Optimization of Nurse Scheduling Problem with a Two-Stage Mathematical Programming Model

Transcription

1 Asa Pacfc Managemen Revew 15(4) (2010) Opmzaon of Nurse Schedulng Problem wh a Two-Sage Mahemacal Programmng Model Chang-Chun Tsa a,*, Cheng-Jung Lee b a Deparmen of Busness Admnsraon, Trans World Unversy, Tawan b Deparmen of Informaon Managemen, Naonal Yunln Unversy of Scence and Technology, Tawan Acceped 6 June Absrac Ths paper consrucs a wo-sage mahemacal programmng model o solve he nurse schedulng problem n order o assgn nurses o shfs over a schedulng perod so ha ceran consrans (organzaonal and personal) are sasfed. In he frs sage, he nurse opmal vacaon schedules are solved by a self-schedule programmng ha can check for any volaon of governmen regulaons, hospal managemen requremens, and schedulng farness. In he second sage, he nurse roser schedule s arranged and a Genec Algorhm (GA) s furher adoped o derve he opmal schedule. An emprcal case sudy s performed and he resuls show ha he proposed approach can solve he nurse schedulng problem effcenly. In addon, can also be easly modfed o su dfferen cases encounered n hospals. Keywords: Inernaonal mahemacal programmng model, genec algorhm, self-schedule, nurse schedulng 1. Inroducon Nursng saff schedulng s an essenal ask n manpower resource managemen. The schedulng qualy drecly nfluences he nursng qualy and workng moral. Nurse schedulng problems represen a subclass of schedulng problems (Ender, 2005). Typcally, personnel schedulng problems are hghly consraned and complex opmzaon problems (Erns e al., 2004). The need o ake no accoun ndvdual preferences furher complcaes he process. In recen years, he emergence of larger and more consraned problems has presened a real challenge. Because obanng good qualy soluons can lead o a hgher level of personnel sasfacon (Burke e al., 2006). Cheang e al. (2003) added nursng saff s preferences no he facors o be consdered when preparng work schedules. In addon, Bard and Purnomo (2007) consdered oher facors, such as nurse workforce, hospal work and hospal schedulng regulaons, o esablsh a schedule-makng decson ree. The consran condons for nurse schedulng are broad, and may dffer from case o case. Some of he consran condons even conflc wh each oher. For nsance, he shf preference of nursng saff may volae he requremen for shf farness. In pracce, he nurse chefs arrange he schedule based on her subjecve experence. To mee he complcaed suaons wh ever-ncreasng paen demands and a lmed nurse workforce, he chefs may requre more me and effor han ever o deal wh he schedulng and sll fal o be far o all he saff. Consequenly, he nurse schedulng ssue remans challengng, and developmen of a more sophscaed approach o solve he problem deserves furher exploraon. Rondeau (1990) and Belzhoover (1994) mananed ha self-schedulng reduces he rao of shf-changes, ncreases opporunes for on-he-job ranng, and ncreases ndvdual * Correspondng auhor. Emal: jmsa@mdu.edu.w 503

2 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) auonomy and self-awareness, whle also helpng saff o beer enjoy her socal and famly lves. Abbo (1995) also noed ha self-schedulng reduces nerference n he personal lves of nursng saff, reducng shf-changes suaons and requess for leave, and ulmaely boosng sasfacon owards work. Though self-schedulng has numerous advanages, communcaon and coordnaon problems are s major lmaons. Consequenly, hs sudy employs a modfed self-schedulng mehod o mprove hese weaknesses. Wh regard o solvng he nurse schedulng problem, Smh and Wggns (1977) proposed hree caegores of approach: cyclcal, heursc, and mahemacal programmng. The cyclcal schedulng approach has shf and vacaon arrangemens se up by he nurse chefs based on he needs of he nurse un, he regulaons of hospal, and he number of nursng saff. The cyclcal schedulng approach normally ulzes a cyclcal schedulng paern on a fxed me range. For he heursc schedulng approach, he nurse chefs ofen consruc a decson ree wh consderaon of nursng saff workforce, nurse servce paerns, hospal schedulng polcy, and oher facors, and hen ulze he resulng schedule on a cyclcal bass. The mahemacal programmng approach s a specal mahemacal model developed o respond o he schedulng problems for dfferen cases. Normally, s consruced wh objecve funcons and consran equaons and hen ulzes approprae algorhms o search for he opmal soluons. The aforemenoned hree nurse schedulng approaches all have her advanages and dsadvanages. The cyclcal schedulng approach can be convenenly execued; however, a new schedule mus be arranged when nurses need o change her job or shfs, whch n s que common n pracce. The heursc schedulng approach needs a decson-makng ree o develop he schedulng rules, bu because he neracon among nursng saff s very complcaed, he ree requred s usually huge. Thus, when he consran condons are numerous, wll generally be dffcul for he heursc schedulng approach o aan a reasonable soluon, and he nurse schedulng acves canno be easly processed (Harvey and Mona, 1998). The mahemacal programmng approach has a subsanal level of dependency on he cases addressed, and when dealng wh dfferen cases requres furher reformulaon, meanng addonal me and effor need o be spen on he projec. A nurse schedulng sysem can be bul up and execued n varous ways. Ahuja and Sheppard (1975) developed a four-module four-sage neracve cyclcal schedulng sysem, whch used a compuer o arrange schedules for dfferen nursng saff. Ther sysem consss of four modules: (a) work paern selecor - denfes cyclcal schedule paerns from he npu nformaon, and dfferen case hospals may have dfferen work paerns; (b) work schedule assembler - combnes nursng saff wh he work paerns generaed by he frs module; (c) work schedule projecor - dsplays he work schedules for boh he ndvduals and he organzaon; (d) work predcon and allocaon of saff - desgns a work load ndex accordng o he requremens of he nursng saff and hen assgns work o saff based on hs work load ndex. Smh and Wggns (1977) developed a bached hree-sage cyclcal schedulng sysem o arrange he nurse schedulng, whch ncludes (a) summarzng he requremens of saff for specfed uns on weekly bases; (b) generang prelmnary schedules and evaluang he schedules wh he consran condons o check f here s any conflc; (c) manually adjusng he prelmnary schedule and creang he fnalzed schedule. Kosreva and Jennng (1991) ndcaed ha a nurse schedulng sysem should nclude survey and scheduler modules. Ineger programmng, mxed neger programmng, goal programmng, lnear programmng, nework programmng, and consran programmng have all been used o solve he nurse schedulng problem. For nsance, Mller e al. (1976), Ozkarahan and Baley (1988) ulzed he neger programmng o search for a schedule wh he lowes averson 504

3 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) devaons; whereas Warner (1976) employed goal programmng wh mulple-choces o solve nurse schedulng problems. In addon, Kosreva e al. (1978) and Bell e al. (1986) developed mxed neger programmng models; whle Arher and Ravndran (1981) used 0-1 goal programmng o solve wo-sage cyclcal schedulng problems. Musa and Saxena (1984) adoped 0-1 goal programmng and heursc search, and Azaez and Sharf (2005) also used 0-1 goal programmng o solve nurse schedulng problem. Baley (1985) developed a cyclcal schedulng model wh neger programmng. Brge e al. (1998) ulzed lnear programmng o oban he soluon ha can smulaneously mnmze he oal paymen, sasfy he saff preference, and level he nurse workforce. Harvey and Mona (1998) employed nework programmng o sudy he cyclcal and non-cyclcal nurse schedulng problems on he bass of 12-hour shfs. Fnally, Meyer (2001) used consran programmng o solve nurse schedulng problems. Oher echnques have also been found, and Randhawa and Sompul (1993) developed a heursc schedulng decson-makng suppor sysem o handle he mulple-goal programmng problems wh bnary varables; whereas Ackeln and Dowsland (2000) used genec algorhms o solve nurse schedulng problems. Dowsland and Thompson (2000) combned Tabu search and nework programmng o esablsh a non-cyclcal schedulng sysem; whle Dmr (2002) used genec algorhms o solve cyclcal schedulng problems. Ackeln and L (2006) used Bayesan opmzaon algorhm o solve nurse schedulng problems. Goldberg (1989) and Sharf (2000) poned ou ha he advanages of GA are numerous. GA can solve he opmzaon problem wh mulple varables; can perform parallel processng, whch can effecvely save processng me. GA does no need o calculae he dfferenal value of he fness funcon, because he naural selecon process s deermned by he fness of chromosomes; s a mulple pons search nsead of a sngle pon search approach, whch has a relavely hgh probably of fndng he value ha s que close o he global opmum raher han beng rapped n a local search. In addon, GA s suable for use n a compuerzed envronmen. In lgh of hese advanages, hs sudy aemps o develop a dfferen approach by negrang a mahemacal programmng approach wh he GA o solve he nurse schedulng problem. When dealng wh dfferen cases, he solvng algorhm does no need o be re-adjused. Insead, we only need o se new objecve funcons and consran condons. Ths sudy wll ncorporae he hospal managemen requremens, governmen regulaons, and nursng saff s shf preferences no a wo-sage mahemacal model. In he frs sage, he nurse opmal vacaon schedules are solved by a self-schedule programmng ha can check for any volaon of governmen regulaons, hospal managemen requremens, and he schedulng farness. In he second sage, he nurse roser schedule s arranged and a Genec Algorhm (GA) s furher adoped o solve he opmal schedule. The remanng pars of hs paper are organzed as follows. Secon wo develops he nurse work and vacaon model and he nurse roser schedule model. Secon hree nroduces he GAs for solvng boh models. In secon four, an Obsercs hospal n Kaohsung, Tawan s suded as our emprcal case, and fnally he concludng remarks are gven. 2. Model developmen Two mahemacal programmng models are proposed n hs sudy. Model I s he holday schedule model based on self-schedulng n order o allow he shf-able o apply nursng saff resources mos effcenly. Model II s scheduled he enre shf-able o oban he mos approprae shf-able. 505

4 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) A blank shf-able wll be offered for he nex monh due o he holday self-schedule mehod. All nursng saffs mark her own desred off days on he blank shf-able, he scheduler has o negoae and se prory o solve he conflc problem. To ease he burden of he scheduler and ensure he farness, Model I s desgned o denfy he opmal soluon of a complee off-shf able schedulng usng LINGO due o he decson-makng varables are deermned. Model II s creaed o resul he enre shf-able by arrangng he shfs for each nursng saff n he second sage. Model II s a mxed neger non-lnear programmng problem (MINLP) usng GA o solve hs parcular ssue. 2.1 Mahemacal model for he holday self-schedule (Model I) The execuon seps for he holday self-schedule mehod are lsed below: Sep 1: Ls he number of off days and holdays avalable for each nursng saff, hen have all of he nursng saff mark her own desred off days. Sep 2: When oo many saff selec he same day, he scheduler has o negoae and se prory, rankng he mporance of he reasons of dfferen ndvduals for wanng he day off. When negoaon s unsuccessful, he dspue s seled by a lucky draw. Sep 3: Calculae he shorage of off days on holdays and weekdays for each saff member. Sep 4: Schedule a complee off shf able. To ease he burden of he scheduler, and ensure he farness and qualy of he off shfs able, hs sudy desgns a mahemacal programmng model o denfy he opmal soluon of Sep 4. Ths sudy uses he followng noaon for problem modelng: (1) Indexng Ses: : nursng saff ndex ; = 1, 2,, N : dae ndex; = 1, 2,, T (2) Parameers: N: oal numbers of nursng saff T: schedule days H : shorage of off days on holdays for he -h nursng saff C : shorage of off-days on weekdays for he -h nursng saff R : maxmum number of nurses avalable for -h day. T sun : ses of Sundays. T sa : ses of Saurdays. T ho : ses of Naonal Holdays. T 1 : ses of Holdays. [ T1 = (Tsun Tsa Thol )] T 2 : ses of weekdays. [ (T - T )] T = 2 1 D E, N ) :number of nursng saff requred on day (evenng, overngh) shf of ( 506

5 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) h day. E (N) :he average days of evenng (overngh) shf on whch flexble shf nursng saff are requred o work n he enre shf able. D E, N ) :accumulaed frequency of -h nursng saff s workng on day (evenng, ( overngh) shf n he enre shf able. (3) Decson Varables ' R 1 = 0 arrange he -h oherwse nurse saff ohave off-dayson -h day n he nalshf able 1 arrangehe -h nurse saff o have off-dayson -h day n heenreshf able. R = 0 oherwse Dum = 0 or 1, Dummy varable ' ' ' 1 D ( E, N ) = 0 oherwse 1 arrangeheh nursesaff = 0 oherwse D E N 1 = 0 1 = 0 oherwse he h nurse saff prearrange day (evenng, overngh) o haveday shf on h day. arrangeheh nursesaff o have evenngshf on h day. arrangeheh nursesaff o haveoverngh shf on h day. oherwse Mn [ Z, Z Z ] 1 2, 3 shf on h day Z = (1) 2 2 Z1 = ( R ) N R N, Tsun (2) 2 N N 2 Z2 = ( R ) N R N, Tsa (3) 2 N N 2 Z3 = ( R ) N R N, Thol (4) N N Subjec o R T 1 T 2 ' = R + Dum, N, Dum = H, N T (5) (6) Dum = C, N (7) R R, T (8) N R + R ( 1) + R ( 2) + R( 3) + R ( 4) + R( 5) + R ( 6) 1, N, T (9) 507

6 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) Equaon (1) s he objecve funcon of he off-shf able n hs sudy, and was derved from formulas (2), (3), and (4). Ths equaon negraes he hree funconal ndexes above no a sngle objecve funcon hrough lnear amalgamaon. The wegh of each funcon s deermned by rankng he saff preferences. Equaons (2), (3) and (4) represen he varance of off-days number on he Saurday, Sunday and naonal holdays. Consran (5) ndcaes he defnon of varable and ensures he raonaly of, whle avodng conradcon beween he orgnal and fnal off-shf ables. Consrans (6) and (7) are desgned o sasfy all ndvdual members of nursng saff wh regard o her off days and off holdays. Consran (8) exss o preven oo many people akng leave on he same day and leadng o an nadequae number saff beng on duy. Fnally, consran (9) s creaed o sasfy he un polcy of no more han sx consecuve work days for any ndvdual. 2.2 Mahemacal model of schedulng for he enre shf-able (Model II) Afer schedulng a complee off-shf able, he enre shf-able can be smply compleed by arrangng he shfs for each nursng saff on her work days. Smlarly, hs sudy proposes a MINLP model as followng: Mn [ Z, Z, Z, Z, Z, Z Z ] Z = (10) , 7 Z 1 = ( E( 3) D ( 2) E( 1) D ) (11) Z 2 = ( D( 2) N( 1) R ) (12) Z 3 = ( E( 2) N ( 1) R ) (13) Z 4 = ( N( 2) R( 1) N ) (14) N T ' ' ' ' ' ' Z5 = ( D + E + N ) ( D D + E E + N N ) / 2 (15) A = N T E E Z = [ A ] [] represens he Gaussan Funcon (16) A 6 = N N Z = [ A ] [] represens he Gaussan Funcon (17) 7 Subjec o N N N D D, T E = E, T N = N, T D + E + N + R =, N, T N 1) + D 1, N, T 508 (18) (19) (20) 1 (21) ( (22) Equaon (10) represens he objecve funcon of hs mahemacal model, whch s derved from equaons (11) hrough (17). Ths sudy negraes he above-menoned sevenfuncons no a sngle arge one. The wegh of each ndvdual funcon s ranked based on nursng saff preference, and he mos favored s assgned he larges ndex value, for example

7 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) seven. Fnally, he ndexes are oaled and he wegh ndex s calculaed based on he percenage aken by each funcon. Equaon (11) expresses he oal number of mes he evenng-day paern occurred durng any four-consecuve days. Moreover, equaons (12), (13) and (14) represen he oal occurrence of day-overngh-off shfs, evenng-overngh-off shfs and overngh-offoverngh shfs n any hree consecuve days, respecvely. Equaon (15) calculaes he number of nfrngemens beween he fnal fxed shf-able and he former saff-preferred shf-able; meanwhle, equaons (16) and (17) express he oal exceed numbers over he requred evenng and overngh shfs for all nursng saff. Consran (18) ensures ha he acual schedulng resul mees he saffng requremens for he daly day-shf, whle consrans (19) and (20) ensure ha s me for he daly evenng and overngh shfs. Furhermore, consran (21) ses he lm ha each nursng saff should no work more han egh hours a day, whle consran (22) prohbs overnghday shfs. 3. Model solvng algorhms and process NCS Ths sudy dvdes nurse schedulng operaon no wo models. The solvng algorhms and process for boh models s brefly nroduced below: (1) Holday self-schedule Durng he execuon seps for he holday self-schedule mehod (secon 2.1), mos decson-makng varables are deermned (from seps 1~3). Hence, hs sudy seeks he soluon wh he sofware package LINGO for he sep 4. (2) Overall schedule Model II s MINLP ha can be solved by he LINGO, a GA s furher adoped o solve he opmal schedule and s wren usng MATLAB language. The soluon-seekng seps are presened below. Sep 1. Producng an ndvdual s decded by codng When applyng he heredy algorhm o solve he problem, he varable desgned mus be ransformed, regardless of wheher s a measurng ndex or a numercal ndex, no code form. Dfferen codng mehods are adoped for dfferen ypes of problems. Genes are used code, and recorded along wh he heredary feaures of ndvdual creaures. The sze of each ndvdual s se as a marx of N x T, wh he horzonal axs as he schedulng dae, and he vercal axs as he labeled number of each ndvdual. The bye value s an neger from 1 o 4. 1 represens day-shfs, 2 denoes evenng shfs, 3 sands for overngh shfs, and 4 represens off-shfs. Three nurses work he evenng shf on he frs day and wll have a day off on he second day, work he day shf on he hrd day, and he overngh shf on he fourh day, (please refer o fgure 1 for deals). Because he off days for each nursng saff are deermned by holday self-schedulng, he locaon of bye value 4 s fxed for each ndvdual. Furhermore, each bye has only four possbles: 1, 2, 3 and 4. Sasfacon of he lmed formula 22 for each nursng saff means ha hey should no work more han egh hours a day. 509

8 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) Fgure 1. The codes are arranged n he form of a srng represenng chromosomes. Sep 2. Defnng approprae funcons Before seekng a soluon usng GA, he funcons of appropraeness mus be defned o assess he effcacy of he soluon. Ths sudy ses he approprae funcon as he objecve funcon (17). Sep 3. Reproducon When selecng an ndvdual for he purpose of reproducon, naural selecon ndcaes ndvduals wh hgher adapaon capacy have a beer chance of beng seleced. The mehod of selecon s he smples and he mos commonly used rao mehod. In hs, he rae of reproducon s deermned by usng he percenage of oal funcon value represened by he funcon value of appropraeness of an ndvdual; namely, he chance of an ndvdual beng chosen depends on s appropraeness. Ths sudy seleced he presenaon mehod of 1 2 N choosng probably, where ψ { pop = X, X,..., X } represens he N pop soluons n generaon, and N pop represens he number of ndvduals n a race. The chance of each soluon X beng seleced as a paren srng s P ( ) : X 2 [ f M ( ψ ) f ( X )] [ f M ( ψ ) f ( X )] P( X ) = 2 (23) X ψ In formula (23), f M ( ψ ) s defned as he poores arge funcon value n he generaon chromosome, whle f ( X ) represens he arge funcon value of every soluon of X n generaon. Sep 4. Crossover The crossover procedure s desgned accordng o he specal feaures of he model, explaned as follows. Before mang s conduced, wo paren ndvduals are seleced by usng he reproducon algorhm, and wo cung pons were deermned randomly. Genes are reproduced from he cung pons of he paren generaon n he same locaon as he chld generaon. Oher genes of he chld generaon are replaced by he genes whn he cung pons of anoher paren, as presened n Fgure 2 (represened by he frs row). Ths sudy pus forward he hypohess ha f wo paren ndvduals proceed o a crossover 510

9 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) Fgure 2. Paren ndvdual before crossover. Fgure 3. Chld ndvdual afer crossover. algorhm, he wo cung pons are 1 and 5, and Fgure 3 llusraes he resuls of he crossover. Sep5. Muaon The heursc muaon of hs sudy s self-desgned based on he specal feaures of model, explaned n he consrans, above, and each ndvdual aemps o sasfy consrans (19), (20), and (21), ensurng ha he schedule resul sasfes he daly requremen for dayshfs and manans he mnmum nursng manpower for evenng and overngh shfs. The muaon rules are hus as follows: (a) each ndvdual s capable of muaonal calculaon and each sasfes consrans (19), (20) and (21). (b) The un of ndvdual muaon s he lne, and he muaonal algorhm reas each lne one by one. 4. Emprcal case analyss 4.1 Case overvew and problem descrpon Ths sudy adoped nursng saff schedulng n he Dep. of Obsercs and Genecology (Ob-Gyn) of a hospal n Kaohsung as a case sudy. Ths case sudy nvolved hree shfs a day, namely day-shf (AM 8:00~PM 4:00), evenng shf (PM 4:00~AM 0:00) and overngh shf (AM 0:00~AM 8:00). A one-monh cycle me was used. The un conaned 14 nursng saff, wh requred manpower of fve saff for he day-shf, and hree and wo saff for evenng and overngh shfs, respecvely. The consans n hs sudy s schedulng sysem are se as follows: 1. Shfs durng annual leave are scheduled hrough loery. Moreover, he head nurse gves he prvlege of off shfs on New Year s Day, Tomb-Sweepng Day, Dragon-boa Fesval and Md-auumn Fesval o hose who dd no ake annual leave; 2. Leave on New Year s Day, Tomb-Sweepng Day, Dragon-boa Fesval and Md-auumn Fesval should no be regsered n advance; 3. The shf-able marks he number of off days and holdays avalable per saff member per monh; 4. Holdays nclude Saurdays, Sundays and Naonal Holdays, bu no New Year s Day, Tomb-Sweepng Day, Dragon-boa Fesval and Md-auumn Fesval; 5. Unless necessary, he schedulng of regular off days on Saurdays, Sundays, or Naonal Holdays s avoded, wh aemps beng made o evenly dsrbue off days a hese mes; 6. The maxmum number of consecuve workng days scheduled for a gven ndvdual s sx; 7. Pror schedulng s permed for specal reasons (brhs, weddngs, engagemens, overseas ravels, or specal long leave.) 511

10 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) For acqurng he opmal resul of Model I and Model II, wo sofware (LINGO and MATLAB) were run on a personal compuer wh a Penum ΙΙI 450MHz CPU. Wh regardng o he sze of he Model II s lsed as followng: he number of varables s equal o 4 * N * T + 3 * T ; he number of consrans s 3 * T + 2 * N * T. The nformaon on off days for each nursng saff n he Ob-GYN should be enered. Ths sudy s conduced n March 2005; he analyss and comparson beween he shf-able. The orgnal manual schedule s as follows. 4.2 Analyss of he holday self-schedule n he OB-GYN Dep. Based on he execuon seps and he mahemacal model (model I) of he holday selfschedule mehod, he off day shf able s shown n he Table 1. Durng he schedule roaon, nursng saff scheduled off days as hey desred (113 days n oal). However, owng o surplus manpower on specfc days, he scheduler had o make some coordnaon and adjusmen. One hundred and wo days of he acual off day shf able were he same as he orgnal schedule from he preferred off day able. Therefore, mos of off days (90.3%) are dencal o he orgnal schedule chosen by he saff. Table 2 gves he resuls of he survey of he nursng saff regardng he holday self-schedule mehod. 4.3 Sysem parameers and he solvng resul In he second sage, GA s adoped o opmze he nurse schedule. Ths sudy esablshed parameers such as populaon sze, probably of muaon and so on for execung GA. Table 3 lss he seleced facors and he level values ha were seleced o ncrease he effcency of problem solvng. A oal of egh (2 x 2 x 2) combnaons of dfferen conrollng facors were used. Thry ess were conduced for each combnaon. From Table 4, he ermnaon condon s se a he 50h generaon, and he chance of obanng he bes soluon s se a he 30h generaon. Therefore, hs sudy se he ermnaon condon a he 50h generaon. From es numbers 5~8 n Table 5, under he ermnaon condon, populaon sze and he ncrease n he probably of mang provded no clear asssance n obanng he opmal soluon, and sgnfcanly ncreased he sysem calculaon me. Therefore, hs sudy ses he populaon sze a 40 and he probably of mang a 0.7. Table 1. The off-day shf able. R: Off-day 512

11 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) Iems Table 2. Survey on holday self-schedulng n he OB-GYN Dep. 1. The pracce of holday selfschedulng ncreases schedulng auonomy and makes easer for me o decde my own holdays. 2. The sysem mproves my chances of geng days off when I wan hem, and enhances my sasfacon regardng schedulng. 3. My chances of geng he days off ha I waned were ncreased, and wh a few excepons mos of he off days were selfscheduled. Srongly Agree 513 Agree No Commen Dsagree Srongly Dsagree 92.86% 7.14% % 14.4% % 14.4% In comparson wh he enre shf-able, Table 5 s derved from usng he GA and he orgnal manual shf-able. The resuls acqured from he model can be easly obaned, and he approach presened n hs sudy s beer han he manually scheduled mehod, as can deermne he mos approprae shf-able n jus fve mnues. Therefore, he model presened here serves as a good reference for any medcal un seekng o solve s nursng shf schedule problem. Table 3. Seleced conrollng facors and her level value. Tes No. Alas Conrollng Facors Level Value A Termnaon Condon 2 nd Level (30,50) B Populaon Sze 2 nd Level (40,80) C Probably of Muaon 2 nd Level (0.3,0.7) Termnaon condon (A) Table 4. Parameers: Combnaon and relaed soluon quales. Sze of populaon (B) Probably of muaon (C) Bes soluonseekng resuls Probably of geng he bes soluon Average solvng me (sec.) (22/30) (22/30) (23/30) (25/30) (27/30) (26/30) (28/30) (27/30) Concludng remarks Ths sudy presens wo effcen mahemacal programmng models and apples GA o deermne he opmal nursng saff shf schedule. Unlke mos exsng approaches, he new approach has he ably o buld schedules. Ths sudy demonsraes a wo-sage effcen and

12 C.-C. Tsa, C.-J. Lee / Asa Pacfc Managemen Revew 15(4) (2010) flexble mahemacal programmng model and arrved wh sold resul raher han radonal schedulng. Upon he needs of he nursng saff, hs approach s able o adjus he objecve funcon, consrans and her weghed value o ncrease he flexbly and generaly of he schedulng model. In parcular, hs sudy proposed wo sgnfcan ssues hghly valued: one s he saff farness prncple, and he oher s he self-deermned schedulng prncple. Hence, he ulmae schedule ruly reflecs he preferences of nurse saff as well as he managemen requremens. The expermenal resuls demonsraed he srengh of hs approach. The conrbuons are as follows. Frsly, hs sudy appled GA o oban he enre shfable. Ths model copes wh dfferen schedulng cases wh he proposed sold core algorhm. I s hghly adapve o oher applcaons. Secondly, hs sudy proposes a heursc muaon mehod o sgnfcanly reduce he schedulng seup me. Thrdly, hs sudy developed a schedulng sysem o handle he schedulng acves. Ths approach can reduce he nurse chef s schedulng workload and also ncrease he nurse saff sasfacon by provdng he saff vacaon farness and more respec on self-deermnaon schedule. Fuure research could be exended o reflec he effec on employee sasfacon and fnancal performance mprovemen. Ths research gves some prelmnary answers of how o nclude human-lke learnng no schedulng algorhms and hs approach mgh be neresed o praconers and researchers n areas of schedulng and evoluonary compuaon. Evaluaon Index Table 5. The comparson beween he manual and enre shf-ables. Manual Shf- Table Enre Shf-Table No evenng-day-evenng-day shfs on any four consecuve days (number of volaons). 0 0 No day-overngh-off shfs on any hree consecuve days (number of volaons). 4 0 No evenng-overngh-off shfs on any hree consecuve days (number of volaons). 3 0 No overngh-off-overngh shfs on any hree consecuve days (number of volaons). 0 0 Machng he preferred enave shf-able (number of volaons). 2 0 The oal exceed numbers over he requred evenng shfs for all nursng saff. 6 0 The oal exceed numbers over he requred overngh shfs for all nursng saff. 6 0 Sasfyng day-shf needs (number of volaons). 0 0 Mananng mnmum labor requremens regardng evenng shfs (number of volaons). 0 0 Mananng mnmum labor requremens regardng overngh shfs (number of volaons). 0 0 Ensurng here are no overngh-day shfs on any wo consecuve days (number of volaons). 0 0 Tme scheduled 2-3 workdays (sec.) 514

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