Sequential Clinical Scheduling with Patient No-shows and General Service Time Distributions

Transcription

1 Sequental Clncal Schedulng wth Patent No-shows and General Servce Tme Dstrbutons Santanu Chakraborty School Industral Engneerng, Purdue Unversty, West Lafayette, IN Kumar Muthuraman McCombs School of Busness, Unversty of Texas, Austn, TX Mark Lawley Weldon School of BoMedcal Engneerng, Purdue Unversty, West Lafayette, IN We develop a sequental clncal schedulng method for patents wth general servce tme dstrbutons. Patents are scheduled durng a call-n perod that precedes the scheduled consultaton sesson. The sesson s dvded nto a set of tme ntervals called slots. When a patent calls, the scheduler assgns the patent to a sutable slot f one exsts. Each scheduled patent has a probablty of not showng up for the appontment, and the scheduler can compensate through overbookng. We show that the proposed schedulng polcy always yelds a unmodal objectve evoluton, whch provdes an optmal stoppng crtera for schedulng and s thus crtcal. We further analytcally explore the specal case n whch servce tmes have a gamma dstrbuton and show how the computatonal complexty of the schedulng algorthm can be sgnfcantly reduced. Fnally, we provde exhaustve computatonal results accompaned by dscusson provdng nsghts nto the practcal aspects of the schedulng approach. Key words : Overbookng, Appontment Schedulng, Patent No-shows, Outpatent Clncs 1. Introducton There s an urgent need to provde more effcent and effectve patent care. From 80-90% of all US patent care s provded n approxmately 200,000 non-psychatrc outpatent clncs (Bodenhemer and Grumback (2002), U.S. Census ). Clncal operatons are drven by the patent schedule, whch determnes the arrval tme of patents to the clnc. Patent schedulng affects all aspects of clncal operaton and s of prmary nterest n any effort to mprove clnc operatons. Clncal managers and physcans are quck to dentfy nadequate patent schedulng as a major source of operatonal neffcency and patent dssatsfacton. Although clncal schedulng has a long research hstory, clncal mpact has been lmted, a major reason beng that publshed methods nadequately 1 Unted States Census 2004, Ambulatory health care servces: 2002 seres 2002 economcs sensus, healthcare and socal assstance, U.S. Census webste 50 (EC ). 1

2 2 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng address routne factors that complcate clncal schedulng, such as sequental patent call-n, patent no-show and walk-n, and on-call physcans. Perhaps the most vexng problem n patent schedulng s patent no-show, that s, patents not showng up for scheduled appontments. No-show rates can be as hgh as 40% n some clncs (Lee et al. (2005)). No-show patents ntroduce sgnfcant uncertanty nto clncal operatons and lmt accessblty to other patents by reservng appontment slots that go unused. Although many factors have been cted as ndcators of patent no-show ncludng patent demographcs, medcal condtons, and envronment (Deyo and Inu (1980)), there s very lttle quanttatve research that uses patent no-show behavor n the schedulng process. Thus, our research focuses on usng noshow behavor to create patent schedules that balance factors such as patent watng tme, staff overtme, physcan utlzaton, and clnc revenue. In ths paper we present a sequental clncal schedulng mechansm for patents wth general servce tme dstrbutons and multple no-show probabltes. We further focus on the specal case of gamma servce tmes and show how the computatonal requrements can be steeply reduced for ths case. Exhaustve computatonal studes and dscussons that llustrate practcal behavor and provde nsghts are also presented. We refer to our schedulng procedure as myopc snce the assgnment decson does not consder the future call-n sequence. We consder a mult-objectve optmzaton formulaton that balances rewards and costs for patent watng and staff overtme at the end of the day. Ths objectve can be convenently nterpreted as the expected proft. We show that the proposed polcy yelds an objectve evoluton that s unmodal. By unmodal, we mean that the expected proft for a schedule s non-decreasng up to the addton of some patent and then monotone decreasng thereafter. Ths unmodalty s crtcal snce t provdes an optmal stoppng crteron wth the guarantee that the objectve wll always decrease thereafter. We note that the optmal sequental schedulng problem can be easly formulated usng a dynamc programmng tree. However, the state space s very complex and computaton s ntractable for all but the smallest problem nstances. Further, t s easy to show that the optmal polcy wll not be unmodal n objectve evoluton, complcatng the dentfcaton of a stoppng crtera. The paper s structured as follows. Secton 2 provdes a bref lterature revew on clncal schedulng, whle Secton 3 presents our model formulaton, the schedulng algorthm, and the dervaton of varous expressons that are necessary for the schedulng algorthm. The specal case of servce tmes havng a gamma dstrbuton s consdered n Secton 4. Secton 5 provdes the theoretcal guarantees that establsh the unmodalty of the objectve evoluton. Secton 6 presents the computatonal results and dscussons. Concludng remarks are made n Secton 7.

3 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 3 2. Lterature Revew Cayrl and Veral (2003) provde an extensve revew of clncal schedulng research, coverng eghty papers from as early as 1952 (Baley (1952), Lndley (1952)) up to They categorze the lterature as statc vs. dynamc; by performance measure; by system desgn; and by methodology. We wll brefly dscuss each of these. In statc schedulng, appontment tmes are not adjusted once the scheduled day begns, whle n the dynamc case, schedule adjustments can be made as the day evolves. Dynamc schedulng s approprate for n-patent stuatons where patents are avalable for early servce or can wat n ther rooms f the schedule s delayed. It can also be appled n outpatent settngs wth same day schedulng. In most outpatent clncs, however, there s lttle opportunty to adjust patent appontment tmes once the schedule s fxed. Most lterature focuses on the statc case, whch typcally nvolves a set of N punctual patents wth d servce tmes to be scheduled for a sngle sesson (day) wth a sngle physcan (sngle server). A representatve set of recent statc papers ncludes Vanden-Bosch and Detz (2000, 2001), Vanden-Bosch et al. (1999), Denton and Gupta (2003), Gupta and Wang (2007), Ho and Lau (1992), Lau and Lau (2000) and Robnson and Chen (2003). Performance measures dctate how a schedule s evaluated and are categorzed as tme, congeston, and farness-based. Tme measures are a functon of patent watng tme, physcan dle tme, and staff overtme; congeston measures capture features such as watng room utlzaton; and farness measures look at how watng tmes are dstrbuted across the day (often, patents arrvng later experence greater expected watng). Gupta et al. (2006) proposes several addtonal measures. A detaled revew s provded n Mondschen and Wentraub (2003). The desgn of an appontment system s specfed by the block (number of patents arrvng at the begnnng of an appontment perod), the ntal block (number of patents arrvng for the ntal appontment), and the nterval (length of the appontment nterval whch s ether fxed or varable). These parameters can be adjusted based on the presence of envronmental complextes such as walk-ns, urgent and emergency patents, and patent specfc servce tme dstrbutons. More detaled dscussons of system desgn n complex envronments s provded n Cayrl et al. (2006), Harper and Gamln (2003), Ho and Lau (1992), Klassen and Rohleder (2004), Lu and Lu (1998) and Rohleder and Klassen (2002). Methodologcal classes nclude analytcal modelng and smulaton. Analytcal approaches use stochastc and determnstc operatons research and focus on basc appontment schedulng wth

4 4 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng lmted consderaton of complcatng factors. Smulaton studes focus on comparng detaled appontment schedulng systems n complex envronments. Representatve analytcal papers nclude Vanden-Bosch and Detz (2000, 2001), Vanden-Bosch et al. (1999), Denton and Gupta (2003) and Robnson and Chen (2003), whle smulaton studes nclude Babes and Sarma (1991), Cayrl et al. (2006), Harper and Gamln (2003), Ho and Lau (1992), Ho et al. (1995), Klassen and Rohleder (2004) and Rohleder and Klassen (2002). Further, Jun et al. (1999) provdes a revew of smulaton studes n healthcare clncs up to Several addtonal papers on clncal schedulng have appeared snce Cayrl and Veral (2003) was publshed. Three of these address clncal schedulng wth patent no-shows and overbookng and thus we wll confne our dscusson to these. Kaandorp and Koole (2007) develop a model for a sngle server system wth exponental servce and a sngle no-show probablty for all patents. The objectve functon mnmzes expected watng tme, server dle tme, and overtme. They prove that the model s mult-modular so that local search can be used to obtan a globally optmal schedule. Ther search algorthm s of super-polynomal complexty snce the neghborhood of a gven schedule s of exponental sze. LaGanga and Lawrence (2007) develop a clncal schedulng model for a sngle server wth determnstc servce tmes and a sngle no-show probablty. Ther objectve functon maxmzes the net offce utlty whch s equal to expected proft mnus expect watng and overtme costs. Cost tems are modeled n both lnear and quadratc forms. Usng numercal experments, they show that overbookng ncreases wth no-show probablty and conclude that schedule overbookng s very effectve at reducng the negatve mpact of patent no-show. Km and Gachett (2006) also propose clncal overbookng to cope wth patent no-show, agan wth a sngle no-show probablty for all patents. Ther model assumes determnstc servce tme and maxmzes expected revenue mnus overtme and penalty costs ncurred when patents leave wthout beng seen. They do not explctly consder patent watng tme as part of the objectve. Ther numercal results ndcate that overbookng can sgnfcantly mprove the revenue as well as provde ncreased access to patents. Although these papers are nterestng and provde some nsghts, they are not applcable and would not perform well n many clncs for at least two reasons. Frstly, they do not consder sequental schedulng, that s, they assume that the complete set of patents s known when the schedule s generated. In our experence wth clncal partners, schedules are rarely bult n ths way. Rather, schedules are constructed as patents call for appontments. Schedulers do not know how many patents wll call for appontments and eventually be added to the schedule. Nor do schedulers

5 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 5 know how many should be added, snce they have no optmal stoppng crtera. Further, there s lttle opportunty to adjust the schedule once completed. Secondly, these papers consder only a sngle no-show probablty, whch does not match well wth realty. The research lterature shows that patent no-show probabltes can be estmated based on patent demographcs, prevalng envronmental condtons, and lead tme to appontment (Dervn et al. (1978), Deyo and Inu (1980), Goldman et al. (1982), Gruzd et al. (1986), Martnez et al. (1987), Cashman et al. (2004), Lee et al. (2005)). Our own unpublshed research shows that sgnfcant mprovements n schedulng effcency can be acheved when more accurate no-show modelng s n place. In Muthuraman and Lawley (2007), we develop a sequental schedulng model wth exponental servce tmes and multple patent no-show probabltes. The objectve maxmzes the expected revenue for patents seen mnus costs for patent watng and staff overtme. The paper leverages heavly on the memoryless property of the exponental dstrbuton to show that expected proft evoluton s unmodal. That s, the expected proft of a schedule s non-decreasng wth the number of patents untl some crtcal number s scheduled and then decreases monotoncally as more patents are added, whch provdes an optmal stoppng rule, after whch patents are rejected for that partcular schedule and can be consdered for the schedules of other days. The work presented n ths paper generalzes ths approach by allowng arbtrary servce tme dstrbutons, whch requres that we nclude varables capturng tme spent n servce. 3. Problem Formulaton Let the perod of nterest (often called a sesson and typcally 4 to 8 hours) be dvded nto I ntervals referred to as slots, and the length of the th slot be denoted by t. Patents callng for an appontment can be scheduled n one of the I slots or rejected. Scheduled patents have a noshow probablty, and each patent arrves (we assume on-tme) ndependently of others. Arrvng patents jon a queue and f they are not servced n ther scheduled slot, they overflow to the next slot. Suppose n patents have been scheduled for a gven sesson. Let X n arrvng at the begnnng of slot and Y n denote the number of patents be the number of patents watng and n-servce at the end of slot (see Fgure 1). Let η be the tme the n-servce patent has spent n servce at the end of slot. We also defne Y n 0 = η 0 = 0. Let L be the random number denotng the number of servce completons provded the queue does not empty. L s drawn from a dstrbuton that mplctly depends on η 1. Then mn(l, Y n 1 + X n ) represents the number of servce completons n slot. The overflow from slot s,y n = max(y n 1 + X n L n, 0). The overflow model s llustrated n Fgure 1.

6 6 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng As stated earler, we assume that each patent has an estmated no-show probablty. We partton the set of patents nto J groups such that a patent belongng to group j has a probablty p j > 0 of arrvng and a probablty 1 p j of not showng. Fgure 1 The Slot Model After n calls, a sesson s schedule s represented by the matrx S n R I J, where S n,j denotes the number of patents of type j scheduled for slot. Thus, N n = j Sn j s the total number of patents scheduled for slot. When the context s clear, n the sequel, we wll often suppress the superscrpt (as n Fgure 1). Let j be an assgnment matrx of sze I J wth a 1 at the, j th poston and zeros elsewhere. The functon Q(.) takes as argument the state matrx S and gves the arrval probablty matrx, Q(S). The, m th element of Q(S) denotes the probablty of m patents arrvng n slot. For notatonal convenence, we take the matrx Q n Q(S n ). Consder the th row of a gven S n and let Φ be the set of all non-negatve, nteger J-vectors π (π 1, π 2,..., π J ) such that J j=1 π j = m and π j S n j for all j. Condtonng on the event that π j number of type j patents show, Q n,m = Pr{X n = m} = π Φ Pr{X n = m (π 1,..., π J )} Pr{(π 1,..., π J )} = π Φ Pr{(π 1,..., π J )} = π Φ j S n,j! π j!(s n,j π j )! pπ j j (1 p j ) Sn,j π j. (1) Further smplfcaton yelds the convenent recurrence relaton Q n 1,m (1 p j ) + (Q n 1,m 1)p j when m 1 and Q n,m = Q n 1,0 (1 p j ) when m = 0. (2)

7 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 7 The functon R( ) represents the over-flow probablty matrx, that s, the, k th element of R(S) represents the probablty of k patents over flowng from slot. As wth Q n, R n R(S n ). Obvously, Q n, R n R I ˆN n where ˆN n = max N n. By defnton, gven S, The calculaton of R n k s dscussed n subsecton 3.1. R n k = Pr{Y n = k}. (3) Suppose the n th patent callng for an appontment s of type j. Lettng U be the set of slots (that s, ntegers from 1 to I), we wsh to select a slot U for the patent so as to maxmze an objectve, f(q n, R n ). That s, we assgn patent n to slot where = arg max f(q(s n 1 + j ), R(S n 1 + j )) and (4) U S n = S n 1 + j. (5) We take r as the reward for each patent served and let c represent the cost or penalty we charge ourselves for makng a patent over flow from slot to slot + 1. Ths provdes suffcent flexblty to model the cost of physcan and staff overtme by assgnng an approprate over flow cost to the end of the consultng perod (assumng that a physcan wll see all patents before leavng for the day). Hence, our objectve wll be to maxmze f(q n, R n ) = r mq n,m c kr n,k m k = E [r X n c Y n ]. (6) 3.1. Calculatng Overflow Probabltes Consder R n,k, that s, the probablty of k patents over-flowng nto slot + 1 from slot, R n,k = Pr{Y = k} = Pr{max(X + Y 1 L, 0) = k} Pr{X + Y 1 L = k} k > 0 = Pr{X + Y 1 L 0} k = 0 (7) Further condtonng yelds, R n,0 = m = m Pr{m + k L }Q n,mr n 1, k k k (1 F L (m + k))q n,mr n 1, k (8)

8 8 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng and smlarly for k > 0, R n,k = m = m Pr{m + k k = L }Q n,mr n 1, k k k f L (m + k k)q n,mr n 1, k (9) where F L (m) = Pr{L m} and f L (m) = Pr{L = m} mplctly depend on the dstrbuton of η 1 and are obtaned from the general servce tme dstrbuton as detaled below. Let f(t) and F (t) represent the probablty densty functon (p.d.f.) and the cumulatve dstrbuton functon (c.d.f.) of the general servce tmes, respectvely. By frst condtonng on the realzaton of η 1, we evaluate the dstrbuton of L. Pr{L = 0 η 1 } = Pr{T,0 η 1 + t } Pr{T,0 η 1 } = 1 F ( t + η 1 ) 1 F (η 1 ) where T,0 s a draw from the servce tme dstrbuton F ( ), representng the total servce tme for the frst patent, who s n servce at the begnnng of slot. For n 1 we have, and lettng U,n = T, T,n, Pr{L = n η 1 } = Pr{L n η 1 } Pr{L n + 1 η 1 } (10) Pr{L n η 1 } = Pr{U,n 1 t + η 1 η 1 } = Pr{η 1 T,0 t + η 1, T,1 t + η 1 U,0,..., T,n 1 t + η 1 U,n 2 } 1 F (η 1 ) 1 t +η 1 t +η 1 u,0 t +η 1 u,n 2 n 1 =... f(t,j )dt,j 1 F (η 1 ) η j=0 (11) where, u,n = t,0 + t, t,n. When n = 1 an ntegral over t,0 alone resdes, that s, only the frst ntegral. In general for any gven n, the above equaton requres the computaton of n ntegrals. Further smplfcaton usng equaton (10) yelds, Pr{L = n η 1 } = 1 1 F (η 1 ) t +η 1 η 1... t +η 1 u,n 2 0 t +η 1 u,n 1 j=0 n f(t,j )dt,j For any gven n, the above equaton requres the computaton of n+1 ntegrals. Fnally, ntegratng wth respect to η 1 we get, f L (n) = 0 (12) Pr{L = n η 1 }dψ 1 (η 1 ) (13)

9 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 9 where Ψ 1 ( ) represents the c.d.f. of η 1. For the case of servce tmes takng an exponental dstrbuton, equaton (13) can be further smplfed to reveal that the dstrbuton of L does not depend on η 1 and s smply a Posson random varable Computaton of Ψ We have from our defnton that η 0 = 0. Note that when no patent servces are completed n the frst slot, η 1 = t 1. Now, for a gven η [0, t + η 1 ), the c.d.f of η s gven by Ψ (η) = 1 where, 0 Pr{η η η 1 }dψ 1 (η 1 ) Pr{η η η 1 } = 1 {η t +η 1 } Pr{L = 0} + Pr{U,n 1 t + η 1 η U,n 1 t + η 1, U,n t + η 1 } Pr{L = n} n=1 = 1 {η t +η 1 } Pr{L = 0} + = 1 {η t +η 1 } Pr{L = 0} + = 1 {η t +η 1 } Pr{L = 0} + n=1 n=1 n=1 Pr{U,n 1 t + η 1 η, T,n η} Pr{U,n 1 t + η 1, T,n η} Pr{U,n 1 t + η 1 η} Pr{T,n η} Pr{U,n 1 t + η 1 } Pr{T,n η} Pr{U,n 1 t + η 1 η} Pr{U,n 1 t + η 1 } Pr{L = n} Pr{L = n} Pr{L = n}. (14) The ndcator functon for event A, n the above, s denoted by 1 {A}. The dstrbuton of U,j and L n equaton (14) can be computed as n subsecton The Schedulng Polcy The schedulng polcy gven enumerates all possble assgnments for the current patent and selects the assgnment that maxmzes the objectve. It s sequental n that t assgns patents as they call and myopc n that t does not consder future arrvals when makng the assgnment. Further, t wll reject the patent and termnate when the objectve declnes. 1. Intalze wth S 0,j = 0, Q 0,0 = R 0,0 = 1 for all = 1,..., I and j = 1,..., J and set n = Wat for n th call. and let the n th call be from a patent of type j. 3. For each U: Set S n = S n 1 + j ; Compute Q n from Q n 1 usng equaton (1); Compute Ψ ( ) and R n as descrbed n sectons 3.2 and 3.1; Compute f n = f(q n, R n ). 4. If max f n f n 1 (a) then = arg max f n, S n = S n 1 + j, Q n = Q n, Rn = R n. Set n = n + 1 and terate by gong to 2. (b) else termnate.

10 10 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng It s lkely that callng patents have tme preferences. To accommodate ths, we can defne U n U as the set of slots that the n th callng patent prefers. Then, step 4 can be modfed to maxmze over U n. Note that a sequence of sngleton U n s would then elmnate any schedulng flexblty. Whle such sequences are unlkely n practce, they are theoretcally feasble. Any arbtrary objectve evoluton can be constructed by such sequences and thus a stoppng crtera as n step 4 cannot guarantee a maxma. Hence, n secton 5, where we seek theoretcal guarantees on the behavor of objectve evolutons, we wll restrct our attenton to U n = U for all n. 4. Gamma Dstrbuted Servce Tmes For general servce tme dstrbutons, the evaluaton of Pr{L = n η 1 } usng equaton (12) nvolves the evaluaton of n + 1 ntegrals. We consder the specal case of gamma dstrbuted servce tmes n ths secton and demonstrate that n ths case the computaton can be sgnfcantly reduced to the evaluaton of a sngle ntegral. The gamma dstrbuton s characterzed by two parameters, the shape and the scale. These two parameters often provde suffcent flexblty to model dfferent emprcal dstrbutons wth varyng skewness. The sum of d gamma random varables n turn s also a gamma dstrbuted random varable. Let the servce tme for a patent be gamma dstrbuted wth parameters α and λ 0. We represent ts c.d.f and p.d.f by G 1 ( ) and g 1 ( ), respectvely. Here e t λ0 g 1 (t) = t α 1, for t 0. (15) λ α 0 Γ(α) The sum of k of these random varables (servce tmes) s then a gamma random varable wth parameter kα and λ 0, the c.d.f. and p.d.f. of whch are represented smply by G k ( ) and g k ( ). Fgure 2 shows examples of gamma dstrbutons for dfferent values of the parameters α and λ 0 as well as an exponental dstrbuton. Now, from equaton (11) we have, for n = 1 and for n 2, Pr{L 1 η 1 } = 1 Pr{L n η 1 } = 1 G 1 (η 1 ) 1 = 1 G 1 (η 1 ) Thus, from equaton (10) we get, 1 1 G 1 (η 1 ) t +η 1 t +η 1 t +η 1 t 0 η 1 t +η 1 0 η 1 g 1 (t 0 )dt 0 g 1 (t 0 )g n 1 (u n 1 )dt 0 du n 1 η 1 g 1 (t 0 )G n 1 ( t + η 1 t 0 )dt 0 Pr{L = 1 η 1 } = t +η 1 η 1 [1 G 1 ( t + η 1 t 0 )] g 1 (t 0 )dt 0 1 G 1 (η 1 )

11 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 11 Fgure 2 Gamma denstes wth equal means (16) and for n > 1, Pr{L = n η 1 } = t +η 1 η 1 [G n 1 ( t + η 1 t 0 ) G n ( t + η 1 t 0 )] g 1 (t 0 )dt 0 1 G 1 (η 1 ) (17) The dstrbuton of η also assumes a smpler form for gamma servce tmes. From equaton (14), Pr{η η η 1 } = 1 {η t +η 1 } Pr{L = 0} + 5. Unmodal Objectve Evaluaton n=1 G n ( t + η 1 η) G n ( t + η 1 ) Pr{L = n} We show that the myopc polcy s unmodal under general servce tmes n ths secton. Theorem 1 shows that f the n th assgnment yelds a decrease n expected objectve then the n+1 st assgnment would yeld a further decrease. By nducton, ths would mply unmodal evoluton of the objectve. As mentoned earler, by unmodal we mean that the expected objectve s non-decreasng up to a partcular call-n patent and then s monotone decreasng thereafter. Theorem 1 If n s such that f(q n, R n ) < f(q n 1, R n 1 ), then f(q n+1, R n+1 ) < f(q n, R n ). Proof: We have, [ f(q n, R n ) f(q n 1, R n 1 ) = E r X n c Y n ] E [ r X n 1 c Y n 1 ]

12 12 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng Now consder the frst expectaton, = re [ X n X n 1 ] E c [ Y n ] Y n 1 (18) E [ X n X n 1 ] = x Pr{X n = x Pr{X n X n 1 = x} X n 1 = 1} = Pr{n th patent arrves for one of the I slots} = p jn. In the above, we let j n denote the type of the n th patent. Condtonng the second expectaton on the event E n, whch denotes that the n th patent shows up, Hence from equaton (18), E[Y n Y n 1 ] = p jn E[Y n f(q n, R n ) f(q n 1, R n 1 ) = p jn [r Y n 1 E n ]. (19) c E [ ] ] Y n Y n 1 E n. (20) Usng the smplfcaton provded n equaton (20), to prove theorem 1, t suffces to show that f r < r < c E [ ] Y n+1 Y n E n+1 (21) c E [ ] Y n Y n 1 E n. (22) To show that (22) mples (21), we need more notaton. Let the n th and the n + 1 st patents be of types j n and j n+1. Let the best assgnment of the n th and the n + 1 st patents, usng the schedulng polcy detal n subsecton 3.3, be to slots n and n+1, respectvely. We have S n = S n 1 + njn and S n+1 = S n + n+1j n+1. Now also consder the schedule S n created by takng the schedule S n 1 and assgnng a patent of type j n to slot n+1 (nstead of slot n whch s the best assgnment). That s, S n = S n 1 + n+1j n. The schedules S n,s n+1, are respectvely constructed by takng S n 1, S n and addng a patent n slot + 1. And note that S n S n 1 = njn. Hence, c E [ Y n Y n 1 E n ] c E [ ] Y n+1 Y n E n+1 (23)

13 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 13 Next compare S n and S n, snce n s the best assgnment of the n th patent, f(q n, R n ) f(q n, R n ). Smplfcaton yelds, c E [ Y n From (23) and (24), we have Y n 1 E n ] c E [ ] Y n Y n 1 E n (24) c E [ Y n Y n 1 The above along wth (22) yelds (21). E n ] 6. Computatonal Results for Emprcal Servce Tme c E [ ] Y n+1 Y n E n+1. (25) The prmary purpose of ths secton s to understand the mpact of and gan nsghts nto the proposed schedulng methodology. To ths extent, frst we evaluate the value of beng able to use general servce tme dstrbutons as opposed to approxmatons lke gamma and exponental servce tmes. Next, we evaluate the performance of our schedulng methodology aganst heurstc schedulng procedures that have been tested and recommended n lterature. Fnally we look at the effect of both mean and varance of servce tme dstrbutons on the expected proft and the number of patents scheduled Qualty of approxmatons One of the prmary questons of nterest s the value that s added n beng able to handle general servce tmes. To answer ths, we frst need a general servce tme dstrbuton that s estmated from actual data. Cayrl et al. (2006) use data collected from a prmary health care clnc n a New York Metropoltan Hosptal whch serves approxmately 300,000 patents per year. Based on the Kolomogorov-Smrnov test (at α = 0.05), they conclude that, a log-normal dstrbuton s the best ft. We wll take the same log-normal dstrbuton (wth mean µ = 15 mnutes and standard devaton σ = 5 mnutes) as our general servce tme dstrbuton. We assume that there are three types of patents, and the no-show probabltes are p = (0.25, 0.5, 0.75). Slot overflow costs are set to c = $100 for = 1,..., I 1 and c I = $300. The reward for each scheduled patent s r = $200. For comparsons, we approxmate ths general servce tme by the gamma and exponental fts shown n Fgure 3. The exponental and gamma are chosen to match the mean and varance of the general servce (log-normal) tmes. For each case, the general servce dstrbuton, the gamma ft, and the exponental ft, we schedule patents accordng to the methodology descrbed n secton 3, secton 4, and Muthuraman and Lawley (2007), respectvely. Fgure 4 shows the evoluton of the expected proft obtaned usng these three servce tme dstrbutons. It s evdent from Fgure 4

14 14 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng that the expected proft from the gamma ft and the general servce tme are very close, whle the expected proft from the exponental ft s sgnfcantly less, where the expected profts are calculated usng equaton 6. Ths s to be nterpreted as the expected proft f the realzed servce tmes are also drawn from the gamma and exponental dstrbutons. Hence, Fgure 4 does not provde a measure of how well a gamma or an exponental based schedule performs on n a general servce envronment, but t does provde a measure of how close the gamma and exponental models are to the general servce model. We now nvestgate how well a gamma or an exponental based schedule wll perform n our general servce envronment. We seek as a measure, the value lost n approxmatng the servce tme dstrbuton for 100 dfferent call-n sequences generated randomly. We use the schedulng methodologes for general, gamma and exponental and prepare 100 schedules each. For each of the 300 schedules we smulate 10 no-show/show and servce tme realzatons (usng general servce tmes) and plot the average realzed profts for each schedule and methodology n Fgure 5. The average realzed proft for all these runs are respectvely $4574, $4578, and $3881 for general servce tme, gamma ft and exponental. The dfferences amongst the general servce tmes and gamma servce tmes are statstcally nsgnfcant. Fgure 5 clearly llustrates the value ganed n beng able to handle general servce dstrbutons as opposed to approxmatng the system wth an exponental ft. Further t also demonstrates the qualty of the gamma approxmaton at least for ths servce tme dstrbuton observed at the New York Metropoltan Hosptal. Fgure 3 PDF of Lognormal and Gamma wth Mean = 0.25 hr and SD = hr.

15 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 15 Fgure 4 Objectve Evoluton for Lognormal and Gamma wth Mean = 0.25 hr and SD = hr. Fgure 5 Average Proft Obtaned from Smulaton for Lognormal wth Mean = 0.25 hr and SD = hr and Usng Schedules from General Servce Tme, Gamma Ft and Exponental Ft. Note that the computatonal requrements for the gamma dstrbutons are much smaller than that for general servce tmes. Calculatng Pr{L = n η 1 } for the Gamma servce dstrbutons requres the computaton of only a one- dmensonal ntegral, whereas general servce dstrbutons requres the computaton of a (n + 1)-dmensonal ntegral. For our ntegral computatons here, we use Gaussan Quadratures wth 8-ponts for each ntegral.

16 16 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 6.2. Comparson wth the Appontment Rules of Cayrl et al. (2006) In ths secton we perform a comparatve study of the Appontment Rules proposed by Cayrl et al. (2006) wth our myopc schedulng polcy under general servce tme. We assume that the servce tme dstrbuton s lognormal wth mean 0.25 hr. and standard devaton hr and schedule dfferent arrval sequences of patents usng each of the seven Appontment Rules as well as the myopc polcy and compare the evoluton of the objectve functon. In the experments descrbed below we assume that the mnmum number of patents to schedule (N) s 20. The Appontment Rules are functons of fve parameters, vz. (β 1, β 2, β 3, k 1, k 2 ). In our experments we use (β 1 = 0.15, β 2 = 0.3, β 2 = 0.05, k 1 = 10, k 2 = 18) (Cayrl et al. (2006)) IBFI, 2BEG Rules: In both the IBFI and 2BEG the slot length s held constant and s equal to the mean of the servce tme. Accordng to IBFI rule only one patent s assgned to each slot whereas n 2BEG rule 2 patents are assgned to the frst slot and 1 patent s assgned to each of the remanng slots. Fgure 6 shows the evoluton of the objectve functons when patents are scheduled usng these two rules as well as the myopc schedulng polcy under general servce tme. Fgure 6 Objectve Evoluton for IBFI & 2BEG and Myopc Polcy under General Servce Tme OFFSET Rule: In OFFSET rule frst (k 1 1) patents are scheduled earler whereas the rest are scheduled later compared to the IBFI rule. Fgure 7 shows the evoluton of the objectve functons when patents are scheduled usng ths rule as well as the myopc schedulng polcy under general servce tme.

17 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 17 Fgure 7 Objectve Evoluton for OFFSET and Myopc Polcy under General Servce Tme DOME and 2BGDM Rules: Accordng to the DOME rule, the frst (k 1 1) patents are scheduled earler, the followng patents up to (k 2 1) are assgned later, and the rest are scheduled earler compared to the IBFI rule. The 2BGDM rule s a combnaton of the DOME and 2BEG rule. Fgure 8 shows the evoluton of the objectve functons when patents are scheduled usng these rule as well as the myopc schedulng polcy under general servce tme. Fgure 8 Objectve Evoluton for DOME & 2BGDM and Myopc Polcy under General Servce Tme.

18 18 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng MBFI Rule: Accordng to the MBFI rule, 2 patents are scheduled at each slot where the length of each slot s twce the mean of the servce tme dstrbuton. Fgure 9 shows the evoluton of the objectve functons when patents are scheduled usng ths rule as well as the myopc schedulng polcy under general servce tme. Fgure 9 Objectve Evoluton for MBFI and Myopc Polcy under General Servce Tme MBDM Rule: The MBDM rule s a combnaton of the DOME and MBFI rule. Fgure 10 shows the evoluton of the objectve functons when patents are scheduled usng ths rule as well as the myopc schedulng polcy under general servce tme. Comparng the objectve functon from each of the seven Appontment Rules and the myopc schedulng polcy under general servce tme we see that maxmum expected proft from the latter s always hgher, n that t s always able to proftably schedule more patents Effect of servce tme varance In ths secton we study the effect the varance of the servce tme dstrbuton has on expected proft. To study the effect of varance of the servce tme dstrbuton on expected proft, we systematcally ncrease the varance of the lognormal servce tme keepng the mean constant at 15 mnutes. Fgure 11 plots the the maxmum of the objectve evoluton as we ncrease the varance of the servce tme. It s clear from Fgure 11 that the value of the maxmum proft decreases as we ncrease the varance of the servce tme dstrbuton.

19 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng 19 Fgure 10 Objectve Evoluton for MBDM and Myopc Polcy under General Servce Tme. Fgure 11 Maxmum Expected Proft wth Increasng SD and Fxed Mean of the Servce Tme. The x-axs unts are n hours. 7. Concludng Remarks In ths paper we develop a sequental clncal schedulng polcy for a clnc wth general servce tme dstrbutons. We show that the polcy developed, though myopc n nature, yelds a unmodal objectve evoluton that provdes a convenent stoppng crtera. The major challenge wth the general servce tme formulaton and schedulng methodology s that t requres the numercal

20 20 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng computaton of several large multdmensonal ntegrals. However, we show how the computatonal needs can be reduced sgnfcantly when servce tmes are approxmated by a gamma dstrbuton. In ths case the multdmensonal ntegrals are shown to reduce to a sngle ntegral. We also demonstrate that for an emprcal servce tme dstrbuton observed n the New York Metropoltan Hosptal, the gamma approxmaton to the general servce tme dstrbuton yelds an extremely good approxmaton as opposed to the smpler memory-less exponental servce tme approxmaton. Fnally, we also compare the performance of our schedulng method to those studed and recommended n lterature. References Babes, M., G. V. Sarma Out-patent queues at the Ibn-Rochd health centre. The Journal of the Operatonal Research Socety 42(10) Baley, N A study of queues and appontment systems n hosptal outpatent departments wth specal reference to watng tmes. Journal of the Royal Statstcal Socety Bodenhemer, T., K. Grumback Understandng Health Polcy: A Clncal Approach. Thrd edton ed. Lange Medcal Books / McGraw-Hll,Medcal Publshng Dvson, New York. Cashman, S. B., J. A. Savageau, C. A. Lemay, W. Ferguson Patent health status and appontment keepng n an urban communty health center. J Health Care Poor Underserved Cayrl, T., E. Veral Outpatent schedulng n health care: a revew of lterature. Producton and Operatons Management 12(4) Cayrl, T., E. Veral, H. Rosen Desgnng appontment schedulng systems for ambulatory care servces. Health Care Management Scence Denton, B., D. Gupta A sequental boundng approach for optmal appontment schedulng. IIE Transactons 35(11) Dervn, J. V., D.L. Stone, C. H. Beck The no-show patent n the model famly practce unt. Journal of Famly Practce 7(6) Deyo, R. A., T. S. Inu Dropouts and broken appontments. Medcal Care 18(11) Goldman, L., R. Fredn, E. F. Cook, J. Egner, P. Grch A multvarate approach to the predcton of no-show behavor n a prmary care center. Archves of Internal Medcne 142(3) Gruzd, D. C., C. L. Shear, W. Rodney Determnants of no-show appontment behavor: the utlty of multvarate analyss. Fam Med Gupta, D., S. Potthoff, D. Blowers, J. Corlett Performance metrcs for advanced access. Journal of Healthcare Management Gupta, D., L. Wang Revenue management for a prmary care clnc n the presence of patent choce. to appear n Operatons Research.

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22 22 Chakraborty, Muthuraman, Lawley: Sequental Clncal Schedulng Rohleder, T. R., K. J. Klassen Rollng horzon appontment schedulng: A smulaton study. Health Care Management Scence 5(3) Vanden-Bosch, P. M., D. C. Detz Mnmzng expected watng n a medcal appontment system. IIE Transactons 32(9) Vanden-Bosch, P. M., D. C. Detz Schedulng and sequencng arrvals to an appontment system. Journal of Servce Research 4(1) Vanden-Bosch, P. M., D. C. Detz, J. R. Smeon Schedulng customer arrvals to a stochastc servce system. Naval Research Logstcs 46(5)