Proceedngs of the 2012 Wnter Smulaton Conference C. Laroque, J.Hmmelspach, R.Pasupathy, O.Rose, and A.M.Uhrmacher, eds CALIBRATION OF A DECISION-MAKING PROCESS IN A SIMULATION MODEL BY A BICRITERIA OPTIMIZATION PROBLEM Crstna Azcárate Fermn Mallor Julo Barado Publc Unversty of Navarre Hosptal of Navarre Campus Arrosada Irunlarrea-Str. 3 31006 Pamplona, SPAIN 31008 Pamplona, SPAIN ABSTRACT In a prevous paper, we developed an accurate smulaton model of an Intensve Care Unt to study bed occupancy level (BOL). By means of accurate statstcal analyss we were able to ft models to arrvals and length-of-stay of patents. We model doctors patent dscharge decsons and defne a set of rules to determne the condtons for earler or delayed dscharge of certan patents, accordng to BOL. For the calbraton of the rule parameters, we proposed a nonlnear stochastc optmzaton problem amed at matchng the model outputs wth the real system outputs. In ths paper, we mprove the calbraton of the rule parameters by ncludng the prncple of mnmum medcal nterventon as a second objectve functon. We replace the prevous objectve functon wth a satsfcng matchng, n order to gan more degrees of freedom n the search for better rules accordng to the new objectve. 1 INTRODUCTION Smulaton has been wdely used to analyze health-care system management problems, whch are characterzed by a stochastc envronment and lmted human and materal resources. Revews and dscusson papers dealng wth the applcaton of smulaton modellng n health care can be found n Bralsford et al. (2009), Eldab et al. (2007), Günal and Pdd (2010) and Katsalak and Mustafee (2011). Many studes use smulaton to analyze hosptal capacty and bed allocaton, but only a few deal specfcally wth ICUs. Worth notng are Km et al. (1999, 2000), n whch ICU admsson and dscharge processes are analyzed through smulaton and several rules for bed allocaton are evaluated; Ltvack et al. (2008) and Rdge et al. (1998) and Costa et al. (2003), whch analyze the problem of ICU capacty; and Kolker (2009), n whch an ICU smulaton model s used to establsh a quanttatve lnk between the daly load levellng of electve surgeres and ICU dverson. The constructon of an ICU smulaton model nvolves fndng the approprate statstcal models for ts stochastc elements: arrval patterns, patent s personal and medcal characterstcs and length of stay (LoS) n the ICU. An overvew of LoS and patent flow modellng technques can be found n Marshall et al. (2005). All these smulaton studes assume that the LoS s ndependent of the ICU workload and bed occupancy level. However, recent research Mallor and Azcárate (2012) has shown that the LoS of some patents can be nfluenced by the ICU bed occupancy level. Doctors may dscharge patents earler when the number of occuped beds threatens the unt s capacty to accommodate new ncomng patents and, conversely, when the ICU bed occupancy s low, patents may be allowed to complete ther recovery n 978-1-4673-4782-2/12/$31.00 2012 IEEE 782
the ICU. Thus a vald smulaton model should nclude doctors patent dscharge decsons. The modelng task s hampered, however, by the lack of any wrtten decson protocol that could be mplemented n the smulaton model. We address the problem of modellng doctors decson-makng by defnng a set of rules dependent on a set of parameters. Model calbraton s the process of determnng the values of unobservable parameters by constranng model output to replcate observed data. Research papers dealng wth the calbraton of smulaton models are not numerous. There are some that deal wth ths topc n the feld of traffc smulaton, e.g. n Park and Q (2005) whch proposes a general procedure for the calbraton of mcroscopc smulaton models. A dscusson on the computatonal complexty of model calbraton appears n Hofmann (2005); and remarks on calbraton wth respect to valdty can be found n Bayarr et al. (2004). To estmate the rule parameters, we formulated a nonlnear stochastc optmzaton problem wth the am of matchng the bed occupancy outputs from the smulaton model to those of the actual system. The am of ths paper s to present an mproved verson of the calbraton process desgned to take nto account not only the match between the smulaton output and the system output but also the opnons of doctors. We consder a bcrtera optmzaton problem n whch the prncple of mnmum medcal nterventon s ncluded as a second objectve functon. We replace the prevous objectve functon wth a satsfcng matchng n order to gan more degrees of freedom n the search for better rules accordng to the new objectve. In secton 2, we present the elements ncluded n the ICU smulaton model. Secton 3 dscusses a way to represent doctors decsons by means of a set of rules. The optmzaton problems defned to calbrate the parameters of ths set of rules are descrbed n Secton 4. In Secton 5, we present the results of the calbraton process appled to the smulaton of the ICU at the Hosptal of Navarre. The fnal secton presents the conclusons. 2 MODELING AN INTENSIVE CARE UNIT An ICU s mathematcally modelled as a queung system n whch the patents are the clents, the beds are the servers and there s no watng room. We consder that any patent arrvng when the ICU s full s transferred to an alternatve ICU (n a neghbourng regon). The smulaton model s therefore structurally qute smple and can be run usng approprate statstcal models for the patents nput pattern and for ther LoS n the ICU. Many studes (see for example, Rdge et al. 1998; Km et al. 2000; Ltvack et al 2008; Oddoye et al. 2009) use a Posson Process as a statstcal model for patent arrval to a health care centre. Ths model holds for outer patents arrvng at the ICU on an ndvdual bass, whose arrval tmes are not nfluenced by pror patent arrvals and are not coordnated to ft n wth any pre-arranged schedule. However, the Posson Process does not apply for patents comng from operatng theatres. These results were confrmed n Mallor and Azcárate (2012), where we found that a Posson Process fts outer patent arrvals very well but does not ft those of the pre-scheduled patents group, whch requred the use of emprcal dstrbutons. The latter group also presented a dfferent arrval pattern for holday perods. One common characterstc of LoS data s the presence of a hgh percentage of extreme values, that s, values that are far enough from the mean to be consdered outlers (see Vaslaks and Marshall 2005). In these cases, the dstrbutons commonly used to represent servce tmes n the health care context do not gve a good ft to real LoS data. Ths problem of poor ft to the orgnal data has been addressed n the lterature n dfferent ways. In our prevous paper, we addressed ths statstcal fttng problem by developng non-normal regresson models ncludng varables wth the power to explan some of the LoS varablty, such as the Apache ndex. 783
3 MODELING THE MEDICAL DECISION MAKING One of our man achevements n studyng ICU smulaton models has been to show that smulaton models that fal to ncorporate the management decsons made by clncal staff can hardly be consdered vald. We reached ths concluson after comparng the bed occupancy outputs from the smulaton model wth those of the ICU (Fgure 1). A vsual nspecton of Fgure 1 suggests that management polces for patent admsson and dscharge affect the bed occupancy dstrbuton. The ICU staff confrmed that some dscharge decsons are made wth vew to keepng bed occupancy from becomng too hgh (thus compromsng the admsson of new patents) or too low (thus wastng valuable resources). ICU dscharge decsons, therefore, can sometmes be drven by current occupancy rates and bed demand, provded the patent s welfare s not compromsed. Fgure 1: Real vs. smulated bed occupancy frequences n a smulaton model wthout manageral decson modellng. Thus, to obtan a vald smulaton model, t s necessary to nclude clncal decson-makng. There s no wrtten protocol for managers to determne patent admsson and dscharge automatcally; these decsons are subject to the judgment of the ntensve care consultant. To model these human decsons concernng patent LoS dependng on the bed occupancy level, we consder two knds of management rules: f the bed occupancy level s hgh and certan condtons are satsfed (the decson must not nvolve a termnal patent and the estmated remanng LoS of the one selected for dscharge must be less than %PR and less than DR days), one patent (the one n the best state of health) leaves the ICU pror to completng treatment. f the bed occupancy level s low, then the LoS of a patent may be ncreased by one day wth certan probablty, PI. There s a maxmum number of days DE by whch the LoS of a patent can be extended. These rules can be dentcally defned for all types of patents or can be made to vary for dfferent groups of patents. For example, we mght dstngush the group of programmed-surgery patents, whose stay tends to be short and consder that t can be shortened by only one day wth certan probablty PC when bed occupancy s. These rules lnk patents LoS to bed occupancy levels. ICU dscharge tme s a value belongng to a set of admssble values defned by the rule parameters (see Fgure 2). 784
% LoS 100 PR Nº Days DR Entry Low Hgh occup. occup. Lmt for early dscharge Extended Stay Recovery Fgure 2: Patent recovery s a contnuous process leadng to ICU dscharge n a perod of tme wth a value that belongs to an nterval of admssble values. Observe that the set of rules defnes nfnte management polces for the ICU: one for each set of parameter values. The smulaton model allows us to assess the performance of only one set of parameters n each smulaton run. By runnng the smulaton model for some reasonable values of the rule parameters, t would be possble to assess ther nfluence on the dstrbuton of the bed occupancy frequences and the degree to whch these smulated results approxmate ICU observed data. Ths brngs us to the queston of how to fnd the best set of parameter values, that s, how to calbrate the smulaton model. We address ths queston n the next secton by defnng dfferent optmzaton problems. 4 OPTIMIZATION PROBLEMS FOR THE CALIBRATION OF THE SIMULATION MODEL Model calbraton s the process by whch we determne smulaton parameters values not observed n the system, such that model output replcates emprcal data. In our case, we need to determne the rule parameter values that produce a bed occupancy output smlar to that observed n a real ICU. In a frst approach, we formulate an optmzaton problem amed at matchng the smulaton model output as closely as possble to the ICU hstorcal data. The decson varables are the parameters PR, DR, PI, DE and PC above defned. Constrants represent realstc monotonous relatonshps nto each set of parameters and upper bounds for ther values (upr, udr, upi, ude and upc). We set as our objectve functon to mnmze the squared dfferences of real and smulated frequency (absolute value of dfferences or maxmum dfference also can be used). Mn n = 0 ( real _ freq( ) smul _ freq( ) ) subject to udr DRn DRn 1... DRn k 2 0 upr PRn PRn 1... PRn k 2 0 ude DE1... DE k1 0 upi PI1... PIk1 0 upc PCn PCn 1... PCn k 2 0 DR,DI j nteger = n k2,...,n j = 1,...,k1, 2 k1 k2 To solve ths problem, we ncorporate these decson rules nto the smulaton model and combne optmzaton and smulaton (ARENA-OptQuest). Ths gves us a bed occupancy dstrbuton that closely matches the observed one, as shown n Fgure 3. (1) 785
Fgure 3: Real vs. smulated (wth and wthout rules) bed occupancy frequences. Management polces obtaned from ths hstorcal data approach can be too aggressve n the sense of allowng extreme shortenng of some stays. An example of such management polces s gven n the frst column of table 1 and commented n next secton. Furthermore, the smulaton model s only a representaton of the real ICU, therefore the results of the smulated system do not have to be dentcal to those of the real system. It s enough that they be smlar. Then we propose a new approach that takes nto account not only the hstorcal data but also the opnons of experts who stated that early dscharges should be kept as low as possble. We nterpret ths dea as the prncple of mnmum medcal nterventon and consder t as our second objectve functon. Observe that a zero value for all parameters would ndcate a stuaton n whch clncans pay no heed to the bed occupancy level, but, as t rses, ts mpact on ther decsons grows stronger. Smlarly, f there s no stuaton n whch patent LoS may be shortened or extended, then clncans wll agan dsregard the bed occupancy level, whle the greater the number of days by whch LoS s shortened or extended, the more the bed occupancy level nfluences clncans decsons. Followng these deas, we consder two dfferent ways of measurng the medcal nterventon level. From a decson-makng pont of vew, we wll try to mnmze the rule parameter values to keep them as close to zero as possble(infl 1 ). From the consequences of the decson-makng process we wll try to mnmze the total number of days of LoS extenson or reducton (INFL 2 ). Specfcally, we defne the followng measure of the nfluence of the bed occupancy level n ICU management: where, and INFL = w INFL + 1 1 w2 INFL 2 * * ( PI + PR + PC + DR + DE ) 1 INFL = 1 k * * DR = DR udr and DE = DE ude INFL 2 = % days' extenson + % days' reducton Observe that, to make parameter values representng a percentage or probablty comparable wth those representng a number of days, we normalze the latter to a maxmum number of days, udr and ude, set as admssble for an early dscharge or extended stay. The value k s the total number of parameters nvolved n the expresson of INFL 1. Then INFL 1 ranges from 0 to 1, where a 0 value ndcates that 786
the bed occupancy level has no mpact on ICU management and ncreasng values of INFL 1 ndcate an ncreasng mpact. The objectve INFL 2 s expressed n percentage terms by dvdng the number of days of LoS extenson or reducton by the total days LoS of all current patents. 5 CASE STUDY AND COMPUTATIONAL RESULTS We have developed a smulaton model that ncludes a representaton of the dscharge decsons made n the ICU of the Hosptal of Navarre, n Span. The Hosptal of Navarra s a general publc hosptal wth reference specaltes n the Communty of Navarra (Neurosurgery, cardac surgery, vascular surgery, oncology, nfectous dseases, etc.). It has 483 beds, 2015 members of staff and 10 operatng theatres. The ICU of ths hosptal has 20 beds and 86 physcans and nurses. It receves patents from 3 sources (emergency, operatng theatres and wards).the data were recorded and provded by the Hosptal admnstraton. We have three fles: a patent fle, a bed occupancy fle and an arrvals fle, wth 9 years of data, from 1/1/2000 to 31/12/2008. The patent fle contans a record of all patents admtted to ICU durng that perod. The known patent varables are as follows: age, arrval date, llness group (8 groups were consdered), dscharge date, APACHE (llness severty), ICU nfectons, and extus (recovered or deceased). The bed occupancy fle s a record of bed occupancy noted daly at 16h. These data are used to valdate and calbrate the smulaton model. The arrvals fle s a record of the number of patents admtted to the ICU each day. We consder the followng bcrtera optmzaton problem, ncludng both the medcal nterventon objectve functon, as descrbed n Secton 4, and a measure of the dfference between the bed occupancy dstrbuton n the smulaton output and n the hstorcal data (observe that ths objectve functon s the statstc used by the Kolgomorov-Smrnov test): Mn INFL Mn subject Max to F_real ()- F_smul() constrants n (1) We estmate the Pareto fronter by usng the ε-constrant method: Mn INFL subject to Max F_real ()- F_smul() constrants n (1) ε (2) Fgure 4 shows the Pareto fronter obtaned by consderng dfferent ε j -values for the data matchng objectve, rangng from 10% to 3%. Representatve optmal solutons of problem (2) are shown n Table 1. As an example, let us consder the soluton for ε j =3.5. PR 17 =45% and DR 17 =4 days mean that a patent s LoS can be reduced a maxmum of 4 days, whenever these 4 days represent less than 45% of the patent s LoS to complete recovery, when occupancy level s 17 beds. j 787
Fgure 4: Pareto fronter for the data-matchng and medcal nfluence objectve functons. Fgure 5 plots the bed occupancy dstrbutons for real data (thck black lne), smulated data wthout dscharge rules (thck red lne), and smulated data wth dfferent dscharge rules scenaros. Dark blue dotted lne represents the smulated bed occupancy dstrbuton wth the dscharge rules obtaned from the optmzaton problem n (1), that s by only optmzng the matchng objectve functon. The set of thn lnes, wth colours rangng from red to blue, represent bed occupancy dstrbutons from smulaton models ncludng the dscharge rules obtaned as solutons of optmzaton problems (2) (one lne for each ε j - value consdered n Table 1). We observe that the closest dstrbuton to the hstorcal bed occupancy dstrbuton s provded by the soluton of the optmzaton problem (1). Nevertheless, the values for the rule parameters of ths management polcy are too aggressve as mentoned n the prevous secton: under ths polcy t s allowed to advance the dscharge of a patent up to 5 days, shortenng ts LoS up to 50%. Settng the approach to the hstorcal data as a constrant and mnmzng the medcal nfluence the dstrbutons move gradually away from the hstorcal one as the value of ε j ncreases. On the other hand the ncrease of ε j leads to softenng the medcal nterventon. We see that boundng the dfference to hstorcal data dstrbuton to a maxmum of 10% all LoS are shortened n no more than one day. Fgure 5. Bed occupancy dstrbutons for the ICU: real data and smulated data wth dfferent dscharge polces. 788
Table 1: Optmal solutons of optmzaton problems (1) and (2) Soluton of optmzaton problem (1) Solutons of optmzaton problem (2) for dfferent ε j -values 3.5% 4% 5% 6% 7% 8% 9% 10% PI hgh PI low PR 15 PR 16 PR 17 PR 18 PR 19 PR 20 PC 15 PC 16 PC 17 PC 18 PC 19 PC 20 DR 15 DR 16 DR 17 DR 18 DR 19 DR 20 6 CONCLUSIONS In ths paper, we have shown that, n order to obtan vald ICU smulaton models, t s necessary not only to ensure accurate statstcal modellng of stochastc elements (arrval process and LoS) but also to nclude clncans dscharge decsons. We have proposed a set of rules to model ths human decson process. Rule parameters are estmated by means of a calbraton process that uses bed occupancy hstorcal data. However, the search for a perfect matchng does not prove fully satsfactory. To mprove the calbraton, therefore, we have ncorporated a second objectve, mnmum medcal nterventon, to complement the classcal data-matchng approach and then formulated a bcrtera optmzaton problem. One of the man problems of the Spansh publc health care system s the length of watng lsts for some specalst surgcal procedures. To reduce watng lsts and mprove servce qualty, the hosptal s extendng some operatng theatre hours, whch ncreases the number of patents from electve surgery. In the past, the lack of beds caused to postpone surgeres. Nowadays, no surgery s cancelled due to the lack of operatng rooms, and the patents are also transferred to other health facltes n the regon, f necessary. Then the smulaton model would beneft from the ncluson of the programmed surgeres. Unfortunately we have no access to the necessary nformaton to model them. More research s needed to select a compromse soluton. We are explorng the possblty of nterpretng the data-matchng objectve functon as the statstc of a dstrbuton fttng test. In that way, ts admssble values are those belongng to the acceptance regon. 789
REFERENCES Bayarr, M. J., J. O. Berger, D. Hgdon, M. C. Kennedy, A. Kottas, R. Paulo, J. Sacks, J. A. Cafeo, J. Cavendsh, C. H. Ln, and J. Tu. 2002. A Framework for Valdaton of Computer Models. In Foundatons 2002 Workshop for Verfcaton, Valdaton, and Accredtaton n the 21st Century, Johns Hopkns Unversty Appled Physcs Laboratory, Laurel, MD. Bralsford, S.C., P.R. Harper, B. Patel, and M. Ptt. 2009. An analyss of the academc lterature on smulaton and modellng n health care. Journal of Smulaton 3:130-140. Costa, A.X., S.A. Rdley, A.K. Shahan, P.R. Harper, V. De Senna, and M.S. Nelsen, 2003. Mathematcal modellng and smulaton for plannng crtcal care capacty. Anaesthesa 58: 320-327. Eldab, T., R.J. Paul, and T. Young. 2007. Smulaton modellng n healthcare: revewng legaces and nvestgatng futures. Journal of the Operatonal Research Socety 58:262-270. Günal, M.M., and M. Pdd. 2010. Dscrete event smulaton for the performance modellng n health care: a revew of the lterature. Journal of Smulaton 4:42-51. Hofmann, M. 2005. On the Complexty of Parameter Calbraton n Smulaton Models. Journal of Defense Modelng and Smulaton 2: 217-226. Katsalak, K., and N. Mustafee. 2011. Applcatons of smulaton wthn the healthcare context. Journal of the Operatonal Research Socety 62:1431 1451. Km, S.C., I. Horowtz, K. Young, and T.A. Buckley. 1999. Analyss of capacty management of the ntensve care unt n a hosptal. European Journal of Operatonal Research 115:36-46. Km, S.C., I. Horowtz, K. Young, and T.A. Buckley. 2000. Flexble bed allocaton and performance n the ntensve care unt. Journal of Operaton Management 18:427-443. Kolker, A. 2009. Process modelng of ICU patent flow: effect of daly load levelng of electve surgeres on ICU dverson. Journal of Medcal Systems 33:27-40. Ltvack, N., M. van Rjsbergen, R.J. Bouchere, and M. van Houdenhoven. 2008. Managng the overflow of ntensve care patents. European Journal of Operatonal Research 185:998-1010. Mallor, F., and C. Azcárate. 2012. Combnng optmzaton wth smulaton to obtan credble smulaton models for Intensve Care Unts. Annals of Operatons Research DOI:10.1007/s10479-011-1035-8. Marshall, A., C. Vaslaks, and E. El-Zard. 2005. Length of stay-based patent flow models: recent developments and future drectons. Health Care Management Scence 8:213-220. Oddoye, J.P., D.F. Jones, M. Tamz, and P. Schmdt. 2009. Combnng smulaton and goal programmng for healthcare plannng n a medcal assessment unt. European Journal of Operatonal Research 193:250-261. Park, B. B. and H. M. Q. 2005. Development and Evaluaton of a Procedure for the Calbraton of Smulaton Models. Transportaton Research Record: Journal of the Transportaton Research Board 1934:208-217. Rdge, J.C., S.K. Jones, M.S. Nelsen, and A.K. Shahan. 1998. Capacty plannng for ntensve care unts. European Journal of Operatonal Research 105:346-355. Vaslaks, C., and A.H. Marshall. 2005. Modellng natonwde hosptal length of stay: openng the black box. Journal of the Operatonal Research Socety 56:862-869. AUTHOR BIOGRAPHIES FERMÍN MALLOR studed mathematcs at the Unversty of Zaragoza, Span. He receved hs doctorate n mathematcs from the Publc Unversty of Navarre, n 1994. Currently, he s a professor n statstcs and operatons research. In addton to more than 20 years lecturng n smulaton, operatons research and statstcs, he has successfully appled hs knowledge n smulaton and statstcal modellng to the analyss of complex real problems n several ndustral companes and nsttutons. Hs research nterests are smulaton modellng, queung theory, functonal data analyss and relablty. Hs emal address s mallor@unavarra.es. 790
CRISTINA AZCÁRATE studed mathematcs at the Unversty of Zaragoza, Span. She receved her doctorate n mathematcs from the Publc Unversty of Navarre, n 1995. Currently, she s an assocate professor n statstcs and operatons research. She lectures n optmzaton and smulaton to cvl engneers. Her research nterests are smulaton modellng and optmzaton wth smulaton. Her emal address s cazcarate@unavarra.es. JULIO BARADO s a physcan at the ICU of the Hosptal of Navarre, Span. Currently, he s a Ph.D. student at the department of statstcs and operatons research of the Publc Unversty of Navarre. Hs research nterests le n the feld of ICU smulaton modellng and n the study of clncal decson-makng processes. Hs emal address s X017905@cfnavarra.es. 791