Mathematcal Problems n Engneerng, Artcle ID 937842, 10 pages http://dx.do.org/10.1155/2014/937842 Research Artcle Modellng a Nurse Shft Schedule wth Multple Preference Ranks for Shfts and Days-Off Chun-Cheng Ln, 1 Ja-Rong Kang, 1 Wan-Yu Lu, 2 and Der-Junn Deng 3 1 Department of Industral Engneerng and Management, Natonal Chao Tung Unversty, Hsnchu 300, Tawan 2 Department of Toursm Informaton, Alethea Unversty, New Tape Cty 251, Tawan 3 Department of Computer Scence and Informaton Engneerng, Natonal Changhua Unversty of Educaton, Changhua 500, Tawan Correspondence should be addressed to Der-Junn Deng; djdeng@cc.ncue.edu.tw Receved 4 October 2013; Revsed 22 February 2014; Accepted 22 February 2014; Publshed 27 March 2014 Academc Edtor: Chng-Ter Chang Copyrght 2014 Chun-Cheng Ln et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. When t comes to nurse shft schedules, t s found that the nursng staff have dverse preferences about shft rotatons and daysoff. The prevous studes only focused on the most preferred work shft and the number of satsfactory days-off of the schedule at the current schedule perod but had few dscussons on the prevous schedule perods and other preference levels for shfts and days-off, whch may affect farness of shft schedules. As a result, ths paper proposes a nurse schedulng model based upon nteger programmng that takes nto account constrants of the schedule, dfferent preference ranks towards each shft, and the hstorcal data of prevous schedule perods to maxmze the satsfacton of all the nursng staff s preferences about the shft schedule. The man contrbuton of the proposed model s that we consder that the nursng staff s satsfacton level s affected by multple preference ranks and ther prorty orderng to be scheduled, so that the qualty of the generated shft schedule s more reasonable. Numercal results show that the planned shfts and days-off are far and successfully meet the preferences of all the nursng staff. 1. Introducton Nurse-schedulng problem s a classcal combnatoral optmzaton problem and has been shown to be NP-hard [1, 2]. The objectve of the nurse-schedulng problem s to determne the rotatng shfts of the nursng staff over a schedule perod (weekly or monthly) [3]. A nurse schedule ncludes the work shfts and days-off of the nursng staff, ensurng that all the combnatons of shfts and days-off meet the manpower requrements of each shft (ncludng total number of staff members, daly mnmum number of staff members,andnumberofsenorstaffmembersrequred),and at the same tme the number of basc days-off of each staff member should be fulflled [3]. In general, there are a lot of types of work shfts and days-off n shft schedules. The most common ones nclude the 2-shft rotaton (.e., 12-hour day shft and 12-hour nght shft) and the 3-shft rotaton (.e., 8-hour day shft, 8- hour evenng shft, and 8-hour nght shft). Regular daysoff allow the nursng staff to rest, and each staff member s enttledto thesame number ofdays-off [4 7]. Due to dverse personal lfestyles and dfferent degrees of physcal tolerance for contnuous workng days, the nursng staff usually have dfferent preferences for work shfts and days-off. The satsfacton of the nursng staff s preference for work shfts and days-off enables them to take the proper rest to ncrease the qualty of medcal servce and reduce medcal cost of the hosptal as well as rsks of occupatonal hazard [8, 9]. Therefore, t has been an nterestng problem n the recent works to take nto consderaton the nursng staff s preferences n plannng the schedule of work shfts and daysoff and adopt maxmzaton of satsfacton and mnmzaton of penalty cost to evaluate the qualty of the shft schedule wth preferences [2, 10 13]. For example, as far as the hard constrants and soft constrants of the nurse shft schedule (ncludng the nursng staff s preferences and the demands of hosptals) are concerned, Hadwan et al. [2] amed to mnmze the penalty cost of a nurse schedule. Ackeln and Dowsland [11] appled a genetc algorthm to solve the nurse shft schedulng problem
2 Mathematcal Problems n Engneerng wth the objectve of mnmzng the penalty cost for not fulfllng the preferences of the nursng staff. Maenhout and Vanhoucke [12] nvestgated the penalty costs wth multple constrants (ncludng the nursng staff s preferences and some specfc combnatons of work shfts and days-off). Topaloglu and Selm [13] consdered a varety of uncertan factors n nurse shft schedulng to propose a fuzzy multobjectve nteger programmng model whch takes nto consderaton the fuzzness of the objectve and the nursng staff s preferences. From the lterature, we dscover that, n the penalty cost (or satsfacton) due to the nursng staff s preferences n the objectve functon, all the prevous works only focused on themostpreferredworkshftandthenumberofsatsfactory days-off of the shft schedule at the current schedule perod and further nvestgated nether dfferent preference ranks (such as three ranks: good, normal, or bad) of the nursng staff toward dfferent work shfts or planned days-off nor the number of tmes n whch ther preferences are satsfed n the prevous shft schedules and days-off. Snce the constrants ofthescheduleleadtothefactthatthemostpreferredwork shfts and days-off of each staff member cannot be fulflled completely, we beleve that f the penalty cost (or satsfacton level) due to preferences of the nursng staff does not consder dfferent preference ranks and the hstorcal data of prevous schedules,thelong-termfarnessandjustceofthenurse shft schedules s affected. For example, f each nursng staff member has three dfferent preference ranks (good, normal, or bad) towards a 3-shft rotaton, those scheduled to a bad work shft would have a larger preference penalty cost (or lower preference satsfacton level) as compared wth those scheduled to work a normal shft and should be assgned a hgher prorty to ther hgher preference rank n ther schedule at the next schedulng perod. In lght of the above, ths paper proposes a bnary nteger lnear programmng model for the nurse shft schedule wth dfferent levels of satsfacton preference and a dfferent prorty orderng of the nursng staff for plannng ther shft schedule. The objectve of ths model s to maxmze the overall satsfacton level of the nursng staff towards ther work shfts and days-off schedule. In addton, some common nurse-schedulng constrants are consdered, ncludng the lmted number of nursng staff members, work shft lmtaton, and day-off lmtaton. Our mathematcal model s capable of effectvely helpng schedule planners to desgn a nurse shft schedule that attempts to, wthn all schedulng constrants, satsfy the shft and day-off preferences of most of the nursng staff members n a far manner and to acheve the hghest overall satsfacton level. The man contrbutons of ths paper are stated as follows. In the proposed mathematcal model for nurse schedule, the work shft and day-off preferences of the nursng staff are categorzed nto dfferent levels and are then ntegrated to be solved.inaddton,nthecase,wherethepreferrednumberof shfts or days-off exceeds the actual shfts or days-off avalable (due to schedule constrants), a prorty orderng mechansm of the nursng staff when plannng ther shft schedule s appled to solve the contradctory stuatons among ther preferences. The rest of ths paper s organzed as follows. Secton 2 gves the lterature revew of our work. Secton 3 frst descrbes our concerned problem and then constructs ts mathematcal programmng model. Secton 4 gves a numerc example to analyze the performance of the shft schedule constructed by our mathematcal model. Fnally, a concluson s made n Secton 5. 2. Lterature Revew The nurse-schedulng problems have been solved by a varety of methods, whch are manly ntroduced by mathematcal programmng, heurstcs, and others n ths secton. Frst, the mathematcal programmng approaches for the nurse-schedulng problem are ntroduced. Maenhout and Vanhoucke [12] proposed an ntegrated analyss method to solve human resource plannng and shft schedulng problem of nurses on the long run. Azaez and Al Sharf [14] solved the nurse-schedulng problem wth a 0-1 lnear programmng model, whch takes nto consderaton the rato of nurses workng nght shfts or havng days-off on weekends and tres to avod unnecessary overtme so that hosptal costs can be reduced. Topaloglu [15] proposedamultobjectve programmng model to tackle the nurse-schedulng problems of house physcans n the emergency room. Based on the AHP method, soft constrants n the model are weghted n accordance wth ther relatve mportance and ths becomes the bass for weghtng objectve functons. Topaloglu [16] proposed a multobjectve schedulng model for plannng the shfts of resdent physcans, n whch senorty of a resdent physcan s used for the weght settng. Emprcal analyss showed that the model s far superor to the manual schedulng method n terms of effcency and tme-savng. Belën and Demeulemeester [17] proposed an ntegrated schedulng method for nurses and surgeons. They appled the branch-and-prce method to perform ntegraton, whch s one of the most common methods to generate explct solutonsamongthecolumngeneratontechnques.glass and Knght [18] dentfed four categores of nurse-schedulng problems and solved them wth mxed-nteger lnear programmng. Results showed that the optmal solutons were produced n all benchmarkng examples wthn 30 mnutes. The characterstc of ths model s that t reduces the collecton space based on the structure of the problem so that the problem-solvng effcency can be enhanced. Valouxs et al. [19] proposed a 2-stage soluton to solve the nurse-schedulng problem, where the workload and days-off of each nurse are frst determned before the shfts are planned. Second, the heurstcs for nurse-schedulng problems are ntroduced. Hadwan et al. [2] proposed a harmony search algorthm for the nurse-schedulng problem, whch was tested n hosptals n Malaysa and was shown to be better than the genetc algorthm approach as well as most heurstc approaches. Ackeln and Dowsland [11] proposed an ndrect genetc algorthm for the nurse-schedulng problem, n whch a chromosome encodng s performed; recombnaton s conducted wth a heurstc approach; evoluton s conducted va mxed-crossover to locate better solutons. The method
Mathematcal Problems n Engneerng 3 has been tested and was found to be superor to some of the already publshed Tabu search methods. Tsa and L [20] proposed a 2-stage programmng model to analyze and solve the nurse-schedulng problem, n whch days-off are planned n the frst stage, and then shfts are determned n the second stage. The two stages were then analyzed wth a genetc algorthm. Sadjad et al. [21]proposed a mxed-nteger nonlnear programmng model to randomly plan shft schedules, n whch the demand for human resource s consdered a varable and s based on a certan probablty dstrbuton. Then, the authors appled the GA and Taguch method to solve the nurse-schedulng problem. Gutjahr and Rauner [22] proposed ant colony optmzaton to solve dynamc regonal nurse-schedulng problem n a publc hosptal n Venna, Austra. Upon verfcaton va smulaton experments, the model was shown to be superor to a greedy assgnment algorthm. Fnally, except for the above two methods, some other solutons for the nurse-schedulng problem are ntroduced. Lü andhao[23] proposed adaptve neghborhood search (ANS) for the nurse-schedulng problem, whch performs a neghborhood search and changes based on three dfferent levels of ntensty and change. The approach was tested on 60 examples and the results were qute mpressve. M. V. Charamonte and L. M. Charamonte [24] proposed to use a compettve agent-based negotaton algorthm for the nurseschedulng problem, whch ams to maxmze preferences of nurses and mnmze the costs. Vanhoucke and Maenhout [25] proposed a set of complexty ndcators for the nurseschedulng problem, whch can ndcate the complexty of the problem and automatcally come up wth a smulaton soluton that meets the level of complexty of the problem. They can be used as the baselne analyss to compare the performance of dfferent approaches. From the above lterature revew, t can be found that many works dd not explore the preference ranks of the nursng staff towards each shft rotaton or day-off. In addton, thepreferredshftandday-offarenotgvenanyprorty orderng accordng to the prevous schedulng perods. Based on ths, ths paper proposes a mathematcal programmng model for the nurse-schedulng problem n order to produce a prelmnary shft schedule that can fulfll the needs of practcalworkandatthesametmesatsfymostnursngstaff members. 3. Methodology In ths secton, we construct a mathematcal model to plan the shft rotaton and day-off based on the preference ranks of the nursng staff. The constructon process of the mathematcal model ncludes descrpton of the problem, dentfcaton of the satsfacton level of the nursng staff, and the development of the mathematcal model. 3.1. Problem Descrpton. In ths subsecton, we explan the actual work scenaro nsde the hosptal and then the problems encountered when plannng for a shft schedule. We frst descrbe the work scenaro, whch ncludes the structure and constrants of shft schedule, combnaton of thenursngstaff,andthepreferenceranksofthenursng staff towards each shft rotaton and days-off. Consder a shft schedule for a 2-week work n whch the shfts of a day start at 0:00 AM and the hosptal runs on a 3-shft rotaton: a day shft (8:00 AM 4:00 PM), an evenng shft (4:00 PM 0:00 AM), and a nght shft (0:00 AM 8:00 AM). Note that only regular days-off are planned n the schedule. The planned schedule has some constrants on shft rotatons: each nursng staff member s only assgned to a fxed type of shft wthn each schedule perod; the number ofnursngstaffmembersrequredforeachshftsfxed (after deductng the number of nursng staff members on regular days-off); each person should have at least an 8-hour rest before contnung on to the next shft. Note that the constrant of a fxed shft for each staff member s reasonable snce, n practce, n order to ensure that the nursng staff enjoy the health wth fxed work and rest, some hosptals assgneachnursngstaffmemberafxedworkshfttypefor all workng days of the schedulng perod. In addton, the planned schedule has some constrants on days-off: the total number of days-off of each nursng staff member wthn the schedule perod s the same, and each nursng staff member s enttled to at least one day-off each week. The maxmum number of the nursng staff members s allowed to be on dayoff,andthenumberofsenorstaffmembersworkngneach shft each day are known and flexble. As for composton of the nursng staff, the qualfed nursng staff members are categorzed nto junor and senor staff members, where the staff members wth at least 2 years of nursng workng experence are consdered senor, and thosebelow2years,junor.also,allthenursngstaffarefull tme. When the total number of staff members s nsuffcent to cover all the shfts, the concerned department has to hre new staff members to fll ths shortage n manpower. Outsourcng nursng staff members from other departments s not allowed. Lastly, the nursng staff are asked to rank ther preferences for each shft and day- off, whch are called preference ranks. The preference ranks of each shft are classfed nto three types: good, normal, and bad shfts. The preference ranks of days-off are classfed nto good and bad daysoffbasedonthe preferred and notpreferred days-off, respectvely. Note that each staff member has a fxed number of days-off wthn each schedule perod. Therefore, we assume that there may be more than one good day-off and no further rank s made among all the good days-off. As the nursng staff have rather dverse preferences, they are asked to fll out a preference form before the schedule s formulated so that the schedule planner has adequate nformaton about the staff s preferences for shfts and daysoffandthetotalnumberofthenursngstaffpreferrngeach shft or day-off. In the preference form, the preference ranks of shfts are expressed n a numercal orderng: 1 ndcates good (most preferred), 2 ndcates normal, and 3 ndcates bad (least preferred). The preference ranks of days-off are expressed as follows: 1 ndcates good (preferred) whle 3 ndcates bad (not preferred). Note that the numbers of
4 Mathematcal Problems n Engneerng the preference ranks for shfts and days-off are three (.e., 1, 2, and 3 for good, normal, and bad, resp.) and two (.e., good and bad), respectvely. We assume that the good and bad preference ranks for shfts and days-off are corresponded wtheachother.therefore,weletthegoodandbadpreference ranks for days-off be 1 and 3, respectvely. Intheactualworkscenaro,eachnursngstaffmember has dfferent preference ranks for shfts and days-off, markedly ncreasng the computng tme and dffculty to formulate a shft schedule. Also, a lot of constrants for schedule make t mpossble for all the staff members to work ther preferred shfts and have ther preferred daysoff. Therefore, t s mportant to formulate a prelmnary shft schedule recommendaton so that schedule planners can perform necessary and flexble adjustments on the recommended schedule, reducng the dffculty and workload of manpower plannng. 3.2. Preference Satsfacton of Shfts and Days-Off. Ths subsecton dscusses the satsfacton of the preference ranks of shfts and days-off, whch s called preference satsfacton. Ths paper ams to maxmze the overall shfts and days-off preference satsfacton of the nursng staff. There are dfferent preference ranks of shfts and days-off, and the preference satsfacton ncreases wth the preference ranks. The more the number of preferred shfts and days-off s satsfed, the hgher the overall satsfacton level towards the shft schedule wll be. However, due to the constrants on shfts and days-off, not all the preferred shfts and days-off could be satsfed. For the staff members not scheduled to ther preferred shfts or days-off for consecutve schedule perods, f ther preferences n the next schedule are satsfed wth a hgher prorty, then ther preference satsfacton must be hgher. Therefore, n desgnng the preference satsfacton, the past schedules are revewed frst (to count the number of tmes n whch a preferred shft of an ndvdual s satsfed n the past few schedule perods) and the weghts of preferred shfts and days-off of each nursng staff member are calculated before the current schedule s formulated. Note that the staff member wth a larger preference weght must be scheduled wth a hgher prorty order. Then, based on these two weghted values, the preference satsfacton of work shfts and days-off of the current schedule can be calculated. 3.3. Mathematcal Model. In ths subsecton, a bnary nteger lnear programmng model s constructed, whch ams to maxmze the overall preference satsfacton of the nursng staff towards the shft schedule by takng nto consderaton thepreferenceranksofthenursngstafffordfferentwork shfts and days-off, despte the constrants of manpower, shfts, and days-off. 3.3.1. Symbols Subscrpt : Index of a nursng staff member j: Index of a shft type k: Index of date. Parameters I: Set of the nursng staff (.e., I) J: Setofshfttypes(.e.,j J;notethatJ = {1 (day shft), 2 (evenng shft), 3 (nght shft)} n ths paper) K: Set of days-off (.e., k K;notethat K =14nths paper) M: Set of preference ranks of shfts, M = {1 (good), 2 (normal), 3 (bad)} N: Set of preference ranks of days-off, N = {1 (good), 3(bad)} P: Number of the past schedule perods consdered α: Coeffcent of the most preferred shft, α>1 β: Total number of days-off of each staff member wthn the schedule perod, β>1 R : The varable to dentfy whether staff member s senor, R = {0 (junor), 1 (senor)} : The preference weght of staff member for work shft W H : The preference weght of staff member for dayoff C: The base of preference weght (C = 2 n ths paper) W S P S,j : Preference satsfacton of staff member n workng shft j P H,k : Preference satsfacton of staff member n takng day k L S,j : Preference rank of staff member for shft j wthn the schedule perod, L S,j M L H,k : Preference rank of staff member for takng day k off wthn the schedule perod, L H,k N T S,m :IntherecentP perods,thenumberoftmesn whch staff member has been assgned to the shft of preference rank m, 0 T S,m P T H,n :IntherecentP perods,thenumberoftmesn whchstaffmemberhas been assgned to the day-off of preference rank n, 0 T H,n (P β) θ S m : Preference score for beng assgned to the shft of preference rank m, θ S m=1 =1<θS m=2 =2<θS m=3 =3 θ H n : Preference score for beng assgned to the day-off of preference rank n, θ H n=1 =1<θH n=2 =3 D j,k : Manpower demand n shft j on day k τ mn j,k : Lower bound of the requred number of senor staff members n shft j on day k N max j,k : The maxmum number of staff members allowed to have day-off n shft j on day k s,j : Whether staff member worked shft j n the prevous schedule perods, s,j {0(no),1(yes)}
Mathematcal Problems n Engneerng 5 h,j,k : Whether staff member had day-off n shft j on day k ntheprevousscheduleperods,h,j,k {0(no), 1(yes)}. Decson Varable s,j : Whether staff member s scheduled for shft j, s,j {0 (off shft), 1 (on shft)}. h,j,k : Whether staff member s scheduled for day-off n shft j on day k, h,j,k {0 (off shft), 1 (on shft)}. 3.3.2. Mathematcal Model. The complete mathematcal model s constructed as follows. Objectve where Max G= Constrants I J j P S {(P S K,j s,j,)+( k (P H,k h,j,k))}, (1),j = { αw S, f LS,j =1 W S {, f LS,j =2 (2) { 0 f L S,j =3,k ={αwh, f L H,k =1 0, f L H,k =3 (3) P H W S =C M m=1 (TS,m θs m ) (4) W H =C N n=1 ((TH,n /β) θh n ). (5) s,j =1, j J I (6) s,j =D j,k, j J, k K I (7) (R (s,j h,j,k )) τ mn j,k, j J, k K (8) I h,j,k =s,j β, k K N max j,k J j k=7 ( k=1 I h,j,k I, j J (9) h,j,k 0, j J, k K (10) k=14 k=8 h,j,k ) > 0, I (11) s,3 s,j h,3,14 h,j,k, I (12) s,j {0, 1}, I, j J (13) h,j,k {0, 1}, I, j J, k K. (14) Table 1: Shft preference weghts of the nursng staff. The assgned shft number n the past Good Normal Bad Preference weght 1 2 0 0 4 2 2 0 0 4 3 2 0 0 4 4 2 0 0 4 5 1 0 1 16 6 1 0 1 16 7 2 0 0 4 8 2 0 0 4 9 2 0 0 4 10 2 0 0 4 11 2 0 0 4 12 1 1 0 8 13 2 0 0 4 14 2 0 0 4 15 2 0 0 4 16 1 1 0 8 17 1 0 1 16 18 1 0 1 16 19 1 0 1 16 20 2 0 0 4 Frst, the objectve functon of the model s explaned as follows. Equaton (1) maxmzes the overall preference satsfacton of the nursng staff towards work shfts ( I J j (PS,j s,j, )) and days-off ( I J j K k (PH,k h,j,k)). As for the overall preference satsfacton towards shfts (resp., days-off), the preference satsfacton of each shft preference rank of each nursng staff s calculated by (2) (resp.,(3)), n whch the preference weght used n the above preference satsfacton calculaton for shft (resp., day-off) s obtaned from (4) (resp., (5)). We contnue to look at the constrants of the model. Nne constrants (from Constrant (6)to(14)) are dentfed based on the shft schedulng constrants, n whch Constrants (6) to (8) are shft constrants, Constrants (9) to(12) are day-off constrants, and Constrants (13) to(14) arebnaryvarable constrants. Constrant (6) enforceseachstaffmemberto be assgned to at most one work shft wthn each schedule perod. Accordng to Constrants (7) and(8), respectvely, manpower demand and number of senor staff members of each shft each day should be met. Constrant (9)enforcesthe fact that the total number of days-off assgned to each staff member n a work shft should be the same wthn a certan schedule perod. Constrant (10)enforcesthemaxmumtotal number of nursng staff members allowed to have day-off on each shft each day. As the number of patents usually fluctuates, the schedule planner may adjust the total number ofnursngstaffmembersallowedtobeonday-offaccordng totheactualnumberofpatentsonthatpartcularday. Constrant (11) enforcesthefactthateachstaffmemberhas
6 Mathematcal Problems n Engneerng Table 2: Shft preference rank and preference satsfacton of the nursng staff. Shft rotaton Day shft Evenng shft Nght shft Preference rank Preference satsfacton Preference rank Preference satsfacton Preference rank Preference satsfacton 1 1 8 2 4 3 0 2 2 4 1 8 3 0 3 1 8 2 4 3 0 4 1 8 2 4 3 0 5 1 32 2 16 3 0 6 1 32 2 16 3 0 7 2 4 1 8 3 0 8 2 4 1 8 3 0 9 1 8 2 4 3 0 10 1 8 2 4 3 0 11 1 8 2 4 3 0 12 1 16 2 8 3 0 13 1 8 2 4 3 0 14 1 8 2 4 3 0 15 1 8 2 4 3 0 16 1 16 2 8 3 0 17 2 16 1 32 3 0 18 1 32 2 16 3 0 19 1 32 2 16 3 0 20 2 4 3 0 1 8 tobegvenatleastoneday-offeachweekandthenterval between two dfferent shfts must be more than 8 hours. Constrant (12) ensures that each staff member has at least an 8-hour rest before contnung on to the next shft. Accordng to Constrants (13) and(14), the decson varables of work shfts and days-off should be ether zero or one. 4. Numerc Analyss The performance and feasblty of the model are evaluated on the obstetrcs and gynecology department of a hosptal. The parameters of the department are stated as follows. The papernvolvesatotalof20nursngstaffmembers,nwhch thosefrom#1to#15aresenor,whlethosefrom#16to#20are junor.eachstaffmembersallowedtohaveatotalof4daysoff over the overall schedule plannng perod (14 days). A total of 8 staff members are requred for day shft, 7 for evenng shft and 5 for nght shft. Durng each shft every day, the number of staff members allowed to have day-off s 3 persons for the day shft, 3 persons for the evenng shft, and 2 persons forthenghtshft,andatleastonesenorstaffmustbeonduty durng each shft. The schedules of the past two perods are collected as hstorcal data, n whch nursng staff members 5, 6, 17, 19, and 20 were on nght shfts n the prevous week whle staff members 6, 7, 8, 9, 14, 15, and 17 were gven day-off on the last day of the prevous week. Frst, the shft preference weghts and the preference satsfacton of three shft preference ranks of each staff memberarecalculatedandlstedntables1 and 2. Table 1 shows the number of tmes n whch each shft preference rank s satsfed n the prevous schedule perods and the preference weght at the current schedule perod. In Table 1, snce staff members 5, 6, 17, 18, and 19 were assgned to bad shfts n the past two schedule perods, ther shft preference weghtsaresettobelargernthecurrentscheduleplannng. To mantan farness of shft rotaton, those staff members are therefore gven hgher chances of beng assgned to good shfts n the current schedule. Table 2 shows each nursng staff member s satsfacton for shft preference ranks. A smaller preference rank number ndcates a more preferred shft; that s, preference rank 1 yelds the hghest degree of preference satsfacton, whle preference rank 3 yelds the lowest degree of preference satsfacton. In general, n solvng a mathematcal model, the model would attempt to satsfy all the shfts wth preference rank 1 n order to enhance the satsfacton of the nursng staff toward the shft schedule. However, n the current schedule plannng perod, 15 staff members consder day shfts as good, 4 staff members consder evenng shfts as good,
Mathematcal Problems n Engneerng 7 and1staffmemberconsdersnghtshftsas good. Sncethe total number of the nursng staff members who prefer day shfts exceeds the total manpower demand of 8 persons for day shft, the good shft preference ranks of 7 nursng staff members cannot be satsfed. Next, each staff member s day-off preference weghts and preference satsfacton for day-off preference ranks are calculated and lsted n Tables 3 and 4. Table 3 shows the number of tmes n whch each day-off preference rank was satsfed n the past schedule perods and the current weghted preference. It s found from Table 3 that staff member 18 was gven the largest number of bad days-off wthn the past two schedule perods, and hence her/hs day-off preference weght s set to be larger than the others n the current schedule. To mantan farness of the schedule, the staff member s therefore prortzed, that s, gven a hgher chance of beng assgned good days-off n the current schedule. Table 4 showseachstaffmember spreferencesatsfactonof the good day-off preference rank. Snce each staff member s enttled to 4 days-off, each has 4 days-off wth a good dayoff preference rank and the same preference satsfacton level. Lastly, the numercal example s solved under IBM ILOG CPLEX Studo 12.4 software on a PC wth an Intel 7-3770 CPU 3.90 GHz and 16 GB RAM, whch operates n Wndows 7 (64x). The optmal soluton s obtaned very frequently. The average computng tme of generatng a soluton s about 3.755 seconds. The generated schedule s lsted n Tables 5 and 6, whch meet the constrants on numbers of senor nursng staff members, shfts, and days-off. Note that, snce both the parameters D j,k for manpower for the maxmum number of staff members allowed to have day-off n shft j on day k can be adjusted accordng to the realstc stuatons, the generated schedule can be used n the realstc nstances wth dfferent szes. For the schedule wth a longer schedule perod (wth multple two-week perods), t can be handled bysplttngthewholescheduleperodtomultpletwo-week perods and then usng the generated schedule of each twoweek perod as the hstory for the next one. Table 5 shows the resultant schedule for shfts, n whch theassgnedshftsfor20nursngstaffmembersareshown; each staff member s assocated wth two entres of 0 (nonassgned shfts) and one entry of 1 (the assgned shft); the entres wth star marks are the shfts that are not preferred. From Table 5, each staff member s assgned to a fxed shft type for all workng days n the schedulng perod of 14 days and the manpower requrement of each shft s satsfed. From Table 5, we also observe that the schedule satsfes the demand of each work shft, n whch eght staff members are scheduled to work on the day shft (staff members 3, 4, 6, 11, 12, 15, 16, and 18), 7 on the evenng shft (staff members 1, 2, 5, 7, 8, 17, and 19), and 5 on the nght shft (staff members 9, 10, 13, 14, and 20). It s also observed from Table 6 that each staff member s also gven suffcent rest tme between two shfts. For example, snce staff member 5 was scheduled to work nght shft n the prevous schedule perod and was gven demandnshftj on day k and N max j,k Table 3: Day-off preference weghts of the nursng staff. The assgned number n the past Preference weght Good Bad 1 7 1 5.6569 ( 5.7) 2 8 0 4 3 5 3 11.3137 ( 11) 4 8 0 4 5 5 3 11.3137 ( 11) 6 8 0 4 7 8 0 4 8 8 0 4 9 7 1 5.6569 ( 5.7) 10 8 0 4 11 8 0 4 12 7 1 5.6569 ( 5.7) 13 6 2 8 14 7 1 5.6569 ( 5.7) 15 7 1 5.6569 ( 5.7) 16 7 1 5.6569 ( 5.7) 17 8 0 4 18 3 5 22.6274 ( 23) 19 8 0 4 20 8 0 4 1 day-off on the last day, he/she s therefore not scheduled for good (day) shft even though he/she has a large shft preference weght. In addton, as for shft preference, 3 staff members are scheduled for the normal shft (staff members 1, 5, and 19), 4 for the bad shfts (staff members 9, 10, 13, and14),andtherestforthe good shft.itcanbefoundthat those scheduled for bad shfts have a small shft preference weght, ndcatng that the schedule s formulated farly. Table 6 shows the resultant schedule for days-off, n whch the four days-off of each staff member are marked wth 1; the entres wth star marks are the days-off that are not preferred. From Table 6, the day-off constrants of all the nursng staff are satsfed and the total number of nonpreferred days-off ssmall.inaddton,notethatthetotalnumberofdays-off of each staff member wthn the schedule perod s the same and each staff member s gven at least 1 day-off each week. Therefore, the number of nursng staff members allowed to take a day-off and the mnmum number of senor nursng staffmembersondutyeachshfteachdayarefulflled.as for day-off preference, only a few staff members are not gven good days-off (staff members 4, 6, 11, and 20), and all these have low day-off preference weghts.
8 Mathematcal Problems n Engneerng Table 4: Day-off preference and preference satsfacton of the nursng staff. Schedule perod 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 5.7 5.7 5.7 5.7 2 4 4 4 4 3 11 11 11 11 4 4 4 4 4 5 11 11 11 11 6 4 4 4 4 7 4 4 4 4 8 4 4 4 4 9 5.7 5.7 5.7 5.7 10 4 4 4 4 11 4 4 4 4 12 5.7 5.7 5.7 5.7 13 8 8 8 8 14 5.7 5.7 5.7 5.7 15 5.7 5.7 5.7 5.7 16 5.7 5.7 5.7 5.7 17 4 4 4 4 18 23 23 23 23 19 4 4 4 4 20 4 4 4 4 Table 5: The resultant schedule for shfts. Number of shfts Day shft Evenng shft Nght shft 1 0 1(normal) 0 2 0 1 0 3 1 0 0 4 1 0 0 5 0 1(normal) 0 6 1 0 0 7 0 1 0 8 0 1 0 9 0 0 1(bad) 10 0 0 1(bad) 11 1 0 0 12 1 0 0 13 0 0 1(bad) 14 0 0 1(bad) 15 1 0 0 16 1 0 0 17 0 1 0 18 1 0 0 19 0 1 (normal) 0 20 0 0 1 5. Concluson Nurse-schedulng s a dffcult and tme-consumng task for schedule planners. Snce there are a lot of hard constrants mposed by the government and the hosptal, and multple preference ranks of the nursng staff for work shfts and daysoff are consdered, t s rather mpossble to obtan a perfect nurse shft schedule manually. Therefore, ths paper performs a mathematcal programmng approach to establsh a model to effectvely make a nurse shft schedule. The results of the schedule show that all hard constrants are flled completely and the preferred shft and days-off of the nursng staff are assgned farly. In the future, we ntend to extend our work along the followng two lnes. Frst, for the objectve functon of the mathematcal model proposed n ths paper, the preference satsfactons of the nursng staff for work shfts and daysoff of the schedule could be ntegrated. Snce the number of days-off of the schedule s greater than the number of shfts, the proposed model s scheduled n two stages n ths paper (frst days-off and then work shfts). Therefore, t would be of nterest to ntegrate them to be solved n a sngle stage to enlarge possble solutons, to ncrease the preference satsfacton of the nursng staff.
Mathematcal Problems n Engneerng 9 Table 6: The resultant schedule for days-off. Schedule perod 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1 1 1 1 2 1 1 1 1 3 1 1 1 1 4 1 1 1 1 5 1 1 1 1 6 1 1 1 1 7 1 1 1 8 1 1 1 1 9 1 1 1 1 10 1 1 1 1 11 1 1 1 1 12 1 1 1 1 13 1 1 1 1 14 1 1 1 1 15 1 1 1 1 16 1 1 1 1 17 1 1 1 1 18 1 1 1 1 19 1 1 1 1 20 1 1 1 1 Each entry wth a star mark n ths schedule ndcates the day-off that s not preferred by the nursng staff member. Second, varous work shft patterns for each nursng staff member could be taken nto account. In spte of most nursng staff members lke a fxed shft type for all workng days of the schedulng perod, some staff members mght want more flexble work patterns, that s, varous work shft patterns n dfferent workng days of the schedulng perod. Therefore, t would be of nterest to deal wth the flexble work patterns. Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper. Acknowledgments The authors thank the anonymous referees for the comments that mproved the content as well as the presentaton of ths paper. Ths work has been supported n part by NSC 101-2628- E-009-025-MY3, Tawan. References [1] J. J. Barthold III, A guaranteed-accuracy round-off algorthm for cyclc schedulng and set coverng, Operatons Research,vol. 29,no.3,pp.501 510,1981. [2]M.Hadwan,M.Ayob,N.R.Sabar,andR.Qu, Aharmony search algorthm for nurse rosterng problems, Informaton Scences,vol.233,no.6,pp.126 140,2013. [3] G. Felc and C. Gentle, A polyhedral approach for the staff rosterng problem, Management Scence,vol.50, no.3,pp. 381 393, 2004. [4]A.T.Ernst,H.Jang,M.Krshnamoorthy,andD.Ser, schedulng and rosterng: a revew of applcatons, methods and models, European Operatonal Research,vol.153,no. 1, pp. 3 27, 2004. [5] J. V. Bergh, J. Belen, P. D. Bruecker, E. Demeulemeester, and L. D. Boeck, Personnel schedulng: a lterature revew, European Operatonal Research, vol.226,no.3,pp.367 385, 2013. [6] C. Valouxs and E. Housos, Hybrd optmzaton technques for the workshft and rest assgnment of nursng personnel, Artfcal Intellgence n Medcne, vol.20,no.2,pp.155 175, 2000. [7] H. W. Purnomo and J. F. Bard, Cyclc preference schedulng for nurses usng branch and prce, Naval Research Logstcs,vol.54, no. 2, pp. 200 220, 2007. [8]P.A.Clark,K.Leddy,M.Dran,andD.Kaldenberg, State nursng shortages and patent satsfacton: more RN better patent experences, Nursng Care Qualty, vol. 22, no. 2, pp. 119 127, 2007. [9] R. M Hallah and A. Alkhabbaz, Schedulng of nurses: a case study of a Kuwat health care unt, Operatons Research For Health Care,vol.2,no.1-2,pp.1 19,2013. [10] J. F. Bard and H. W. Purnomo, Preference schedulng for nurses usng column generaton, European Operatonal Research,vol.164,no.2,pp.510 534,2005. [11] U. Ackeln and K. A. Dowsland, An ndrect genetc algorthm for a nurse-schedulng problem, Computers and Operatons Research,vol.31,no.5,pp.761 778,2004.
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