A GENETIC ALGORITHM-BASED METHOD FOR CREATING IMPARTIAL WORK SCHEDULES FOR NURSES

82 Internatonal Journal of Electronc Busness Management, Vol. 0, No. 3, pp. 82-93 (202) A GENETIC ALGORITHM-BASED METHOD FOR CREATING IMPARTIAL WORK SCHEDULES FOR NURSES Feng-Cheng Yang * and We-Tng Wu Insttute of Industral Engneerng Natonal Tawan Unversty Tape (067), Tawan ABSTRACT The head nurse s generally responsble for creatng work schedules for nurses n a nursng unt. However, snce versatle constrants, shft requrements, and leave requests are mposed n the problem, the generated schedule usually ncurs complants or crtcsms from nurses on ts mpartalty. Ths study nvestgates a complex real nurse schedulng problem and constructs an optmzaton model for the problem. The model conssts of fve mnmzaton goals subject to fve hard constrants. Then a genetc algorthm-based optmzaton method s developed for solvng the optmzaton model. The method proposes a dedcated GA codng scheme for the soluton to ths problem. Based on the scheme the method presents talored crossover and mutaton operators for soluton evoluton. In the GA method, four selecton operaton modes are proposed to round n superor chromosomes for the next generaton. A personnel data set from a nursng unt of a real hosptal s used n numercal tests. In addton, a modfed hll clmbng greedy method and a well-performed varable depth search method from the lterature are mplemented to solve the same problem for numercal comparson. Numercal results suggest that a determnstc selecton mode outperforms others and our method generates better schedules than other methods on every run. Keywords: Genetc Algorthm, Impartal, Work Schedules, Nurses *. INTRODUCTION An employee schedulng problem s to create a pertnent or fxed workng schedule that dstrbutes manpower evenly and farly to routne jobs or tasks. Typcally, an employee schedules s stable, snce ordnary employees have relatvely stable work routnes. Generally ordnary employees work day shfts on weekdays and rest on weekends. In contrast, n hosptals, nurses work very dynamc day-and-nght shfts. Nurses n the same nursng unt must share around-the-clock nursng work durng both weekdays and weekends. Employee schedulng for nurses s generally a challengng and laborous task that must be routnely completed by the head nurse n each nursng unt. Complants about the schedule generally outwegh complments, when the schedule s announced. An automatc optmal scheduler s needed to evenly and farly dstrbute the workload and mpartally grant leave requests for the nurses n a nursng unt. The goal of ths study s to create a genetc algorthm based schedulng method for creatng mpartal schedules for nurses. In partcular, the study defnes a complex schedulng problem for nurses n a domestc hosptal. Then ths study use the developed * Correspondng author: efcyang@ntu.edu.tw meta-heurstc algorthm to creatng optmal mpartal schedules for the problem. 2. LITERATURE REVIEW Over the past decade, a varety of methods have been used to create work schedules for nurses. Yh [0] showed that heurstc methods, such as modfed hll clmbng, can be used to create work schedules for any type of employee. The modfed hll clmbng method uses a determnstc greedy search technque to repeatedly assgn the most favorable jobs to the most favorable employees. Azaez and Sharf [] used a goal programmng approach to creatng work schedules for nurses. The goal programmng method creates work schedules to meet the expectatons of the nurses. Brucker et al., [2] used an adaptve constructve method to create work schedules for nurses. The adaptve constructve method creates rosters and then uses a greedy local search technque to create work schedules. Dowsland [8] dentfed requrements for creatng day and nght shft work schedules for nurses. Dowsland and Thompson [7] used a three-stage method to create complex work schedules for nurses. The frst stage uses a knapsack technque to create schedules that fulfll human resource requrements. The second

F. C. Yang and W. T. Wu: A Genetc Algorthm-based Method for Creatng Impartal Work Schedules for Nurses 83 stage uses a tabu search technque to create schedules that fulfll day and nght shft work schedule requrements. The thrd stage dvdes the day shfts nto early and late shfts. L and Ackeln [9] enhanced the three-stage method to create work schedules for nurses. The method uses four schedulng rules and a Bayesan optmzaton algorthm to create feasble work schedules. They used the method to create work schedules for 2 test cases. Burke et al., [3-] used several dfferent methods to create work schedules for nurses. They also created a webste that contans several test cases, whch can be used for benchmark testng. Ther tme varable depth search method, whch was mplemented n the C# programmng language, outperformed the other methods on the benchmark tests. They also proposed other methods, such as a scatter search method, whch could be used to create better work schedules. Charamonte [6] used an agent-based method to create work schedules for nurses. The method uses nurse agents that smulate ntellgent human behavors. The nurse agents bd for work shfts and work leaves untl all of the agents are satsfed wth ther work schedules. Pror studes created methods that mprove work schedules for nurses. The methods used to create work schedules nclude local search technques, heurstc search technques, stochastc search technques, or ntellgent agents. Most of the optmzaton technques can conduct local searches effectvely. However, they cannot conduct extensve searches effectvely. Meta-heurstc algorthms, on the other hand, can conduct extensve searches effectvely. Genetc algorthms (GAs), ant colony optmzaton algorthms (ACOs), and partcle swam optmzaton algorthms (PSOs) have been used to solve many dfferent types of optmzaton problems. They can be used to fnd near-optmal solutons to complex optmzaton problems wth nonlnear constrants and multple goals. Meta-heurstc algorthms have many advantages over tradtonal optmzaton technques. They can search the optmal soluton wthout gradent nformaton; they can conduct effectve populaton search for a wder soluton space; they can use heurstc nformaton for advantageous optmum search; and they are generally controlled or governed by evoluton laws or natural ntellgence technques. Ths study develops a genetc algorthm-based method to create mpartal work schedules for nurses. The evoluton method adopts natural ntellgence technques to create work schedules that meet varous nonlnear constrants and multple goals. 3. THE GENETIC ALGORITHM- BASED METHOD The goal of ths study s to model a work schedulng problem for nurses at a target hosptal and to create a genetc algorthm-based method that can generate mpartal work schedules for the problem. The hosptal uses a shft bundled technque to create work schedules for nurses n ordnary nursng unts. The hosptal monthly assgns nurses to ther dedcated shfts for an entre month. The head nurse s responsble for creatng new work schedules for each new month at the mddle of each current month. To create the new work schedules, the head nurse must revew prevous shft assgnments, prevous leave request approval records, and all future leave requests. It usually takes about one week to complete the new work schedules for each new month. The process s dffcult and tme consumng, manual schedulng s subject to actual or perceved bases, and manually created records can be ncomplete, naccurate, or naccessble. The goal of ths study s to create an automatc method that can be used to reduce the tme and effort needed to create work schedules for nurses. The method must create mpartal work schedules for the p nurses n a nursng unt P = {, 2, L, p}. The skll level of nurse, α, s classfed nto one of four skll levels α {, 2, 3, 4}. Hgher skll levels ndcate hgher senortes and hgher level nursng sklls. A good schedule should not let the nurses on the same shft are all wth the lowest skll levels. A good schedule must adequately staff a nursng unt on both weekdays and weekends. Nurses cannot always take days off on weekends. They must submt leave requests to the head nurse. The head nurse must consder all leave requests as well as the approval records to approve or deny the leave requests. Normally, each nurse s gven eght days off per month. However, a head nurse may ask nurses to forfet days off when there are unexpected needs. On the other hand, a head nurse may ask nurses to take extra days off when there are reduced needs. A head nurse may also grant leave requests that exceed eght days per month. Leave approvals must be recorded and used for create new work schedules. For each nurse, f ε s the number of leaves granted to nurse, a good schedule must prortze leave requests to gve nurses wth lower values of ε hgher opportuntes for ther leave requests. A good schedule should grant leave requests to eventually balance the values of ε for all of the nurses. Exceeded or shorted number of days off must also be recorded and used to create work schedules. For nurse, f γ s the cumulated shortage of leave days taken, when γ s postve, the nursng unt owes the

84 Internatonal Journal of Electronc Busness Management, Vol. 0, No. 3 (202) nurse γ days off; when γ s negatve, the nurse owes the nursng unt γ workng days. The target hosptal has three work shfts for the nursng unts. Shft s a day shft that runs from 8 am to pm, shft 2 s an evenng shft that runs from 3 pm to pm, and shft 3 s a nght shft that runs from pm to 8 am. K = {, 2, 3} s the set of all work shfts. The nght shft, Shft 3, s generally consdered to be an undesrable work shft. A good schedule must mpartally assgn nght shfts to nurses. However, the target hosptal has assgned the type of work shft to each nurse based upon the senorty. If nurse s dedcated to work shft β, then β K. Work shfts are scheduled for q days n a month Q = {, 2, L, q }. Days are ndexed sequentally from Monday to Sunday. If R s j the week ndex of day j, then R j {, 2, L, 7 }. Therefore, f R j =, day j s Monday; f R j = 7, day j s Sunday. Nurses submt ther leave requests for each new month by completng leave request forms. The leave requests are classfed nto eleven categores. Category s for regular leaves; 2 s for annual vacatons; 3 s for medcal leaves; 4 s for weddng leaves; s for maternty leaves; 6 s for mscarrage leaves; 7 s for funeral leaves; 8 s for personal leaves; 9 s for offcally requred leaves; 0 s for work-related njury leaves; and s for offcal n-servce tranng leaves. U = {, 2, L,} { 0} s the set of all leave categores, ncludng no leave request, whch s category 0. For nurse and day j, Hj s the leave request of nurse for day j. Hj U. The number of nurses that are needed for each day and each shft s known or determned n advance. For day j and shft k, ρ jk s the number of nurses needed on day j and shft k. The number of nurses that can 3 take a leave day s p ρ k = jk. When the number of nurses that submt a leave request for day j exceeds the number of nurses that can take a leave day on day j, the head nurse must determne whch nurses can take a leave day, based on the nformaton of H j, ε, and γ. For nurse and day j, x j s the leave request result for nurse and day j. Intally, x j s set to -. Durng schedulng, xj s set to 0 f nurse requests a leave day on day j and the request s approved. A request approvng operaton evaluates values of H j, ε, and γ to approve or declne the leave requests. After leave requests are completed, the GA-based schedulng algorthm replaces x j that are - wth a shft ndex ( x j K ) or a value 0 for a day-off. After all of the x j are set, nurse s gven a leave day on day j f x j = 0 ; nurse s assgned to shft xj = β K on day j. In practce, after a schedule s created, mnor changes can be made to the schedule. Generally, nurses may want to trade shfts or leave days wth other nurses. Changes may need to be made throughout the scheduled month. The changes must be approved by the head nurse. If the changes are approved, they must be recorded for future reference. The schedule must be checked before t can be used. The schedule mght not be a good schedule, mght not be feasble, or mght not meet regulatons. For example, the schedule mght declne too many leave requests, assgn too many sngle-day leaves rather than two-day leaves, assgn a nurse to two shfts on the same day, or assgn a nurse to more workng days than labor laws allow. The goal of the schedulng process s to create mpartal work schedules that meet the expectatons of the nurses. However, the schedules must be feasble, and satsfy all knds of regulatons. In ths study, the expectatons of the nurses and the target hosptal are modeled as optmzaton objectves and regulatons are modeled as hard constrants. Fve objectves are dentfed and weghted and ther weghted values are added to the sngle objectve functon. In general, meta-heurstc optmzaton technques do not deal wth hard constrants drectly. Instead, they ether count the occurrences of constrant volatons or measure the degree or amount of the volatons. Then the values of constrant volatons are regarded as penalty to the objectve value. In ths study, fve hard constrants are establshed to meet the requrements and regulatons from the nursng unt, hosptal, and the labor law. The counts of constrant volatons are added to the objectve functon as penaltes by multplyng a large penalty value. The objectve functon for the problem s ( ) s= s s s= s T x = z f + M v. () In Equaton, z s s a user specfed weght for optmzaton objectve value f s ; M s a fxed weght for hard constrant volaton count v s ; and x s the generated schedule. The goal of the optmzaton process s to mnmze the overall value of the objectve functon. 3. Optmzaton Objectves When nurses request leave days for offcal n-servce tranng, they prefer to have the days before and after tranng scheduled as leave days, to get adequate rest before and after tranng. The objectve functon contans a term that counts the number of

F. C. Yang and W. T. Wu: A Genetc Algorthm-based Method for Creatng Impartal Work Schedules for Nurses 8 tmes that an n-servce tranng request s granted ( Hj = xj = 0 ) and ether the day before or after the n-servce tranng sesson s not scheduled as a leave day ( x ( ) 0 j > or x ( j+) > 0 ): f = Count ; j=2,3,..., q ( H 0 ( ( ) 0 ( ) 0 j = xj = x > x > j j+ )). (2) For regular days off, nurses don t expect to get weekend leaves every week. However, they prefer to have some weekend leaves every month. The objectve functon contans a term that counts the number of nurses who do not receve any weekend leave: 2 ( j ( ) ) j R = 6 j f = Count Count x = 0 x + = 0 < j. (3) For regular days off, nurses prefer to have one two-day leave, rather than two sngle-day leaves to get enough rest. The objectve functon contans two terms that count the number of tmes nurses receve sngle-day leaves and nurses have two days off separated by a workng day: f = Count 3 ; j= 2,3,..., q ( H 0 ( ) 0 ( ) 0 j xj = x > x > j j+ ) 4 ( j 0 ( ) 0 ( ) 0) ; j=2,3,..., q j j+, (4) f = Count x > x = x =. () A good schedule should approve all the leave requests to meet nurses expectatons. However, t s nearly mpossble. Therefore, based on the avalable manpower and averaged workload, each nursng unt has set up a target rato of granted leave requests to the total number of leave requests. The head nurse should try to meet the target δ when creatng a schedule. The objectve functon, therefore, contans a term that measures the dfference between the target and the rato of granted leave requests to the total number of leave requests: f Count H j = max 0.0,00 δ Count H j ( j > 0 xj = 0) ( j > 0).(6) 3.2 Hard Constrants The number of nurses assgned to each shft on each day must be equal to the predetermned number of nurses that the hosptal needs. The objectve functon contans a term that counts the number of tmes the number of nurses assgned to each shft on each day does not match the predetermned number of nurses that the hosptal needs: jk, ( ( j ) jk ) v = Count Count x = k ρ. (7) The work schedule of a nurse s regulated by labor laws and hosptal rules. If labor laws or hosptal rules lmt the number of consecutve workng days of a nurse to D days, a feasble schedule must lmt the number of consecutve workng days of the nurse to no more than D days. The objectve functon contans a term that counts the number of tmes that assgned workng days exceed the lmt: v 2 r, =,2,..., q D,,..., ( Count ( xj > 0) D j= r r+ r+ D ) = Count. (8) In general, nursng unts need a lower number of nurses durng nght shfts. Low skll level nurses are usually assgned to nght shfts because they are young, they have less senorty, and they are generally sngle. However, f all of the nurses have low skll levels, servce qualty s low. As a result, hosptal rules requre that at least one nurse must have a skll level greater than. Therefore, the sum of the levels of nurses on the nght shft should not smaller or equal to the number of nurses. The objectve functon contans a term that counts the number of tmes that a nght shft does not meet the requrement: ( α ( 3) x j ) j = v = Count Count x =. (9) 3 j, 3 Nurses can be asked to work more days n a month than usual. However, workng too many days n a month can affect the nurses physcal health. As a result, hosptal rules requre that nurses have at least C leave days every month. The objectve functon contans a term that counts the number of tmes that the number of leaves does not meet the requrement: ( ( j ) j ) v = Count Count x = < C. (0) 0 4 Hosptal rules also requre that nurses have at least eght hours of break between two consecutve shfts. As a result, nurses cannot work an evenng or nght shft and a subsequent day shft. The objectve functon contans a term that counts the number of tmes that assgned shfts do not meet the requrement:

86 Internatonal Journal of Electronc Busness Management, Vol. 0, No. 3 (202) ( j { 2,3} ( ) j ) v = Count x x + =. (), j=,2,..., q 3.3 The Genetc Algorthm The genetc algorthm repeatedly evolves a set of solutons toward optmalty usng smulated genetc operators ncludng genetc selecton, crossover, and mutaton operators. The genetc algorthm-base nurse schedulng method conssts of a dedcated genetc codng and customzed genetc operators to generate near-optmum schedules for the nurses. In ths model, a schedule represented by a matrx of ntegers x p q= x j where xj K {} 0. The value of xj ndcates whether nurse s assgned to a shft or gven a leave day on day j. The genetc operators do not evolve x j values that are ntally set to 0 n leave approval, whch mean nurse has requested and been granted a day-off on day j. The proposed method uses an Approve_Leaves() operaton to set some x j values to 0 to grant or turn down the leave requests. The operaton grants leave request based the nformaton of cumulated shortage of leave days γ and the number of leave days granted ε for each nurse and the number of nurses needed on each shft ρ jk. The Approve_Leaves() operaton sets x j to - f the leave request s not approved: Approve_Leaves ( ρ,h,x ) xj ;, j 2 for j to q 3 for k to 3 allowed Count β = k ρ 4 ( ) =,2, L, p V { Hj > 0 β = k; } 6 f V > allowed 7 Sort nurses n V n descendng orders of γ and ascendng orders of ε, such that v { v, v2, L }, γ v γ k v ε k v ε + k vk+ 8 xj 0; { v, v2,..., vallowed } 9 else xj 0; V 0 end f 2 end for 3 end for If the number of leave requests for a shft exceeds the number of leaves that can be allowed, the nurses are sorted by the cumulated shortage of leave days taken and the number of leave days gven. Nurses that jk have greater cumulated shortages of leave days taken and smaller numbers of leave days gven are gven hgher prortes for leave approval. 3.4 Intal Populaton The ntal populaton of schedule chromosomes () (2) ( ϕ ) s { x, x,..., x }. The populaton sze ϕ s specfed by the user. An Intalze_Populaton() operaton s used to ntalze the chromosome values to create an ntal populaton of schedules: Intalze_Populaton ( ρ, x ) for j to q 2 Assgn_Daly_Shfts ( j, ρ, x ) 3 end for The Intalze_Populaton() operaton executes an Assgn_Daly_Shfts() procedure to assgn shfts for all of the q work days: Assgn_Daly_Shfts ( j, ρ, x ) for k to 3 2 v { xj = β = k} = { v, v, L } 3 for s to ρ jk 4 Random( v ) xj k v v 7 end for 8 xj 0; v 9 end for 6 { } 2 In lne 2, the Assgn_Daly_Shfts() procedure dentfes the set of nurses that dd not submt leave requests for day j and s bundled wth shft k. In lnes 3-8, the procedure randomly assgns ρ jk nurses from the set to shft k on day j and then gves other nurses leave days. Ths procedure calls a Random() operaton to randomly select a nurse from the set of nurses that dd not submt leave requests for day j and shft k. 3. Crossover Operator Ths study develops a dedcated crossover operator to breed offsprng from a par of parent chromosomes. The operator randomly selects θ days and swaps thers shfts between parents. θ s a user specfed number and 7 θ s suggested. For a par of () (2) parent chromosomes x and x, the operator clones and modfes x () (2) and x to generate offsprng chromosomes x () and x (2) va the mplemented operaton: Date_Scattered_Crossover ( x (), x (2), x (), x (2),θ )

F. C. Yang and W. T. Wu: A Genetc Algorthm-based Method for Creatng Impartal Work Schedules for Nurses 87 2 3 { a a2 aθ } ( Q) A,,...,, as Random ; s =,2,..., θ (2),f () xj j A x j,, j () xj,otherwse (),f (2) xj j A x j,, j (2) xj,otherwse In lne, the Date_Scattered_Crossover() operaton randomly selects θ days from the current month. In lnes 2 and 3, the operaton exchanges the schedules of the parent chromosomes for the selected days to create offsprng chromosomes. 3.6 Mutaton Operator Ths study also develops a correspondng mutaton operator to yeld new chromosome. As notced, T column j, xj x2j L x pj, n chromosome matrx x s a schedule for day j that meets the nursng unt s need for nurses. A mutaton operaton that smply swaps two arbtrary columns s lkely to yeld an nfeasble soluton. Therefore, the mutaton operaton proposed s conducted wthn a column (day) by randomly reassgnng the work shfts for the nurses. For chromosome matrx x, the mutaton operator randomly selects ω days to reassgn the day shfts and clones the schedules of other days to yeld a new chromosome matrx ˆx : Mutaton ( ( c ρ, x ), x ˆ, ω) { a a2 aω } ( ) A,,...,, a Random Q ; m=, 2,..., ω m 2 for j to q 3 f j A 4 Assgn_Daly_Shfts ( j, ρ, x ˆ ) else 6 xˆ j xj, =,2, L, p 7 end f 8 end for In lne 4, the mutaton operator creates a new schedule for each of the ω randomly selected days. Conversely, the operator copes the parent schedule for each of the unselected days. 3.7 Selecton Genetc selecton consttutes a new set of chromosomes and evolves to the next generaton of genetc evoluton. The selecton pool of our method conssts of parent and offsprng chromosomes. Four selecton modes are proposed: () determnstc selecton, (2) stochastc selecton, (3) hybrd selecton, and (4) grouped stochastc selecton. All of the selecton modes select parent or offsprng chromosomes based upon ther ftness values (objectve values of the schedules). For the combned set of parent and offsprng schedule chromosomes G = {, 2, L, m}, m > ϕ, T c s the objectve value of chromosome c n G, c m. The goal of the optmzaton process s to mnmze the value of the objectve functon. Therefore, the ftness of schedule chromosome c s set to T c. The determnstc mode chooses the ϕ chromosomes that have the hghest ftness values to consttute the next generaton U : Determnstc_Selecton(G). Sort chromosomes n G n descendng orders of ther ftness as G { G, G 2,..., G m},, c =, L, m TG T c Gc + 2. U { G, G 2,..., G ϕ } The stochastc selecton mode chooses ϕ schedule chromosomes from G by callng a Probablstc_Selecton() operaton: Stochastc_Selecton(G) U Probablstc_Selecton ( G,ϕ ) The Probablstc_Selecton() operaton stochastcally selects n chromosomes from chromosome set U to consttute a subset U based on the ftness values of the chromosomes: Probablstc_Selecton ( U,n) n pc, c U Tc c U T 2 U { } c 3 cumulated 0 4 μ RealRandom(), 0 μ for c to n cumulated cumulated + p 7 whle cumulated > μ 8 U U { c} 9 μ μ + 0 end whle end for 2 return U 6 c In lne, the operaton sets the relatve probablty of each schedule chromosome n U. Each p c s pro-

88 Internatonal Journal of Electronc Busness Management, Vol. 0, No. 3 (202) portonal to the ftness value T c. In lne 4, the operaton calls a RealRandom() functon to obtan a random number between 0 and.0. In lne 8, the operaton adds chromosome c to U, one or more tmes. The hybrd selecton mode chooses the best ϕ schedule chromosomes determnstcally, wth ϕ < ϕ. The mode then choosesϕ ϕ chromosomes stochastcally from the remanng chromosomes. The grouped stochastc selecton mode groups the schedule chromosomes n G nto three groups, based on ther ftness values: hgh ftness, medum ftness, and low ftness. In each selecton step, the mode stochastcally chooses a group frst and then stochastcally chooses a schedule chromosome from the group. The procedure repeats ϕ tmes to complete the selecton: Grouped_Stochastc _Selecton(G) Sort chromosomes n G n descendng orders of ther ftness as G { G, G 2,..., G m},, c =, L, m TG T c Gc + 2 m 0.6m 3 m 0.9m 4 () G { G, G 2,..., G m } (2) G { G m, G + m + 2,..., G m } (3) G G, G,..., G 6 { m + m + 2 m} 7 U {} 8 for c to ϕ 9 μ RealRandom(),0 μ () G,f 0 μ 0.6 (2) 0 G% G,else f 0.6< μ 0.9 (3) G,otherwse U U Probablstc _ Selecton G %, 2 end for ( ) In lnes 4-6, the Grouped_Stochastc _Selecton() operaton groups the schedule chromosome nto three groups. In lnes 8-2, the operaton stochastcally chooses chromosomes from these three groups wth probabltes 60%, 30%, and 0% separately. 3.8 Stoppng Condtons The proposed genetc algorthm-based method can use dfferent stoppng condtons to complete the searchng process: teraton lmts, CPU executon tme lmts, and lmt of number of objectve functon evaluatons. After completng the searchng process, the genetc algorthm gves the best (hghest ftness) * * schedule chromosome x = x j. After the schedule s used, the number of leaves granted ε and the cumulated shortages of leaves taken γ, for each nurse s updated accordngly: * ε +, f xj = 0 Hj = 0 ε, ; (2) ε,otherwse * ( Count ( x ) j ) γ γ + 8 = 0,. (3) j 4. TEST RESULTS The method was mplemented n a software program, the Impartal Schedule Targeted Nurse Schedulng System (ISTNSS). The ISTNSS program was mplemented n the C# programmng language on a McroSoft.Net platform. Input data concernng the nurses, nursng unt needs, and prevous schedules can be drectly accessed from databases or manually entered wth the software nterface shown n Fgure. The ISTNSS program was used to create work schedules for nurses n a nursng unt of the target hosptal. Tests were conducted to determne whch of the four selecton modes gve the best work schedules. The best selecton mode was then used to compare the ISTNSS program to a program usng the varable depth search method and a program usng the modfed hll clmbng method. The two programs were reconstructed from descrptons n prevous studes [3,0]. 4. Data The nursng unt of the hosptal conssts of nurses. Table shows data for the nurses. Table 2 shows the nursng unt s needs for nurses. Table 3 shows the leave requests fled by the nurses for the new month. Table : Data for the nurses Nurse 2 3 4 6 7 8 9 0 2 3 4 Rank α 3 3 3 3 2 3 2 3 3 2 2 3 4 Shft β 2 2 2 2 3 3 3 Leave Shortages γ -3 0 3-2 0-0 - 3 0 2 Leaves Granted ε 9 3 9 9 0 0 8 9 4 3 The weght values for the fve optmzaton objectves f- f were all set to ;.e., z s =, s =, 2, K,. The fxed penalty value M for the hard constrant counts v -v was set to 00. The genetc algorthm populaton sze ϕ was set to 20; the crossover rate was set to.0; the mutaton rate was set to 0.; and the stoppng condton was set to elapsed CPU tme for 30 seconds.

F. C. Yang and W. T. Wu: A Genetc Algorthm-based Method for Creatng Impartal Work Schedules for Nurses 89 Fgure : The ISTNSS software nterface Table 2: The nursng unt s needs for nurses Shft k Monday Tuesday Wednesday Thursday Frday Saturday Sunday 4 4 2 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 Table 3: Leave requests for the new month j 2 3 4 6 7 8 9 0 2 3 4 6 7 8 9 20 2 22 23 24 2 26 27 28 29 30 R j 2 3 4 6 7 2 3 4 6 7 2 3 4 6 7 2 3 4 6 7 2 3 H j 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 To determne the best selecton mode, the ISTNSS program was run ten tmes on the sample problem for each selecton mode. Table 4 shows the best and the average objectve functon values obtaned n each selecton mode. Table 4: Objectve functon values Technque 2 3 4 Best value 36 43 39 2 Average value 4.2 236.6 6.0 97.6 The weght values for the fve optmzaton objectves f- f were all set to ;.e., z s =,

90 Internatonal Journal of Electronc Busness Management, Vol. 0, No. 3 (202) s =, 2, K,. The fxed penalty value M for the hard constrant counts v-v was set to 00 The genetc algorthm populaton sze ϕ was set to 20; the crossover rate was set to.0; the mutaton rate was set to 0.; and the stoppng condton was set to elapsed CPU tme for 30 seconds. To determne the best selecton mode, the ISTNSS program was run ten tmes on the sample problem for each selecton mode. Table 4 shows the best and the average objectve functon values obtaned n each selecton mode. Fgure 2 shows the progresses of the objectve functon values durng the runnng process of the ISTNSS program. The determnstc mode outperformed the other modes. As a result, determnstc mode was used to complete the rest of the tests. The same data was used to create work schedules usng the ISTNSS program, the program mplementng the varable depth search method, and the program mplementng the modfed hll clmbng method. The frst two programs were run 0 tmes. Snce the modfed hll clmbng method s a determnstc method only one run was conducted. Fgure 2: The objectve value progresses for dfferent selecton modes Table lsts objectve functon values obtaned from the three programs. The results show that the ISTNSS program created better schedules than the other two programs. The objectve functon values obtaned from ISTNSS never exceed 00. Ths ndcates that the schedule does not volate any hard constrant because the penalty for hard constrants M was set to 00. On the contrary, the other two programs dd not create schedules that met all hard constrants and the obtaned schedules are nfeasble. Table 6 shows the best schedule that was created by the ISTNSS program, whch had an objectve functon value of 36. For the test schedule, f = 0, f 2 = 7, f 3 = 23, f 4 = 6, f = 0, and v s = 0 s =, 2, K,. Although the soluton s a good schedule, but t s not a perfect one. There are seven nurses, whose ndces are double underlned n Table 6, unable to have at least one full weekend leave. Moreover, sx workng shfts break two successve leaves and 23 sngle-day leaves le between two workng shfts. These shfts or leaves are shown n bold-talc face and underlned n the Table 6. Notce that obtanng a perfect schedule that has objectve value T = 0 s nearly mpossble, f the avalable nurse resources are nadequate for what are requred. The resource condton on the evenng and nght shfts for the sample unt s crtcal n the sample problem: 4 nurses servng for evenng shfts that requre 3 nurses daly and 3 for nght shfts that requre 2. Although, the obtaned schedule mght not please all of the nurses, but the ISTNSS program offers the head nurse alternatves to try dfferent results of objectves f to f by settng dfferent weghts on them. The provded soft computng method s therefore able to let user adjust the types and counts of unsatsfed cases for a crtcal schedulng envronment. Once the mportance degrees of the goals are set, the ISTNSS wll try ts most to meet as much the requrement from head nurse, nurses, and the hosptal. Table : Objectve functon values for the three programs Program 2 3 4 6 7 8 9 0 Best Average ISTNSS 4 4 47 38 42 46 0 4 36 8 36 4.2 Varable Depth Search 22 32 48 6 232 8 6 4 7 6 29 23.0 Modfed Hll Clmbng 27 - - - - - - - - - 27 27.0. CONCLUSIONS Hosptals must create mpartal work schedules for nurses every month. Creatng good mpartal schedules helps to mantan an effcent and pleasant workng envronment. Manual schedulng s dffcult and tme consumng. As a result, manual schedulng can lead to dssatsfacton and complants. Ths study creates a genetc-algorthm-based method for creatng mpartal work schedules for nurses. The method can creates work schedules that consder optmzaton objectves, hard constrants, and the expectatons of the nurses. The method can be used to create stable mpartal workloads that are based upon the ranks and skll levels of the nurses n a nursng unt, as well as hosptal rules and government regulatons. The study shows that the genetc algorthm-based method can fnd optmal or near-optmal work schedules that are feasble and acceptable for real hosptal nursng unts.

F. C. Yang and W. T. Wu: A Genetc Algorthm-based Method for Creatng Impartal Work Schedules for Nurses 9 Table 6: The best ISTNSS schedule j 2 3 4 6 7 8 9 0 2 3 4 6 7 8 9 20 2 22 23 24 2 26 27 28 29 30 R j 2 3 4 6 7 2 3 4 6 7 2 3 4 6 7 2 3 4 6 7 2 3 3-3 0 0 0 0 0 0 0 0 0 2 3 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2-9 0 0 0 0 0 0 0 0 6 3 9 0 0 0 0 0 0 0 0 0 0 0 7 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 8 3 0 0 0 0 0 0 0 0 0 0 0 0 9 3 2-2 2 2 2 0 2 2 2 2 2 0 2 2 2 2 2 2 0 2 2 2 2 0 2 2 2 2 2 0 0 0 2 2 0 8 0 0 0 2 2 2 2 0 2 2 2 2 0 0 2 2 2 2 2 2 0 2 2 2 0 2 2 2 2 2 2-2 2 2 0 2 2 2 2 0 0 2 2 2 2 2 2 0 2 2 0 2 2 2 0 2 2 2 2 2 2 2 2 2 3 9 2 2 2 2 2 0 0 2 2 2 2 0 2 2 0 0 2 2 0 2 2 0 2 2 2 0 0 0 2 2 3 3 3 4 3 3 3 0 0 3 3 3 0 3 3 3 0 0 0 3 3 3 3 0 0 3 0 3 3 0 0 0 3 3 4 4 3 0 3 3 3 3 3 0 0 3 3 3 3 0 3 3 3 3 3 0 0 3 3 0 3 3 3 3 3 3 0 0 3 2 3 0 0 0 3 3 3 3 0 3 0 0 3 3 3 3 0 0 3 3 3 3 3 3 0 0 3 3 3 3 3 H j REFERENCES. Azaez, M. and Al Sharf, S., 200, A 0- goal programmng model for nurse schedulng, Computers and Operatons Research, Vol. 32, No. 3, pp. 49-08. 2. Brucker, P., Burke, E. K., Curtos, T., Qu, R. and Vanden Berghe, G., 200, A shft sequence based approach for nurse schedulng and a new benchmark dataset, Journal of Heurstcs, Vol. 6, No. 4, pp. 9-73. 3. Burke, E., Curtos, T., Qu, R. and Berghe, G., 2007, A tme pre-defned varable depth search for nurse rosterng, Computer Scence Techncal Report, No. NOTTCS-TR-2007-6. Unversty of Nottngham, Jublee Campus, UK. 4. Burke, E., Curtos, T., Qu, R. and Berghe, G., 2009, A scatter search methodology for the nurse rosterng problem, Journal of the Operatonal Research Socety, Vol. 6, pp. 667-679.. Burke, E., De Causmaecker, P., and Vanden Berghe, G., 999, A hybrd Tabu search algorthm for the nurse rosterng problem, Smulated Evoluton and Learnng, 87-94. 6. Charamonte, M. V. and Charamonte, L. M., 2008, An agent-based nurse rosterng system under mnmal staffng condtons. Internatonal Journal of Producton Economcs, Vol. 4, No. 2, pp. 697-73. 7. Dowsland, K. and Thompson, J. M., 2000, Solvng a nurse schedulng problem wth knapsacks, networks and Tabu search, Journal of Operatonal Research Socety, Vol., No. 7, 82-833. 8. Dowsland, K., 998, Nurse schedulng wth tabu search and strategc oscllaton, European Journal of Operatonal Research, Vol. 06, No. 2/3, pp. 393-407. 9. L, J. and Ackeln, U., 2004, The applcaton of Bayesan optmzaton and classfer systems n nurse schedulng, In: Yao X et al. (Eds). Parallel Problem Solvng from Nature. Sprnger Lecture Notes n Computer Scence, 3242, 8-90. 0. Yh, Y., 200, Handbook of Healthcare Delvery Systems. CRC Press ABOUT THE AUTHORS Feng-Cheng Yang s currently an assocate professor n the Insttute of Industral Engneerng at Natonal Tawan Unversty. He receved the Bachelor s degree n Mechancal Engneerng from Natonal Tawan Unversty, n 980, and the Ph.D. degree n Mechancal Engneerng from the Unversty of Iowa, USA, n 99. Hs present research nterests nclude varous heurstcs technques for solvng ndustral optmzaton problems. Recently, hs focus s the newly developed Bandwdth Restrcted Transmsson-Smulated Optmzaton Algorthm for varous optmzaton problems. He has developed several software systems coverng varous heurstcs: ant colony optmzaton algorthms, electromagnetsm-lke mechansm, genetc algorthms, wrapped-round self-organzng maps, water flow-lke algorthm etc. These systems are developed to facltate academc research and teachng. Varous benchmark problems frequently used for new algorthm verfcaton are ncluded n these systems.

92 Internatonal Journal of Electronc Busness Management, Vol. 0, No. 3 (202) We-Tng Wu s currently a junor producton engneer of the Tawan Semconductor Manufacturng Co., Tawan. He receved hs MS degree from Insttute of Industral Engneerng, NTU, n August 200. Hs specaltes nclude algorthms, soft computng, and optmzaton technques. (Receved May 202, revsed August 202, accepted September 202)

F. C. Yang and W. T. Wu: A Genetc Algorthm-based Method for Creatng Impartal Work Schedules for Nurses 93 護理師公平排班問題的遺傳演算求解法楊烽正 * 吳威霆國立臺灣大學工業工程學研究所臺北市羅斯福路四段一號摘要無法常態休假的護理師排班問題它的複雜度和困難度遠高於一般的雇員排班問題原因是有多樣的限制條件和相互衝突的排班排休需求因此排班工作通常由護理長以人工方式進行然而排班結果的公平性卻常遭護理師抱怨指責和批評本文剖析一個本地醫院的病房排班問題並建立一含有五個目標和五個硬式限制條件的最佳化求解模式此外並研擬一個考量公平性的給假演算法結合遺傳演算求解最佳班表 : 一個沒有違反限制條件且高度迎合排班目標的結果本文分別研擬特定的遺傳編碼法以及搭配的交配和突變演算子, 並提供四種遺傳篩選法文中具體展示各求解程序細節, 同時以一個實例的病房資料進行排班試驗為比較成效, 同時實作文獻中的修正式貪婪爬坡法以及變動式深度搜尋法求解該範例並進行成效比較試驗結果顯示確定型篩選模式求解結果較佳此外, 本文研擬的求解法產生的排班結果都較其他兩個方法優良關鍵詞 : 護理師排班問題公平性包班遺傳演算法 (* 聯絡人 :efcyang@ntu.edu.tw)