Pareto Set-based Ant Colony Optimization for Multi-Objective Surgery Scheduling Problem

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1 Send Odes fo Repns o epns@benhamsene.ae The Open Cybenes & Sysems Jounal, 204, 8, Open Aess Paeo Se-based An Colony Opmzaon fo Mul-Objeve Sugey Shedulng Poblem Xang We,* and Gno Lm 2 Fauly of Mehanal Engneeng and Mehans, Nngbo Unvesy, Nngbo, 52, P.R. Chna; 2 Depamen of Indusal Engneeng, The Unvesy of Houson, Houson, TX, 77204, USA Absa: Sugey shedulng deemnes he ndvdual sugey s sequene and assgns equed esoues. Ths ask plays a desve ole n povdng mely eamen fo he paens whle ensung a balaned hospal esoues ulzaon. Coneng seveal eal lfe onsans assoaed wh mulple esoues dung he omplee -sage sugey flow, a sugey shedulng model s pesened wh mulple objeves of mnmzng makespan, mnmzng oveme and balanng esoue ulzaon. A Paeo ses based an olony algohm wh oespondng an gaph, pheomone seng and updae, and Paeo ses onsuon s poposed o solve he mul-objeve sugey shedulng poblem. A es ase fom MD Andeson Cane Cene s bul and he shedulng esul by hee dffeen appoahes s ompaed. The ase sudy shows ha he Paeo se-based ACO fo mul-objeve poposed n hs pape aheved good esuls n shoenng oal end me, edung nuses oveme and balanng esoues ulzaon n geneal. I ndaes he advanage by sysemaally sugey shedulng opmzaon oneng mulple objeves elaed o dffeen shaeholdes. Keywods: An olony opmzaon, mul-objeve opmzaon, opeang oom managemen, paeo se, sugey shedulng.. INTRODUCTION One of he man hallenges n healh ae sysems n een yeas s o delve hgh qualy seve unde lmed avalable esoues. Wh he nease of agng populaon, soal demands fo sugal seve have been onsanly neased []. As a val hospal omponen, he opeang oom (OR) dvson aouns fo appoxmaely moe han 40% of a hospal s oal evenues and expenses [2]. Hene, s essenal o mpove paen flow and opmze OR managemen n ode o povde mely eamens fo he paens and o maxmze ulzaon of he avalable esoues. Sugey shedulng plays a ual ole n he OR managemen. I deemnes he opeaon me and alloaes he esoues o sugees o be pefomed n dffeen sugal speales ove a shedule peod. The oveall sugey poess nvolves hee man sages,.e. pe-sugey, sugey, and pos-sugey. The esoues equed o pefom a sugey ompse pesonnel (sugeons, anaeshess, nuses, e.) as well as fales (spealzed equpmen, pe-opeave holdng uns (PHUs), mulple ORs, pos anaeshesa ae uns (PACUs), and nensve ae uns (ICUs)). Moeove, all he equed esoues have o be ouped smulaneously. Sugey shedulng poblem has been a wdely suded op and hee exss a vas amoun of leaue n he medal and opeaon eseah. Revews abou OR plannng and shedulng have been ondued mos eenly [-6]. They povded dealed lassfaons of eseahes based on poblemspefed, opeaons eseah mehodology, and deson levels X/4 Mulple sugey flow sages and mulple esoues nvolved n OR managemen no only neases he omplexy of sugey shedulng poblem, bu also alls fo he moe ompehensve evaluaon of shedulng esul. The evaluaon on he effeny of sugey shedulng should one seveal pefomanes nludng paens sasfaon, saffs sasfaon and OR managemen effeveness. The paens sasfaon depends dely on he mely sugey aangemen, less wang me, and qualy eamen. Saffs sasfaon s usually expessed as less oveme and a moe balaned ask. The effeveness of OR managemen an be fuhe desbed as managemen os and esoues ulzaon. Theefoe, s neessay o ake mulple objeves no aoun fo sugey shedulng. Seveal sugey shedulng poblems desbed n he leaue one mulple objeves. Ogulaa and Eol developed a se of heahal mulple ea mahemaal pogammng models o geneae weey opeang oom shedules. The objeves oneed n hs sudy ae maxmum ulzaon of opeang oom apay, balaned dsbuon of opeaons among sugeon goups and mnmzaon of paen wang mes [7]. Cadoen e al. pesen a mul-objeve opmzaon model fo shedulng ndvdual ases n he sugal dayae ene of a lage Belgan hospal [8]. Sx dffeen objeves (hlden and pozed paens as ealy as possble, paens havng a lage avel dsane as muh as possble befoe a paula hou, mnmzng oveme n eovey and levellng bed oupany n boh eovey phase and eovey phase 2) ae oneed n he model. Belën e al. pesened a deson suppo sysem o develop mase sugey shedules. The sysem s bul on dffeen opmzaon poedues ha am a levellng he esulng bed oupany, onenang sugeons of he same goup n he same ooms, 204 Benham Open

2 22 The Open Cybenes & Sysems Jounal, 204, Volume 8 We and Lm Table. The noaons. Ses I Se of sugees J Se of sages of sugey, I C S Se of esoue ypes Se of sugey speales M Se of esoues n esoue ype, C C Se of esoue ypes equed fo sage j of sugey, I, j =,2, J Se of sugey sages ha an be pefomed by esoue m of esoue ype, C, m D Se of sugey omplexy (: smple; 2: Regula; : Complaed ) TW Se of avalable me wndows of esoue m of esoue ype. 2 TW TW { TW, TW, TW, TW } =. When =SG, epesens sugeon s non-ln hou; when = NC/NS, epesens nuse s wokng shf; C, m Paamees SG,NC,NS epesen sugeon, ulao and sub n esoue ype. SG, NC, NS C n The numbe of equed esoue n esoue ype fo pefomng sage j of sugey. I, j =,2,, The duaon of sage j of sugey by esoue m of esoue ype. If epesens sugeon, he sugey duaon s dffeen aodng o dffeen m. I, j =,2,,, m M T C M M C T, C w T The duaon of seup and eovey sage of sugey. The me wndow of a wokng hou day w, C [ T, T ] w =, whee Sw Ew T Sw and of a wokng day w The h avalable me wndow of esoue m of esoue ype. TW [ STW, ETW ] TW H T espevely epesen he sa me and he end me Ew =, whee STW and ETW espevely epesen he sa me and he end me of he h avalable me wndow, + N, An abay lage posve numbe P = f sugey belongs o sugey spealy s and s n omplexy d, P = 0, ohewse, I, s S, d D TW Q f esoue m of esoue ype an deal wh he sugey n sugey spealy s and n omplexy d, Q s = 0, ohewse. C, m, s S, d D M RT The egula wok me of esoue m of esoue ype. C, m Deson vaables ST The sa me of sugey. I M x = f esoue m of esoue ype s assgned o sage j of sugey, x = 0, ohewse, I, j =,2,,, m M C Z = f sugey sage k peedes sugey sage l on a same esoue, Z = 0, ohewse, k, l J. w = f esoue m of esoue ype pefom sage j of sugey whn s avalable me wndow TW, w = 0, ohewse. I, j =,2,,, m M, C TW and keepng he shedules onssen fom week o week [9]. Je and Fguea addessed a mul-objeve shedulng poblem n a Coaan hospal and fomulaed as a BIP model. Coneng he dsee seah spae, numeous vaables, onsans and mulple objeves, hey poposed seveal mea-heuss,.e. a vaable neghbohood seah, sae seah and a non-domnaed song gene algohm fo shedulng medal eamens [0]. Alhough no neessaly, he pefomanes ofen onfl wh eah ohe, meanng ha buldng an opmal shedule wh espe o one objeve goes a he os of he ohe objeves. Fuhe moe, he ombnaoal naue and he nonlneay n on-

3 Paeo Se-based An Colony Opmzaon fo Mul-Objeve Sugey The Open Cybenes & Sysems Jounal, 204, Volume 8 2 sans of he sugey shedulng poblem make exemely dfful o opmze. In hs pape, a Paeo Se-based an olony algohm s poposed o solve suh mul-objeve sugey shedulng poblem. Mos leaue eseahes solved mul-objeve sugey shedulng poblem by weghng objeves. Ou appoah negaes he Paeo se soluons o he ACO so as o make moe effen even when he onflng mulple objeves exs. The es of he pape s oganzed as follows. In Seon 2 he mul-objeve shedulng poblem s oulned. In Seon we nodue he Paeo Se-based ACO algohm and pesen he deal mehansms n suh appoah. In Seon 4, we povde he ompuaonal expemens o valdae and evaluae ou appoah. We lose n Seon 5 wh summay and suggesons fo fuue eseah. 2. MULTI-OBJECTIVE SURGERY SCHEDULING MODEL We assume ha hee s a se of sugees, epesened as I, o be pefomed n an opeang sysem wh dffeen ypes of esoues (nludng boh pesonnel esoues and fales assoaed wh sugey goups). Eah sugey has s sugey demand and esoue demand. Sugey demand deemnes a sugey an only be pefomed by seveal sugeons n a speal sugey goup, and s opeang me vaes by sugeon. Resoue demand ndaes all esoues equed fo he omplee sages of a sugey. Whehe a sage an sa suessfully s esed no only by he vaey of esoues, bu also by he pefomane of eah pevous sage. The shedulng goal s o sele he bes esoues, deemne he vaous sugees opeang sequenes, and ge he shoes makespan, he mnmum oveme and he balaned ulzaon, oneng he dveses of ypes and quany of esoues and he muual onsan of he avalable me. The noaons used fo ou model ae lsed n Table. The mul-objeve mahema model s desbed as followng: The objeve funon s o mnmze he hee objeve f, f 2 and f. mn F = ( f, f f ) 2, The hee objeve funons nlude: ) The s objeve s o mnmze he me o fnsh all sugees, so-alled makespan. f = mn max ET (2) I 2) The 2 nd objeve funon s o mnmze he vaaon oeffen of esoues wokng me (VCWT). f 2 = mn max CV C CV s he VCWT of esoues n esoue ype and s defned as he ao of he sandad devaon o mean as shown n equaon (4). I s used o evaluae he balane of esoue ulzaon. CV () () = (4) μ μ = T x (5) M = mm jj I M mm jj I T x 2 μ ) The d objeve s o mnmze he oal oveme of all esoues: f ( ) = mn max ET x RT (7) I, jj C m M The onsans nlude: ) The end me of sage j of sugey s deemned by he sa me and he maxmum duaon of all assgned esoues. ET = ST + C, mm { T x } (6) max, I, j =,2, (8) 2) Fo any wo onseuve sages j and j+ of a sugey, j+ sas mmedaely when j has fnshed s poessng sep. ET, I, j =, 2, (9) = ST ( j +) ) Any wo sages of a sugey an no o be pefomed a he same me. ET q ET T, p < q J, I (0) p q 4) A esoue an only be assgned o one sugey sage a a me. ET el ET + H ) k ( Z T, el k l J,,, e I, () ETk ETel + HZ T, k k, l J,, e I, e, (2) 5) The sugey mus be pefomed whn he same wokng day..e. ST ET C, w TSw ST TEw T T max{ T 2 x }, + C, m M, =,2, n, w N () 6) The exaly equed numbe of esoues n esoue ype s assgned o pefom sage j of sugey. mm x = n, C (4) 7) The sugey mus be saed and fnshed whn he assgned esoue s avalable me wndow. ST ET x x TW ( w STW ) = TW ( w ETW ), M (5) = I, j =,2,, C, m (6) 8) The esoue (hee manly efe o sugeon, ulao and sub) assgned o a sugey mus have he ably n e-

4 24 The Open Cybenes & Sysems Jounal, 204, Volume 8 We and Lm qued sugey spealy and equed qualfaon of dealng wh sugey omplexy. x P Q, I, =,2, dd ss { SG, NC NS} j, C,, m M (7) 0) Followngs ae bnay vaables { 0,} x, I, j =,2,, C, m M (8) { 0,} z, k, l J, (9) { 0,} w, I, =,2, j, C, m M, (20) TW. PARETO SET BASED ANT COLONY ALGO- RITHM FOR MOB SURGERY SCHEDULING The ombnaoal naue and he nonlneay n onsans of he above sugey shedulng poblem make obanng an opmal shedulng esul hallengng. Insead, we amed a a mea-heus appoah fo sub-opmal soluon and developed a Paeo se based ACO algohm o solve he mul-objeve sugey shedulng poblem. Sne he omplee sugey shedulng deemnes boh he esoues alloaon fo eah of he sugey sages and he sequenng of sugees n me peod, a wo-level an gaph s desgned. In ou pevous eseah, an an olony algohm wh suh wo-level an gaph was poposed fo solvng he sugey shedulng poblem wh sngle objeve of makespan (Lae n seon 4, we name as ACO-SOB) []. The oue level gaph s defned as a sugey gaph. The nodes n he sugey gaph epesen he sugees, and he deonal a ndaes he peedene sequene. The an foagng pah s he shedulng sequene of he sugees [2]. The nne level gaph s defned as esoues gaph. The nne-gaph node epesens he oal avalable esoues of he OR managemen sysem along he hee-sage sugey poedue. An foagng pah n nne level gaph deemnes he esoues seleon fo eah spef sage dung a sugey... Pheomone defnon and updae saegy As o ACO fo sngle objeve opmzaon, he pheomone defnon and updae saegy ae desgned o effenly eod and enhane he pheomone o a spef (an opmal) soluon []. Howeve, n he mul-objeve opmzaon, hee may no longe exs suh opmal soluon, bu a Paeo se of soluons. Theefoe, a spef pheomone seng, alled sngle-pah-mul-pheomone (SPMP), and he oespondng pheomone updae saeges ae nodued o ake no aoun he mpa of seveal Paeo opmal soluons due o mulple objeves. SPMP allows layng mulple pheomone value on a sngle an pah aodng o he numbe of objeves. A pheomone veo onsss of hee pheomone values whh ae assoaed o hee objeves (=,2,) s defned on eah pah. In oue an gaph, he sequene-elaed pheomone ( ) s defned n equaon (2) o ndae he sengh of sequene fom node o node j. Is value s deemned by he ndvdual objeve sengh and he assoaed weghs, =,2,. = (2) = = ) ( s ) ( + (22) Q, f an s goes hough (, j) n hs eaon (2) ( s ) = L s 0, ohewse As o he pheomone updae saegy (equaon (22) and (2)), an eaon-paeo-opmal (IPO) updae saegy s adoped o luse ans o pahs wh Paeo opmal soluons. We se Q as he pheomone sengh veo assoaed wh hee objeves. L s he h objeve value of a s soluon s. Only he pheomone value assoaed o he ndvdual bes objeve s enfoed, he ohe wo pheomone value n veo keep unhanged. Howeve, evapoaon wll be happened fo all pheomones n pheomone veo, denoes he pheomone evapoaon ae. The nne sugey-elaed pheomone ( In ) s defned o assoae sugey wh esoue m of esoue ype. Is deal defnon and updae saegy ae he same o above sequene-elaed pheomone, and ae desbed n equaon (24) o (26). In In = (24) In = In In ( ) + ( s ) = (25) = (26) k The nne esoue-elaed pheomone In s defned fo m eodng he nfomaon elaed o esoue ulzaon dung an an onsung nne esoue alloaon soluon. I sas o be effeve as long as an an enes o nne esoue gaph, and ends o be empy when an an goes ou o oue gaph. In any sugey sage, one a esoue s seleed, s oppouny o be seleed by ohe sugey should be edued so as o balane he esoue ulzaon. Theefoe, s updaed loally afe vsng eah node (.e. on a-sep ompleon) as equaon (27), whee q 0 s he deemened pheomone value. By loal pheomone updang, he possbly of ans awlng hough he same pah deeases, hus an effevely avod he uneven ulzaon of esoues. In k In k = (27) m m q0.2 Consu Paeo Se Indvdual an aveses he wo-level an gaph o buld a feasble shedule soluon whh nludes boh he sugees sequene and he esoues alloaon fo ndvdual sugey. The deal pobabls anson ule used n an soluon onsuon an be found n ou pevous wok []. Usually

5 Paeo Se-based An Colony Opmzaon fo Mul-Objeve Sugey The Open Cybenes & Sysems Jounal, 204, Volume 8 25 n mul-objeve opmzaon, ypally hee exss no feasble soluon ha mnmzes all objeve funons smulaneously. Theefoe, Paeo opmal soluons,.e. soluons ha anno be mpoved n any of he objeves whou mpamen n a leas one of he ohe objeves, ae poposed. Sugey shedulng poblem n hs wok s desbed as a mul-objeve opmzaon poblem wh hee objeves: mn F = ( f, f 2, f ). Assume S s he feasble soluon se. In mahemaal ems, a feasble soluon s S s sad o domnae anohe soluon s 2 S, epesened as ( s s2 ), f ) f ( s) f( s2), {,2,} and 2) f j ( s) < f j ( s2), j {,2,}. Soluon s s alled Paeo opmal, f hee does no exs anohe soluon ha domnaes. The se of Paeo opmal s ofen alled he Paeo se. Paeo se based an olony algohm fo mul-objeve opmzaon (ACO-MOB) s o onsu suh Paeo se fo eah an yle, o keep updae Paeo se along wh he eaon, and o fnally appoah o Paeo opmal se. The fnal deson an be he one based on deson make s pefeene n suh Paeo opmal se. Assume a feasble soluon se S wh N soluon, and a Paeo se PS, nally se as nl. A hallenge-lke algohm s desgned n ACO o onsu Paeo se as followng: Algohm: Consu Paeo se by hallenge-lke algohm Se Paeo se: PS = whle ( S ) do Randomly sele a soluon fom S as he wnne : = ; Updae feasble soluon se: S = S ; Se unvsed se: V=S; Whle ( V ) do Randomly sele a soluon fom V as he hallenge; f ( ) {Updae feasble soluon se: S = S } else f ( ) {Replae wnne and updae se: = ; S = S } Updae unvsed se: V = V ; End whle PS = PS ; Consu Paeo se: { } End whle End.. ACO-MOB Algohm Despon The dealed poedue of he poposed ACO-MOB algohm fo he mul-objeve sugey shedulng poblem s explaned as follows: Algohm: he Paeo se based ACO-MOB fo sugey shedulng Sep: Buld an an gaph model Whle (eaon emnaon ondon no me) do Sep 2: Pu m ans on abay node Sep : Inalze pheomone al Sep 4: Consu a feasble soluon se S by ans avesng oue and nne an gaph Whle (an emnaon ondon no me, k<m) do Sep 4.: Inalze soluon abu= ; and sugees se I Sep 4.2: Consu an an soluon by vsng a node n oue gaph based on he sae anson ule; hen updae abu = abu { I } and I = I \{ I} Sep 4.: An enes no he nne gaph, onsu esoue se G Sep 4.4: Consu a esoue alloaon soluon fo eah demandng esoue ypes do Sep 4.4.: Consu an an soluon by vsng a node n nne gaph based on he sae anson ule Sep 4.4.2: Loal updae nne esoue-elaed pheomone End fo Sep 4.5: Deodng & Updae esoues me wndow Sep 4.6: f I, go o Sep 4.. else add an k s soluon abu o feasble soluon se S: S = abu S End Whle Sep 5: Consu a Paeo se PS now by hallenge-lke algohm. Sep 6: Fom an eaon feasble se S eaon by ombnng Paeo se PS now wh pevous eaon Paeo se PS eaon: Seaon = PSeaon PSnow Sep 7: Consu a fnal eaon Paeo se PS eaon by e-apply hallenge-lke algohm Sep 8: Updae sequene-elaed and sugey-elaed pheomone n boh oue and nne an gaph based on IPO updae saegy. End Whle Sep 9: Sele a soluon S * fom a fnal Paeo se PS eaon aodng o deson make s pefeene. End 4. EMPIRICAL STUDY The poposed ACO-MOB algohm s mplemened wh Malab and s un on a PC unnng Wndows XP wh Inel and GB of memoy. 4.. Tes Case Despon Daa fo he expemen was exaed fom MD Andeson Cane Cene, one of he wold s mos espeed ane eamen fales. A daly shedulng daa se was bul aound 28 ases n 7 ORs o be saffed by 2 RNs and 4 subs eah wokng one of he fve shfs sang a 6:0 AM and endng a :0 PM. Sugal elaed nfomaon and daly ose nfomaon ae gven n efeene [4]. Fou pefomane measuemens: end me, VCWT, maxmum oveme of nuses, and oal oveme of nuses ae used fo ompason. End me,.e. he fnshng me of all 28 sugees o be sheduled, s o measue he effeny of he shedulng. VCWT s defned as he ao of he sandad devaon o mean and s used o evaluae he balane of esoue ulzaon. The smalle value n VCWT means he bee ulzaon among esoues. A zeo value means a fully balaned assgnmen n esoues ulzaon. Sne n eal-lfe OR managemen, nuses ae assgned o wok n dffeen shfs, heefoe nuses ae fuhe dvded no goups aodng o shfs when alulang he VCWT of

6 26 The Open Cybenes & Sysems Jounal, 204, Volume 8 We and Lm Table 2. The opmal ACO-MOB paamees. M Q Q 0 NC_max w:w2:w [50,.5,60] :0.2:0.6 Table. Sugey shedulng esuls by ACO-MOB algohm. No. OR Culao Sub No. OR Culao Sub nuses. Oveme (OT) eods he oal addonal me equed n addon o he egula wokng hous and dely effes he os of OR managemen. Suh OT measuemen s fuhe evaluaed by boh he maxmum OT among nuses and he oal OT among nuses. An auhozed oveme s an addonal me ha nuses an be assgned f a sugey s no fnshed by he end of he egula shf The Opmal ACO-MOB Paamees Seng The bas ACO-MOB paamees nlude he numbe of ans (m), pheomone fao (), heus fao (), evapoae ae (), pheomone nensy (Q), deemened pheomone value (q0), and weghs of mulple objeves ( ). Those paamees have mpa on he an exploaon and an followng known pheomone, whh bngs dffeene n algohm s onvegene and soluon qualy. Seveal expemens ae bul o denfy he opmal paamees by dffeen adjusmens, and Table 2 lss he opmal ACO- MOB paamees aheved. 4.. Compuaonal Resul Dsusson A ompason expemen s bul o evaluae he pefomane of hee dffeen shedulng appoahes. These hee appoahes ae eepvely, he manual shedulng n MD Andeson Cane Cene (named as Manual ), he bas ACO appoah wh sngle objeve of makespan (named as ACO-SOB ), and he poposed ACO-MOB n hs wok. The fnal sugey shedulng esul wh esoues assgnmen solved by ou ACO-MOB s lsed n Table. The ompason esuls by hee dffeen shedulng appoahes n seveal measuemens ae shown n Table 4. Fom Table 4, boh ACO-MOB and ACO-SOB aheve a shedule wh all 28 sugees fnshed by 8:0 and has.5 hou eduon ompaed o manual shedulng. Suh one and half hou eduon n end me an be a good poof ha sugey shedulng by ACO has advanage n shedulng effeny. As o he esoues ulzaon, exep VCWT value n shf nuse s 0 beause of only one nuse on duy, he VCWT of ohe esoues nludng ORs and nuses n shf and shf2 all show vayng mpovemens. The VCWT of ORs n ACO-SOB and ACO-MOB s 0.2 and 0.2, whh has a eduon aound 6%. Boh VCWT of nuses n shf and shf2 by ACO appoah have a muh bee value of 0.8(0.2) and 0.05(0.06) ompaed o hose by manual shedulng of 0.5 and o.7. The mpovemen s 68% (6%) fo shf and 68% (6%) fo shf2. Thee s no muh dffeene n VCWT of nuses and ORs by ACO-MOB o ACO-SOB. OT means he neasng of os, whh s no waned n OR managemen. The hd obsevaon fom ompason esuls s he mpovemen n OT measuemens n boh

7 Paeo Se-based An Colony Opmzaon fo Mul-Objeve Sugey The Open Cybenes & Sysems Jounal, 204, Volume 8 27 Table 4. Compason on hee appoahes. End Tme Vaaon Coeffen of Resoues Wokng Tme: VCWT Nuse OR Shf Shf 2 Shf Indvdual Nuse Maxmum OT (h) Nuses Toal OT (h) Manual 9: ACO-SOB 8: ACO-MOB 8: maxmum OT of ndvdual nuse and he oal OT of all nuses. Indvdual nuse s maxmum OT s edued o 2.75hous and has a 9.25hous mpovemen ompaed o 2 hous n manual wok. Toal OT of nuses s also edued fom 2.5hous o hous by ACO-SOB, and o 7hous by ACO-MOB. I eods aound 6% mpovemen by ompang ACO-MOB wh ACO-SOB. In sum, he ompason esul shows ha he poposed Paeo se-based ACO algohm fo sugey shedulng has good pefomane on shoenng oal me and alloang esoues fo sugey shedulng. 5. CONCLUSION AND ONGOING WORK We have developed a Paeo se-based ACO appoah fo solvng mulple objeve sugees shedulng poblem ha ases n lage opeang sues. The poblem s omplaed beause of nvolvng he omplee sages of sugey flow and mulple esoues and onsans n OR managemen. The mahema model of suh sugey shedulng poblem wh mulple objeves of mnmzng makespan, mnmzng oveme and balanng esoue ulzaon s poposed. Due o he ombnaoal naue of he poblem, he nonlnea onsans nvolved, and he onflng objeves oneed, a Paeo se-based ACO algohm by amng a ahevng sub-opmal soluons s poposed n hs pape. A Paeo ses based an olony algohm wh oespondng an gaph, pheomone seng and updae, and Paeo ses onsuon s poposed o solve he mul-objeve sugey shedulng poblem. A wo-laye nesed ACO suue as well as he elave mehansms (.e. he SMSP pheomone defnon, he pheomone updang saegy, and he Paeo se soluon) s pesened. The an avel gaph s desgned as an oue sugey gaph and mulple nne esoue gaphs. We emphaszed boh he nfomaon elaed o he good soluon n makespan and he balaned ulzaon of he esoues. The global (eaon-bes) and loal (on a-sep ompleon) pheomone updae ules ae adoped. A es ase fom MD Andeson Cane Cene s olleed o valdae ou mehod. I povdes a oal 28 ases n 7 ORs o be saffed 2 RNs and 4 subs whn one day. Sugeons and nuses ae defned wh spealy, ole, qualfaon and avalably onsans. The shedulng esul of he poposed Paeo Se-based ACO algohm s ompaed wh he eal lfe manual sugey shedulng esul, and fom he bas ACO appoah wh sngle objeve of makespan. Fou measuemens,.e. end me, ndvdual nuse s maxmum oveme, oal oveme of nuses and he vaaon oeffen of wokng me of esoue, ae evaluaed. Compason esuls ndae a supeo pefomane fo he poposed Paeo se-based ACO n geneal. I an be onluded ha he algohm an solve he mulple objeve sugey shedulng poblem effevely whn a easonable alulaon me, whle a he same me povde a shoenng end me and a elave balaned esoue alloaons. Fuue eseah wll be n he deon of exendng ou ACO algohm o solvng he sugey shedulng poblems wh uneanes and moe eals onsans ase n aual OR managemen n hospal, lke sugeons/nuses pefeene onsans n medal eam. CONFLICT OF INTEREST The auhos onfm ha hs ale onen has no onfl of nees. ACKNOWLEDGEMENTS The Poje s suppoed by Zhejang Povnal Naual Sene Foundaon of Chna (LY2G0007), Nngbo Naual Sene Foundaon (20A6009) and K.C.Wong Magna Fund n Nngbo Unvesy. REFERENCES [] D. A. Ezon, J. H. Lu, M. A. Maggad, and C. Y. Ko, The Agng Populaon and Is Impa on he Sugey Wokfoe, Annals of Sugey, vol. 28, pp , 200. [2] B. Denon, J. Vapano, and A. Vogl, Opmzaon of sugey sequenng and shedulng desons unde uneany, Healh Cae Managemen Sene, vol. 0, pp. -24, [] B. Cadoen, E. Demeulemeese, and J. Belën, Opeang oom plannng and shedulng: A leaue evewognal, Euopean Jounal of Opeaonal Reseah, vol. 20, pp , 200. [4] S. A. Edogan, B. T. Denon, J. J. Cohan, L. A. Cox, P. Kesknoak, J. P. Khaoufeh, and J. C. Smh, Wley Enylopeda of Opeaons Reseah and Managemen Sene John Wley & Sons, Denon, 200. [5] F. Gueeo, and R. Gudo, Opeaonal eseah n he managemen of he opeang heae: a suvey, Healh Cae Managemen Sene, vol. 4, pp. 89-4, 20. [6] J. H. May, W. E. Spangle, D. P. Sum, and L. G. Vagas, The Sugal Shedulng Poblem: Cuen Reseah and Fuue Oppounes, Poduon and Opeaons Managemen, vol. 20, pp , 20 [7] S. Ogulaa, and R. Eol, A Heahal Mulple Cea Mahemaal Pogammng Appoah fo Shedulng Geneal Sugey Opeaons n Lage Hospals, Jounal of Medal Sysems, vol. 27, pp , Jun 200. [8] B. Cadoen, E. Demeulemeese, and J. Belen, Opmzng a mulple objeve sugal ase sequenng poblem, Inenaonal Jounal of Poduon Eonoms, vol. 9, pp , Jun 2009.

8 28 The Open Cybenes & Sysems Jounal, 204, Volume 8 We and Lm [9] J. Belën, E. Demeulemeese, and B. Cadoen, A deson suppo sysem fo yl mase sugey shedulng wh mulple objeves, Jounal of Shedulng, vol. 2, pp. 47-6, Apl [0] S. V. Je, and J. R. Fguea, Mul-objeve shedulng and a esoue alloaon poblem n hospals, Jounal of Shedulng, vol. 5, pp. 5-55, O 202. [] W. Xang, J. Yn, and G. Lm, Modfed an olony algohm fo sugey shedulng unde mul-esoue onsans, Advanes n Infomaon Senes and Seve Senes, vol. 5,no. 9, pp , 20. [2] J. Yn, and W. Xang, An Colony Algohm fo Sugey Shedulng Poblem., Leue Noes n Compue Sene, vol. 7, pp , 202. [] M. Dogo, and T. Süzle, An Colony Opmzaon MIT Pess, Cambdge, [4] A. Mobashe, G. Lm, J.F. Bad, and V. Jodan, Daly shedulng of nuses n opeang sues, IIE Tans aons on Healhae Sysems Engneeng, vol., pp , 20. Reeved: Sepembe 6, 204 Revsed: Deembe 2, 204 Aeped: Deembe, 204 We and Lm; Lensee Benham Open. Ths s an open aess ale lensed unde he ems of he (hps://eaveommons.og/lenses/by/4.0/legalode), whh pems unesed, nonommeal use, dsbuon and epoduon n any medum, povded he wok s popely ed.