Dynamic Scheduling of Emergency Department Resources

Transcription

1 Dynamc Schedulng of Emergency Department Resources Junchao Xao Laboratory for Internet Software Technologes, Insttute of Software, Chnese Academy of Scences P.O.Box 8718, No. 4 South Fourth Street, Zhong Guan Cun Bejng , Chna [email protected] Leon J. Osterwel Department of Computer Scence Unversty of Massachusetts Amherst, MA USA [email protected] Qng Wang Laboratory for Internet Software Technologes, Insttute of Software, Chnese Academy of Scences P.O.BOX8718, No.4 South Fourth Street, Zhong Guan Cun Bejng , Chna [email protected] ABSTRACT The processes carred out n a hosptal emergency department can be thought of as structures of actvtes that requre resources n order to execute. Costs are reduced when resource levels are kept low, but ths can lead to competton for resources and poor system performance. Careful allocaton can mprove performance by enablng more effcent use of resources. Ths paper proposes that resource schedulng be done n a seres of dynamc reschedulngs that use precse, detaled nformaton about emergency department processes and avalable department resources to mprove the qualty of schedulng results. Reschedulng s done over a small set of actvtes, and uses a genetc algorthm. Smulatons are used to evaluate ths approach, and results ndcate that t can be effectve. Categores and Subject Descrptors D..9 [Management]: Software process models; I..8 [Problem Solvng, Control Methods, and Search]: Schedulng General Terms Algorthms, Management, Performance, Human Factors. Keywords Incremental resource schedulng, genetc algorthm, process smulaton, healthcare process analyss 1. Introducton The processes used to delver care n hosptal emergency departments are very complex, but are of central mportance. Such systems are typcally comprsed of a group of actvtes, each of whose executons requres dfferent enttes that may be humans (e.g. doctors), equpment (e.g. MRI devces), or software (e.g. electronc patent records). In ths work we refer to any and all such enttes that are needed n order to enable the performance of an actvty as the actvty s resources. Because resource avalablty s usually lmted, resource contenton problems often arse durng process executon, sometmes leadng to delays and neffcences. Thus, for example, a doctor may be needed to treat a low acuty Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. IHI 10, November 11-1, 010, Arlngton, Vrgna, USA. Copyrght 010 ACM /10/11...$ patent mmedately, but wll very shortly also be needed to treat a patent that s n urgent need of care. Assgnment of the doctor to the patent wth mmedate needs mght delay or deprve the patent havng urgent needs of tmely care. Careful resource schedulng can help to mtgate the negatve effects of such nevtable contenton, and can reduce delays, neffcences, and patent watng tme [30]. In a typcal hosptal resource schedulng s done nformally by humans, and there s consderable evdence that t s often done very poorly resultng n neffcences and delays that can cause sufferng, needless cost, and even death. Accordngly there s nterest n explotng resource schedulng research that has been appled n other domans. Ths work has focused on determnng optmal schedules of assgnment of resources to actvtes. One approach s statc resource schedulng, n whch a complete schedule of resource assgnment s computed n advance based on advance knowledge of the sequence of actvtes to be performed and the sze and duraton of all these actvtes [6, 8, 1]. However, a hosptal emergency department s a dynamc place, wth great uncertanty about the future course of the executon of any realstc process. Uncertantes such as the sudden arrval of new patents, unexpectedly slow task performance, and unplanned lack of resources [11, 0] all change the executon envronment creatng the potental for consequent schedule dsruptons [1]. Because of the nevtablty of such uncertantes n the emergency department, dfferent knds of dynamc resource schedulng approaches, such as reactve schedulng, and proactve schedulng need to be consdered [1]. These methods seek to schedule only actvtes that are wthn a restrcted part or phase of system executon. They address only a reduced set of actvtes usng extensve or exhaustve searchng approaches to compute optmal or near-optmal schedules. But the scale of the schedulng effort can stll be qute large f the schedule covers an extensve part of the system s actvtes. In addton, dsruptve events may stll nvaldate the assumptons of the schedulng effort, necesstatng further reschedulng (ths s especally problematc as the part of the system beng scheduled becomes large). These ssues are partcularly troublesome n healthcare, where patent care systems must contnually adapt n response, for example, to new patent arrvals and medcal emergences. Ths ndcates the need to fnd new ways to mtgate the problems nherent n ncremental reschedulng. Our approach explots detaled specfcatons of emergency department actvtes, ther needs for resources, and the characterstcs of the resources themselves to acheve better resource schedulng. We decompose the overall resource schedulng problem nto a seres of dynamc reschedulngs 590

2 at selected tmes, coverng sets of actvtes for whch access to detaled nformaton could be the bass for more effectve resource schedules. To pursue ths we have explored: (1) Usng very complete and precse nformaton about emergency department process actvtes and resources. Ths should enable schedulng schemes to produce hgh qualty results that should reman accurate over most or all of the actvtes for whch resources have been scheduled. () Keepng the actvty set for whch resources are to be scheduled relatvely small thereby keepng analyss costs relatvely modest and enablng relatvely quck response to changng emergency department envronment condtons. (3) Enablng dynamc changes n successve reschedulngs. Earler resource allocaton decsons and unexpected events can alter the choce and mportance of later actvtes, affectng how resources mght be allocated to them. Thus we use schedulng parameters (e.g. constrant sets) that may vary to make t easer to compensate for the effects that prevous actvtes have on resource allocaton for upcomng actvtes. Ths paper explores these approaches by proposng a tme wndow based ncremental resource schedulng method. In ths method, resource schedulng and reschedulng s performed ncrementally at selected ponts durng system executon. Our approach reles upon detaled specfcatons of both system actvtes and resources provded by well-defned languages capable of supportng specfcatons that are both very precse and very detaled. Ths causes the characterstcs and behavors of the actvtes n the wndow, and the resources allocated to those actvtes, to be relatvely predctable. Our expectaton s that ths should help us generate very hgh qualty results. Though relatvely small, our reschedulng wndows wll stll contan quanttes of actvtes and resources that are suffcently large to requre consderable schedulng computaton. Thus we use a genetc algorthm (GA) [13] n our schedulng approach. GA algorthms are fast and can also readly ncorporate constrants nto the defnton and soluton of the schedulng problem. We acknowledge that actual deployment of our schedulng system wll pose addtonal challenges, such as assurng that computatons are completed quckly enough that they do not slow the fast pace of an emergency department, and communcatng schedulng nformaton to the rght people at the rght tme. Before addressng these challenges we elected frst to determne whether the basc approaches and algorthms showed promse of beng effectve. Thus, ths approach was evaluated by runnng smulatons of processes that are representatve of some of the ways that emergency department resources are deployed and used. The smulatons used dfferent detals of processes and resources, dfferent constrants, and dfferent GA parameters to compute dfferent resource allocatons. The results obtaned suggest that ths approach shows promse of beng effectve n actual use. The paper s organzed as follows. Secton descrbes some related work. Secton 3 presents our tme-wndow based ncremental schedulng method. Secton 4 presents some detals of the components and technologes used. Secton 5 descrbes a smulaton of a process n the doman of emergency health care and reports on some case studes amed at evaluatng the approach. Secton 6 summarzes the observed benefts of the approach, and Secton 7 presents conclusons and suggests future work.. RELATED WORK A number of projects have attempted to use understandngs of resource utlzaton to mprove the effectveness of health care processes. Connelly and Bar [5] presents a dscrete event smulaton system that predcts actual patent care tmes usng smulaton. Ther work does not model, however, the consderable dynamsm nherent n ths doman. Draeger [7] used medcal staff personnel models to support smulatons of nurse staffng approaches and alternatves for mprovements. McGure [18] used resource and process models to support smulatons amed at reducng the length of stay for ED patents. Rossett [4] used smlar smulatons to test alternatve ED attendng physcan staffng schedules and to analyze the correspondng mpacts on patent throughput and resource utlzaton. Samaha [5] used ED smulatons to do what-f analyss of the effect of process and staff level changes on LOS. But, none of these studes consdered the fundamental dynamc nature of ED resources, whch seems essental for accurate and effectve resource schedulng. As noted above, resource schedulng research nvestgates two man approaches: statc and dynamc. But the key assumpton of statc schedulng, that the executon envronment s relatvely fxed over the entre system executon [6], does not hold n the healthcare doman, where uncertanty about the key parameters needed to support resource schedulng s a major concern [17]. To address dynamc change n uncertan envronments, researchers have proposed two approaches: reactve schedulng and robust schedulng [1]. Reactve schedulng deals wth uncertantes arsng durng system executon by dong complete or partal reschedulng as soon as unexpected events or uncertantes are recognzed [3, 31]. Ths seems effectve n addressng some reschedulng problems, but ts effectveness s reduced when actvty estmates are unrelable, uncertantes are numerous, and when t attempts to reschedule large numbers of actvtes. Under such crcumstances reschedulng may take consderable amounts of tme, yet stll necesstate frequent new reschedulngs. Robust schedulng ams to antcpate the effects of possble dsruptons whle stll generatng schedules that support a hgh level of performance [1, 10, 17, 6]. Robust schedulng s most effectve when there are lmted and predctable dsruptons n system executons. If actual dsruptons exceed expectatons, excessve reschedulng may stll be needed. Ths approach should beneft greatly from access to system specfcatons that are as clear, complete, and as precse as possble about system executon dsruptons. Our own work adopts ths approach. Consderable research has also addressed the need for good resource schedulng algorthms, because these problems have hgh complexty tme bounds, and even relatvely smple heurstcs have been shown to be NP-hard []. Genetc algorthms (GAs) [13] have often been used n resource schedulng [9, 14]. But because they are heurstc, and cannot guarantee optmal, or even near optmal results, much attenton has been drected to seekng approprate parameters and evoluton methods that mprove convergence and avod local optma. Fnally we note that smulaton seems to be a popular and effectve method for evaluatng schedulng approaches [8, 15, 16], and ndeed we also have evaluated our approach by applyng t to smulatons of processes that defne the use of complex systems. 591

3 3. INCREMENTAL RESOURCE SCHEDULING METHOD The work addressed n ths paper uses the ncremental resource schedulng method descrbed n [9]. The approach combnes the strengths of the robust ncremental schedulng approach and the GA technology, wth the explotaton of more complete and precse nformaton about uncertanty that we derve from the analyss of partcularly detaled and precse defntons of both the system beng executed and the resources avalable for allocaton. Currently our ncremental reschedulng s carred out at fxed ponts n tme. However, ths approach also lends tself to support reschedulng ether 1) reactvely, when events occur that are beyond the scope of what we have been able to antcpate, or preferably, ) proactvely, at tme ponts that may be dctated by hstorcal data or recognton of upcomng uncertanty derved from analyss of system defntons. Each reschedulng actvty covers only the tasks that wll occur wthn a specfed wndow. A key goal of our research s to study how to determne the optmal sze and shape of ths wndow. If the wndow s too small more frequent (but perhaps more accurate), reschedulngs may be needed. If the wndow s too large, schedulng may be less frequent, but schedulng cost may be hgh, and accuracy low. Determnng the rght wndow sze and schedulng approach s facltated by the avalablty of a system defnton specfcaton that contans clear ndcatons of such uncertantes as locatons of exceptons, possbltes for human decson-makng, and the dosyncrases of executon agents. Ths nformaton s used n the desgn of GA chromosomes that are more completely and precsely specfed, thereby standng a greater chance of convergng on more optmal results at lower cost. The archtecture of the ncremental tme-wndow reschedulng system that we have bult s shown n Fgure 1, whch shows the followng major components (as descrbed n [9]): Fgure 1. Incremental Resource Schedulng Framework Reschedulng ndcator component, whch determnes when reschedulng should be done. Reschedulng s trggered when the reschedulng ndcator determnes that executon s about to proceed past the wndow over whch the last reschedulng had been computed. Ths component could also be used to dentfy when certan types of unexpected events, such as lowprobablty exceptons, sudden unavalablty of resources, and unexpectedly long task executon tmes occur, makng reschedulng desrable or necessary. Schedulng actvty set constructor. Ths component assembles the reschedulng problem, whch s prncpally a specfcaton of the actvtes that may possbly be executed n the near future, ther resource requrements, and the resources avalable for use by those actvtes. Scheduler component, whch uses the output of the schedulng actvty set constructor and a Genetc Algorthm (GA) to dentfy the specfc resources to be used to support the executon of each actvty. System executon component, whch provdes executon events needed to update the system executon state upon whch the reschedulng ndcator and the scheduler rely. We now descrbe the system used to evaluate our approach and archtecture. 4. THE SYSTEM USED FOR OUR EVALUATION 4.1 Process Actvty Defnton To enable us to evaluate one of our central research hypotheses, namely that a more complete, precse, and detaled system defnton can mprove the qualty of the resource schedulng approach, we used a powerful process defnton language, Lttle-JIL, to defne the processes that use the system for whch we wll do our schedulng. Lttle-JIL [4, 7] was orgnally developed to support the defnton of the processes by whch software s developed. More recently t has been used to defne processes n such domans as healthcare, government, and scence. Wse [7] provdes full techncal detals of the language. Here we outlne the features that seem most relevant to our schedulng work. A Lttle-JIL process defnton conssts of a specfcaton of three components, an artfact collecton (not descrbed here due to space constrants), an actvty specfcaton, and a resource repostory. A Lttle-JIL actvty specfcaton s a herarchy of steps, each of whch represents an actvty to be performed by an assgned resource (referred to as ts agent). Each step has a name and a set of badges to represent control flow among ts sub-steps, ts nterface, the exceptons t handles, etc. A leaf step (one wth no sub-steps) represents an actvty to be performed by an agent, wthout any gudance from the process. Each step may also specfy the need for resources n addton to ts agent. Each such request s specfed by the followng defnton. Req = (ResName Defnton 1. 1,Capablty 1,SkllLevel 1,...,, ResName r,capablty r,sklllevel r ) where, ResName s the type of the resource beng requested, (e.g. doctor, nurse, bed). Capablty s the specfc capablty that the resource s beng asked to provde. SkllLevel s the mnmum level of skll n Capablty that s requred. Fgure shows a Lttle-JIL actvty defnton that defnes at a hgh level of abstracton part of a process by whch a sngle patent s treated n a typcal hosptal Emergency Department. Note that ths process s nstantated for every new patent, and thus the workngs of an actual ED are represented by the concurrent executon of several of these processes. Each process needs the same types of resources, whch must be managed by one central resource 59

4 repostory. Ths sets up resource contenton. The entre process s represented by the top step, TreatOnePatent, whose three substeps provde elaboratve detal about how a patent s treated. A more complete and detaled process defnton would be needed to support schedulng n a real-world context. Such a defnton would use such more powerful language features as concurrency, the throwng and handlng of exceptons, step knds that allow human agents to make choces, and pre- and post-requstes that functon as guards for the performance of steps. At present we can only conjecture that these language features wll suffce to capture the needed detals. Further research s needed to ascertan ths. Table 1. Sze and resource requests for leaf steps n Fgure Step Request Sze ResName Capablty SkllLevel TragePatent 11 TrageNurse Trage 3 RegsterPatent 11 Clerk Regster RNAssessment 11 Nurse Assessment MDIntalAssessment 11 Doctor Assessment 3 PerformTests 31 AutoAgent Test RNProcedure 16 Nurse Assessment MDProcedure 16 Doctor Assessment 3 MDFnalAssessment AndDecson 6 Doctor Assessment 4 RNPaperwork 6 Nurse Paperwork 3 Defnton. Res = ( ID, ResName,At trbutes,s chedulable TmeTable, Capablty,SkllLeve l,productv ty, Capablty, 1 SkllLevel, Productv ty,...) where, 1 1 Fgure. Process descrbed by Lttle-JIL In Fgure the rght arrow n TreatOnePatent specfes that, n sequental order, the ED patent s frst traged by a trage nurse, then regstered by a clerk, and then placed n a bed for assessment and treatment. Ths last step s further decomposed nto two sequental substeps, each of whch s decomposed stll further. The executon of each step n a Lttle-JIL process requres one or more resources, whch can be ether human or non-human. In the ED process descrbed n Fgure, PatentInsdeED needs a bed resource whle the other steps do not need physcal resources. But most steps need human resources. Note that non-leaf steps are used essentally to create scopes, and real work s done only by leaf steps. Thus, the sze (namely an estmate of the relatve length of tme an actvty takes to execute) and resource requests are shown only for the leaf steps n ths process. Note that mean and standard devaton data mght be used to estmate the sze of each step. A large standard devaton for a step mght ndcate that the step s executon creates relatvely greater uncertanty, and greater need for antcpatory reschedulng. 4. Resource Repostory The resource repostory component of a Lttle-JIL process defnton s also needed to support our reschedulng approach. The resource repostory contans the resources avalable for assgnment to tasks specfed n the Lttle-JIL actvty dagram. Thus, ResourceRepostory = {Res1,Res,...,Resl }, where each element of ths set has certan capabltes and avalabltes. A resource s defned as follows: ID s a prose dentfcaton of the resource. ResName s the type of the resource, whch s an mplct specfcaton of the capabltes that ths resource has. Attrbutes s a set of (name, value) pars that descrbe the resource. Some example attrbute names mght be Age, Experence_Level, Pay_Rate, and Model_Number SchedulableTmeTable represents the tmes when a resource s avalable to be assgned to an actvty. Ths s a set of tme ntervals, defned by a start tme (st) and end tme (et), when the resource can be assgned to an actvty. Thus, SchedulableTmeTable = {[st 1,et 1 ],[st,et ],...,[ st s,et s ]} Capablty ( = 1, ) s the th knd of capablty that the resource has to offer. Two examples of capabltes of a resource that s a doctor or a nurse are 1) the capablty to trage patents and ) the capablty to assess patents. SkllLevel ( = 1, ) s the level of qualty at whch the resource s able to perform Capablty. Productv ty ( = 1, ) s the productvty that the resource s able to acheve n performngcapablty. In the above, SkllLevel and Productvty are attrbutes of Capablty, and are used to determne whether a gven resource has both the skll to perform a certan actvty and the quantty of avalable capacty needed to complete the actvty. Thus, specfcally, assume that an actvty specfes that S s the quantty of Capablty requred n order to complete the actvty. Then S/ Productv ty, s the tme resource R needs to do the actvty, where Productv ty s R s productvty n dong Capablty. Only f ths amount of tme s contaned wthn R s SchedulableTmeTable attrbute, can R be assgned to that actvty. 593

5 Table s an example of how the resources needed to support executon of the process n Fgure mght be specfed. Note that both human and non-human resources can be specfed, although because of space lmtaton, we do not specfy bed resources or explore ther allocaton n ths example. Moreover, for smplcty tme s specfed usng hypothetcal tme unts rather than actual wall clock tmes, and we set the productvty of all resources to 1. ID Name Table. Avalable resource descrptons Human Name Schedulable Tme Table (Capablty, Skll Level, Productvty) 1 TrageNurse TrageNurse1 [0, 10000] (Trage, 4, 1) Doctor Doctor1 [0, 10000] (Assessment, 5, 1) 3 Nurse Nurse1 [0, 10000] (Assessment, 4, 1), (Paperwork, 5, 1) 4 Nurse Nurse [0, 10000] (Assessment, 5, 1), (Paperwork, 3, 1) 5 Clerk Clerk1 [0, 10000] (Regster, 3, 1) 6 AutoAgent AutoAgent1 [0, 10000] (Test, 4, 1) 7 AutoAgent AutoAgent [0, 10000] (Test, 4, 1) 8 AutoAgent AutoAgent3 [0, 10000] (Test, 4, 1) 4.3 Reschedulng Indcator The reschedulng ndcator collects such runtme state nformaton as the actvtes currently beng executed, the resources supportng those actvtes, resource capacty avalable, new arrvals, changes n prortes, and constrant changes. The followng are examples of crtera that could be used n determnng whether a reschedulng should be performed: If an actvty that needs to be executed has not been allocated resources, a reschedulng should be carred out. If resources have been scheduled to an actvty, yet the resources are not avalable when the actvty should begn, a reschedulng should be carred out. If key attrbutes of some resources (e.g. cost or avalablty) have changed, a reschedulng should be carred out. Research should determne the reschedulng crtera to be used for any resource allocaton problem. Some crtera (e.g. the need to perform an actvty for whch no resource has prevously been dentfed) seem unversally applcable. Other crtera may be doman or applcaton specfc. And, ndeed, dfferent crtera may trgger reschedulngs based upon tme wndows of dfferent szes, and reschedulng decsons may be made dfferently under dfferent executon crcumstances. Fnally, note that n the work descrbed n ths paper reschedulng s done only at fxed ponts n tme, wth the more dynamc reschedulng trggers suggested n ths secton beng left to be expermented wth n future work. 4.4 Schedulng Actvty Set Constructor When the reschedulng ndcator determnes that a reschedulng should be carred out, the Schedulng Actvty Set Constructor s used to assemble all of the nformaton needed to make schedulng decsons. Ths functon determnes whch upcomng actvtes fall wthn the schedulng wndow, and assembles the actvtes nto a graph called the Dynamc Flow Graph (DFG). The sze of ths reschedulng wndow s an mportant parameter to determne because a large wndow may enable consderaton of more uncertanty, perhaps leadng to better schedulng results, but probably ncurrng greater computaton cost. Smaller reschedulng wndows may ncur less computaton cost, but may perhaps lead to schedulng results that are unable to take nto account enough uncertanty to produce good resource utlzaton. The DFG s derved from an analyss of another graph called the resource utlzaton flow graph (RUFG), whch s derved from a Lttle-JIL actvty dagram, and represents all possble process executon sequences. When a reschedulng s needed the statc RUFG and dynamc state nformaton are used to generate the DFG that s the bass for the reschedulng. The sze and shape of the DFG s determned by a specfcaton of the tme wndow, whch dctates how many of the future executon possbltes are to be consdered n the reschedulng. At present we defne the schedulng wndow W to consst of CURRACT, the set of actvtes that are currently beng performed, CURRACT = actvty, actvty,..., actvty }, { 1 n as well as all nodes, NODE for whch, for some, 1 n, there s a path, P, n the RUFG P = ( actvty, n1, n,..., nk, NODE) such that k s less than some fxed nteger, L. Each node n DFG contans two runtme attrbutes. One s the collecton of resources that are canddates for assgnment to the actvty represented by the node. Ths set s drawn from the collecton of avalable resources n the resource repostory. The other attrbute enumerates the resources that have actually been allocated at the concluson of the schedulng process. Further detals about the defnton of the RUFG and DFG can be found n [8] and are omtted here due to space constrants 4.5 Resource Reschedulng by Usng a GA Though a small wndow sze can reduce the magntude of the schedulng problem, the problem stll has very hgh computatonal complexty. Many approaches, such as constrant satsfacton programmng [], smulated annealng [19], and genetc algorthms (GA), have been used to address ths problem, Because the GA approach offers the advantages of hgh effcency, ncorporaton of varous knds of constrants, and ndependence from specfc doman characterstcs, we felt that GA was well suted for use durng ths prelmnary stage of our research where the prmary goal was determnng the feasblty of the approach. Other optmalty approaches mght offer greater advantages (e.g. greater speed), and should be consdered n subsequent work. The GA approach descrbed n [9] was our schedulng algorthm. The frst step n usng the GA approach s to represent the schedulng problem as an ntal populaton of chromosomes. Through populaton evoluton over a number of subsequent generatons, ncreasngly optmal schedulng results can be obtaned. Ths GA process s specfed more precsely as follows. (1) Generate ntal populaton that contans a certan number of chromosomes. Each chromosome s encoded to represent a possble soluton to the schedulng problem. () For each generaton, decode each chromosome n the populaton as a schedulng problem soluton, applyng 594

6 constrants to elmnate some, and evaluatng the qualty of those remanng usng some predefned measure of soluton qualty to determne the ftness of the chromosome. (3) Select chromosomes wth the hghest ftness value(s) as the seed(s) for the next generaton. (4) Make crossovers and mutatons to the selected chromosomes thus generatng a new generaton. (5) Return to () and contnue untl satsfyng some stoppng crteron (e.g. completng some number of generatons). The chromosome wth the hghest ftness n the fnal generaton s selected and decoded to yeld the schedulng result Encodng and decodng We used the bnary representaton of ntegers to help encode the reschedulng problem as a chromosome as descrbed n [9]. Note that because the DFG nodes to be scheduled changes durng process executon, new chromosomes must be bult for each reschedulng. Each chromosome encoded by ths method can, subject to the applcaton of constrants, be decoded as a schedulng scheme, namely the assgnment of a specfc resource to each of the requests made by each of the actvtes n the tme wndow. Decodng s done by reversng the encodng process Schedulng constrants Full detals about how encodng and decodng are done are omtted due to space lmtatons. But the role of constrants s partcularly mportant. Thus we now ndcate how three types of constrants are used to enhance the effcency and qualty of our GA-based schedulng approach n ED. Capablty constrant: Only resources wth needed capablty and skll levels can be scheduled to satsfy a resource request. Durng the encodng process, none but such resources are determned as canddate resources for a request. Ths nvolves searchng the resource repostory to dentfy resources that have the capablty to satsfy the request, usng the Capablty and SkllLevel attrbutes descrbed n secton 4.. Avalablty constrant: A resource can be assgned to a step for a certan tme perod only f the resource s avalable at ths tme perod, and has the capacty to provde enough effort to complete the step. Ths constrant s enforced durng the decodng process by frst determnng the tme perod requred usng the Capablty attrbute of the step and the Productvty attrbute of the canddate resources, and then examnng the ScheduledTmeTable of each assgned resource. Step executon order constrant: Steps can be executed only after all of ther precedng steps have completed. Thus resources must be assgned to steps n a tme wndow n an order dctated by the executon sequencng defned by the DFG. Ths constrant s appled durng the decodng process. In partcular, the start of the executon of a step must begn at a tme after the tme of completon of all of ts predecessor steps. If a resource allocated to a step s no longer avalable because t has been allocated to another step (e.g. one executng n parallel), the schedule defned by ths chromosome s rejected and ths chromosome s not carred over to the next generaton Ftness functon The role of the ftness functon s to evaluate the relatve desrablty of each of the chromosomes as a soluton to the resource reschedulng problem. Chromosomes wth hgher ftness are selected for the next generaton of the GA, thereby movng the GA towards optmal solutons. The ftness functon reflects an optmzaton goal for the resource allocaton. Thus, for example, one possble goal of resource allocaton n an ED s to mnmze total patent watng tme. In ths case, the ftness functon must quantfy the watng tme expected for each of the resource assgnments specfed by a chromosome. Ths mght be done as follows. Suppose the set of steps n the tme wndow s: Schedulng StepSet = { Step1, Step,..., Step N } A schedulng scheme set SSS for SchedulngStepSet s the set of all the schedulng schemes correspondng to a set of chromosomes that represent possble resource allocatons for SchedulngStepSet. Now suppose that the fnshng tme for the latest-fnshng of all of the steps that mmedately precede a step s tme P. Then, P s defned as the Can be started tme of Step. Assume that analyss of the avalablty of resources assgned by the schedulng scheme to Step determnes that Step cannot be started untl tme S. Then the watng tme for Step s defned as ( S - P ). If schedulng scheme SSk s the one that has the mnmum total watng tme, then SS satsfes the followng equaton: k (( SS SSS) ( ( Sa Pa ) < ( S a SS b SS k b P ))) Note that ths ftness functon does not attempt to mnmze the total watng tme for all steps, only the total watng tme for the steps that are to mmedately follow the currently executng steps. Thus ths example s only one of many possble ftness functons, some of whch wll be harder to compute than others, and some of whch wll mnmze overall watng tme more effectvely. Expermentaton (perhaps doman specfc), wll be needed to determne whch ftness functons are most cost-effectve Runnng GA Before runnng GA, the followng parameters must be set: Populaton scale (PS) s the number of chromosomes n each generaton. When PS s larger the computaton of each generaton wll take longer. Crossover rate (CR) s the number and possblty of crossover among chromosomes n a populaton. If CR s large, chromosomes wth hgher ftness mght be destroyed. If CR s small, evoluton and optmzaton rates may be slower. Mutaton rate (MR) s the probablty that a chromosome wll be subject to mutaton. If the mutaton rato s hgh unstable evoluton may result. If t s low, there s less chance of avodng local optma and fndng a global optmum. Generaton number (GN) s the number of generatons (teratons) that the GA s to compute. Fewer generatons wll take less tme, but may not come close to an optmum. Research s needed to establsh relable gudelnes for specfyng how these parameters should be set. We wll present the results of usng some specfc choces of parameters n our expermentaton. b 595

7 5. EVALUATION To support analyss of the effectveness of our approach, we used t to allocate resources durng smulatons of processes that represent how hosptal emergency departments (EDs) perform some actvtes and utlze ther resources. A hosptal ED requres the use of many dfferent knds of resources--human, mechancal, and automated--to support the treatment of patents. Snce the costs of most of these resources (e.g. doctors, MRIs) are hgh, only lmted numbers of them are avalable. Snce many patents are typcally beng treated n an ED concurrently, contenton for these resources can be expected. Ths contenton can lead to excessve patent watng tme. Watng tme can be reduced by provdng more resources, but there s a retcence to ncur the szeable expenses of these resources unless t can be shown that ths wll lead to worthwhle reductons n watng tmes. Smulatons such as those descrbed here can suggest what the magntude of those reductons mght be. 5.1 The Smulaton Settng The process used as the prncpal bass for the case studes presented here s the Lttle-JIL process shown n Fgure. Ths process s a very hgh level representaton of some aspects of a process that specfes how a typcal ED goes about treatng patents. The resources requred by each step n the process are descrbed n Table 1. And the resources avalable to ths process are descrbed n Table. The complete set of nputs requred n order to run a smulaton of the ED process comprses 1) a process descrpton, ) a resource repostory, 3) a specfcaton of patent arrval rates and dstrbutons of types, and 4) parameters needed to specfy the executon of the GA. For our evaluatve work we vared each of these nputs n order to support analyss of how senstve the results obtaned are to these varatons. The settngs and parameters we used ntally are lsted n Table 3. Table 3. Intal smulaton settngs and parameters Settngs and Parameters Value GA populaton scale 3 GA crossover rate 1 GA mutaton rate 0.1 Patent number 50 Frst patent arrval tme 5. Smulaton Case Studes 5..1 Case Study 1: The effect of process detal on schedulng effectveness. One hypothess of ths paper s that more complete and precse system specfcatons can support the computaton of better schedulng schemas. To evaluate ths hypothess, we compared the results obtaned from runnng smulatons of the process defned n Fgure, but usng resource schedulng results obtaned based on analyss of a less precse process defnton. To do ths we supposed that the assessment work done by the nurse and doctor s done n some unspecfed way, rather than sequentally, as n Fgure. A step named Assessment descrbes ths actvty. It ncludes requests for two resources, a doctor and a nurse. The AssessAndTest sub-tree s then as shown n Fgure 3. Fgure 3. ED process wth less precse detals We set the schedulng tme wndow to and used a patent arrval nterval of 0. We estmated the executon tme of the Assessment step to range from the tme that would be taken f assessment s done sequentally, down to 11, for the extreme case where assessment s done completely concurrently by the doctor and nurse. Other lengths of tme between 11 and are possble for cases where the overlap of the efforts of the doctor and nurse s not complete. The total smulated patent watng tme obtaned for all these lengths of tme s shown n Fgure 4. The addtonal detal, namely that Assessment s the sequental performance of two substeps, leads to substantal watng tme reducton and there s ncreasng reducton as the concurrency of the actons of the doctor and nurse are decreasngly complete. For completeness we also show the results of usng the process shown n Fgure 4 both as the bass of schedulng and as the bass for the smulaton used to compute watng tme. The results of usng ths less complete and detaled process n ths way are stll less satsfactory, gvng stll more support to our hypothess that greater process detal seems to provde mportant mprovements n schedulng qualty. Total patent watng tme Executon tme of Assessment Usng the process from Fgure 4 for schedulng and smulaton Usng the process from Fgure 4 for schedulng, and usng Fgure for runnng smulaton Usng the process from Fgure for schedulng and smulaton Fgure 4. Total watng tme of less precse process under dfferent executon tme of assessment Improvement s most dramatc n the case where the elaboraton of the step s as sequental executon, suggestng the partcular value of ths type of elaboratve detal. Interestngly, doman experts say that assessment s ndeed usually performed sequentally by a doctor and a nurse. Thus, the greater detal n the defnton shown n Fgure seems to support the possblty of schedulng that could reduce watng tme n a real-world ED. 5.. Case Study : The effect of resource specfcaton detal on schedulng effectveness. Another hypothess of our approach s that complete and precse resource avalablty and capablty specfcatons are the bass of better schedulng schema. To evaluate ths, we executed our reschedulng approach usng resource specfcatons that dd not nclude the SchedulableTmeTable attrbute descrbed n Secton 4, and compared the results to those obtaned when ths attrbute was specfed. We appled a frst come frst serve dscplne for resource 596

8 assgnment, and compared results for patent arrval ntervals rangng from 5 to 34. The results are shown n Fgure 5. Total patent watng tme Patent arrval nterval (tme unt) Schedule wth complete and precse resource avalablty and capablty nformaton Schedule wth less complete resource avalablty and capablty nformaton Fgure 5. Total watng tme usng precse and less precse resource descrptons These results suggest that when the patent arrval rate s hgher resource contenton ncreases and more precse resource descrptons provde better support for schedulng. Decreasng patent arrval rates reduce resource contenton, and less precse resource descrptons produce schedules that are ncreasngly close to those obtaned wth more precse resource descrptons Case Study 3: Schedulng cost varaton wth changng wndow sze Other case study was amed at determnng the wndow sze that represents a good compromse between lower costs of schedulng over smaller wndows vs. better schedules resultng from larger wndows. Fgure 6 shows the effect of dfferent wndow szes on the number of reschedulngs, total smulaton tme, and schedulng qualty obtaned wth patent arrval set at 0 tme unts. Schedulng tme or total watng tme devded by Schedulng wndow sze Number of Reschedulng Schedulng tme Total patent watng tme dvde by 10 Number of reschedulng Fgure 6. Schedulng tme and number of reschedulng under dfferent wndow sze Note that when the sze of the schedulng wndow ncreases from 1 to, the number of reschedulngs decreases sharply and the total tme for all schedulngs also decreases. As the wndow sze keeps ncreasng, the number of reschedulngs decreases far more slowly, but total tme spent schedulng ncreases markedly, presumably because the number of steps n each reschedulng s large, makng the cost of each reschedulng large as well. Interestngly, note that when the wndow sze reaches the number of patents beng processed concurrently some reschedulngs wll be trggered whle sgnfcant amounts of schedulng nformaton from the prevous reschedulng has not yet been used. Reschedulng thus causes some prevous data to be superseded, thereby wastng effort. Moreover, the dagram shows that schedulng qualty (as measured by total patent watng tme) does not necessarly mprove as wndow szes ncreases. Thus ths case study suggests that wndow sze selecton should be carefully consdered, and n fact mght well best be determned dynamcally, based upon the state of process executon Case Study 4: GA cost and accuracy Because GA s essentally a heurstc, t s not possble to be sure that the results obtaned are optmal, or even near-optmal. To help us gan confdence n the qualty of the results obtaned usng GA, we compared them to results obtaned usng an exhaustve search (ES) of the space of all schedulng possbltes. As the computatonal complexty of ES s exponental, ES s possble only for relatvely small schedulng problems. But we used these small schedulng problems to form a bass for comparson wth results obtaned usng GA. We ran a number of smulatons wth the number of patents set to 8, patent arrval nterval set as 40 tme unts, settng the GA generaton number to be 100. We noted that GA consstently obtaned the exact same schedulng results as ES, ndcatng that GA found the global optmum for all of these small problems. Indeed GA nvarably found the global optmum wthn the frst 10 generatons. On the other hand, GA offers substantal speed advantages, as expected. Fgure 7 shows the tme requred to do a seres of schedulng problems. In ths fgure, the X-axs represents the number of nodes n a reschedulng wndow. The prmary Y-axs represents the amount of tme consumed n the correspondng schedulng (n seconds) by ES and the secondary Y-axs represents the amount of tme consumed n the correspondng schedulng (n seconds) by GA. The value of each pont s gotten from the average of several runs. Tme spent by ES (Second) Number of nodes wthn a schedulng Fgure 7. Schedulng tme comparson of GA and ES 6. ANALYSIS AND DISCUSSION The tme-wndow ncremental reschedulng approach that we have proposed seems to promse the followng advantages: The approach seems to be able to use suffcently complete and precse specfcatons of processes and resources to delver effectve schedulng results. The case studes n secton 5..1 and 5.. show that complete and precse specfcatons can mprove schedulng results, although these case studes also suggest that some detals seem to be of more potental value than others. More research s needed to understand better whch detals are worth specfyng. The wndow sze used matters. The case studes n sectons 5..3 and 5..4 suggest that f the wndow sze s approprately set, the benefts of lower schedulng cost and hgher schedulng qualty can be both obtaned. Ths research s stll qute prelmnary, but t suggests that ths wndow sze may be context dependent and that more research s needed to understand better what features and state nformaton should be used (and how) to suggest optmal wndow sze. Contnuous schedulng decson support can be provded n a process envronment where frequent changes lead to contnuous uncertanty. Our case studes suggest that relatvely small tme Tme spent by GA(Second) ES GA 597

9 wndows are lkely to be most effectve, perhaps because they enable relatvely rapd reacton to changes (e.g. the sudden arrval of a new patent) and ther attendant uncertantes. The GA schedulng heurstc seems effectve. Our case study showed that GA can produce optmal results quckly for small schedulng problems. Whle ths makes no assurance of GA effcacy for larger problems, the ntal results are encouragng. 7. SUMMARY AND FUTURE WORK Ths paper has presented a tme wndow based ncremental resource schedulng method that uses a genetc algorthm. We used ths method to develop a schedulng tool that was ntegrated wth an exstng dscrete event smulaton system n order to study the effectveness of the approach n creatng good resource allocaton schedules n affordable tme. We used ths system to support a varety of smulatons of hosptal emergency department processes. These ntal case studes suggest that ths approach can be effectve. Numerous drectons of future work are suggested. Some specfc drectons are: Explorng realstc emergency department processes, resource mxes, and resource allocaton strateges: The work done so far rests on very hgh level process defntons that lack detals of real ED processes. Approprately detaled processes must be elcted, and ndeed research s needed to determne how effectvely languages such as Lttle-JIL wll be able to capture the needed detals. In addton, the process presented here, and the optmzaton goal used, are only examples of the knds of ED processes and problems that need to be explored. More dversty and more detals n processes, resources, and goals should be specfed and explored. Whch detals matter: The prevous secton suggested the need for careful study of whch process and resource detals are actually valuable n ncreasng the effectveness of ths schedulng approach. We have seen evdence, for example, that more detals about process step sequentalty can lead to better schedules, but that elaboratng the detals of concurrently runnng steps may be less valuable. We need to determne whch detals are worthwhle, and whch seem to be less useful so that approprate attenton can be focused on ncludng n resource optmzaton studes the detals that matter the most. Dynamc trggerng of reschedulng: In ths work reschedulng was trggered at fxed, predetermned ntervals. But our archtecture s desgned to support dynamc determnaton of when to reschedule based upon varous runtme parameters. Future work should explore when to carry out such dynamc reschedulng, and how to use runtme parameters to defne the reschedulng problem parameters (e.g. the reschedulng wndow). Analyss of dfferent processes and parameters: ths paper manly focuses on changng parameters of wndow sze and patent arrval nterval n the specfc hosptal ED process descrbed here. Dfferent processes should be studed as well, and for each of these dfferent processes GA parameters, such as crossover rate, mutaton rate, and generaton number should also be the subjects of further study to determne whch combnatons of these parameters are most effectve. Further, a mechansm should be sought for dynamcally adjustng these varous parameters dependng upon the process and state of ts executon. Combne dfferent value objectves n one schedulng: In ths work schedules suggested by dfferent chromosomes were evaluated usng a sngle fxed objectve functon. But objectves may change durng the runnng of a system (especally a long-runnng system). Thus t seems mportant to evaluate our approach usng dfferent objectve functons, weghted dfferently at dfferent tmes durng process executon. Pragmatc ssues n usng ths approach n a real ED: Ths research has suggested that the proposed approach could be effectve n supportng better schedulng of ED resources. But brngng the advantages of ths approach to a real ED requres addressng numerous problems. It s not suffcent only to create an optmzed resource allocaton. It s also necessary to be sure that t s communcated to approprate medcal professonals n clear and tmely ways that are consstent wth current communcaton patterns and vehcles. Other research must address how to support mantenance of the needed resource repostores. Research s also needed to dentfy the knds of actual ED process events that should lead to the knds of dsruptons that are of most mportance n trggerng reschedulng. In addton, t wll be essental to carry out research amed at determnng whether reschedulng algorthms are ndeed suffcently fast to be used n the hectc real-tme envronment of a busy ED. 8. ACKNOWLEDGMENTS We would lke to thank Dr. Phlp L. Henneman for hs nsghts nto the workngs of a hosptal ED and for provdng detals about both the actvtes and the resources nvolved n provdng care n an ED. We also thank Bn Chen and Heather Conboy for ther help wth the transformaton from Lttle-JIL to RUFG, and Prof. Lor A. Clarke, Dr. M. S. Raunak, and Sandy Wse for ther valuable feedback about ths work. Ths paper s supported by the Natonal Natural Scence Foundaton of Chna under grant Nos , the H- Tech Research and Development Program (863 Program) of Chna under grant No. 007AA010303, 007AA01Z186, as well as the Natonal Basc Research Program (973 program) under grant No. 007CB Ths work was also supported by the Natonal Scence Foundaton under Awards No. CCR , CCR , and IIS Any opnons, fndngs, and conclusons or recommendatons expressed n ths publcaton are those of the author(s) and do not necessarly reflect the vews of the Natonal Scence Foundaton. 9. REFERENCES 1. Al-Fawzan, M.A., Haouar, M. A b-objectve model for robust resource-constraned project schedulng Internatonal Journal of Producton Economcs 96 (005) Barreto, A., Barros, M.d.O., Werner, C.M.L. 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