduing (OA-AS) system with exponentiay distributed service time. A queueing mode is formuated for the traditiona AS system with no-show probabiity. The OA-AS modes assume that a patients who ca before the session begins wi show up for the appointment on time. Two types of OA-AS systems are considered: with a samesession poicy and with a same-or-next-session poicy. Numerica resuts indicate that the superiority of OA-AS systems is not as obvious as those under deterministic scenarios. The samesession system has a threshod of reative waiting cost, after which the traditiona system aways has higher tota costs, and the same-or-next-session system is aways preferabe, except when the no-show probabiity or the weight of patients waiting is ow. It is concuded that open-access poicies can be viewed as aternative approaches to mitigate the negative effects of no-show patients. Keywords: heathcare management, appointment scheduing, no-show. INTRODUCTION.. Background With the rapid deveopment of the heathcare industry, how to reduce operationa costs and improve service quaity is becoming an essentia task for service providers in outpatient services []. It is common for patients to book an appointment in advance. It has made the appointment scheduing (AS) system an active research issue over the past six decades (Cayiri and Vera [], Gupta and Denton [3], LaGanga [4], * Corresponding author: Prof. Jiafu Tang, Coege of Management Science & Engineering, Dongbei University of Finance and Economic, Shahekou, Daian, 66, China. Phone: (86) Fax: (86) Emai: jftang@mai.neu.edu.cn. Other authors: yanchongjun67@63.com, ddxx@sina.cn, meykfung@cityu.edu.hk.

2 346 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service Rinder et a. []). A we-designed AS system can achieve a baance between the efficiency of service providers and the satisfaction of patients. The objective of AS is to find an effective poicy to optimize its performance measures, e.g., the waiting time, ide time and overtime. Two widey used AS systems are the traditiona AS system and the open-access appointment scheduing (OA-AS) system [6]. In the traditiona AS system, a patient who books an appointment in advance may not show up. On the other hand, a random number of patients ca to make appointments at the beginning of the session, and a patients are served during that session or the next session in the OA-AS system. In the traditiona AS system, a considerabe number of patients who have made an appointment do not show up. This phenomenon cannot be ignored because it resuts in ide time of physicians and other expensive physica assets and equipment in some intervas. Cinics usuay overbook additiona patients in expectation of patients no-shows, and it may ead to undue congestion and dissatisfaction of patients. This situation indicates that no-shows waste substantia operationa costs and resources in outpatient AS systems [7]. Athough Huang and Zuniga [8] showed that the poicies aowing canceation up to severa hours in advance coud reduce the negative impact of no-shows, these approaches coud not entirey eiminate noshows for cinics. Most practitioners resort to overbooking or compressing the appointment intervas to offset no-shows. However, it eads to onger waiting times for patients. Hence, a reasonabe tradeoff decision is needed to baance the interests of patients and those of physicians. The so-caed OA-AS system was introduced by Murray and Tantau [9] as an aternative approach to reduce no-show probabiities and improve patients satisfaction. The target of the OA-AS system is to accept appointments on the caing session or the session after rather than keeping the sick waiting for severa days before the appointment time. The OA-AS systems have received more attention from both practitioners and academics. Two widey used OA-AS poicies are investigated, i.e., same-session poicy and same-or-next-session poicy, depending on whether the patients can be postponed to the next session. Under the same-session poicy, a patients who ca before the session begins wi be schedued to the same session. On the contrary, some of the patients have to wait unti the next session in order to smooth the arriva rate fuctuation under the same-or-next-session poicy. There are both successfu and faiure stories reported about the OA-AS systems. Therefore, quantitative comparison is needed to hep cinic administrators answer some essentia questions, such as how effective the OA-AS systems are compared to the traditiona AS system, what the most infuentia factor is on the performance measures of OA-AS systems, and how these poicies shoud be better impemented. For this purpose, Robinson and Chen [] compare the traditiona AS and the OA-AS systems under deterministic service time. Their performance measures incude the waiting cost, the ide time cost during working hours, and the overtime cost to serve the remaining patients beyond the norma working hours. The resuts show that the open access poicies outperform the traditiona one substantiay, but they aso concude that the differences may be ess significant if other types of variabiity are expoited.

3 Journa of Heathcare Engineering Vo. 6 No Objectives and Significance of the Study By extending the work of Robinson and Chen [], the goa of this paper is to make a comparison of the performance measures of AS and OA-AS under exponentiay distributed service time. Both systems aim to minimize the tota weighted costs of waiting time of patients, ide time and overtime of the heathcare systems. To this end, a queueing mode is first formuated in this paper for the traditiona AS system with a noshow probabiity, and the stochastic programming mode under a no-gof poicy is aso presented. By no-gof poicy, it means the physician is not aowed to go off duty even if he has no patients waiting. This research extends the work by Hassin and Mende [] to consider overtime cost []. Thereafter, the mathematica modes for the OA-AS systems under the same-session poicy and same-or-next-session poicy are deveoped. To faciitate the comparison, it is assumed that a patients who ca before the session begins wi show up for the appointment on time in the OA-AS queueing modes, whie the expected workoad remains the same as the traditiona AS mode. The comparison of system performance between the traditiona AS and the OA-AS is made numericay. The effects of factors incuding session-ength, expected work-oad, ratios of overtime cost and no-show probabiity on system performance measures are examined. This is the first paper to compare different AS systems under exponentiay distributed service time. The resuts indicate that the superiority of open-access poicies is not as expicit as those under deterministic assumptions. The same-session AS system outperforms the traditiona AS system in terms of the tota costs beyond a threshod of reative waiting cost in most of the cases considered. The same-or-next-session system is aways preferabe, except when the no-show probabiity or weight of the patients waiting is ow..3. Literature Review on Appointment Scheduing The components invoved in the AS queueing systems are the arriva process (distribution, punctua arriva and no-show), the service process (number of servers, service rues and service time), patients and providers preferences, incentives and performance measures criteria (Gupta and Denton [3]). The basic factors considered in optimizing AS systems are the number of servers, deterministic or stochastic service time and the no-show probabiity. Cayiri and Vera [] performed a comprehensive review of environmenta factors and performance measures that have been investigated in previous studies and categorized soution methodoogies and decision variabes in the design of an appointment system. The most reevant studies incuding the modeing approaches of different appointment systems and methods to mitigate the impact of noshows are summarized as foows. From the perspective of the modeing approach, anaytica studies based on queuing modes (Creemers and Lambrecht [3, 4]) are used to obtain the steady-state distribution of performance. Researchers aso pay attention to provide a better representative of outpatient queue in a fixed ength of a day (Kassen and Yoogaingam [], Kong et a. [6], De Vuyst et a. [7], Mak et a. [8]). Meanwhie, some researchers deveoped quantitative modes for OA-AS systems (Qu and Shi [9], Wang and Gupta [], Huang et a. []).

4 348 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service There is a significant body of evidence that iustrates how no-show becomes a crucia factor decreasing the performance of heathcare services a over the word. Cayiri et a. [] report that the average no-show probabiity amounts to 38% and varies from % for coonoscopies to 67% for pediatric neuroogy. The first to address the appointment system with no-show is Mercer [3], who provides an anaytica approach based on queueing theory. Previous studies on appointment systems with noshows fa into three categories: ) estimating the no-show probabiity and the socia characteristics of such patients based on empirica data, e.g., Cayiri et a. [], Kim and Giachetti [4]; ) anayzing the impact of no-show patients on the performance measures of outpatient cinics, e.g., Green and Savin [], Cayiri et a. [6]; and 3) proposing a series of recommendations to mitigate the negative impacts of no-show, e.g., Erdogan and Denton [7], Chakraborty et a. [8, 9], Berg et a. [3]. Hassin and Mende [] found the optima schedue in terms of a weighted sum of patients waiting time and doctors working hours for exponentiay distributed service times without consideration of overtime; the impact of no-show on the system performance was aso investigated. Finay, attention is moved to the situation of equa-spaced interva engths. Tang et a. [] extended a previous work by integrating overtime cost into the numerica optima soution. Zeng et a. [3] studied the cinic overbooking probem for patients with heterogeneous no-show probabiities, identified the properties of optima schedue and designed a heuristic search agorithm to yied a oca optima soution. Begen and Queyranne [3] determined an optima appointment schedue for a given sequence of patients with the objective of minimizing the expected tota operationa costs for discrete distribution service time. Their mode can be easiy adjusted to hande overtime, no-show and emergencies as we. Dome pattern is introduced and demonstrated as an effective rue in many situations [33, 34]. Kassen and Yoogaingam [, 3, 36] showed that the dome and pateau dome pattern perform quite we even if both service interruption and doctor ateness are considered at the same time. Cayiri et a. [37] formuated a dome appointment rue as a function of the environmenta factors and used simuation and noninear regression to derive the panning constant by parameterization. The subsequent procedure expicity minimizes the negative impacts of no-shows and wak-ins. A these papers propose approaches to mitigate no-show effects under certain conditions by adjusting the interva ength aocated to the patients or overbooking more than one patient in the same interva. There are exampes of successfu appication of open-access poicy in the iterature using short ead time to avoid the wasted capacity caused by no-shows. Kopach et a. [38] appy discrete simuation to investigate the impacts of four different parameters, incuding the owest percentage of open appointments, the time horizon for fixed appointments, the provider care group and the overbooking horizon, on the performance of an open-access AS system in terms of continuity of care provided to patients under various settings. Green and Savin [] proposed a singe-server queueing system to anayze the AS system with state-dependent no-show probabiity, identifying the patient pane size for the cinic impementing the open-access poicy. Liu et a. [39] proposed a heuristic procedure that outperforms a other benchmark

5 Journa of Heathcare Engineering Vo. 6 No poicies with no-shows and ate canceations when the workoad is high. Their simuation resuts indicate that the open-access poicy is appropriate when the patient oad is reativey ow. Dobson et a. [4] reserved time intervas for urgent patients in order to maximize the revenue of a physician with two service quaity measures, i.e., the average number of urgent patients who are not handed during norma hours and the queue ength of routine patients. Wang and Gupta [] designed a new outpatient AS system with patient choice, matching the random arrivas of appointment requests with doctors capacity reservation in a way that maximizes the cinics revenue and patients satisfaction simutaneousy. LaGanga and Lawrence [7] proposed a gradient search method to find quaity soutions for joint capacity and scheduing probems considering no-show patients based on the submoduarity of the objective function. Qu et a. [4] deveop a mean-variance mode and an efficient soution approach to determine the Pareto-optima open appointment percentages, increasing the average number of patients seen in a session whie reducing the uncertainty caused by noshows. Over a wide range of scenarios and cinics, Patrick [4] demonstrated that the MDP methods perform as we as or better than the OA-AS poicy in terms of the cinic revenue. Lee et a. [43] compared the open access poicy and the overbooking poicy under some commony used rues by simuation, but optimization is not appied in the comparison. The current iterature survey shows that there is a need to compare these traditiona methods with the open-access poicy in terms of commony used performance measures in order to determine the condition under which open-access performs more effectivey than the dome pattern.. METHODS.. Assumption and Notations for Traditiona AS System The prime objective of most of the traditiona AS mode is to search for an optima schedue to minimize the weighted sum of patients waiting cost and doctors ide cost and overtime cost. Common medica practice aows doctors to have their own waiting ists, as it can provide a one-on-one doctor-patient experience. Therefore, the appointment system is formuated as a singe-server queueing system with schedued arrivas. The number of patients to be schedued in a working session is determined in advance by the statistics of historica operation data, and unschedued arrivas are not taken into consideration []. It is assumed that each patient in the queue shows up punctuay with a probabiity. The no-show probabiity is the same for a patients and is obtained by statistics, and the arriva processes of individua patients are independent of one another. The service time of patients foows an independent and identica exponentia distribution (Kopach et a. [38] justify this assumption by simuation). Patients are served in the order of their schedued appointments. If some patients are schedued to arrive simutaneousy, the one with a ower schedued position in ine is served first if he/she arrives. Athough the working hours of a doctor are panned in advance by the cinic, he/she can eave the office ony after seeing a patients that show up. Hence, the doctor has to be avaiabe unti the ast schedued appointment. The notations for the traditiona AS system are the foowing:

6 3 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service c i : Doctor s ide cost per unit time (doar/min). c o : Doctor s overtime cost per unit time (doar/min). c w : Patients waiting cost per unit time (doar/min). I: Expected ide time (min) that a doctor wastes during a standard working session. N: Number of patients to be schedued in a standard working session. N i : Number of patients in the queue just before the arriva time of the ith schedued patient. O: Expected overtime (min) of the doctor in a standard working session. p: No-show probabiity of each patient. T: The predetermined ength (min) of a standard working session. t i : (min) of the ith schedued arriva, i ti = xj. j= w i : Expected waiting time (min) of the ith patient if he/she shows up, w =. W: Expected waiting time (min) of a patients schedued in a session if a N patient shows up, W = w i. x: A vector of schedued intervas, x = (x, x,, x N ). x i : interva (min) between the ith and (i + )th schedued patient. α = c w /c i : The reative cost of patients waiting, as a fraction of the ide time cost per unit time (min). β = c o /c i : The reative cost of doctor s overtime, as a fraction of the ide time cost per unit time (min). /: Mean of exponentiay distributed service time. By definition, t i, time of the ith schedued arriva, is the function of schedue x, vector of schedued intervas; the foowing subsection shows that variabes w i (expected waiting time of the ith patient if he shows up), N i (number of patients in the queue just before the arriva time of the ith schedued patient), W (expected waiting time of a patients schedued in a session if a patient shows up), O (expected overtime of the doctor in a standard working session), I (expected ide time that a doctor wastes during a standard working session) are a functions of schedue x... The Genera Mode for Traditiona AS System The objective is to find an optima schedue x that can minimize the weighted sum of expected tota patients waiting time vaued at α (reative cost of patients waiting), the doctor s ide time standardized at, and the doctor s overtime vaued at β (reative cost of doctor s overtime): C( x)= cw( p) W + ci i + coo= ci( α( p) W + I + βo) The expected waiting time of a show-up patient in the queue is determined by the number of patients upon his/her arriva []: ()

7 Journa of Heathcare Engineering Vo. 6 No. 3 3 i j wi = { = } Pr Ni j, i j= () The probabiity that j patients remain in the system just before t i, denoted by Pr{N i = j}, is derived recursivey from system state just before t i (i ): For j < i, i j { } { } ( x ) i Ni = j = p Ni = j+ k Pr Pr k= i ( xi ) { Ni = } = ( p) { Ni = k } Pr Pr i k= ( xi ) { i } + p Pr N = k k= k= i j { } k xi + p Pr Ni = j+ k e k! = k! = k e k! xi! xi k e e xi xi (3) (4) Ide time is defined as the difference between the expected working time and the actua cosing time of a session. Given that the former is a constant (-p)n/, the expected ide time is derived by subtracting it from the expected off duty time: = N [ ] + p N I xi E service time after tn = N + + p p N xi wn () Overtime is defined as a positive deviation between the working hours and the actua cosing time. The expected overtime is categorized into two cases based on the appointment time of the ast patient and is denoted by O and O, respectivey:. If a patients are schedued before the end of session, i.e., x, i T the expected overtime is given as: N N Pr { } ( ) O = N = k p o + po N k k k= (6) In this case, the doctor has to remain avaiabe unti t N to see whether the ast patient shows up for the appointment. Using the memoryess property of exponentia distribution, the expected overtime when the ast patient shows up, represented by o k, is the probabiity of departures before the cosing time mutipied by (k + )s

8 3 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service expectations of service time. The expected overtime when the ast patient does not show up, o k, is computed in a simiar way, except that there are ony (k )s expectations of service time. Thus, O is derived by: N O = Pr { NN = k} p k= N k k + T t + N ( T t ) k N p e! { N k} = Pr = k= N ( p) + k = = k = = ( T t )! ( T t )! k N ( T t ) N e! e N T tn T tn e T tn k (7). If not a patients are schedued before the end of session, i.e., N xi > T the expected overtime is given as: = n + = N O x E service time after t T x + w + p T i [ n] i N (8) Thus, the traditiona AS mode for an arbitrary schedue is formuated as foows: Φ ( x) = c ( α( p) W + I + βo ) i = α + N β + + p p N ci + ( p) W xi wn O N st.. x T i Φ = ( α( ) + + β ) = α( ) + N x c p W I O c p W x + i i i w + N p p N + β + + p xi wn T N st.. x > T i N (9) ()

9 Journa of Heathcare Engineering Vo. 6 No The objective functions are simpified by negecting the constants ( p)( N)/, and c i : Φ = α( ) + N x p W β x + w + i N O N st.. x T i () Φ = α( ) + N N β p x p W xi wn xi wn T N st.. xi > T ().3. Traditiona AS Mode under the No-Gof Poicy This section considers a more reaistic scenario, the so-caed no-gof poicy, in which the doctor is not free to eave before the cosing time of a session even if he finishes seeing a patients appointed. This assumption is more reasonabe when auxiiary jobs are taken into consideration, such as going through records of the patients of the next session. Neither the expected waiting time of each show-up patient nor the doctor s overtime is infuenced by this assumption. However, the expected ide time is determined by the cosing time of the doctor, which is the arger of the actua off duty time and the predetermined end of the session. If not a patients are schedued before the end of session, the doctor certainy has to work overtime to see a the patients. In this case, the expected ide time is the same as eqns.. If a patients are served before the end of session, the ide time is N Pr { } ( ) I = N = k p I + pi k= N k k (3) where I k, and I k represent the cases when the ast patient shows up and doesn t show up, respectivey: I k = + = k = k+ ( T t )! ( T t ) + + e T k n ( T t ) T t N! n n e ( n ) T p (4)

10 34 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service I k = + = k k = ( T t )! ( T t ) + e T k n ( T t ) T t N! n n e ( n ) T p () Substituting eqns. 4 and into eqns. 3 eads to the expected ide time: N I = T p + O (6) The expected ide time is equa to subtracting the expected working hours from the actua time off duty, which is consistent with the intuition. The objective function under no-gof poicy is rewritten as foows after omitting the negative constant ( p)n/: Φ ( x) = α( p) W + T + O( + β) N st.. xi T Φ = α ( ) + N + + p x p W xi wn N + β + + N p xi wn T st.. x i > T (7) (8).4. Notations and Assumptions for OA-AS System To faciitate the comparison, it is assumed that a patients who ca before the beginning of a session wi show up for the appointment on time under open access poicy, and the expected workoad is the same as that of the traditiona AS system. As Robison and Chen [] noted, it is theoreticay sensibe to use a Poisson distribution to mode the number of booking requests in the session. From a reaistic point of view, an upper bound is set for the maximum number of patients schedued in a session. As for the same-or next-session poicy, the number of patients who can be deayed to the next session is no more than the expected workoad because deferring too many patients provides no benefit in smoothing the demand fuctuation between different sessions. The formuation is given ony for the case in which the doctor can ony eave the office after seeing a of the patients on his/her schedue. The notations for the OA-AS system are the foowing: d: The maximum number of patients that can be postponed to the next session. I(m): The ide time (min) of the doctor if m patients are schedued in a session. M: The maximum number of patients to be schedued in a session.

11 Journa of Heathcare Engineering Vo. 6 No. 3 3 n: The expected workoad in a session, n = N p. O(m): Expected overtime (min) of the doctor if m patients are schedued in a session. q i : The probabiity of starting a session with i patients deayed from the previous session. w i (m): Expected waiting time (min) of the ith patient if he/she shows up on the condition that m patients are seen in a session. w(m): Expected waiting time (min) if m patients are schedued in a session. ϕ(m): The probabiity mass function that m patients ca for appointments before the session begins. Φ(m): The cumuative distribution function of f(m). φ(m): The probabiity distribution that m patients are seen in a session. Other notations and assumptions are the same as the traditiona mode... Mode for Same-Session Poicy Under the same-session poicy, the number of patients schedued in a session foows a Poisson distribution, and any requests exceeding the upper bound of the capacity wi be ignored; a appointments are schedued to the session the patients request. When M is arge enough, the abandonment has itte impact on the expectation of the distribution. ϕ m m ( n) = φ = m e n, m,! m ( n) m = m e n,! m,..., M M ( n) n e, m= M k! k= k (9) The performance when m patients ca before the beginning of the session is evauated in the same way as the traditiona AS mode, except N = m, p =. The probabiity distribution of system state at t i can be derived from eqns. 3 and 4: i j xi xi For j < i, Pr N = j = Pr N = + () i i j k e k! { } { } k= i { = } = { = } xi Pr Ni Pr Ni k! k= xi The overa performance measures W, I, O are the weighted sums of w(m), I(m), O(m), where = k e k () i m j wi( m) = { = } Pr Ni j, W m = wi m, j= M W = W( m) φ ( m) m= ()

12 36 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service M O = O m m m = m = + + M m I m xi wm, m, I = I m φ m φ m m x = i + w m + T, if x > i = i T i O( m) = m ( ) { } + = = k T t m T t N m k e m k m Pr, if x = i T k! m= (3) (4) The objective of the same-session poicy is to find an optima schedue to minimize the tota costs: C( x)= c W + ci + c O= c ( αw + I + βo) w i o i ().6. Mode for Same-Or-Next-Session Poicy As for the same-or-next-session poicy, the performance measures are cacuated by the same approach described in eqns. 9-4, but the number of patients seen in a session need to be re-evauated. As demonstrated by Robinson and Chen [], the probabiity mass function of the number of patients deayed to the next session can be obtained by soving the transition equation of the Markov chain. No deferred patients means that the workoad of the session before is no more than T : d q = qj Φ ( T j) j= i deferred patients means T + i patients arrive in the ast session: The sum of a probabiity is equa to : d qi = qj ϕ T + i j, i d j= q j = With the soution to the above-mentioned equations, the number of patients seen in a session is obtained as foows: d j=

13 Journa of Heathcare Engineering Vo. 6 No φ ( m) = min { dm, } j= j= d d j= d j= q ϕ m j, m T j q ϕ T j+ k, m= T j d k= q ϕ d+ m j, T + m< M j q ϕ d+ m j, m= M j M (6) 3. RESULTS 3.. Soution for the Genera Traditiona AS Mode The traditiona appointment scheduing probem is a convex minimization probem [44] of which the optima numerica soution can be found by the fmincon function in the commercia software Matab impementing sequentia quadratic programming agorithm [, ]. The effects of the parameters are anayzed based on a medium-sized probem. In particuar, the expected service time is normaized at, and the basic probem assumes N = 6, T =. That is to say, the day ength is times as ong as the expected service time. In practice, this case corresponds to the situation where 6 patients are schedued to a three-hour working session with an average service time of fifteen minutes. A performance measures in the foowing experiment are presented by means of this normaized dimensioness time for simpicity. Other input parameters are given as foows: the no-show probabiity p is., reative cost of overtime β is, and the reative cost of waiting α ranges from to. Because of the scarcity of the heathcare resources, the situation that waiting cost is more important is not considered. Figure shows the commony dome-pattern interva engths under different vaues of α, with the interva numbers abeed on the horizonta axis [36]. It is found that the interva ength increases in the earier part of session, and then remains reativey steady afterwards. Finay, it decreases in the atter part of session under each α considered except because the doctor compresses the first intervas in order to reduce the possibiity of ide time at the beginning of the session. The ast few patients (e.g., 3) are schedued coser in order to mitigate ide time and overtime incurred by no-show patients at the end of session. Appointment intervas are kept amost the same in most of the intervas in order to maintain steady workoad. It is aso noted that the more patients waiting are weighted, the arger the interva engths are aocated. Intuitivey speaking, this phenomenon occurs in that the objective is to strike a baance between the interests of patients and the doctor. Figure presents the expected waiting time of each patient showing up, the expected ide time and overtime of the doctor. Figure dispays that the impacts of no-shows

14 38 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service Figure. x i α =. α =.. α = i α = Schedued intervas of traditiona poicy under different α vaues. 8 6 W 6 4 overtime W 3 ide time W α Figure. System performance of traditiona poicy under different α vaues. accumuate in the queue, and the deay of each show-up patient increases with the position in the waiting ist. Another intuitive resut is the monotonicay non-increasing curve of waiting for each patient in ine, refecting that the more weight the cinic gives to waiting time, the ess time the patients have to spend. On the contrary, the ide time and overtime curves are monotonicay increasing with the weight of waiting, refecting that the objective function is aimed at baancing the efficiency of doctors and the satisfaction of patients. More attention paid to the waiting time of the patients means that the reative weight of ide time and overtime in the objective function wi decine. From the standpoint of the appointment system, the soid ine in Figure 3 depicts the trend of the objective function under various vaues of α in traditiona AS system. The fact that tota costs monotonicay increases with the weight given to waiting time is not

15 Journa of Heathcare Engineering Vo. 6 No traditiona poicy Cost same-day poicy same-or-next-day poicy Figure α Tota operationa costs of AS systems under different α vaues. ony because of the higher coefficient of the patients, but aso due to the decrease of doctor s efficiency caused by the attempts to reduce patients waiting time. The superiority of open-access poicies is not as expicit as the resuts under the deterministic service time. The same-session system has a threshod of α in terms of tota operationa costs and the same-or-next-session system is aways preferabe. 3.. Numerica Soution for the Open-Access AS Mode The parameters under open-access poicies are consistent with those under the traditiona poicy for a reasonabe comparison. As for the same-or-next-session poicy, the vaue of d is set equa to T in a of the subsequent experiments. The key point for finding the optima interva ength is to determine how many appointments can be schedued within the working hours. Therefore, the origina probem is decomposed into M + sub-probems as foows:. a of the patients can be schedued before the end of session, i.e., ;. ony M- patients can be schedued before the end of session, i.e., M M x T, x > T i i ; M + ) a of the patients are schedued in the overtime, i.e., x i > T. In each sub-probem, ony one overtime expression of eqn. hods for any given m. The minimum of a oca optima soutions derived by the same approach described in section 3. is the goba numerica optima soution for an open-access poicy. Figures 4 and dispay the OA-AS system performance measures under the samesession and same-or-next-session poicies, respectivey. The curves in red represent the

16 36 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service O s Figure 4. O t I t W t I s W s α System performance of same-session poicy under different α vaues. performance of the same-session and same-or-next-session systems with the same expected workoad. The phenomenon that the average expected waiting time of a patients is monotonicay non-increasing coincides with the traditiona resuts. Higher waiting parameters decrease the deays in the queue and increase both the ide time and overtime because the objective is to baance the waiting time of patients and the working time of doctors. Another observation is that the OA-AS poicy is superior to the traditiona poicy in the basic probem in terms of average waiting time when α is arger than.. The samesession poicy gives rise to higher overtime for a vaues of α and higher ide time when α >.8, whereas the same-or-next-session poicy brings about ess ide time a the time. When α <., the overtime is amost the same for the both traditiona and same-or-nextsession poicies; otherwise, the same-or-next-session poicy requires sighty ess overtime when α grows arger. From the standpoint of a cinic, the same-session poicy owers the tota operationa cost when α.4, whie the same-or-next-session poicy maintains ower cost for a vaues of α considered as shown in Figure 3. The source of uncertainty for OA-AS poicies is the random number of patients. The above observations impy that the negative effect of random patients is ess than that of no-show patients for the open-access poicy when greater importance is attached to the waiting time of patients. Contrasting Figure 4 and Figure, it is noted that the same-or-next-session poicy obviousy outperforms the same-session poicy in terms of overtime and ide time. As noted by Robinson and Chen [9], the fuctuation of patients seen in each session under the same-or-next-session poicy is much smaer than that of same-session poicy Numerica Soution under No-Gof Poicy This section examines the situation when the doctor cannot eave the office before the predetermined end of the session. Compared with Figure, Figure 6 shows that the nogof poicy ony affects the ast few intervas in the traditiona AS system. It is observed

17 Journa of Heathcare Engineering Vo. 6 No O t O sn I t I sn W t W sn Figure α System performance of same-or-next-session poicy under different α vaues. x i. α = α =..6 α = α = i Figure 6. Schedued intervas of traditiona no-gof poicy under different α vaues. that the ast few patients are schedued scarcey because there is no benefit to finish consutation ahead of time. However, increasing the interva ength cuts down the patients waiting time significanty without infuencing the ide time of doctors. When the waiting time is given more weight, the schedued intervas are in the same pattern as Figure in order to mitigate the patients waiting by onger ide time and overtime. Figure 7 shows that the waiting time of each patient increases with his/her position in the queue, and the expected ide time is equa to the expected overtime for each vaue of α because of the parameters satisfying the equation T = N( p). Otherwise, the curve of overtime and ide time wi be parae. Since the session ength is just enough for the

18 36 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service 8 6 Figure 7. 4 W 6 W W 3. ide time/overtime α System performance of traditiona no-gof poicy under different α vaues. 6 4 I s /O s 3 I t /O t I sn /O sn Figure 8. W sn. W t W s α System performance of open-access no-gof poicy under different α vaues. expected workoad, the above equaity reveas that ide time often causes overtime. This phenomenon is aso viewed as an expense of the cinic to trade off the uncertainty of service time. Compared with Figure, it is found that onger working hours reduce deays in the waiting ist. In addition, considering the possibiity that the doctor may finish the examination of the ast patient in advance, ide time is inevitabe even if patients waiting is negigibe. Figures 8 and 9 dispay the resuts of the same-session no-gof poicy and the sameor-next-session no-gof poicy, respectivey. The trends of the curves are the same as

19 Journa of Heathcare Engineering Vo. 6 No traditiona poicy Cost same-day poicy same-or-next-day poicy Figure α Tota operationa costs of no-gof AS systems under different α vaues. those in Figures 6 and 7 except when the ide time is onger than usua. Ide time is neary the same as overtime for any given α because the physician is not aowed to go off duty in advance, and this wi cause irreducibe ide time in a particuar session. However, the reason for the equaity is quite different from that depicted in Figure 7. In fact, the overtime and ide time is not exacty the same if the precision of the numerica computing is improved. For a given number of arrivas, they are not necessariy the same; therefore, the anaogy of the utimate resuts is viewed as a coincidence. Figure 9 shows a simiar concusion as Figure 4; i.e. the open-access poicy owers the tota operationa costs when α is arger than a certain threshod, e.g.,.7, for the samesession poicy; the same-or-next-session poicy is aways preferabe Effects of the Session Length The session ength is varied from to in order to anayze its effects on performance measures, whie the expected workoad, no-show probabiity and reative overtime cost are kept the same as those in the basic probem. Figures to dispay the average expected waiting time with a back soid ine, expected overtime with a bue dotted ine and expected ide time with a red dashed ine as functions of T for the traditiona poicy, same-session poicy and same-or-next-session poicy, respectivey. To carify the variation trends of the performance, the curves are potted ony with α equa to,.,.4,.6,.8,. It is noted that the average waiting time and expected overtime decrease, the expected ide time increases with the session ength for any specific α under a poicies. Overtime cannot be eiminated thoroughy due to the stochastic characteristics of service time, even if the session ength is adequate for a patients to show up. The average waiting time is amost the same for different T vaues when α is zero because increasing the session ength reduces the overtime instead of deays in the queue when the waiting cost of patients is negigibe. Simiar to the basic case, it is observed that the average waiting time and ide time are reduced by OA-AS poicies when α is arge, and

20 364 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service 7 6 α = 4 3 α = α = α = Figure. α = α = T System performance of traditiona poicy under different T vaues. 7 6 α = 4 3 α = α = Figure. α = α = α = T System performance of same-session poicy under different T vaues. more overtime are incurred by the same-session poicy for any given T. Regarding the tota operationa costs shown in Figure 3, the same-or-next-session poicy (denoted by the bue dashed ine) has ower costs than those of the traditiona poicy (denoted by the back soid ine) except when α is sma. For a of the T vaues considered, the samesession poicy (denoted by the red dotted ine) reduces the tota costs when α is arger than a certain threshod. 3.. Effects of the No-Show Probabiity To anayze the impacts of no-show probabiity, the expected workoad n is kept constant at, whie the number of patients schedued per session varies from to 4. The no-show probabiity can be cacuated as p= n / N between % and %; the session ength and reative cost of overtime are the same as those in the basic probem.

21 Journa of Heathcare Engineering Vo. 6 No α = 4 3 α = α = Figure. α = α = α = T System performance of same-or-next-session poicy under different T vaues. α = Cost α = α = Figure 3. α = α = α = T Tota operationa costs of AS systems under different T vaues. Figures 4 to 7 show the average expected waiting time, expected ide time, expected overtime and tota operationa cost as a function of α under different no-show probabiities, respectivey. The performance measures under the traditiona poicy are potted with back soid ines, whie those under the same-session and same-or-nextsession poicies are represented by the red dashed ines and purpe dotted ines, respectivey. If a no-show patients notify the cinic of their canceation of appointment in advance, the AS system with the same expected workoad (e.g., p = ) can reduce the patients waiting time as we as the doctor s ide time. It is aso noted that a three operationa costs increase monotonousy with no-show probabiity. The arger the no-show probabiity, the worse the performance measure. As intended by the administrator of the cinics, the OA-AS poicy can be viewed as an aternative way to

22 366 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service 6 4 same-day poicy 3 same-or-next-day poicy p = p =. Figure α Average waiting time of AS systems under different p vaues p =... p =. same-day poicy same-or-next-day poicy α Figure. Ide time of AS systems under different p vaues. ower operationa costs when the no-show probabiity is high. Simiar to the basic cases, the same-or-next-session poicy is preferabe to the same-session poicy in terms of tota operationa costs. Once again, Figure 7 indicates that the same-session system outperforms the traditiona system beyond a certain threshod of α, and the same-ornext-session system is aways preferabe except when the no-show probabiity or the weight of the patients is ow Effects of the Expected Workoad To investigate the impacts of the expected workoad, the number of patients schedued in a session is varied from 4 to 8, whie the expected workoad is sti cacuated by

23 Journa of Heathcare Engineering Vo. 6 No same-day poicy 3. p =. p = Figure 6.. same-or-next-day poicy α Overtime of AS systems under different p vaues. 3 same-day poicy p =. p = same-or-next-day poicy Figure α Tota operationa costs of AS systems under different p vaues. n = N p. To faciitate comparison, the session ength remains the same as the expected workoad to eiminate the infuences of other factors, e.g., T = n ; other parameters are kept the same as those in the basic probem. Figures 8 to present the system performance measures as a function of N under different vaues of α for traditiona, same-session and same-or-next-session poicies, respectivey. Athough the vaue of α is tested from to with an increment of., the curves are depicted ony with α vaues of,.,.4,.6,.8, for iustration. As shown in the figures, the trends of average waiting time denoted by the back soid ines, ide time by the red dashed ines and overtime by the bue dotted ines are the same as those in the basic probem for any given N. For any given α, the ide time and overtime increase with the number of patients schedued per session because the negative

24 368 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service 8 7 α = 6 Figure 8. α = 4 3 α = α = α = α = N System performance of traditiona poicy under different N vaues α = 4 3 α = α = α = Figure 9. α = α = N System performance of same-session poicy under different N vaues. infuence of no-show is accumuated through the waiting ist, just ike the buwhip effect in suppy chain. However, when α., the average waiting time does not change significanty when N is arge enough; this can be viewed as the preiminary period before the steady state. This phenomenon aso reveas that the cinic has to make additiona efforts in maintaining the same service eve as the expected workoad increases. Figure presents the average operationa cost per patient for traditiona, same-session and same-or-next-session poicies with the back soid, red dashed and bue dotted ine, respectivey. It is noted that the average operationa costs increase with the expected workoad except when α =, given that ony overtime is incurred in this situation with ower increment than that of the expected workoad.

25 Journa of Heathcare Engineering Vo. 6 No α = 4 3 α = α = α = α = α = N Figure. System performance of same-or-next-session poicy under different N vaues.. α = α = Cost. α =. 8 α = α = α = N Figure. Average operationa costs of AS systems under different N vaues Effects of the Reative Overtime Cost This subsection iustrates the effects of the overtime ratio on the system performances. β is varied from to 4 with an increment of., whie other parameters remain the same as those in the basic probem. Figures to 4 present the system performance under different vaues of β by a back soid, red dashed and bue dotted ine representing the average waiting time, ide time and overtime, respectivey. As with the previous session, the curves are potted ony with α vaues of,.,.4,.6,.8,. As shown in the figures, higher overtime ratios require more efficient use of the working hours in order to reduce the overtime as much as possibe, eading to shorter ide time and onger waiting time. In other words, higher overtime ratios decrease the reative weights given to the patients in

26 37 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service 7 6 α = 4 α = 3 α = α =. α = α = β Figure. System performance of traditiona poicy under different β vaues α = α = α = α = α = α = β Figure 3. System performance of same-session poicy under different β vaues. the tota costs. Consequenty, the optima schedue aeviates the reativey high overtime cost by increasing patients waiting time. Figure presents the tota operationa costs for a poicies under different vaues of β. The same-or-next-session poicy (represented by the bue dotted ine) reduces the tota cost except when α is sma, whereas the samesession poicy does not outperform the traditiona poicy beyond certain threshods of β for a cases. The same-session poicy has a significant increase in tota costs when β is arge owing to the higher overtime than that of other poicies.

27 Journa of Heathcare Engineering Vo. 6 No α = 4 α = 3 α = α = α = α = β Figure 4. System performance of same-or-next-session poicy under different β vaues. Cost 3 α = α = α = α =. α = α = β Figure. Tota operationa costs of AS systems under different β vaues. 4. DISCUSSION Sequentia quadratic programming methods are empoyed to search for the numerica optima soutions for different AS systems. Session ength, no-show probabiities, expected workoad and reative overtime cost are examined separatey to investigate their infuences on the comparison. It is noted that the same-session AS system has a threshod of reative waiting cost, beyond which it outperforms the traditiona system; and the same-or-next-session system is aways preferabe except when the no-show

28 37 Comparison of Traditiona and Open-Access Appointment Scheduing for Exponentiay Distributed Service probabiity or the weight of the patients is ow. According to the numerica resuts, the no-gof poicy has significant impacts on the ength of the ast intervas, and the expected waiting time is reduced especiay when reative waiting cost is sma. This indicates that when patients satisfaction is not vaued highy, enarging the ast intervas wi improve the efficiency of the cinic, and thus reduce the tota operationa costs. Increasing the session ength certainy resuts in ess average waiting time, ess overtime and more ide time. Increasing the penaty coefficient for overtime forces the AS system to offset the overtime by more patients waiting. Consistent with previous studies, no-show is identified as a key factor infuencing the system performance. Athough traditiona overbooking poicies of compressing the interva ength baances the negative effects of no-show to some extent, the open-access poicy is a more effective aternative when the no-show probabiity is high. Athough the current comparisons are focused on the impacts of exponentiay distributed service time on the differences of two widey used appointment scheduing poicies, some assumptions may restrict the generaity of the resuts. First, the service time may foow genera distribution, such as ognorma or Gamma distribution [8]. It remains unknown to what extent this wi change the resuts of the comparisons. Second, the rea appointment processes are more compicated than the scenarios studied here. Patients usuay have preferences of the appointment time based on their convenience, and undesirabe time intervas obviousy increase the possibiities of noshows. Taking patients choice into consideration is definitey an effective way to reduce no-shows [4]. Third, ate canceations are not entirey the same as no-shows. If the confirmation phone cas or emais revea potentia no-shows, the time intervas concerned can be reeased to other urgent patients who hope to see the doctor in the caing session [8]. Fourth, a patients are assumed to be homogenous in this research. Cayiri et a. [6] evauated the effects of patients cassification on the efficiency of the AS system, and their simuation resuts indicate that appropriate sequencing and interva adjustment according to patient types significanty reduce patients waiting time, physician s ide time and overtime. Further optimization methods need to be deveoped to accommodate the heterogeneous characteristics of service time and wakin patients [46]. Finay, physicians can cooperate and share medica appointments in case one physician is overoaded in a particuar session whie other physicians have free intervas [47]. Furthermore, when patients arrive at the cinic, they usuay go through registration, examination by the doctor, X-ray, aboratory and checkout []. These impy that mutipe-server and mutipe-stage modes are needed to study the transition probabiities between different stages.. CONCLUSIONS In this paper, two types of appointment scheduing poicies are compared under exponentiay distributed service times. The sequentia quadratic programming method is empoyed to search for a soution to minimize the weighted sum of the expected waiting time, ide time and overtime under the traditiona poicy. An open-access poicy is proposed as an aternative approach to mitigate the negative effects of no-shows. Numerica experiments show that an open-access poicy can reduce the operationa cost