Capacity allocation and flexibility in primary care

Transcription

1 Capacty allocaton and flexblty n prmary care Har Balasubramanan, Ana Murel, Asl Ozen, Lang Wang, Xaolng Gao, Jan Hppchen Abstract We dscuss capacty allocaton for prmary care practces at three dfferent plannng levels: the strategc, the tactcal and the operatonal. The goal n each case s to maxmze two mportant but often conflctng metrcs: (1) tmely access and (2) patent-physcan contnuty. Tmely access focuses on the ablty of a patent to get access to a physcan as soon as possble. Patent-physcan contnuty refers to buldng a strong relatonshp between a patent and a specfc physcan by maxmzng patent vsts to that physcan. Each prmary care provder (PCP) has a panel of patents for whose long term holstc care the PCP s responsble. At the hghest or strategc level, the desgn of physcan panels, we demonstrate the mpact of case-mx, or the type of patents n a physcan s panel, and show how panels can be redesgned effectvely. Panel redesgn, however, nvolves changng exstng patent-physcan relatonshps. A vable alternatve s managng the nherent flexblty of PCPs to see patents of other physcans. At the tactcal level, ths requres allocatng the flexble capacty to two types of appontments: 1) prescheduled appontments whch are booked n advance and requre contnuty; and 2) same-day appontments. Usng a 2-stage stochastc optmzaton model, we show that greedy algorthms fnd the optmal capacty allocaton, and fnd that a partally flexble practce provdes a good compromse between tmely-access and contnuty. Fnally, at the operatonal level, the mplementaton of flexblty durng a workday has to be made under partal demand nformaton, as patent calls arrve over the course of a day. We dscuss the mpact of flexblty and suggest heurstcs that practces can use n ths dynamc case. 1 Introducton Prmary care provders (PCPs) are typcally the frst pont of contact between patents and health systems. Broadly they nclude famly physcans, general nternsts, geratrcans and pedatrcans. From a patent s perspectve, PCPs provde the ma- 1

2 2 Balasubramanan et al. jorty of care they receve durng ther lfetme. They are responsble for a varety of health servces ncludng preventve medcne, patent educaton, routne physcal exams, and referrals to medcal specaltes for secondary and tertary care. The benefts of a strong prmary care system are well documented n the clncal lterature. Papers n the health servces lterature, [33] for example, have establshed that ncreased access to prmary care 1) mproves access to health servces for relatvely deprved populaton groups; 2) has a strong postve relatonshp wth preventon and early management of health problems; and 3) leads to ncreased famlarty wth patents and, consequently, to less wasteful expendtures due to napproprate specalst care. Fg. 1 Dedcated versus pooled physcan capacty llustratng the best cases for contnuty and tmely access respectvely. Despte ts pvotal role, prmary care physcans receve lower salares than specalsts, whch has the effect of drvng medcal students away from pursung prmary care careers. Ths s one of the man reasons for shortages n prmary care, whch are common n many countres. A recent study by the Amercan College of Physcans, [1], reports that prmary care n the U.S. s at grave rsk of collapse due to a dysfunctonal fnancng and delvery system. They also emphasze the growng demand for prmary care; by 2015, an estmated 150 mllon Amercans wll have at least one chronc condton. Tmely access to care and patent-physcan contnuty, the two metrcs mportant to prmary care practces, have been adversely affected due to these broader trends. The focus on tmely access, or the ablty to secure an appontment quckly, s well known n the operatons research lterature. The nablty to get a tmely appontment to a prmary care physcan ncreases the lkelhood of patents vstng the

3 Capacty allocaton and flexblty n prmary care 3 emergency room (ER), [32]. Ths hnders the approprate management of chronc dseases that could have been effectvely treated n a prmary care settng. It also exacerbates the problem of long patent wat tmes n crowded emergency rooms. Patent-physcan contnuty s one of the hallmarks of prmary care and promotes a long-term relatonshp between the patent and the physcan. Numerous studes have documented the mportance of contnuty to patents (see for example [28], [3]). Several studes [9] show that patents who regularly see ther own provders are 1) more satsfed wth ther care; 2) more lkely to take medcatons correctly; 3) more lkely to have problems correctly dentfed by ther physcan; and 4) less lkely to be hosptalzed. The lnk between lack of contnuty and ncreased ER use has been shown n [10]. Contnuty s especally mportant for vulnerable patents wth a complex medcal hstory and mx of medcatons [36] patents wth long standng chronc condtons (dabetes, hypertenson and coronary artery dsease for example). Ths forms a large percentage of the U.S. populaton. Contnuty s also benefcal for physcans, snce ther workloads are more focused [27]. How are tmely access and patent-physcan contnuty related to capacty plannng and allocaton n prmary care? When t comes to access to appontments, the two measures are often n conflct. It may be possble for a patent to get a sameday appontment but not necessarly wth her own physcan. Alternatvely, a patent may get to see her own physcan but only weeks later. The two extremes are llustrated n Fgure. 1. Fgure 1(a) shows the stuaton where patents see only ther own physcan whle n Fgure. 1(b) all physcan resources are pooled. In the former contnuty s perfect but tmely access may be straned, whle the latter results n hgh levels of tmely access but patents may end up seeng unfamlar physcans. The focus of ths chapter s on capacty plannng and allocaton for prmary care practces at varous levels of the plannng herarchy, to balance tmely access and contnuty. For the purposes of ths chapter, the term capacty refers to the number of appontment slots a physcan has n a workday. Capacty allocaton refers to the assgnment of patent requests for an appontment to avalable appontment slots. At the strategc level, we consder the mpact of sze and composton of a physcan s panel on the ablty to provde tmely access and contnuty. In prmary care, a physcan s responsble for the long-term, holstc care of the patents n her panel. The sze and composton of a physcan panel determnes a physcan s daly appontment burden. The management of panels thus determnes capacty plannng at the hghest level. Next, at the tactcal level, we llustrate how a mult-physcan practce can manage the nherent flexblty of prmary care physcans to see patents of other physcans to balance tmely access and contnuty. We study physcan flexblty at the tactcal level where two uncertan demand streams: prescheduled (non-urgent) appontments and open access (same-day) appontments must share the same capacty. Prescheduled appontments are booked n advance of the workday whle open access appontments have to be fulflled the same day. Fnally, at the operatonal level, we also nclude a dscusson of the dynamc context, where decsons about capacty allocaton under flexblty have to be made as same-day

4 4 Balasubramanan et al. requests for appontments arrve over the course of the day, that s, under ncomplete demand nformaton. The remander of ths chapter s organzed as follows. In Secton. 2, we dscuss the lterature relevant to capacty allocaton n prmary care. Secton 3 provdes the background on capacty plannng at the strategc level, the desgn of physcan panels, and quantfes the mpact of case-mx usng an example. In Secton 4, we dscuss flexblty n prmary care and how t dffers from flexblty n manufacturng and other servce contexts. In Secton 5 we provde a modelng framework for testng the mpact of flexblty at the tactcal level where physcan capacty has to accommodate both prescheduled as well as same-day (open access) appontments. We also provde computatonal results. In Secton 6, we provde an outlne of the decsonframework for usng flexblty n the dynamc case (as same-day requests come n over the course of a workday), and dscuss other drectons for future research. We conclude by summarzng our man fndngs n Secton 7. 2 Lterature Revew The applcaton of operatons research to appontment schedulng n healthcare s a growng area of research. We focus here only on the papers most relevant to our research n the prmary care context. For a detaled dscusson of the mpact of prmary care access on populaton health, we pont the reader to key references from the health servces lterature, such as [33], [22] and [21]. Over the last decade the adopton of open access [24], a schedulng polcy whch urges practces to provde same-day appontments rrespectve of the urgency of the request, has brought to the forefront questons regardng appontment system desgn. What should physcan panel szes be to allow open access? What f patents prefer to have appontments at some future tme rather than see a doctor the same day? These questons have necesstated the use of queueng and stochastc optmzaton approaches that provde gudelnes to practces. For nstance, [13] nvestgated the lnk between panel szes and the probablty of overflow or extra work for a physcan under open access. They proposed a smple probablty model that estmates the number of extra appontments that a physcan can be expected to see per day as a functon of her panel sze. The prncpal message of ther work s that for advanced access to work, supply needs to be suffcently hgher than demand to offset the effect of varablty. In [12] a queueng model was used to determne the effect of no-shows on a physcan s panel sze. They developed analytcal queueng expressons that allow the estmaton of physcan backlog as a functon of panel sze and no-show rates. In ther model, no show rates ncrease as the backlog ncreases; ths results n the paradoxcal stuaton where physcans have low utlzaton even though backlogs are hgh ths s because patents that had to wat for long, do not show up. In [15] results of emprcal study of clncs n the Mnneapols metropoltan area that adopted open access are presented. They provded statstcs on call volumes,

5 Capacty allocaton and flexblty n prmary care 5 backlogs, number of vsts wth own physcan (whch measures contnuty) and dscuss optons for ncreasng capacty at the level of the physcan and clnc. In [19] a dscrete event smulaton s used to study the effects of clncal characterstcs n an open access schedulng envronment on varous performance measures such as contnuty and overbookng. One of ther prmary conclusons s that contnuty n care s affected adversely as the fracton of patents usng open access ncreases. The authors mentoned provder groups (or physcans and support staff) workng n teams as a soluton to the problem. [31] compared the performance of open access wth a tradtonal appontment schedulng system. In the open access system, a practce has to deal wth day to day varablty but very few no shows, whle n the tradtonal appontment system, patents book ther appontments well n advance wth the result that day to day varablty s smoothed but patents have a hgher probablty of no-show. Ther numercal analyss reveals the open access generally outperforms the tradtonal appontment system when the objectve functon s a weghted average of patents watng tme (lead tme to appontment, the physcan s dle tme, and the physcans s overtme). Only when the patent watng tme s held n lttle regard or when the probablty of no-show s small does the tradtonal system work better than the open access system. [20] proposed new heurstc polces for dynamc schedulng of patent appontments under no-shows and cancelatons. They fnd that open access works best when patent load s relatvely low. The papers most closely related to the topc of ths Chapter are by [30] and [14]. [30] derved condtons under whch a soluton for the number of prescheduled appontments to reserve s locally optmal. In Secton. 5, we show a stronger result, guaranteeng global optmalty, by frst showng that our revenue maxmzaton functon has dmnshng returns under mld assumptons. [14] explctly modeled many of the key elements of a prmary care clnc. They consdered schedulng the workday of a clnc n the presence of 1) Multple physcans 2) Two types of appontments: same-day as well as non-urgent appontments 3) Patent preferences for a specfc slot n a day and also a preference for physcans. The objectve s to maxmze the clnc s revenue. They use a Markov Decson Process (MDP) model to obtan bookng polces that provde lmts on when to accept or deny requests for appontments from patents. In terms of flexblty, ther clnc s fully flexble wth regard to both non-urgent and urgent appontments. The prncpal dfference between ther model and the capacty allocaton framework proposed n Secton. 5 s that patent preference drves the schedulng of prescheduled appontments n [14], whle we try to balance pre-scheduled demand and same-day demand through physcan flexblty and an explct consderaton of ts effect on tmely access and contnuty.

6 6 Balasubramanan et al. 3 Background on Prmary Care and the Impact of Case-mx 3.1 Patent types At the strategc level of the three part herarchy defned n the Introducton, a physcan bulds a panel of patents. The physcan s appontment burden depends on the 1) sze; and 2) case-mx or composton of the panel. A physcan workng full tme may have patents. Case-mx refers to the type of patents n the panel, and can be characterzed by varous patent attrbutes, such as age and gender and the chronc condtons afflctng the patent, whch play an mportant role n determnng the dstrbuton of vsts. For example, a panel where the majorty of patents are young and healthy wll have a dfferent appontment demand profle compared to a panel consstng mostly of elderly patents wth chronc condtons. Patent classfcaton can be useful for clncs because they enhance a practce s understandng of ts populaton and dsease trends, and allow t to desgn ts care models effectvely. Furthermore, Barbara Starfeld s semnal work about ambulatory care groups (ACGs) [35] argued that understandng the role of patent clncal complexty n care utlzaton forms the cornerstone for effectve resource plannng and determnng payment methods n healthcare. Fg. 2 Hstograms of the percentage (or fracton) of total patents requestng appontments n a week for two dfferent patent age and gender categores. Age and gender s the smplest patent classfcaton n leu of more dseasespecfc data. It has also been found to be generally effectve ( [24], [4]). Fgure 2 shows the dstrbuton of the fracton of total patents requestng appontments n a week for two categores - males (48-53 years old) and women (73-78 years old), based on on hstorcal data from (156 weeks), from the Prmary Care Internal Medcne Practce (PCIM) at Mayo Clnc, Rochester, MN. The two dstrbutons show how appontment request rates can vary wth gender and age. There are 708 males y.o (48-53 M) and 986 females y.o (73-78 F)

7 Capacty allocaton and flexblty n prmary care 7 empanelled n the practce. 8.4% of all F patents request for appontments on average n a week as opposed to 4.8% of all M patents. The standard devaton F (3.07%) s more than double that of M (1.1%). Fg. 3 Mean and standard devaton of yearly vsts for groups wth dfferent counts of comorbdtes In [25] the authors show that more than an ndvdual chronc condton such as dabetes or hypertenson, t s the number of smultaneous chronc condtons (or comorbdtes) that predcts the consumpton of healthcare costs. Furthermore, for prmary care physcans focusng on all comorbdtes of a patent s more holstc than focusng n solaton on specfc chronc condtons. Fgure 3 shows mean and standard devaton of vst rates as a functon of the number of patents under varous counts of comorbdtes. The data was smulated based on hstorcal vsts of 20,000 patents empanelled n the PCIM practce [29]. Clearly, not only does the mean number of vsts ncrease wth the number of comorbdtes, the varance does as well. For nstance f a physcan has 50 6-comorbdty patents then she wll have 450 appontment requests on average each year. If she has the same number of 0-comorbdty patents she wll have only 75 yearly vsts on average. Table 1 Four physcans at PCIM, Mayo Clnc and ther patent case-mx based on comorbdty count. Physcans Panel sze Physcan Physcan Physcan Physcan

8 8 Balasubramanan et al. 3.2 Example: Four physcans We now consder an example of four physcans wth approxmately the same panel sze (1050 patents), but dfferent case-mxes, based on comorbdty counts. These panel compostons are shown n Table 1. The case-mx can be used to smulate the dstrbuton of daly vsts for each physcan, by samplng for each comorbdty count from hstorcal data. Once the daly vst dstrbuton s obtaned, the overflow for a gven daly appontment capacty can be calculated. Overflow s smply the fracton of total samples n whch the patents vst requests exceed the avalable capacty of the physcan. Patents that are not seen ether vst an unfamlar physcan or an ER, or may choose to wat to see the physcan on another day. Thus, f overflow s hgh, both tmely access and contnuty are adversely affected. Overflow can also be modeled n the followng way. Consder a practce n whch there are J physcans and M patent classes. Frst a practce determnes p, the probablty that a patent of class = 1...M wll request an appontment on any gven day. Ths can be obtaned by calculatng the total vsts generated by all patents of the class n the practce over a perod of tme for example two years and dvdng t by the number of unque class patents as well as the number of workdays n the two year perod. The method s smlar to the one proposed n [13]. Next, suppose n j denotes the number of class patents n physcan j s panel. If we assume that each patent requests ndependently of others, then the total requests from each patent class for each physcan can be modeled as a bnomal random varable, wth mean n j p and varance n j p (1 p ). Gong further, the mean demand for the entre panel s gven by µ j = M =1 p n j and standard devaton σ j = M =1 p (1 p )n j. We wll use the normal approxmaton of a sum of bnomal random varables. Then O j, the overflow for physcan j s related to the percentle of the standard normal dstrbuton, denoted by Φ, n the followng way: O j = 1 Φ( C j µ j σ j ). Here C j s the capacty of the physcan, that s the total daly slots that she has avalable n a day. Note that ths analyss s at the aggregate level t does not consder the actual duraton of appontments once patents are n the clnc, but tests whether the number of appontment slots (typcally 20-mnute slots) a physcan plans to have avalable n a day s suffcent. It also assumes that all appontments are of the same type. In realty, some appontment requests (such as follow-up appontments) are for a future day, whle some are same day requests. Nevertheless, f overflow s hgh for all appontments, then t s guaranteed that the tmely access for both same-day as well as non-urgent future appontments wth one s own PCP wll be adversely affected. The overflow for the four physcans of Table 1 as a functon of the total capacty (daly appontment slots) s shown n Fg. 4. We calculated these overflow profles usng the bnomal approxmaton descrbed above, but t s also possble obtan the same curves by samplng from hstorcal vst data. For the same capacty, Physcan 3 and Physcan 1 have relatvely hgh levels of overflow. Ths s because there are more patents wth two or more comorbdtes n ther panels (see Table 1), and

9 Capacty allocaton and flexblty n prmary care 9 Fg. 4 Overflow for the physcans as a functon of the daly capacty (appontment slots) these patent groups generate a hgher number of vsts. Ths graph shows that t s napproprate for clncs to make capacty decsons based solely on panel sze. Case-mx s also an mportant consderaton. It s also clear that to keep overflow levels down to manageable levels, 20 or more appontment slots may be needed for each of the 4 physcans. Such analyss allows practces to dentfy whch physcans are overburdened. In the above case, t s clear that physcans 3 and 1 need to have ther capacty enhanced ether by workng extra hours n a day, or by addtonal nurse practtoner support, or a reducton n the sze of ther panels. The long-term opton for practces s to acheve a better balance among physcans, by movng hgh demand, hgh varablty patents from an overburdened physcan to a physcan wth avalable capacty. More detals about the panel redesgn approach are presented n [4]. The paper shows that t s possble to reduce the wat tme and the number of redrectons to unfamlar physcans by more than 35%. The dffcultes of redesgnng panels are also dscussed n [4]. Reallocatng patents abruptly could damage exstng patent-physcan relatonshps. Rather the approprate strategy would be to redesgn panels when opportuntes present themselves. Many prmary care panels are dynamc. Patents enter and leave them all the tme as they age, are dagnosed wth new condtons, move out of the geographc area and many other reasons. A useful by-product of ths constant state of flux s that t affords contnuous opportuntes to make ncremental changes to patent panels wthout dsruptng the vst patterns of patents who already have strong tes to ther PCP. For example, practces can leverage patents who have yet to decde on a PCP, new patents, and the turnover of exstng patents. Patent surveys can be used to determne preferences and nclnaton towards change. In some cases, to mnmze dsrupton, reassgnment may smply be to another physcan, whom the patent has seen almost as often as her own PCP, or to a physcan wthn the same care team (f the care team conssts of multple physcans).

10 10 Balasubramanan et al. Another vable alternatve to panel redesgn s carefully managng physcans ablty to see patents of other physcans, dependng on whether the requests are urgent/same-day requests or non-urgent requests. Ths management of physcan flexblty forms the content of the next two sectons. 4 Flexblty n Prmary Care The nherent flexblty of PCPs to see patents from other panels gves practces another lever to provde tmely access to care. Usng ths flexblty, of course, comes at a cost: the resultng loss of contnuty when a patent sees unfamlar physcans. How should practces be desgned and managed to use ths flexblty to better balance tmely access and contnuty? Fg. 5 Fgure llustratng dfferent flexblty confguratons that tradeoff contnuty of care wth tmely access. A practce can acheve maxmum contnuty of care by mandatng that patents should see only ther own provder. Ths, however, hampers tmely access to care. At the other extreme, a practce may allow patents to see any provder. Ths s deal for tmely access, but hampers contnuty of care. The two extremes are shown n Fgure 5(a) and (b). In the frst case, the provders are dedcated whle n the second the provders are fully flexble. Fgure 5(c), (d) and (e) show partally flexble confguratons that offer a mddle ground between (a) and (b). In each of them, a patent sees only one physcan other than her own PCP. Fgure 5 (c) s referred to as the 2-chan n the manufacturng flexblty lterature [Jordan and Graves, 1995] and allows demand varaton to be absorbed effectvely by the entre practce. In a 2-chan all physcans are drectly or ndrectly lnked to each other and ths property can be exploted to cope wth varablty n demand. If demand s hgh for a partcular physcan, the allocatons can be desgned such so that the excess demand s shfted

11 Capacty allocaton and flexblty n prmary care 11 to a physcan who has capacty avalable, even though the latter physcan may not be drectly lnked to the former. Whle the 2-chan s a concept new to healthcare, practces do use the subgroup confguraton (Fgure 5(d)). Physcans here may be dvded nto ndependent, self-contaned teams (such as n the PCIM practce at Mayo Clnc and other academc prmary care practces). The dedcated wth overflow confguraton of Fgure 5 (e) s also common - here f the patent s PCP s unavalable, the patents tend to see an overflow physcan or nurse practtoner (we have observed ths settng at a small prvate practce as well as a communty clnc n Western Massachusetts; academc medcal centers also use ths model). In the deal open access world, there are no appontment types, such as urgent and non-urgent. All appontments are treated dentcally and scheduled the same day wth the patent s PCP. However, the realty s that clncs have a fracton of ther schedule avalable for open access or urgent appontments. Such appontments, because of the perceved mmedacy of need, are typcally seen the same day, but not always by the patent s personal physcan. The rest of the clnc s schedule conssts of appontments booked a week or more n advance. We call these appontments prescheduled appontments. These non-urgent appontments are typcally physcals or follow-up appontments for patents wth chronc condtons. Whle the loss of contnuty has to be mnmzed for all appontments, t can be approprately sacrfced for urgent appontments needng mmedate attenton by ntroducng some form of flexblty. We wll thus assume that flexblty apples only to urgent appontments. Non-urgent or prescheduled appontments are always seen by the patent s own provder. We approach the desgn and management of the flexble practce at two dfferent levels. At the tactcal level n prmary care, the desgn of flexblty needs to consder capacty allocaton for the two streams of uncertan demand, non- urgent (prescheduled) and urgent (open access), each wth dfferent requrements for tmelness and contnuty; whle at the operatonal or dynamc level, at whch same-day, urgent appontments are booked as patents call over the day, allocaton decsons have to be made n real tme wthout full knowledge of demand. In the remander of ths secton, we frst summarze the man lessons from the flexblty lterature as they apply to the prmary care context, and then consder the tactcal and operatonal cases n detal. 4.1 Flexblty Lterature Perspectve Our study of flexblty n prmary care practces bulds upon the extensve lterature on manufacturng flexblty and ts more recent applcaton to servce systems and worker tranng and allocaton. There are, however, key operatonal dfferences that make the applcaton of flexblty to prmary care worthy of further analyss: (1) two demand streams assocated wth each resource, where one (prescheduled demand) s realzed before the other (open access demand); (2) two conflctng objectves, tmelness and contnuty of care; (3) no fxed cost assocated wth nstallng flex-

12 12 Balasubramanan et al. blty, but a loss n contnuty for usng t; (4) appontments are booked over tme and thus future resource capacty s sequentally beng allocated under partal demand nformaton. As n the case of cross-tranng n seral producton lnes [16], flexblty mproves effcency n two man ways n the prmary care envronment. The frst beneft s n what they refer to as capacty balancng: If physcan panels are mbalanced wth respect to the nduced average number of vsts to a physcan per day, flexblty wll allow the load to be shared between physcans, therefore mprovng overall tmelness of care and physcan utlzaton. The second s n varablty bufferng: Even f the average workloads are balanced between physcans, varablty n patent requests for a partcular day/tme wll be better accommodated by a flexble envronment. Reference [16] compares a strategy that balances capacty usng the mnmum amount of cross-tranng wth the channg of sklls n the sequence of the seral lne. They fnd that skll-channg strateges are more robust, and more effectve n varablty bufferng. The concept of channg has receved much attenton snce t was frst ntroduced n the semnal work of [18]. In a sngle-perod, mult-product, mult-plant producton network, they show that the 2-chan (Fgure 2(c)) results n ncreased sales and capacty utlzaton, relatve to the dedcated confguraton (Fgure 2(a)), comparable to those acheved by a fully flexble system (Fgure 2(b)). That s, a few lnks, confgured n the rght way (2-chan) provde almost the same performance as the complete, fully flexble network. Furthermore, ths strategc analyss has been extended recently to mult-stage supply chans [11], and to a make-to-order envronment where flexblty s also used to hedge aganst operatonal varablty [23]. Reference [8] dstngushes between range (the dfferent demand scenaros that can be accommodated) and response (the cost of dong so; that s, the cost of usng secondary rather than prmary resources for producton/servce) of flexble systems. They show that upgradng system response (.e., buldng systems where physcans can handle other physcan s panels at lower addtonal cost) outperforms mprovng system range (creatng systems that can accommodate ever more extreme patent demand scenaros). Ths result suggests that n the prmary care settng, the benefts of restrctng the number of doctors that can see a partcular patent (resultng n lower cost of servce because of famlarty and thus ncreased response) s lkely to outwegh the hgher range provded by a fully flexble team care practce where any doctor can see the patent. A number of computatonal reports n the lterature (e.g. [18]) pont out an ncrease n the margnal beneft assocated wth addng one more flexblty lnk (.e. allowng one more panel to see a second physcan) n formng the 2-chan, culmnatng wth a markedly hgher ncrease when the last lnk that closes the chan s put n place. Recently, [34] prove that that s always the case and show that long chans are always superor to any other strategy where each product (panel) can be produced at two plants (can be assgned to two physcans). Ths suggests that the larger practces wll beneft most by managng ther nherent flexblty n the form of a long chan.

13 Capacty allocaton and flexblty n prmary care 13 5 Example of Flexblty n Prmary Care In ths secton we provde a specfc example of a model for evaluatng the nfluence of flexblty n a prmary care practce. We focus at the tactcal level. We provde theoretcal results relevant to the model and numercal experments that llustrate the relatve benefts of flexblty. The basc queston addressed by the model s: How much of the physcan s total daly workload should be dedcated to prescheduled versus urgent appontments? Ths wll depend on how much flexblty the practce allows when allocatng urgent patent demand. We thus need to address ths queston under dfferent flexblty confguratons as llustrated n Fgure 2. Ths wll also allow us to compare ther resultng performance n terms of system revenue, contnuty and tmely access. For that purpose, we develop a two-stage stochastc nteger program that can accommodate any flexblty confguraton, and greedy, but exact, algorthms to quckly calculate the optmal capacty allocatons n dedcated and fully flexble systems. The analytcal and expermental results and conclusons summarzed here are from [4]. Two-stage capacty allocaton model: We solve the capacty allocaton problem for a sngle workday usng a two-stage stochastc nteger programmng model. We consder a general prmary care practce wth M physcans, ndexed by = 1,2,...,M, each wth N avalable appontment slots. Let A be the set of all possble panel-physcan lnks (, j) such that the open access (same-day) requests of patents n panel (.e., physcan s panel) can be served by physcan j. The set A represents the partcular flexblty confguraton under consderaton. Let R p be the revenue assocated wth physcan seeng one of hs pre-scheduled patents, and R o j be the revenue assocated wth physcan j seeng an open-access patent of panel, for any (, j) A. The demand for prescheduled and open access appontments s represented by a random vector D = (D p 1,Do 1,...,Dp M,Do M ) where the superscrpt p refers to prescheduled and o to open access, and the subscrpt ndcates the prmary care physcan. Vector D follows a dscrete dstrbuton that assgns a probablty q s to each possble realzaton of demand, ndexed by s, s = 1,2,...,S. That s, P[D = (d p 1s,d 1s) o,...,d p Ms,do Ms )] = q s. We ntroduce the followng capacty allocaton varables: N p : Number of slots allocated for pre-scheduled demand of physcan. : Number of patents allocated to physcan under demand realzaton s x p s x o js : Number of open access patents of panel assgned to physcan j under demand realzaton s, for all = 1,2,...,M and (, j) A. The objectve s to maxmze the expected revenue of satsfyng prescheduled and open access appontments. We use bnary decson varables to capture whether the prescheduled demand for a physcan s less or greater than the correspondng N p value. Inequalty 3 ensures that φ us = 1 f d p s < N p. Inequalty 4 ensures that φ us = 0 f d p s > N p. Equatons 5 and 6 lmt the number of pre-scheduled appontments to the allocated capacty and the realzed demand, respectvely. Inequaltes 7 and 8 ensure that the total open access appontments for any physcan j do not

14 14 Balasubramanan et al. exceed remanng capacty, when φ us = 1 and φ us = 0 respectvely. Inequalty 9 lmts the total number of open access appontments scheduled from a panel to the realzed demand for such appontments from that panel. Inequalty 10 s the bnary constrant. (SIP) max{ S m s=1 =1 q s {R p xp s + {, j} A R o jx o js}} (1) :(, j) A N p N (2) d p s + N φ us, (,s) (3) d p s φ u s, (,s) (4) N p N p x p s N p (,s) (5) x p s d p s (,s) (6) x o js N j d p js φ ju js ( j,s) (7) x o js N j N p j + φ ju js N j ( j,s) (8) :(, j) A :(, j) A x o js ds, o (,s) (9) φ us (0,1),, u s = 0,1,...,N (10) N p,xp s,xo js 0, (, j) : (, j) A, s (11) We note n the above revenue optmzaton that R o j > Rp j. Ths s because n a relatve sense, open access appontments are more valuable than prescheduled appontments, as explaned n [6]. Frst, we note that open access appontments, because they have such short lead tmes, tend to have much lower no-show rates. Second, f a prescheduled appontment results n a no-show, t can be substtuted by an open access appontment, whle the reverse s not possble at such a short notce. Thrd, because prescheduled appontments are made generally a week or more n advance, the patent s lkely to be flexble about choce of the appontment day, and thus ths may result n postponed but not lost demand f dened tmely access. An open access patent, on the other hand, needs to see a physcan mmedately and hence s flexble n provder choce. In the next two sectons, we present analytcal solutons to the capacty allocaton problem for dedcated practces, where physcans can only see patents n ther own panel, and fully flexble practces where open-access patents can be seen by any of the physcans n the practce. For large practces usng partal flexblty such as the 2-chan confguraton, unfortunately, the above stochastc program s too large to solve effcently n practce. Whle the number of bnary and nteger varables s qute manageable, the sheer number of possble demand realzatons makes the

15 Capacty allocaton and flexblty n prmary care 15 problem ntractable. To overcome ths ssue, we wll solve the problem usng a computatonally effectve sample average approxmaton method proposed by [36] for two-stage stochastc nteger programmng problems (see Secton 5.1). The Dedcated Case: In a dedcated practce, physcans can only serve the prescheduled and open access patents from ther own panel. They need to decde, however, a maxmum number of appontment slots to make avalable to prescheduled patents, N p, so that enough capacty s reserved for the more lucratve open access ones. The system confguraton s shown n Fgure 6. Fg. 6 System confguraton for a dedcated practce. Let E[R (N p )] be the total expected revenue from the panel of physcan, as a functon of N p 0,1,2,...,N. Our goal s to fnd the optmal value of N p. The condtons for local optmalty presented n [30] for the problem of maxmzng the expected number of patents consulted n a sngle-physcan practce can be easly adapted to our revenue maxmzng objectve. In [6], we show a stronger result, guaranteeng global optmalty, by frst showng that the objectve functon has dmnshng returns under mld assumptons. Proposton 1. If P[D o N N p Dp N p p ] s non-decreasng n N the dfference n revenue assocated wth ncreasng the number of prescheduled slots by one, E[R (N p + 1)] E[R (N p p )], s non-ncreasng n N The above condton holds when the demand for prescheduled and open-access appontments are ndependent of each other. Furthermore, t wll be satsfed n most practcal scenaros. Intutvely, for t to be volated, the probablty of open access demand beng large would need to sgnfcantly decrease as the demand for prescheduled appontments grows; that s, the demand for open access and prescheduled appontments would need to be heavly negatvely correlated. As a result of Proposton 1, we have that the expected revenue functon exhbts dmnshng returns, an analog of concavty for a dscrete functon, and thus ts global maxmum must occur at the largest nteger N p N such that E[R (N p )] E[R (N p 1)] 0 leadng to the followng theorem. Theorem 1. If P[D o N N p Dp N p p ] s non-decreasng n N, the optmal soluton to the Dedcated Problem s the largest non-negatve nteger N p N such that P[D o N N p Dp N p ] Rp /Ro.

16 16 Balasubramanan et al. The optmal soluton can thus be easly obtaned by calculatng that probablty startng at N p = 0 and ncreasng one unt at a tme untl t exceeds the threshold R p /Ro. A bnary search could also be used. Observe that n the case of ndependent open-access and prescheduled demands, the optmal value of N p does not depend on the dstrbuton of pre-scheduled demand for physcan. The Fully Flexble Case: In a fully flexble practce, open access patents can be seen by any avalable physcan. In ths case, the optmal number of slots to make avalable to prescheduled demand of the physcans, N p 1 and N p 2 n the case of two physcans, can stll be found wth a smple greedy algorthm. Ths s because the revenue functon agan exhbts dmnshng returns as the number of slots offered to prescheduled patents s ncreased. For ease of exposton, we assume that all physcans have the same capacty of N slots, and that the revenue of an open access appontment s dentcal for all physcans and panels and denoted by R o. We frst consder the case of two physcans, and j. See Fg. 7. Fg. 7 System confguraton for a fully flexble practce. Proposton 2. If P[D o + Do p j > 2N (N + mn(d p j,np j ) + 1) Dp N p + 1] s nondecreasng n N p and N p j, the dfference n revenue assocated wth ncreasng the number of prescheduled slots of physcan by one, E[R (N p +1,N p j ]) E[R (N p,np j ]), s non-ncreasng n N p and N p j. Observe that, as n the dedcated case, the condtons wll hold when open access and prescheduled demands are ndependent, and n any practcal scenaro except for contrved cases where the demands for prescheduled and open access appontments are sgnfcantly negatvely correlated. Snce the revenue functon exhbts decreasng returns n both N p and N p j under those mld condtons, whch can be nterpreted

17 Capacty allocaton and flexblty n prmary care 17 as concavty of the dscrete revenue functon, a greedy algorthm that keeps ncreasng one appontment slot at a tme to the physcan where t produces the hghest system revenue wll provde the optmal capacty allocaton scheme. Proposton 2, and therefore the optmalty of a greedy algorthm, can be easly extended to the general case of M physcans that fully share open access demand. Full detals of the theorems and the proofs are gven n [6]. 5.1 Computatonal Experments The exact greedy algorthms allow us to fnd the optmal capacty allocaton and system revenues for dedcated and fully flexble practces. To test the performance of partal flexblty confguratons (see Fgure 5), whch promote contnuty by restrctng the number of doctors that a patent can be assgned to, we use the two stage stochastc nteger program (SIP). In what follows we present a summary of the results emphaszng (1) the value of the 2-chan to mprove open-access whle keepng acceptable levels of contnuty, and (2) how the optmal porton of clnc capacty reserved for open access changes as more flexblty s allowed when allocatng open access demand; for full detals, please see [6]. Value of Partal Flexblty: Followng the fndngs of [7], we assume a typcal no show rate for pre-scheduled demand of 25%, and a 10% no show rate for open access demand. Thus, an appontment slot gven to an open access patent brngs hgher expected revenue, 0.9, as compared to revenue of 0.75 for schedulng one pre-scheduled patent. To encourage contnuty n the system, we assume that there s a 0.05 cost of seeng patents from another physcan s panel (the revenue of gvng an appontment slot to one open access patent not from a physcan s panel s therefore = 0.85). Whle the no-show rates for the two types of appontments can be estmated from past data, the cost of dvertng an open access patent to a non-pcp physcan s very dffcult to quantfy. Furthermore, n a lmted flexblty envronment, where the patent only sees at most one physcan beyond her PCP, the actual cost of redrecton s mnmal, very dfferent from that occurrng n a large, fully flexble practce where care s sgnfcantly more fragmented and much harder to coordnate. For that reason, rather than comparng the expected revenues obtaned under the dfferent confguratons, we focus here on the resultng tmely access rates (TAR). We defne TAR as the percentage of all patents, both prescheduled and open access, who get access to an appontment on the gven workday. Fgure 8 shows the gans, relatve to a dedcated practce, of mplementng partal flexblty (confgured as a 2-chan) and full flexblty to share open access demand as system workload ncreases n practces wth 3 and 6 physcans. Workload s defned as the rato of the expected total demand for the clnc and total avalable capacty. Each physcan has 24 appontment slots avalable n the day. The left graph, or symmetrc case, nvolves a practce where all physcans face dentcal

18 18 Balasubramanan et al. Fg. 8 Comparson of tmely access rate (TAR) mprovement between 3 and 6 physcans n the symmetrc (left fgure) and asymmetrc (rght fgure) cases panel demand dstrbutons (Posson demands wth a rate of 10 for prescheduled appontments and 14 for open access demand). The rght graph, or asymmetrc case, consders a practce where physcans have varyng panel compostons and therefore varyng appontment burdens. Ths s common n practce. Senor and well establshed physcans may have hgher workloads snce ther panels are larger and nclude older, more complex patents, whle physcans who have been recently hred may have lower workloads. In partcular, we test a practce where: Physcan 1 has an expected prescheduled demand of 6 and an expected open access demand of 12 (low workload); Physcan 2 has an expected prescheduled demand of 8 and an expected open access demand of 16 (balanced or full workload); Physcan 3 has an expected prescheduled demand of 10 and an expected open access demand of 20 (hgh workload). For the sx physcan case, we merely double the 3 physcan case, thus retanng the mbalances. The tmely access rates of 2-chan flexblty and full flexblty are nearly the same no matter what the sze and workload level of the system are. Ths s consstent wth the results reported n the lterature on flexblty n manufacturng settngs. The dfference s even lower n the healthcare settng that forms our test case, snce we assume that prescheduled demand cannot be shared between physcans; flexblty can only be used for open access demand. We also observe, as n the manufacturng lterature, that the gans accrued through flexblty ncrease sgnfcantly as: (1) the number of physcans ncreases from 3 to 6; and (2) the physcans have dfferent workloads,.e. n the asymmetrc case, when flexblty helps not only to accommodate demand varablty but also to balance physcan workloads. These results suggest that flexblty provdes an mportant lever for practces to ncrease ther ablty to accommodate open access demand. Furthermore, the 2- chan confguraton allows them to do so wthout severely compromsng contnuty

19 Capacty allocaton and flexblty n prmary care 19 and patent/physcan bonds. Capacty Allocaton: The results above llustrate the value of flexblty. But how are capacty allocaton decsons affected by the flexblty confguraton used? What trends do they follow, f at all, and can the trends provde clues to capacty allocaton decsons n practce? In our model, the capacty allocaton s decded wth the optmal frst stage varables, N p, whch represent the capacty made avalable to prescheduled appontments. Fgure 12 shows the average values for the entre clnc (that s for all the physcans) under dfferent workloads and the three flexblty confguratons for the 6 physcan asymmetrc case. We see the same trends by lookng at the ndvdual physcans values (rrespectve of the number of physcans, symmetry and prescheduled to open access demand ratos). Thus the Fg. 9 summarzes our conclusons about N p values concsely. Fg. 9 Trends n N p values summed over all 6 physcans wth Prescheduled demands [6,8,10,6,8,10] and open access demands [12,16,20,12,16,20] In general, for the case of very low system workload, the total N p values for the dedcated and flexblty confguratons, not surprsngly, are very close. Snce the demands are so low, the values are lkely to be farly robust at ths level. As the system or clnc workload ncreases to 80% and 100%, the clnc as a whole reserves more prescheduled appontments n the flexblty cases than the dedcated case. Ths s a drect consequence of flexblty: open access appontments can be absorbed effectvely by poolng the (lower) remanng capacty of all physcans together. The effect s especally strong n the case of 100% workload: the dedcated case ncreases the capacty reserved for the more proftable and now more abundant open access patents (N N p ) relatve to the lower workload cases, whle the flexble confguratons decrease t to allow for more of the now plentful prescheduled patents and stll meet open access demand through sharng any unused capacty. In contrast, n the hgh system workload cases (120% and 140%), there s enough demand for the hgh revenue open access appontments to lower the total of the clnc. The flexblty cases have a lower total N p value than the dedcated case,

20 20 Balasubramanan et al. reservng more capacty for open access. Ths s because there s a hgher probablty of usng the addtonal capacty when physcans are able to see each others open access appontments. Thus, usng the easly computable dedcated case N p as a reference, practces can heurstcally determne ther capacty allocaton to be above or below the dedcated value, dependng on ther flexblty confguraton and overall system workload. 6 Future Research Opportuntes In ths secton, we frst outlne the decson-framework for usng flexblty n the dynamc case, report our prelmnary fndngs, and then dscuss other drectons for future work. 6.1 Flexblty n the Dynamc Case The dscusson of flexblty so far assumed that demands are nstantly realzed and fulflled. In practce, however, allocaton decsons for open access and same-day appontments have to be made wthout full realzaton of demand. At the begnnng of the day all non-urgent appontments scheduled on physcan calendars are known n advance, but calls for same-day appontments come throughout the day and have to be dynamcally assgned to avalable physcan slots. As before, the challenge s to balance tmely access (mnmze the number of dened same-day appontment requests) wth contnuty (ensurng patents see ther own physcan as much as possble). Fg. 10 Vsual llustraton of the model at decson epoch t. The dark slots correspond to prescheduled appontments; the gray slots to already booked same-day appontments. Slots that are not shaded are avalable to be assgned.

21 Capacty allocaton and flexblty n prmary care 21 Consder agan a clnc wth M physcans, each wth a panel of patents for whch he/she s the prmary care physcan (PCP). Dependng on the flexblty confguraton patents wll be allowed to see one or more physcans. The tme horzon begns when the clnc opens ts open access appontments, and ends when the clnc stops takng further appontments. Ths wll typcally mean the entre duraton of a day (7-8 hours). Calls for a gven physcan s slots come wth a certan probablty (p for physcan ) at every tme pont durng the horzon. Each physcan s calendar conssts of successve 20-mnute slots that are booked as calls come n durng the day. In the deal stuaton, all open access appontments are contguous and occur durng the same tme of the workday. In practce, because of patent preferences for tme slots, avalable same-day slots wll be nterspersed wth prescheduled slots. The stuaton s shown n Fg. 10. The decson framework can be modeled n a fnte horzon stochastc dynamc program. For the mathematcal detals see Hppchen (2009) [17]. At each tme pont t, f there s a request from physcan panel, the decson facng the clnc s whether ths request should be 1) assgned to her PCP; 2) to some other physcan (as allowed by the flexblty confguraton) or 3) dened. If there are no requests at tme t for any of the physcans then the acton s to do nothng. The optmzaton problem s to choose the best acton at each decson epoch to mnmze total cost of dened requests and mssed contnuty (measured by the number of non-pcp dversons) for the day. The state of the system at any decson epoch t s represented by the number of open access patents booked n the future n each physcan s calendar. Denyng a request ncurs a cost dened requests are a reflecton of the lack of tmely access to prmary care, or the costs needed to provde care to these patents outsde the regular hours of clnc operatons. What mpact does flexblty have n the operatonal or dynamc case? Recall that n the tactcal case, flexblty was benefcal n hedgng aganst varablty of demand. In the dynamc case, there s an addtonal component of varablty, snce appontment requests arrve randomly over tme. There s therefore greater opportunty for flexblty to meet demand mbalances at dfferent ponts n tme. On the other hand, patent calls requre an mmedate appontment allocaton decson, under only partal demand nformaton avalable at that pont; ths decreases the mpact of flexblty, snce allocaton decsons that can be made optmally n the tactcal case may not be as effectve n the dynamc case. These counteractng effects may be the reason why the benefts of flexblty are mostly dentcal n both the tactcal and dynamc cases. Our computatonal experence wth the stochastc dynamc program [17] shows that the benefts of full and partal flexblty n the dynamc case produce the same percentage mprovements n tmely access rate shown n Fgure 8. Whle the stochastc dynamc program has been used llustrate the mpact of flexblty, prmary care offces requre easly mplementable polces or heurstcs that can be put nto practce as calls for same-day appontments come n. Consder two contrastng polces n the fully flexble case: Prmary Frst (PF) and Most Slots (MS). PF assgns ncomng same-day calls to the patent s PCP frst, so long as slots are avalable. If PCP slots are not avalable for the day, t assgns the patent to the