Multi-objective Nurse Scheduling Models with Patient Workload and Nurse Preferences

Transcription

1 Management 01, (5): DOI: 10.59/.mm Mult-obectve urse Schedulng Models th Patent Workload and urse Preferences Gno J. Lm 1,, Areou Mobasher 1, Murray J. Côté 1 Department of Industral EngneerngThe Unversty of HoustonHouston, TX, 7704, USA Department of Health Polcy and Management Texas A&M Health Scence Center College Staton, TX , USA Abstract The purpose of ths paper s to develop a nurse schedulng model that smultaneously addresses a set of multple and oftentmes conflctng obectves n determnng an optmal nurse schedule. The obectves e consder are mnmng nurse labor costs, mnmng patent dssatsfacton, mnmng nurse dle tme, and maxmng ob satsfacton. We formulate a seres of mult-obectve bnary nteger programmng models for the nurse schedulng problem here both nurse shft preferences as a proxy for ob satsfacton and patent orkload as a proxy for patent dssatsfacton are consdered n our models. A to-stage non-eghted goal programmng soluton method s provded to fnd an effcent soluton that addresses the multple obectves. umercal results sho that consderng patent orkload n the optmaton models can make postve mpacts n nurse schedulng by (1) mprovng nurse utlaton hle keepng hgher nurse ob satsfacton and () mnmng unsatsfed patent orkloads. Keyords urse Schedulng, Optmaton, Goal Programmng, Mathematcal Model 1. Introducton 1.1. Problem Descrpton The cost of health care s rsng every year n the Unted States. Accordng to the Toers Perrn health care survey, health care costs have ncreased on average 6% n 008 compared to 007[1]. The cost of developng ne technologes and treatments, Amerca's agng populaton, rsng personal ncome and generally ncreasng demand for health care are some of the reasons for ths rapd groth n health care cost. Operatons research technques can help reduce health care costs by mprovng resource utlaton, hosptal patent flo, medcal supply chan, staff schedulng, and medcal decson makng, to name a fe. In ths paper, e focus on staff schedulng, especally, nurse schedulng. urse shortage s a despread problem[]. ot havng enough sklled nurses n clncal settngs has caused a sgnfcant negatve mpact on patent outcomes, ncludng mortalty[]. In ths context, hosptal admnstrators and nurse managers are n dre need to optmally utle and retan currently avalable nurses thout eopardng ther ob satsfacton. When one schedules nursng staff, t s knon that consderng ther shft preferences can ncrease ob Correspondng author: gnolm@uh.edu(gno J. Lm) Publshed onlne at Copyrght 01 Scentfc & Academc Publshng. All Rghts Reserved satsfacton, hch leads to savngs n labor costs due to reduced nurse turnover and the subsequent other ssues that are related th feer nurses[4, 5]. Schedulng nurses to meet a hosptal's daly demand and satsfy staffng polces such as those dctated by a unon contract and regulatons mandatng specfc nurse-to-patent ratos s an extremely co mplex task to perform. Confoundng ths envronment s the fact that the nurses are non-homogeneous th respect to ther skll set, experence, employment type,.e., part tme versus full tme, and general avalablty. Further, the demand for nurses vares n accordance th patent census. Because some of these obectves may conflct th each other, nurse schedulng must be carefully done to capture approprate trade-offs among the dfferent obectves. In ths paper, e develop a nurse schedulng model that smultaneously addresses a set of multple and oftentmes conflctng obectves n determnng an optmal nurse schedule; 1) mnme nurse labor costs, ) mnme patent dssatsfacton, ) mnme nurse dle tme, and 4) maxme ob satsfacton. We formulate a seres of mu lt-obectve bnary nteger programmng models for the nurse schedulng problem (SP) here e use nurse shft preferences as a proxy for ob satsfacton (e.g., f the nurse gets the shft s/he ants, ob satsfacton s ncreased) and patent orkload as a proxy for patent dssatsfacton (e.g., hgher patent orkload mples ncreased patent dssatsfacton). We use a to-stage, non-eghted goal programmng soluton method to fnd an effcent soluton that addresses the multple obectves. Unlke prevous

2 150 Gno J. Lm et al.: Mult-obectve urse Schedulng Models th Patent Workload and urse Preferences research[6], e defne patent orkload to be more than smply a care unt's census. Rather, patent orkload s veed as a factor for the care unt's case mx for a gven shft. If the case mx s hgh, there ll be a correspondngly hgh orkload placed on nurses and, n turn, patent orkload ll be hgh. In effect, a hgh patent orkload mples a hgher orkload for the nurses. Smlarly, f the case mx s lo, there ll be lo orkload for the nurses. For example, f a patent requres an MRI, he may need to receve a prelmnary examnaton, a dye necton, and the MRI procedure. Consequently, ths patent requres three dfferent servces hch can be consdered as three separate orkloads. If patent orkload s less than nursng capacty, there ll be dle nurses. Therefore, e defne nurse dle unts as the number of patent needs that could be handled by the current nursng capacty. By defnton, nurse dle unts ll alays be greater than or equal to the number of dle nurses. 1.. Lterature Reve The research on nurse schedulng and staffng s vast and e make no attempt to reve all of the lterature n ths doman. Rather, the lterature that s relevant to our manuscrpt s as follos. The artcles by Cheanget. al. [7]and Burke et. al.[8] mght be to of the most comprehensve surveys n nurse schedulng and rosterng problems. Burke et al descrbed a general nurse schedulng and rosterng problem, and evaluated varous models and soluton approaches found n more than 140 artcles and PhD dssertatons. Some of these papers can be found n[9,10,11,1]. Snce then, most researchers have focused on heurstc approaches that provde an effcent soluton to SP due to the complexty of modelng and solvng optmaton models[1,14,15,16]. For example, Burke et. al.[17] developed a decson support system for the nurse rosterng problem. They ntroduced a hybrd varable neghbourhood search (VS) algorthm that can effectvely handle problems th feer than 0 nurses. Integer programmng (IP) has been dely used for solvng the SP problem[1,18,19,0,1,].purnomo and Bard[]ntroduced a ne methodology usng Lagrangan penalaton technques to solve an nteger programmng model th complex constrants to satsfy nurse shft preferences and mnmng costs consderng nurses demands. Recently Belen and Demeulemeester[] have developed an ntegrated nurse and surgery schedulng problem usng IP and column generaton methods to solve the model. A nonlnear nteger programmng approach as also reported n the lterature[4].they hghlghted ho nurse-to-patent ratos and other polces mpact schedule cost and schedule desrablty. Most of the proposed optmaton models deal th a sngle obectve functon. Hoever, SP s truly a multple obectve optmaton problem[14,5,6,7,8]. Parr and Thompson[7] developed a mult-obectve frameork for a nurse schedulng problem usng a eghted cost functon here smulated annealng as used to solve the problem. Recently, Maenhout and Vanhoucke[5], presented a branch and prce algorthm for solvng a mult-obectve nurse schedulng problem ncorporatng some penalty scores assocated th schedulng neffcences. Other relevant nurse schedulng papers can be found n[9,0]. Some papers have consdered patent demand [16,1,]. Ogulata et al[1] used an IP model to assgn patents n a hghly demanded hosptal to physcans. They defned the patent demand and the hosptal prortes thout consderng staff preferences. Although there s a vast amount of lterature n SP, none of the optmaton models have addressed both the mpact of patent orkload and nurse preferences smultaneously. Therefore, our purpose here s to ntroduce a general and easy-to-mplement mult-obectve nurse schedulng problem that consders patent orkloads as ell as nurse preferences. Our man goal s to present the mportance of addng patent orkload n nurse schedulng problems. The rest of the paper s organed as follos. In Secton, e formulate to nteger programmng models. The frst one s the base model for nurse shft preferences and the second model ncludes patent orkload. Both models are formulated as mult-obectve optmaton problems. We provde a soluton algorthm for solvng the models n Secton. umercal results are presented n Secton 4 to sho the effect of addng patent orkload to the model. We provde concludng remarks and drectons for future research n Secton 5.. urse Schedulng Models Hosptals have ther on rules and regulatons for assgnng nurses to shfts. Such rules and regulatons can be thought of as constrants n an optmaton model. Constructng a mathematcal model can be qute complex f e consder all possble constrants for a gven model. Hoever, some constrants are nherently common n all hosptal staffng. Ths leads us to develop a basc structure for a general nurse schedulng problem. One such constrants s based on shft lmtatons. Hosptal staffng must cover 4 hours a day.therefore, hosptals hre nurses to fulfl multple shfts per day and these nurses fall nto one of to categores of nurse types: full tme and part tme. Part tme nurses ork based on ther eekly contracted ork hours. ote that a hosptal cannot force a nurse to ork more than the regular contract hours. Therefore, nurse shortages (.e., not havng suffcent nurses to meet demand) and dled nurses (.e., havng more nurses than needed to meet demand) can add substantal labor expense to the operatng costs of hosptals. Havng these constrants n mnd, e defne to models: assgnment model and patent orkload model. The Assgnment Model ncludes nurse preferences, but not patent orkload hle the Patent Workload Model

3 Management 01, (5): consders both of these constrants..1. Assgnment Model (AM).1.1. Assgnment Model (AM) Our am s to assgn dfferent types and grades of nurses for hosptal staffng n order to mnme operatng costs. In addton to nurse types, nurses are further classfed based on ther skll level, ork experence, certfcaton, or other qualfcaton crtera. The nurse scheduler decdes ho many shfts the hosptal needs per day. We defne a shft as a consecutve 4-hour perod such that e have 6 shfts per day. Sometmes, due to nurse shortages, hgher sklled nurses can be assgned to shfts that loer sklled nurses are often assgned to. Hoever, the opposte s not approprate because t equates to poor qualty of care or an napproprate or llegal use of resources. A master schedule can be developed for each schedulng perod such as a eek, month, or quarter. Based on ths nformaton, e defne the follong ndexes n Table 1. otaton F P Table 1. otaton Descrpt on Set of all full tme nurses Set of all part tme nurses ={ F P } Set of all nurses G Set of all avalable grades of nurses S = {1, K, 6} Set of all shft s T Plannng horon Set of eeks n the plannng horon.1.. Input Parameters There are several nput parameters to the optmaton model. Frst, the nurse schedule ll be constraned by the number of avalable nurses ( a ) that a hosptal has for a specfc shft. If there s a need for more nurses than ths number ( a ), the hosptal ll experence a shortage of nurses for the shft. As stated before, ths shortage of nurses can add to the hosptals operatng cost. By the same token, any dle nurses ll contrbute to ths cost because the hosptal s not fully utlng ts orkforce. We defne a gst as the number of avalable nurses for each grade g G,shft s Sand day t T. Second, hosptals assgn dfferent nurses to dfferent tasks based on ther skll level (grade). Assumng that nurse grades are gven as nput, e defne nurse grades n the model as follos: b g = 1, If the grade o f nurse s hgher or equal g G ; 0, otherse. Snce each type of nurse has a dfferent compensaton structure per shft per day, the thrd nput s the cost of assgnng a nurse to a shft; C. The next nput s the type st of contract such as full tme and part tme. We assume that each nurse orks up to ther contracted hours,.e., e do not consder overtme. We address ths ssue by hrng part tme nurses so that overtme s not necessary n the model. All nurses can be assgned to shfts that nclude days, nghts, and eekends. Other addtonal nput parameters are defned as follos: h : umber of shfts nurse mustork n a eek, : Maxmumnumberoflate nghtshfts a fulltme nursecanbeassgned to, n the plannng horon, W : Maxmum number of eekend shfts a fulltme nurse can be assgned to, n the plannng horon, r : Mnmum number of nurses assgned to each shft, m : Penalty score of assgnng a nurse s to a shft s Sbased on nurses preference, m : Penalty of utlng part tme nurse P n the schedulng, 4 m gs : Penalty of assgnng hgher grade nurses to late nght and eekend shfts, 5 m : Penalty of assgnng fulltme nurses to late nght and s eekend shfts..1.. Decson Varables Our task s to determne hch nurse should be assgned to hch shft of each day. Therefore, the decson varables of the optmaton model are defned as follos: y gst = 1, f nurse th grade s assgned to shft s Sn day t T ; 0, otherse Constrants urse Grades: Each nurse has a grade that reflects the skll sets one has. Ths grade ll determne hat knd of tasks the nurse has the authorty and capablty to perform. Typcally, a nurse th a hgher grade can perform tasks that a nurse th a loer grade performs. Equaton (1) rtes ths constrant: ygst bg,",, s S, t T. (1) o More Than One Grade Assgnment per urse per Shft: Constrant () ensures that, for each shft, nurses must be assgned to only one of the grades they are authored to perform. ote that ths constrant ll prevent a nurse th a hgher grade from beng assgned to more than one grade level per shft. ygst 1, ", s S, t T. () urse Avalablty: Constrant () ensures that the nurse schedule s constraned by the number of nurses that the hosptal currently has. ygst agst,", s S, t T. () Consecutve Shft Lmtaton 1: Constrants (4), (5), (6),(7) and (8) make sure that a full tme nurse cannot ork on more than to consecutve shfts per day.

4 15 Gno J. Lm et al.: Mult-obectve urse Schedulng Models th Patent Workload and urse Preferences y y, gst yg ( s+ 1) t + yg ( s-1) t g ( s+ 1) t g ( s-1) t 1, " s { S -{1,6}}, F, t T. (4) + y " s { S-{1,6}}, F, t T. (5) y,,. g (1) t yg () t " F t T (6) y,,. g (6) t yg (5) t " F t T s S y, " t T, F. gst Consecutve Shft Lmtaton : Furthermore, f a full tme nurse s assgned to a late nght shft, the nurse must not be assgned to an early mornng mmedately after the prevous nght shft. ote that part tme nurses can be assgned to any shft and day. ( y + y + y + y ), g (5) t g (6) t g (1)( t + 1) g ()( t + 1) " F, t T. (9) urse Workng Hours: Each nurse must ork based on the contracted hours ( ) per eek. Ths means that a full tme nurse s expected to ork 40 hours per eek and part tme nurses can ork based on the contract one has such as hours or 0 hours. Full tme nurses cannot ork more than a specfc amount of late nght and eekend shfts n a month. We assume that the frst day of the month s Monday thout loss of generalty. Constrants (10), (11), (1) and (1) rte these condtons: 7 gst t=7( - 1) + 1s S 7 gst t=7( - 1) + 1s S s { S-{,4,5}} t T y = h, ",. (7) (8) F (10) y h, ",. y, " F. gst gst t={7,7-1} s S P (11) y W, " F. (1) (1) Ths set of constrants s equvalent to the mnmum number of consecutve orkng days as found n[7]. Mnmum urse Assgnment per Shft: Each shft must have at least r nurses assgned,.e., a shft cannot be defned thout r nurses ( r 1). ygst r, " s S, t T (14) Ths set of constrants s to ensure that no empty shft ll be alloed n the fnal soluton..1.5.obectve Functons There can be many goals n schedulng nurses such as mnmng the nurse assgnment cost, maxmng nurse ob satsfacton, or any other goals that the hosptal has set n terms of schedulng nurses. Some of these goals may naturally conflct th each other. In ths subsecton, e categore to groups of obectves. One of these groups s common to perhaps all hosptals,.e., mnmng assgnment cost and maxmng nursng ob satsfacton, hle the second group addresses specfc hosptal requrements or condtons. Mnmng Assgnment Cost: Cost of assgnng nurses to each shft s one of hosptal's man concerns. Therefore, e defne our frst obectve functon to mnme nurse assgnment cost. The total cost ll be based on the hosptal's compensaton structure that ncludes salary, benefts, and any other extra expenses that the hosptal must be responsble for hrng nurses: mn =. 1 Cst ygst s S t T (15) Ths s a common obectve functon and t has appeared elsehere[19]. Maxmng ursng Job Satsfacton: Full tme nurses are vtal components n hosptal operaton. Hoever, nursng shortages have been an ncreasng orldde problem due to factors such as a lack of traned nurses and lo ob satsfacton[]. Many approaches have been developed to address ob satsfacton problems such as the score card approach to dentfy and aggregate nurse preferences on shft assgnment[19] or the aucton approach to trade shfts n an aucton[4]. The score card approach s one the most ell-knon and easy to mplement approaches n lterature. In ths approach, each nurse s gven a sheet of upcomng empty shft assgnments. Based on personal preference, the nurse s asked to assgn penalty scores n such a ay that a smaller penalty should be assgned to a preferred shft hle a hgher penalty should be gven to an undesrable shft. Some master schedulers may share ths round of schedule preferences among nurses and resolve some conflctng shfts that no one ants to take or that everyone ants to take. Ho to assgn penaltes to dfferent shfts s an mportant ssue. Hoever, t s beyond the scope of ths paper. Therefore, e assume that such data are gven to us n advance. There are many papers dealng th assgnng penalty scores n lterature[5]. urse preferences have been consdered n many papers ether by defnng obectve functons[6] or constrants[0]. We use the score card approach to consder nurse preferences. All nurses are requred to fll out the penalty score sheet based on ther preferences. For example, a nurse ho prefers nght shfts ould gve a loer penalty on nght shfts hle nurses ho do not prefer nght shfts ould put don hgher penalty scores for the shfts. Therefore, our obectve s to mnme the summaton of these penalty scores that reflect nurse preferences: mn =. ms ygst (16) F s S t T Although t s mportant that shft preferences of nurses

5 Management 01, (5): should be a consderaton n formulatng a desrable schedule, ths goal may conflct th other constrants such as contracted orkng hours and shft lmt regulatons f adequate coverage over the schedulng tme horon s not met. Therefore, there should be a reasonable balance beteen shft preferences and hosptal requrements n schedulng ts staff. Hosptals often hre part tme nurses n order to make up for the shortages n full tme nursng staff. The cost of hrng part tme nurses ll vary dependng on the type of shfts, e.g., eekdays versus eekends, day tme versus late nght tme, etc. So far, e have dscussed to obectves that are common for most hosptals. Hoever, hosptals also operate based on ther nternal rules or regulatons. There can be many dstnctve requrements a hosptal may have and have been modelled usng soft constrants[9]. We do not attempt to address all possble hosptal specfc goals n ths paper. Hoever, the follong three goals are ntroduced to demonstrate the optmaton modellng aspects n staff schedulng. These goals are selected based on the cost of assgnng nurses to shfts. Full Tme over Part Tme: There has been an ncreasng trend for hosptals to hre part tme nurses because t s harder to fnd full tme nurses. Part tme nurses are more flexble n shft assgnments and are generally less expensve. Hoever, the core of nurse schedulng s stll based around full tme nurses. Therefore, our thrd obectve s to gve a hgher prorty to full tme nurses n assgnng shfts over part tme nurses. mn y =. ms gst s S t T (17) Hgher Grade Full Tme urses to Regular Day Shfts: We prefer to assgn hgher grade full tme nurses to regular day shfts due to ncreasng rates of assgnng them to late nght or eekend shfts. Ths s knon as the shft dfferental. Shft dfferental, outlned fully n Cvl Servce Rule 6.8 (a) and (c), s used to recrut ob applcants and retan current employees by provdng hgher pay for shft ork and non-standard ork hours pay006. Therefore, salary rates for nght and eekend shfts s normally pad more than regular hour shfts. Ths means that hgher grade nurses ll receve a hgher hourly rate by orkng late nght shfts and eekends. Ths goal s formulated as follos: mn y = 4. 4 mgs gst s S t T (18) Full Tme urses to Regular Day Shfts: We prefer to assgn full tme nurses to regular day shfts as much as possble. Ths s due to the pay rate ncrease for assgnng a full tme nurse to late nght or eekend shfts. These rates are dfferent n varous hosptals based on shft dfferentals hossal. Ths goal s formulated as follos: mn y = 5. 5 ms gst s S t T (19) Obectves (17) and (18) can also be consdered as cost functons. Hoever, n many real-lfe stuatons, t s often dffcult to estmate these costs. Therefore, these goals are expressed as penalty functons relatve to the compensaton functon (15). Mult-Obectve Optmaton Model: Based on the obectve functons descrbed n Secton.1.5 and constrants descrbed n Secton.1.4, our mult-obectve optmaton model s formulated as: ( Mn 1, Mn, Mn, Mn 4, Mn 5) s/ t: (0) constrants: {(1), K, (14)} y {0,1}, "(,, s S, t T ). gst.. Patent Workload Model (PWM) We extend our assgnment model to ncorporate patent orkload n the optmaton model. Ths model s ntended to sho ho patent orkload can affect the outcome of the model, (.e., nurse schedulng). It s bascally the assgnment model th an extra obectve functon and auxlary constrants. For the orkload model formulaton, e need to kno ho many patent orkloads a nurse can handle n each day ( k ); namely, a nurse-to-patent rato. Based on k, e can determne the number of patent orkloads that a nurse should be assgned to per shft. We defne satsfed customers as the total number of patent orkloads ho receve hosptal care durng the schedulng perod. In addton, e need to kno the estmated patent orkload based upon the hosptals census ( D ) over the plannng horon. Patent Workload Requrement Constrant: We may not be able to meet all demands that patents request. Hoever, our goal s to provde the hosptal care to at least b percentage of the total patent orkload to ncrease customer satsfacton. 7 s S t=7( - 1) + 1 ky bd, ". gst (1) Addtonal Obectve Functon: We defne an addtonal obectve functon that measures the dfference beteen the eekly patent orkload and the number of patents ho receved hosptal care. The obectve functon () s to mnme ether the patents ho dd not receve the requested care or the number of nurse dle unts durng the perod. 6 7 mn = D - k y. gst s S t=7( - 1) + 1 () By addng the obectve functon and the constrant to the assgnment model (0), our master mult-obectve optmaton model that addresses patent orkload s formulated as follos:

6 154 Gno J. Lm et al.: Mult-obectve urse Schedulng Models th Patent Workload and urse Preferences Mn 1, Mn, Mn, ŁMn 4, Mn 5, Mn 6 ł s/ t: constrants: {(1), K, (14),(1)} y {0,1}, "(,, s S, t T ). gst. Soluton Method () The optmaton models that e attempt to solve have multple obectves. Many algorthms have been reported for solvng mult-obectve optmaton (or goal programmng) models (MOOM)[7]. To man categores of soluton approaches are the Weght method and the Preemptve method[8]. Both of these methods requre that the decson makers prorte the obectve functons. The most common soluton approach s the eghted sum method. In ths approach, a ne obectve functon s developed based on a lnear combnaton of the goals: n = l, here l > 0, " = 1, K, n. (4) =1 Algor thm 1 T o-stage non-eghted GP: ormaled Weght Met hod I. DIVIDE and ORMALIZE: Intale: =1, n = Total number of obectve functons, < n Whle < n do 1. Solve a sngle obectve ( ) BIP problem: mn = c y Ay = b, By b, y {0,1}. 1. Record the optmal obectve value. ormale obect ve funct on. If % e then - =. Else + e % =, e >0 e end f end hle II. AGGREGAT E: n 5. Defne a ne obectve functon: = l =1 %. 6. Solve the follong BIP model: mn Ay = b1, By b, y {0,1}. Decson makers ll frst rank the goals (.e., obectve functons) based on ther perceved mportance of each goal aganst other ones. Then eghts ll be assgned to the obectve functons n such a ay that the relatve mportance ll be accounted approprately. The process of selectng eghts ( l ) can be a dauntng task. Many approaches have been developed for addressng ths problem[9]. We note that a eghted obectve functon th carefully selected eghts sometmes may not guarantee that the fnal soluton ll be acceptable. In ths case, the decson maker needs to redesgn the eghts based on the outcome of the prevous tral. The second ssue s that the obectve functons often have dfferent scales n magntude. Ths dfference n scale makes t more dffcu lt to select approprate eghts. We address these ssues by applyng the ormaled Weght Method as shon n Algorthm. Ths method provdes a non-dmensonal obectve functon to make eght selecton easer. Our method s comprsed of to stages: 1) Dvde and ormale and ) Aggregate. In stage 1, sngle obectve optmaton problems th a set of constrants n Secton.1.4 are solved one at a tme: mn = c y s.. t Ay = b1, (5) By b, y {0,1}. Once an optmal soluton of the sngle obectve problem s found, the obectve functon s normaled by frst canterng, -, and then dvdng, -, by the optmal obectve value.ote that may take a value of ero. Ths ll cause the specfc dvson not to be defned. We fx ths problem by addng a s mall value e to both the numerator and the denomnator[40]. In stage, the mu lt-obectve functon s regrouped by a lnear combnaton of the normaled functons and solves the follong optmaton problem to fnd an optmal schedule: mn = l % s.. t Ay = b1, (6) By b, y {0,1}. Based on the obectve functon transformaton, e make to observatons. - Observaton 1: % = 0, = 1, K, n, n the optmal soluton of (6). Observaton : In the BIP model (6), the obectve functon n n =1 < mn = l % > (7) =1 s equvalent to < mn = n lx, here x = >. =1 (8) Therefore, e use n our optmaton models.

7 Management 01, (5): Table. urse Informaton urse. employment type urse ID Grade k h Cost ($) Per shft W Full Tme 1,, ,, Part Tme A A 1, A A Part Tme 14, 15, A A Table. Soluton of the Assgnment Model Obectve % % - ( ) % ( / ) We do not clam that our method ll resolve all the nherent ssues that the eghted sum method has, such as fndng approprate eghts that reflect the relatve mportance of each goal, producng solutons that are not Pareto effcent. But, e attempt to ease the process of rankng (or ratng) dfferent obectves by normalng the scale of the obectve functon values, and then assgn the eghts f t s necessary. umercal results are presented n Secton 4 to sho the effect of addng patent orkload to the model. 4. umercal Results In Secton, e have dscussed to mult-obectve optmaton models. The assgnment model s a typcal nurse schedulng model that does not consder patent orkload. Snce one of our ams s to understand the effect of consderng patent orkload n the model, e have proposed the patent orkload model. In Secton, e have provded an mproved method for solvng mult-obectve optmaton problems. Therefore, numercal experments are desgned n ths secton to analye the behavor of these models th and thout patent orkload, and to test computatonal performance of our proposed soluton algorth m. Frst, e defne the shfts over consecutve four hour nterval startng at mdnght. Ths means that there are sx shfts per day: (mdnght to 4 a.m.), (4 a.m. to 8 a.m.),, and (8 p.m. to mdnght). Based on ths assumpton, e generated to data sets to test our models and to sho the effcency of the soluton algorthm n to dfferent nursng capacty confguratons. The frst set of data (Secton 4.1) contans 10 full tme and 6 part tme nurses hle the second set of data (Secton 4.) conssts of 0 full tme and 6 part tme nurses. In order to understand the behavor of our model, multple patent orkload scenaros are generated. For each scenaro a set of 10 experments have been mplemented and the soluton tme s calculated based on the average soluton tme of all the experments. Optmaton models are formulated n GAMS and solved usng CPLEX 1.1 on a Workstaton th.8 GH Intel Xeon quad CPU runnng Wndos Server 008 R1 th 16GB RAM Test Data Set I Consder a care unt that can have at most 10 full tme and 6 part tme nurses (see Table ). The full tme and three part tme nurses have to dfferent grades hle the other three part tme nurses have only one grade. Ther eekly orkng hours and the compensaton per shft s gven n the table. Other nput data ncludes the number of eekly patent orkload, the number of patent orkloads that each nurse can handle, k, the number of eekly shfts that each nurse must take, h, the maxmum number of eekly allo able late nght shfts,, and the eekend shfts, W. Frst, e calculate the hosptals nursng capacty (.e., the maxmum amount of patent orkload that the hosptal can handle) n our case study. We then calculate the total number of patents that a nurse can handle per eek, tp : tp = k h. As a result, the current nursng capacty s a orkload of 55 (= tp ). If the orkload exceeds 55, e ll experence patent dssatsfacton. Otherse, there ll be dle nurses, hch are consdered a non-productve resource n the hosptal operaton. Therefore, our optmaton models attempt to mnme ether the dle nurses or patent dssatsfacton.

8 156 Gno J. Lm et al.: Mult-obectve urse Schedulng Models th Patent Workload and urse Preferences Results on the Assgnment Model Assgnment model (0) has fve obectves. The soluton output of ths model s summared n Table. The second column contans the optmal obectve values of the sngle obectve optmaton models descrbed n Secton. The thrd column contans obectve values of all fve obectve functons hen the mult-obectve optmaton model (0) s solved usng the soluton method descrbed n Secton. It s easy to verfy that %, " = 1, K,5. The optmal value of the fnal obectve (See equaton (8) th l =1) s approxmately 6.. The model as solved n.78 seconds. If all the obectve functons could reach ther respectve optmum soluton, then ould have been 5. But the mult-obectve nature of ths problem forces some obectves to reach sub-optmalty. In our example, the soluton has met nurse preferences ( and 4 ) hle 5 e may need to hre more part tme nurses ( ). Of course, f there s a need to decrease the number of part tme nurses, e ould assgn a hgher value of l n the optmaton model to acheve ths goal Results on the Patent Workload Model We contnue our experments th the patent orkload model descrbed n Secton.. The obectve functon conssts of sx obectves. The patent orkload model s solved usng Algorthm 1 th l =1for to cases; one th a total of 400 patent orkloads and the other th 55 patent orkloads. The model as solved n.16 seconds for the case th 400 patent orkloads and n.74 seconds for the second case. The results are shon n Table 4. In order to compare to dfferent models, e have added a comparson measure formu la usng equaton (9) n each table and t s defned as 4 P = ( + ), (9) =1 P here, s the amount of orkload unsatsfed n eek due to the lack of nursng staff and t s expressed as 7 P - gst s S t=7( - 1) + 1 = max 0, D k y. (0) Ł ł Smlarly, s the total nurse dle unts per eek due to too many nurses and the lack of patent orkloads to fully utle the nursng staff and t s expressed as: 7 s S t=7( - 1) + 1 gst - = max 0, sum k y D. (1) Ł Ł łł These ndexes ( and ) can be useful hen the P scheduler ants to kno f more nurses are needed or dle nurses can be assgned to dfferent care unts n the hosptal. Tables 4a and 4b sho a comparson beteen the patent orkload model and the assgnment model on fve obectves. In the case th 400 eekly patent orkloads, the fourth goal (.e., to assgn hgher grade nurses to regular day shfts, 4 ) reached ts optmum value hle 1 and attaned near optmal solutons. The normaled obectve value of the thrd obectve functon (.e., x = % / ) s close to, h ch s far fro m 1 (the optmal). We reason that ths fndng s because the pay rates of the part tme nurses are much loer than the full tme nurses n ths partcular case study. Table 4a. Comparson beteen PWM and AM % PWM D = 400, % ( / ) AM % A p p Table 4b. Comparson beteen PWM and AM % PWM D = 55, % ( / ) AM % A p p ote that the orkload model has a value of 1 hch s smaller than 45 of the assgnment model. Snce there s a smaller orkload than the nursng capacty of 55,

9 Management 01, (5): both models sho ero values of AM shos hgher values of does not nclude. On the other hand, than PWM because AM n the obectve functon. Ths ndcates that addng the patent orkload goal to the problem mproves the nursng staff utlaton. We solved the model once agan for the case th 55 eekly patent orkloads for a month and the results are shon n Table 4b. Snce total patent orkloads are equal to the nursng capacty, the sxth obectve reached ts optmum. Hence, 6 p values are ero for both of the models. Ths has all orkload satsfed for PWM but a defct o f 9 for AM. We make the follong to observatons based on these results. Frst, PWM has a loer number of unsatsfed orkloads and a loer number of nurse dle unts. We speculate that ths happens because PWM consders mnmng patent orkloads as ell as mnmng nurse dle unts n the obectve functon. Second, n order to satsfy orkloads, PWM favors hrng more part tme nurse s. As a result, PWM has a hgher value of % than AM. 4.. Test Data Set II We no consder a care unt that can have at most 0 full tme and 6 part tme nurses. The full tme and three part tme nurses have to dfferent grades hle the other three part tme nurses have only one grade. We generated ths data based on Table and have doubled the full tme nurses. Therefore, our nursng capacty s 100 (= tp ) total number of patent orkloads Results on the Assgnment Model The soluton output of ths model s summared n Table 7 n Appendx. The optmal value of the fnal obectve ( ) s 5.4 and the model as solved n seconds. The results sho that the behavor of the model s smlar for both data sets. It s easy to see that e do not need to hre many part tme nurses because e have more full tme nurses. Hence, value of ( % / ) n data set II s smaller than that of data set I. Overall, ncreasng the number of full tme nurses dd not affect much on nurse preferences Results on the Patent Workload Model To scenaros are run for the patent orkload model; one th 800 total patent orkloads and the other th 100. The model as solved n 60.8 seconds for the frst case and n seconds for the second case. The results are shon n Table 8 n Appendx. The results from the second data set sho a smlar performance compared to the frst data set. One thng to note s that data set II had a smaller value of % 6 /, hch means that t s easer to 6 satsfy patent orkloads snce e have more avalable nurses. In both cases, values of PWM are loer than those of AM. Ths ndcates that addng the patent orkload goal to the problem ndeed mproves nursng staff utlaton n both data sets. 4.. Senstvty Analyss We have shon that addng patent orkload to optmaton models can make postve mpact n nurse schedulng n the prevous secton. We no analye the behavor of our models under dfferent patent orkload scenaros and make performance comparson beteen the models. An deal nurse schedule should satsfy both the nurses and the patents. In realty, they are often conflctng goals. So the queston s hch model performs better n both of these measures? We perform a senstvty analyss of these models on patent orkloads to anser ths queston Customer Satsfacton Customer satsfacton can be consdered for both the nurses and the patents n the hosptal. ursng staff ob satsfacton can be accommodated by tryng to schedule them accordng to ther preferences. From the hosptal pont of ve, the scheduler shes to utle ts nurses at ts optmal level,.e. IU = 0. Therefore, f the orkload s ell belo ts nursng capacty, the hosptal has an opportunty to mprove ts operatng cost by reducng part tme nurses or re-assgnng certan nurses to dfferent care unts that experence hgh volume of patent orkloads. On the other hand, the scheduler may also sh to satsfy as many of the antcpated orkloads as possble so as to mnme patent dssatsfacton. urse shortage s a concern that many hosptals are facng noadays. In many cases, hosptals may not be able to satsfy ts eekly patent orkloads. Therefore, our orkload model has constrant (1) that sets an upper bound on hat percent of patent orkloads must be satsfed. One can change the value of based on the nursng capacty. Furthermore, ths model can be useful n estmatng ho much orkload ll not be satsfed gven that the nursng capacty s fxed. Based on ths, e conducted a senstvty analyss usng dfferent values of b and the results are dsplayed n Table 9 n Appendx. We calculate the amount of unsatsfed orkloads (UW ) and nurse dle unts ( IU ) for orkloads rangng from 00 to 650 for the assgnment model and the patent orkload model th b = 1.0, b = 0.9, b = 0.8 and b = 0.7 for test data set I.Fgure 1 shos the comparson beteen AM and PWM for b = 0.9. Fgure 1 shos the comparson on IU, here the horontal axs represents patent orkload and the vertcal axs s for nurse dle unts. In Fgure 1, the vertcal axs represents unsatsfed patent orkloads ( UW ). For both UW and IU, PWM performs better than AM by provdng feer IU s hen the orkload s less than 55 and feer UW s hen the orkload s hgher than 500. Evdently, ncreasng orkload ll lead to a smaller nurse dle unts and more unsatsfed orkloads. But, t s clear that the orkload model has smaller nurse dle unts as ell as smaller unsatsfed orkloads than the assgnment model n all cases tested.

10 158 Gno J. Lm et al.: Mult-obectve urse Schedulng Models th Patent Workload and urse Preferences Table 5. umber of nurses and the shfts assgned to full tme and part tme nurses Demand F# P1# P1S1 AM P1S P1S P# PS1 PS PS PWM, b= nf nf nf nf nf nf nf nf nf 600 nf nf nf nf nf nf nf nf nf 65 nf nf nf nf nf nf nf nf nf 650 nf nf nf nf nf nf nf nf nf PWM, b= nf nf nf nf nf nf nf nf nf 650 nf nf nf nf nf nf nf nf nf Table 5 shos a nursng schedule comparson beteen the assgnment model and the orkload model for test data set I. Ths table shos ho many full tme nurses ( F # ), part tme nurses of type 1 ( P1# ), and part tme nurses of type ( P# ) are scheduled to ork based on dfferent patent orkload levels. Snce there are 10 F s, P1 s, and P s, e add three columns for P1 s (namely, PS 1, 1 PS 1 and PS 1 ) and another three colu mns for s P (namely, PS, and ) that ndcate ho many 1 PS PS hours each part tme nurse s assgned to ork. A cell th nf ndcates that the soluton s nfeasble for the specfc parameter settng. As e can see from the table, the assgnment model soluton remans the same for all levels of patent orkload. It s because AM does not consder patent orkload. Due to the same reason, the AM schedule assgns all full tme nurses and part tme nurses of type 1 for all o rkload levels. Ths s clearly not the case hen e consder patent orkload n the model. In PWM, dfferent types of nurses are scheduled to ork th dfferent hours n order to reduce IU s hle meetng the patent orkload. For example, hen the orkload s 500 and b = 1.0, n addton to 10 full tme nurses, P1 s are hred to ork 1 hours and 11 hours and all three P s are hred to ork 17, 17, and 18 hours, respectvely. If e decrease the value of b to 0.9, the schedule ncludes P1 s th hours and 1 P th 4 hours. We notce that some part tme nurses ork less than or 0 hours. Ths s an example of a rotatng nurse ho orks for several dfferent dvsons n the hosptal. If e force the schedule to satsfy 100% patent orkload ( b = 1 ), PWM can fnd a feasble soluton hen the orkload level s less than 55. As the level of customer satsfacton requrement decreases ( b = 0.9 ), the orkload model fnds feasble solutons for orkloads up to 600, hch s about 10% more than the current nursng capacty. In realty, ths strct constrant can be easly removed to fnd a feasble soluton, hch s a compromse beteen patent dssatsfacton and the lmted nursng capacty. (a) IU results (b) UW result s Fgure 1. Comparson beteen AM and PWM th IU results; (b) UW results b = 0.9 : (a) ursng Staff Satsfacton urse ob satsfacton s another concern n our model. Ths s treated by consderng ther preferences n the optmaton model. We compare the performance of to models to see hch one s more lkely to meet nurse preferences. Snce the optmaton models behave smlarly on both test data sets, results are reported based on the test

11 Management 01, (5): data set I. For ths comparson, e construct a base model that has a sngle obectve to satsfy nurse preferences,.e., Base Model = {sngle obectve model th ( ) constrants (1),, (14), and (1)}. Fgure. Total daly shft assgnment comparson of the three optmaton models th T = 8, D = 55, b = 1 Table 6. Obectve value comparson among Base Model, PWM and AM: Dat a Set I - 10 full tme and 6 part tme nurses Optmal obectve value Base Model AM PWM Consder a case th 55 eekly orkloads, hch s the current nursng capacty, and t b=1.the results are shon n Fgure and Table 6. Fgure depcts the performance comparson for the total daly shft assgnments among three optmaton models. The horontal axs represents the shft number and the vertcal axs shos frequency of each shft assgnment based on a four eek schedule. The bars have to patterns. The darker (bottom) part s the count for full tme nurses and the gray (upper) porton s for the part tme nurses. Evdently, shfts and 4 are most preferred n our test data set. Thus, assgnng more full tme nurses to shfts and 4 ll be deal. Ths s clearly the case for the base model that consders only the nurse preference. In shft, AM has a loer frequency than the base model because AM has mult ple ob ectves to compromse. Table 6 shos ths trade-off. The base model has the smallest value of 1, hch s the smallest devaton from the nurse preference. But the rest of obectve values of the base model are hgher than those of AM. Ths confrms that meetng nursng staff k preference only can come at the cost of unsatsfed patent orkloads. That s one of the man reasons hy PWM has hgher frequency for all shfts n Fgure. Securng more (especally part tme) nurses ll not only make t easer to process more patent orkloads, but also helps reduce the dle nurse unts. As a result, PWM generates a schedule th s maller UW and IU. 5. Conclusons We have developed a seres of to mult -obectve nurse schedulng models: the assgnment model consdered labor costs and nurse preferences hle the patent orkload model added patent orkload nformaton to the optmaton model.because solvng the assgnment model s computatonally expensve, a to-stage non-eghted goal programmng soluton method as developed to fnd an effcent soluton quckly for these models. In order to understand the effects of addng patent orkloads n the model, e ntroduced a comparson ndex that captured unsatsfed patent orkloads and the dle nurse unts. Based on the representatve case studes, t s clear that addng patent orkload nto the optmaton model can not only mprove patent satsfacton by decreasng the unsatsfed patent orkloads, but also can mprove nurse staff utlaton by mnmng nurse dle unts. These goals can be acheved thout creatng hgher levels of nursng staff dssatsfacton. We also shoed that our soluton algorthm can fnd optmal solutons n a reasonable tme: t took about 1 mnute to fnd a schedule for a hosptal th 0 full tme and 6 part tme nurses. aturally, havng patent orkload nformaton can help estmate an optmal nursng staff capacty. When there s a seasonal orkload pattern (e.g., flu seasons), schedulers can easly apply a forecastng approach to estmate the orkload. Then our optmaton models can be modfed to produce ho many nurses they need to acqure n order to maxme staff utlaton for the schedulng perod. Our future ork ll consder such nurse and patent orkload uncertanty. REFERECES [1] T. Perrn, "008 health care cost survey," 008. [] C. Ulrch et al., "The nursng shortage and the qualty of care," The e England Journal of Medcne, vol. 47, no. 14, pp , 00. [] L. H. Aken, S. P. Clarke, D. M. Sloane, J. Sochalsk, and J. H. Slber, "Hosptal nurse staffng and patent mortalty, nurse burnout, and ob dssatsfacton," The Journal of Amercan Medcal Assocaton, vol. 88, no. 16, pp , 00. [4] M. Bester, I. euoudt, and J.H. V. Vuuren, "Fndng good nurse duty schedules: a case study," Journal of Schedulng, vol. 10, no. 6, pp , 007. [5] J. Blythe, A. Baumann, I. Zeytnoglu, M. Denton, and A. Hggns, "Full-Tme or Part-Tme Work n ursng:

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