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Computer Science Technical Report No. NOTTCS-TR-2007-4

Component Based Heuristic Search method with Adaptive Perturbations for Hospital Personnel Schedu Jingpeng Li, Uwe Aickelin and Edmund K. Burke

First released: Nov 2006

© Copyright 2007 Jingpeng Li, Uwe Aickelin and Edmund K. Burke

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A C om ponentBased H euristic Search m ethod w ith A daptive Perturbations for H ospitalPersonnelScheduling Jingpeng Li,U w e A ickelin and Edm und K .Burke {jpl,uxa,ekb}@ cs.nott.ac.uk SchoolofCom puterScience and Inform ation Technology The U niversity ofN ottingham N ottingham ,N G 8 1BB U nited K ingdom

A bstract- N urse rostering is a com plex scheduling problem thataffects hospitalpersonnelon a daily basis all over the w orld. This paper presents a new com ponent-based approach w ith adaptive perturbations,fora nurse scheduling problem arising ata m ajorU K hospital. The m ain idea behind this technique is to decom pose a schedule into its com ponents (i.e.the allocated shift pattern of each nurse), and then m im ic a natural evolutionary process on these com ponents to iteratively deliver better schedules. The w orthiness of allcom ponents in the schedule has to be continuously dem onstrated in order for them to rem ain there. This dem onstration em ploys a dynam ic evaluation function w hich evaluates how w elleach com ponentcontributes tow ards the finalobjective.Tw o perturbation steps are then applied:the firstperturbation elim inates a num ber ofcom ponents thatare deem ed notw orthy to stay in the currentschedule;the second perturbation m ay also throw out,w ith a low level of probability, som e w orthy com ponents.The elim inated com ponents are replenished w ith new ones using a set of constructive heuristics using local optim ality criteria. Com putationalresults using 52 data instances dem onstrate the applicability of the proposed approach in solving real-w orld problem s.

K eyw ords:N urse Rostering,Constructive H euristic,LocalSearch,A daptive Perturbation

1 Introduction Em ployee scheduling has been w idely studied for m ore than 40 years. The follow ing survey papers give an overview of the area: Bradley and M artin,1990;Ernst etal.,2004a and 2004b. Em ployee scheduling can be thoughtofas the problem ofassigning em ployees to shifts orduties over a scheduling period so thatcertain organizationaland personalconstraints are satisfied. It involves the construction of a schedule for each em ployee w ithin an organization in order for a setoftasks to be fulfilled. In the dom ain ofhealthcare,this is particularly challenging because of the presence ofa range ofdifferentstaffrequirem ents on differentdays and shifts. U nlike m any other organizations,healthcare institutions w ork tw enty-four hours a day for every single day of the year. Irregularshiftw ork has an effecton the nurses’w ellbeing and job satisfaction (M ueller and M cCloskey, 1990). The extent to w hich the staff roster satisfies the staff can im pact significantly upon the w orking environm ent. A utom atic approaches have significant benefits in saving adm inistrative staff tim e and also generally im prove the quality ofthe schedules produced. H ow ever,untilrecently,m ostpersonnel scheduling problem s in hospitals w ere solved m anually (Silvestro and Silvestro, 2000). Scheduling by hand is usually a very tim e consum ing task. W ithoutan autom atic toolto generate

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schedules and to test the quality of a constructed schedule, planners often have to use very straightforw ard constraints on w orking tim e and idle tim e in the recurring process. Even w hen hospitals have com puterized system s,testing and graphicalfeatures are often used butautom atic schedule generation features are stillnotcom m on. M oreover,there is a grow ing realisation that the autom ated generation of personnel schedules w ithin healthcare can provide significant benefits and savings. In this paper,w e focus on the developm entofnew techniques forautom atic nurse rostering system s. A general overview of various approaches for nurse rostering can be found in Sitom puland Randhaw a (1990),Cheang etal.(2003)and Burke etal.(2004). M ost real w orld nurse rostering problem s are extrem ely com plex and difficult. Tien and K am iyam a (1982),forexam ple,say nurse rostering is m ore com plex than the travelling salesm an problem due to the additionalconstraintof total num ber of w orking days w ithin the scheduling period. Since the 1960’s,m any papers have been published on various aspects ofnurse rostering. Early papers (W arnerand Praw da,1972;M iller,Pierskalla and Rath,1976)attem pted to solve the problem by using m athem aticalprogram m ing m odels. H ow ever,com putationaldifficulties exist w ith these approaches due to the enorm ous size of the search space. In addition,for m ostreal problem s,the goal of finding the ‘optim al’ solution is not only com pletely infeasible, but also largely m eaningless. H ospital adm inistrators norm ally w ant to quickly create a high quality schedule thatsatisfies allhard constraints and as m any softconstraintsas possible. The above observations have led to a num berofotherattem pts to solve realw orld nurse rostering problem s. Severalheuristic m ethods have been developed (e.g.,Blau,1985;A nzai and M iura, 1987). In the 1980’s and later, artificial intelligence m ethods for nurse rostering, such as constraintprogram m ing (M eyerauf’m H ofe,2001),expertsystem s (Chen and Y eung,1993)and know ledge based system s (Beddoe and Petrovic,2006) w ere investigated w ith som e success. In the 1990’s and later,m any of the papers tackle the problem w ith m eta-heuristic m ethods,w hich include sim ulated annealing (Brusco and Jacobs,1995),variable neighbourhood search (Burke et al.,2004),tabu search (D ow sland 1998;Burke,D e Causm aecker and V anden Berghe,1999)and evolutionary m ethods (Burke et al., 2001; K aw anaka et al., 2001). In very recent years, there have been increasing interests in the study of m athem aticalprogram m ing based heuristics (Bard and Purnom o, 2006 and 2007; Beliën and D em eulem eester, 2006) and the study of hyperheuristics (Burke etal.,2003;Ross,2005) for the problem (Burke,K endalland Soubeiga,2003; Ö zcan 2005). This paper tackles a nurse rostering problem arising at a m ajor U K hospital (A ickelin and D ow sland,2000; D ow sland and Thom pson,2000). Its target is to create w eekly schedules for w ards ofnurses by assigning each nurse one ofa num berofpredefined shiftpatterns in the m ost efficient w ay. Besides the traditional approach of Integer Linear Program m ing (D ow sland and Thom pson,2000),a num ber of m eta-heuristic approaches have been explored for this problem . For exam ple, in (A ickelin and D ow sland, 2000 and 2003; A ickelin and W hite, 2004) various approaches based on genetic algorithm s are presented. In (Li and A ickelin,2004) an approach based on a learning classifier system is investigated. In (Burke,K endalland Soubeiga,2003) a tabu search hyperheuristic is introduced, and in (A ickelin and Li, 2006) an estim ation of distribution algorithm is described. In this paperw e w illreporta new com ponent-based heuristic search approach w ith adaptive perturbations,w hich im plem ents optim ization on the com ponents w ithin single schedules. This approach com bines the features of iterative im provem ent and constructive perturbation w ith the ability to avoid getting stuck atlocalm inim a. The fram ew ork ofournew algorithm is an iterative im provem entheuristic,in w hich the steps of Evaluation, Perturbation-I, Perturbation-II and Reconstruction are executed in a loop until a stopping condition is reached. In the Evaluation step, a current com plete schedule is first

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decom posed into assignm ents for individual nurses, and then the assignm ent for each nurse is evaluated by a function based upon both hard constraints and soft constraints. In the Perturbation-I step,som e nurses are m arked as ‘rescheduled’and theirassignm ents are rem oved from the schedule according to the evaluating values oftheirassignm ents. In the Perturbation-II step,each rem aining nurse stillhas a sm allchance to be rescheduled,disregarding the evaluating value of his/her assignm ent. Finally, in the Reconstruction step, a refined greedy heuristic is designed to repaira broken solution and the obtained com plete solution is fed into the Evaluation step again to repeatthe loop. O ur proposed m ethod belongs to the general class of local search.In particular,itis som ew hat sim ilarto the Iterated LocalSearch algorithm (Lourenco,M artin and Stutzle,2002):they include a solution perturbation phase and an im provem ent phase. H ow ever, they differ in the w ay in w hich these tw o phases are im plem ented:The purpose ofperturbation in Iterated LocalSearch is to transform one com plete solution into another com plete solution. This serves as the starting pointforthe localheuristics w hich follow .H ow ever,the aim ofthe perturbation in ourm ethod is to transform one com plete solution into a partialsolution w hich is then fed into the reconstruction heuristics forrepair. The restofthis paperis organized as follow s. Section 2 gives an overview ofthe nurse rostering problem , and introduces the general fram ew ork of our m ethodology. Section 3 presents our algorithm for nurse rostering. Benchm ark results using real-w orld data sets collected from a m ajorU K hospitalare presented in section 4. Concluding rem arks are in section 5.

2 Prelim inaries 2.1 The N urse R ostering Problem The nurse rostering problem tackled in this paperis to create w eekly schedules forw ards ofup to 30 nurses at a large U K hospital. These schedules have to m eet the dem and for a m inim um num ber of nurses of different grades on each shift, w hilst being seen to be fair by the staff concerned and satisfying w orking contracts. The fairness objective is achieved by m eeting as m any of the nurses’ requests as possible and considering historical inform ation (e.g. previous w eekends) to ensure thatunsatisfied requests and unpopular shifts are evenly distributed. In our m odel, the day is partitioned into three shifts: tw o types of day shift know n as ‘earlies’ and ‘lates’,and a longernightshift. D ue to hospitalpolicy,a nurse w ould norm ally w ork eitherdays or nights in a given w eek (but not both), and because of the difference in shift length, a full w eek’s w ork w ould norm ally include m ore days than nights. H ow ever,som e specialnurses w ork otherm ixtures and the problem can hence notsim ply be decom posed into days and nights. H ow ever,as described in D ow sland and Thom pson (2000),the problem can be split into three independentstages. The firstuses a knapsack m odelto ensure thatthere are sufficientnurses to m eetthe covering constraints. Ifnot,additionalnurses(agency staff)are allocated to the w ard,so that the problem tackled in the second phase is alw ays feasible. The second stage is the m ost difficult and involves allocating the actual days or nights a nurse w orks. O nce this has been decided,a third phase uses a netw ork flow m odel(A huja etal.,1993)to allocate those on days to ‘earlies’and ‘lates’. Since stages 1 and 3 can be solved quickly,this paperis only concerned w ith the highly constrained second step.

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The days or nights thata nurse could w ork in one w eek define the setof feasible w eekly w ork patterns (i.e.shiftpatterns) for thatnurse. Each shiftpattern can be represented as a 0-1 vector w ith 14 elem ents, w here the first 7 elem ents represent the 7 days of the w eek and the last 7 elem ents the corresponding 7 nights ofthe w eek. A ‘1’ or ‘0’ in the vector denotes a scheduled day/night“w orked” or“notw orked”. Forexam ple,(1111100 0000000)w ould be a pattern w here the nurse w orks the first5 days and no nights. In total,the hospitalallow s justunder 500 such shiftpatterns. A specific nurse’s contract usually allow s 50 to 100 of these.D epending on the nurses’ preferences, the recent history of patterns w orked, and the overall attractiveness of the pattern,a preference costis allocated to each nurse-shiftpattern pair. These values w ere setin close consultation w ith the hospitaland range from 0 (perfect)to 100 (unacceptable),w ith a bias to low er values. D ue to the introduction of these preference costs w hich takes into account historic inform ation (e.g. w eekends w orked in previous w eeks), w e are able to reduce the planning horizon from the originalfive w eeks to the currentone w eek w ithoutaffecting solution quality.Furtherdetails aboutthe problem can be found in D ow sland (1998). The problem can be form ulated as follow s. D ecision variables: xij =1 ifnurse iw orks shiftpattern j,0 otherw ise. Param eters: m = N um berofpossible shiftpatterns; n = N um berofnurses; g = N um berofgrades; ajk =1 ifshiftpattern jcovers period k,0 otherw ise; qis =1 ifnurse iis ofgrade s orhigher,0 otherw ise; pij = Preference costofnurse iw orking shiftpattern j; Rks = D em and fornurses w ith grade s on period k; A(i)= Setoffeasible shiftpatterns fornurse i. Targetfunction: n

∑ ∑p x

M in

ij ij

.

(1)

i=1 j˛ A(i)

Subjectto:

∑x

ij

= 1," i˛ {1,...,n},

(2)

j˛ A (i)

n

∑ ∑q a is

x ‡ Rks," k ˛ {1,...,14},s˛ {1,...,g}.

jk ij

(3)

j˛ A (i) i=1

The constraints outlined in (2) ensure that every nurse w orks exactly one shift pattern from his/her feasible set. The constraints represented by (3) ensure that the dem and for nurses is fulfilled forevery grade on every day and nightand in line w ith hospitalpolicy m ore nurses than necessary m ay w ork during any given period. In practise,there is an acute shortage ofnurses and actualoverstaffing is very rare. N ote thatthe definition of qis allow s thathigher graded nurses can substitute those at low er grades if necessary. This problem can be regarded as a m ultiplechoice set-covering problem . The sets are given by the shiftpattern vectors and the objective is to m inim ize the costof the sets needed to provide sufficientcover for each shift ateach grade. The constraints described in (2) enforce the choice of exactly one pattern (set) from the alternatives available foreach nurse.

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2.2 G eneral D escription of the C om ponent Based H euristic M ethod w ith A daptive Perturbation (C H A P)

The basic m ethodology iteratively operates the steps ofEvaluation,Perturbation-I,PerturbationII and Reconstruction in a loop on one solution (see the pseudo code presented in Figure 1). A t the beginning of the loop,an Initialization step is used to obtain a starting solution and initialize som e input param eters (e.g. stopping conditions). In the Evaluation step, the fitness (i.e. the degree of suitability) of each com ponentin the currentsolution is evaluated under an evaluation function. Then, the fitness m easure is used probabilistically to select com ponents to be elim inated in the Perturbation-I step. Com ponents w ith high fitness have a low er probability of being elim inated. Furtherm ore, to escape local m inim a in the solution space, capabilities for uphill m oves m ust be incorporated. This is carried out in the Perturbation-II step by probabilistically elim inating even som e superior com ponents of the solution in a totally random m anner. The resulting partial solutions are then fed into the Reconstruction step, w hich im plem ents application specific heuristics to derive a new and com plete solution from partial solutions. Throughoutthese iterations,the bestsolution is retained and finally returned as the finalsolution. This algorithm uses a greedy search strategy to achieve im provem ent through iterative perturbation and reconstruction. ______________________________________________________________________________ CHAP ( ) { t=0; Create an initial solution S(0) with an associate cost C(0); Cbest= C(0); While (stopping conditions not reached) { /* Decompose the solution into its component (i.e. shift Patterns of individual nurses) */ S(t)={s1, s2,..., sn}; /* The Evaluation step Use an evaluation function to assign each component a score; /* The Perturbation-I step Eliminate some well-arranged components from S(t); Obtain an incomplete solution S’(t); /* The Perturbation-II step Randomly eliminate some components from S’(t); /* The Reconstruction step Add new components into S’(t) to make it complete; S(t)=S’(t); If (C(t) is better than Cbest) Cbest=C(t); t = t+1; } Return the best solution with the cost Cbest; }

Figure 1:The pseudo code ofthe basic algorithm . In sum m ary,our m ethodology differs from som e other local search m ethods such as sim ulated annealing (K irkpatrick,G elattand V ecchi,1983)and tabu search (G lover,1989)in the w ay thatit does notfollow one trajectory in the search space. By system atically elim inating com ponents of a solution and then replenishing w ith new com ponents,this algorithm essentially em ploys a long

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sequence of m oves betw een iterations,thus perm itting m ore com plex and m ore distantchanges betw een successive solutions. This feature m eans thatour m ethod has the ability to jum p quite easily outoflocalm inim a. Furtherm ore,unlike population-based evolutionary algorithm s w hich need to m aintain a num ber of solutions as parents for offspring propagation in each generation, this m ethod operates on a single solution ata tim e. Thus,itelim inates the extra CPU -tim e needed to m aintain a setofsolutions.

3 A C om ponent Based H euristic procedure w ith A daptive Perturbation for N urse R ostering The basic idea behind the m ethod is to determ ine,for each currentschedule,the fitness of shift patterns assigned to individualnurses. The process keeps the shiftpatterns of som e nurses that are w ellchosen (having high fitness values) in the currentschedule and tries to replace the shift patterns of other nurses that have low fitness values. To enable the algorithm to execute iteratively,ateach iteration,a random ly-produced threshold (in the range [0,1])is generated,and all shift patterns w hose fitness values exceed the threshold are labelled as “good patterns” and survive in the currentschedule.The rem aining shiftpatterns are labelled as “bad patterns” and do notsurvive (becom e extinct). The fitness value therefore corresponds to the survivalchance ofa shiftpattern assigned to a specific nurse. The “bad” shiftpatterns are rem oved from the current schedule and the corresponding nurses are released, w aiting for their new assignm ents by a constructive heuristic. Follow ing this,the above steps are iterated. Thus the globalscheduling procedure is based on iterative im provem ent,w hile an iterative constructive process is perform ed w ithin. 3.1 Initialization

In this step,an initialsolution is generated to serve as a seed for its iterative im provem ent. Itis w ell know n that for m ost m eta-heuristic algorithm s, the initialization strategy can have a significant influence on perform ance. Thus, norm ally, a significant effort w ill be m ade to generate a starting pointthatis as good as possible. For nurse rostering,there are a num ber of heuristic techniques thatcan be applied to produce good starting solutions. For our m ethodology, due to the fact that the replacem ent rate in its first iteration is relatively high,the perform ance is generally independentofthe quality of the initialsolution. H ow ever,if the seed is already a relatively good solution,the overallcom putation tim e w illdecrease. Since the m ajorpurpose ofthis paperis to dem onstrate the perform ance and generalapplicability ofthe proposed m ethodology,w e deliberately generate an extrem ely poor initialsolution by random ly assigning a shiftpattern to each nurse. The steps described in section 3.2 to 3.5 are executed in sequence in a loop until a stopping condition (i.e.solution quality or the m axim um num ber of iteration)is reached. 3.2 Evaluation

In this step, the fitness of individual nurses’ assignm ents, based on com plete schedules, is evaluated. The evaluation function should be norm alized and hence can be form ulated as 2

F (E i)= ∑ w k fk (E i), " i˛ {1,...,n},

(4)

k=1

subjectto

6

2

∑w

k

= 1.

(5)

k=1

W here Ei are the shiftpattern assigned to the i-th nurse,n is the num ber of nurses, f1 (E i) and f2 (E i) is the contribution of Ei tow ards the preference and the feasibility aspectof the solution respectively. f1 (Ei) evaluates the shiftpattern assigned to a nurse in term s of the degree to w hich itsatisfies the soft constraints (i.e. this nurse’s preference on his/her assigned shift pattern). It can be form ulated as pm ax - pij f1 (Ei)= , " i˛ {1,...,n}, (6) pm ax - pm in w here pij is the preference cost of nurse i w orking shift pattern j and pm ax and pm in are the m axim um and m inim um cost values am ong the shift patterns of all nurses on the current schedule,respectively. f2 (E i) evaluates how far the shiftpattern assigned to a nurse satisfies the hard constraints (i.e. coverage requirem entand grade dem ands). Thiscan be form ulated as cij - cm in f2 (E i)= , " i˛ {1,...,n} , (7) cm ax - cm in w here cij is the coverage contribution of nurse iw orking shiftpattern jand cm ax and cm in are the m axim um and m inim um coverage contribution values am ong the shiftpatterns of allnurses on the currentschedule,respectively. In a currentschedule,the coverage contribution ofeach nurse’s shiftpattern is its contribution to the cover of all three grades, w hich can be calculated as the sum of grade one, tw o and three covered shifts that w ould becom e uncovered if the nurse does not w ork on this shift pattern. Therefore,w e form ulate cij as 3

14

s=1

k=1

cij = ∑ qis (∑ ajk dks ),

(8)

W here qis = 1 ifnurse iis ofgrade s orhigher,0 otherw ise; ajk = 1 ifshiftpattern jcovers period k,0 otherw ise; dks = 1 ifthere is a shortage ofnurses during period k ofgrade s(i.e.the coverage value w ithoutconsidering shiftpattern jis sm allerthan dem and Rks),0 otherw ise.

3.3 Perturbation-I This step is to determ ine w hetherthe i-th nurses’assignm ent(denoted as Ei, " i˛ {1,...,n}) should be retained for the nextiteration or w hether itshould be elim inated and the nurse placed in the queue w aiting for the nextrescheduling. This is done by com paring his/her assignm entfitness F(Ei)to a random num berrs generated foreach iteration in the range [0,1]. IfF(Ei)≤ rs,then Ei w illbe rem oved from the currentschedule;otherw ise Ei w illsurvive in its presentposition. The days and nights that the nurses’ shift pattern covers are then released and updated for the next Reconstruction step (see below ). By using this step,an assignm entEi w ith a larger fitness value F(Ei)has a proportionally higherprobability ofsurvivalin the currentschedule. This m echanism perform sin a sim ilarw ay to roulette w heelselection in genetic algorithm s.

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3.4 Perturbation-II Follow ing the Perturbation-I step,the shiftpattern of each rem aining nurse stillhas a chance to be elim inated from the partial schedule at a given rate of rm . The days and nights that an elim inated shiftpattern covers are then released for the nextReconstruction step. A s usualfor m utation operators,com pared w ith the elim ination rate in the Perturbation-I step,the rate here should be relatively sm aller to facilitate convergence. O therw ise, there w ill be no bias in the sam pling,leading to a random restarttype algorithm . From a series ofexperim ents w e found that rm ≤5.0% yields good results and hence is the value adopted by us for our experim ents. This process is analogous to the m utation operator in a genetic algorithm . N ote thatour m ethod uses its Perturbation-II step to elim inate som e fitter com ponents and thus generate a new diversified solution indirectly.

3.5 R econstruction The Reconstruction step takes a partialschedule as the input,and produces a com plete schedule as the output. Since the new schedule is based on iterative im provem ent from the previous schedule,allshiftassignm ents in the partialschedule should rem ain unchanged. Therefore,the Reconstruction task is reduced to assigning shiftpatterns to allunscheduled nurses to com plete a partialsolution. Based on the dom ain know ledge ofnurse rostering,there are m any rules thatcan be used to build schedules. Forexam ple,A ickelin and D ow sland (2003)introduce three building rules:a ‘Cover’ rule,a ‘Contribution’ rule and a ‘Com bined’ rule. Since the lasttw o rules are quite sim ilar,in this paper w e only apply the ‘Cover rule and the ‘Com bined’ rule to fulfil the Reconstruction task. The ‘Cover’ rule is designed to achieve the feasibility of the schedule by assigning each unscheduled nurse the shift pattern that covers the m ost num ber of uncovered shifts. For instance,assum e thata shiftpattern covers M onday to Friday nightshifts. Further assum e that the currentrequirem ents forthe nightshifts from M onday to Sunday are as follow s:(-4,0,+1,-3, -1,-2,0),w here negative sym bolm eans undercover and positive m eans over-cover. The given shiftpattern hence has a cover value of 3 as itcovers the nightshifts of M onday,Thursday and Friday. N ote thatfornurses ofgrade s,this rule only counts the shifts requiring grade s nurses as long as there isa single uncovered shiftforthis grade. Ifallshifts ofgrade s are covered,shifts of grade (s-1)are counted. This operation is necessary as otherw ise highergraded nurses m ightfill low ergraded dem and first,leaving the highergraded dem and m ightunm etatall. The ‘Com bined’rule is designed to achieve a balance betw een solution quality and feasibility by going through the entire setof feasible shiftpatterns for a nurse and assigning each one a score. The one w ith the highest(i.e.best)score is chosen. Ifthere is m ore than one shiftpattern w ith the bestscore,the firstsuch shiftpattern is chosen. The score of a shiftpattern is calculated as the w eighted sum ofthe nurse’s preference costpij forthatparticularshiftpattern and its contribution to the coverofallthree grades. The latter is m easured as a w eighted sum of grade one,tw o and three uncovered shifts that w ould be covered if the nurse w orked this shift pattern, i.e. the reduction in shortfall. M ore precisely and using the sam e notation as before,the score Sij ofshift pattern jfornurse iiscalculated as 3

14

s=1

k=1

Sij = w p (100 - pij)+ ∑ w sqis (∑ a jk eks ),

(9)

8

w here w p is the w eightof the nurse’s preference costpij for the shiftpattern and w s is the w eight of covering an uncovered shiftof grade s. qis is 1 if nurse iis of grade s or higher,0 otherw ise. ajk is 1 if shiftpattern jcovers day k,0 otherw ise. eks is the num ber of nurses needed to atleast satisfy the dem and Rks ifthere are stillnurses in shortage during period k ofgrade s,0 otherw ise. (100−pij)m ustbe used in the score,as higherpij valuesare w orse and the m axim um forpij is 100. U sing the above tw o rules atthe rates of p1 and p2 respectively,the Reconstruction step assigns shiftpatterns to allunscheduled nurses untilthe broken solution is com plete. In addition,to avoid stagnation atlocaloptim a,random ness needs to be introduced into the Reconstruction steps. This is achieved by allow ing each unscheduled nurse to have an additional sm all rate p3 to be scheduled by a random ly-selected shiftpattern. N ote thatthe sum of p1,p2 and p3 should be 1. A lso note thatbecause w e solve the problem w ithoutrelying on any priorknow ledge aboutw hich nurses should be scheduled earlier and w hich nurses later,the indexing order of nurses given in the originaldata setw illbe applied throughoutthe Reconstruction step. A fter a broken solution is repaired, the fitness of this com plete solution has to be calculated. U nfortunately, due to the highly-constrained nature of the problem , feasibility cannot be guaranteed. H ence, the follow ing penalty function approach is used to evaluate the solutions obtained 14 g n m   + p x w m ax R  ks ∑ ∑ qisa jk xij;0 , ∑ ∑ ij ij dem and ∑ ∑ i=1 j=1 i=1 j=1 k=1 s=1   n

M in

m

(10)

w here constantw dem and is the penalty peruncovered shifts in the solution,and a “m ax” function is used due to the penalization ofundercovering.

4 C om putationalR esults This section describes the com putationalexperim ents used to testourproposed algorithm . Forall experim ents,52 realdata sets (as provided by the hospital) are available. Each data setconsists ofone w eek’s requirem ents (i.e.14 tim e periods)forallshiftand grade com binations and a listof nurses available together w ith their preference costs pij and qualifications. Typically,there w ill be betw een 20 and 30 nurses per w ard,3 grade-bands and 411 differentshiftpatterns. They are m oderately sized problem s com pared to other problem s reported in the literature (Burke et al., 2004). The data w as collected from three w ards over a period of several m onths and covers a range of scheduling situations,e.g.som e data instances have very few feasible solutions w hilst others have m ultiple optim a. A zip file containing allthese 52 instances is available to dow nload athttp://w w w .cs.nott.ac.uk/~jpl/N urse_D ata/N urseD ata.zip.

4.1 A lgorithm D etails Table 1 lists detailed com putationalresults of various approaches over 52 instances. The results listed in Table 1 are based on 20 runs w ith differentrandom seeds. The second lastrow (headed ‘A v.’) contains the m ean values ofallcolum ns,and the lastrow (headed ‘% ’)show s the relative percentage deviation values of the above m ean values to the optim al solution values. W hen com puting the m ean,a censored cost value of 255 has been used if an algorithm fails to find a feasible solution (denoted as N /A ).The follow ing notations are em ployed in the table:

• IP: optim al or best-know n solutions found by X PRESS M P, a com m ercial integer program m ing solver(D ow sland and Thom pson,2000);

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• G A -1:bestresultoutof20 runs ofa basic genetic algorithm (A ickelin and W hite,2004). • G A -2: bestresultoutof 20 runs of an adaptive G A ,w hich is the sam e as the basic genetic algorithm revision,butitalso tries to self-learn good param eters during the runtim e starting from the values given below (A ickelin and W hite,2004). • G A -3:bestresultoutof20 runs ofa m ulti-population genetic algorithm ,w hich is the sam e as the adaptive one,butalso features com peting sub-populations (A ickelin and W hite,2004). • G A -4:bestresultoutof20 runs ofthe hill-clim bing genetic algorithm ,w hich is the sam e as the m ulti-population genetic algorithm ,butitalso includes a localsearch in the form ofa hillclim beraround the currentbestsolution (A ickelin and W hite,2004). • G A -5:bestresultoutof 20 runs of an indirectgenetic algorithm ,w hich m aps the constraint solution space into an unconstrained space, then searches w ithin that new space and eventually translates solutions back into the originalspace (A ickelin and D ow sland,2003). U p to fourdifferentrules and a hill-clim berare used in this algorithm . • ED A :bestresultoutof 20 runs of an estim ation of distribution algorithm (A ickelin and Li, 2006); • LCS:bestresultoutof20 runs ofa Learning ClassifierSystem (Liand A ickelin,2004); • Con-heu:bestresultoutof20 runs ofourm ethod w ithoutthe tw o steps ofperturbation; • CH A P:ourfullCom ponentbased H euristic m ethod w ith both A daptive Perturbation steps; • Best:bestresultoutof20 runs ofCH A P; • M ean:average resultof20 runs ofCH A P; • Inf:num berofruns term inating w ith the bestsolution being infeasible; • #:num berofruns term inating w ith the bestsolution being optim al; • ≤3: num ber of runs term inating w ith the best solution being w ithin three cost units of the optim um . The value of three units w as chosen as it corresponds to the penalty cost of violating the least im portant level of requests in the original form ulation. Thus, these solutions are stillacceptable to the hospital. Set

IP

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18

8 49 50 17 11 2 11 14 3 2 2 2 2 3 3 37 9 18

GA -1 9 57 51 17 12 7 N /A 18 N /A 6 4 14 3 4 6 40 12 19

GA -2 9 57 51 17 11 7 N /A 18 N /A 6 4 14 3 4 6 40 12 19

GA -3 8 50 50 17 11 2 11 15 3 4 2 2 2 3 3 38 9 19

GA -4 8 50 50 17 11 2 13 14 3 2 2 2 2 3 3 38 9 19

GA -5 8 51 51 17 11 2 12 15 4 3 2 2 2 3 3 39 10 18

ED A

LCS

8 56 50 17 11 2 14 15 14 2 2 3 3 4 4 38 9 19

9 60 68 17 15 2 31 43 17 5 2 4 5 17 5 38 22 33

Con -heu 31 100 94 20 22 20 45 41 N /A 13 N /A N /A 103 21 5 159 N /A 125

CH A P (20 runs) Best M ean Inf # 8.0 0 20 8 54.9 0 2 49 51.9 0 12 50 17.0 0 20 17 11.5 0 19 11 2.1 0 18 2 11.5 0 12 11 16.0 0 10 14 8.5 0 12 3 3 3.6 0 0 2.0 0 20 2 2.4 0 15 2 2.3 0 14 2 19.2 0 3 3 3.0 0 20 3 37.2 0 16 37 9.2 0 18 9 18.1 0 19 18

≤3 20 3 17 20 19 20 20 15 12 20 20 19 20 5 20 20 20 20

10

19 1 20 7 21 0 22 25 23 0 24 1 25 0 26 48 27 2 28 63 29 15 30 35 31 62 32 40 33 10 34 38 35 35 36 32 37 5 38 13 39 5 40 7 41 54 42 38 43 22 44 19 45 3 46 3 47 3 48 4 49 27 50 107 51 74 52 58 A v. 21.1 % 0

5 10 7 43 8 4 6 N /A 17 66 20 44 N /A 51 N /A 42 36 N /A 8 N /A 9 14 N /A 41 24 36 N /A 17 N /A 9 36 N /A N /A N /A 79.8 278

5 1 1 1 10 8 8 7 7 0 0 0 35 26 25 25 8 0 0 0 3 1 1 1 5 0 0 0 N /A 48 48 48 17 4 2 2 66 64 63 63 20 141 17 15 44 42 38 35 284 166 95 65 51 99 41 42 N /A 12 12 10 42 48 40 39 36 35 36 35 36 41 33 32 8 5 5 5 N /A 14 16 15 8 5 5 5 10 8 8 7 65 55 54 54 41 39 38 38 24 39 24 23 36 48 25 19 9 3 3 3 10 6 6 3 5 4 3 3 9 6 4 4 36 30 29 30 N /A 211 110 110 N /A N /A 75 74 N /A N /A 75 58 65.0 37.1 23.2 22.0 208 76 10 4

10 7 1 26 1 1 0 52 28 65 109 38 159 43 11 41 46 45 7 25 8 8 55 41 23 24 6 7 3 5 30 109 171 67 29.7 41

32 7 6 38 3 1 0 93 19 67 56 41 123 42 15 70 64 54 12 30 13 15 57 80 58 34 15 28 3 18 37 110 125 85 35.5 68

N /A 36 23 150 N /A N /A 4 148 N /A N /A N /A 97 N /A N /A N /A N /A N /A 198 62 121 118 26 121 51 N /A N /A 111 N /A N /A N /A N /A N /A N /A N /A 157.4 646

1 7 0 25 0 1 0 48 3 63 15 35 66 40 11 38 36 32 6 14 5 7 54 40 22 19 3 3 3 5 27 107 89 58 21.7 2.7

1.6 14.2 0.1 26.9 0.1 1.0 1.1 68.6 17.7 63.3 62.4 43.3 69.5 45.7 12.0 42.7 43.5 41.7 7.0 46.5 5.9 8.2 54.2 41.1 23.6 28.7 4.5 5.8 3.0 12.9 38.3 107.5 180.9 85.7 28.6 35.5

0 11 20 0 8 8 0 18 20 0 6 16 0 19 20 0 20 20 0 15 20 0 8 16 0 0 2 0 11 20 1 9 11 0 5 5 0 0 0 0 8 15 0 0 18 0 5 14 0 0 2 0 4 5 0 0 16 0 0 10 0 5 20 0 18 18 0 18 20 0 0 16 0 16 17 0 1 4 0 4 19 0 2 13 0 20 20 0 0 5 0 1 2 0 12 20 3 0 0 1 3 4 0.1 9.6 14.4

Table 1:Com parison ofresults by various approachesover52 instances. Foralldata instances,w e used the follow ing setoffixed param eters in ourexperim ents: • Stopping criterion: a m axim um iteration of 50,000, or an optim al/best-know n solution has been found;

• Rate ofPerturbation-IIin Section 3.4:rm =0.05. • Rates ofReconstruction in Section 3.5:p1 =0.80,p2 =0.18,p3 =0.02; • W eightsetin form ula (9):w p =1,w 1 =8,w 2 =2 and w 3 =1; • Penalty w eightin fitness function (10):w dem and =200; N ote that som e param eter values (i.e.the m axim um num ber of iterations,rm ,p1,p2 and p3) are based on our experience and intuition and thus w e cannot prove they are the best for each instance.The rest of the values (i.e.w p,w 1,w 2,w 3 and w dem and ) are the sam e as those used in previous papers solving the sam e 52 instances,and w e are continuing to use them forconsistency.

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O ur m ethod w as coded in Java 2,and allexperim ents w ere undertaken on a Pentium 4 2.1G H z m achine under W indow s X P. To test the robustness of the proposed algorithm , each data instance w as run tw enty tim es by fixing the above param eters and varying the pseudo random num ber seed at the beginning. The execution tim e per run and per data instance varies from several m illiseconds to 20 seconds depending on the difficulty of the individual data instance. Table 2 lists the average runtim es of various approaches over the sam e 52 instances:the firstsix (i.e.IP,G A -1,G A -2,G A -3,G A -4 and G A -5) w ere run on a differentPentium III PC,w hile the follow ing tw o (i.e.ED A and LCS)on a sim ilarPentium 4 2.0G H z PC. O bviously,the IP is m uch slow erthan any ofthe above m eta-heuristics.A m ong these m eta-heuristic m ethods,ouralgorithm takes no m ore tim e although an accurate com parison in term s of runtim e is difficult due to the different environm ents (i.e. m achines, com pilers and program m ing languages) in use. For exam ple,the genetic algorithm s are coded in C and the ED A is coded in C++. The com parison in term s of the num ber of evaluations is also difficult because the other algorithm s evaluate each candidate solution as a w hole,w hile ouralgorithm evaluates partialsolutions as w ell. IP G A -1 19 Tim e (sec) >24hours

G A -2 23

G A -3 13

G A -4 15

G A -5 12

ED A 23

LC S 45

CH AP 12

Table 2:Com parison ofthe average runtim e ofvarious approaches.

4.2 A nalysisofR esults The results of all the approaches in Table 1 are obtained by using the sam e 52 benchm ark test instances,w ith the bold figure representing the optim alsolution found by a com m ercialsoftw are package. Com pared w ith the results of the m athem aticalprogram m ing approach w hich can take up to 24 hours runtim e (show n in the ‘IP’ colum n),our results (show n in the ‘Best’ colum n) are only 2.7% m ore expensive on average but they are all achieved w ithin 20 seconds. Com pared w ith the best results of various m eta-heuristic approaches, in general the CH A P results are slightly better than those of the best-perform ing indirect genetic algorithm (w ith a relative percentage deviation value of 4% ) and are m uch better than the others (w ith deviation values from 10% to 278% ). Since ourproposed m ethodology uses a ‘Cover’rule and a ‘Com bined’rule in its Reconstruction step for schedule repairing, it m ay be interesting to know if the good perform ance of our algorithm is m ainly due to these tw o delicate building rules. To clarify this,w e perform ed an additionalsetof experim ents by skipping the tw o perturbation steps,i.e.only im plem enting the Reconstruction step to build a schedule from an em pty solution. This m ethod does not yield a single feasible solution for 24 instances, as the ‘Con-heu’ colum n show s. This underlines the difficulty of this problem , and m ost im portantly it underlines the key roles played by the tw o elim ination steps in our full m ethodology, as the Reconstruction step alone is not capable of solving the problem . Figures 2 and 3 show the results ofourm ethod and the bestindirectgenetic algorithm graphically in m ore detail. The bars above the y-axis representsolution quality outof20 runs:the black bars show the num berofoptim alsolutions found (i.e.the value of‘#’in Table 1),and the dotted bars represent the num ber of good feasible solutions w hich are w ithin 3 cost units of their optim al solutions (i.e.the value of ‘≤3’ in Table 1). The bars below the y-axis representthe num ber of tim es the algorithm failed to find a feasible solution in these 20 runs (i.e. the value of ‘Inf’ in Table 1). H ence,the less the area below the y-axis and the m ore above,the betterthe algorithm ’s perform ance. N ote that ‘m issing’ bars m ean that,in 20 runs,feasible solutions are obtained at leastonce,butnone ofthem are optim alorofgood quality (w ithin 3 units ofoptim alvalues).

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20 15 10 5 0 -5 -10 -15 -20 N o.infeasible

N o.optim al

N o.w inthin 3

Figure 2:Results from CH A P. Figure 2 show s that for CH A P, 21 out of 52 data instances are solved w ell (i.e. w ith 100% solutions being w ithin 3 units ofoptim alvalues),42 instances are solved optim ally atleastonce, and overallthere are 5 infeasible solutions for3 instances. Forthe bestindirectgenetic algorithm (show n in figure 3),the results are slightly w orse:15 data instances are solved w ell,28 are solved to optim ality atleastonce,and in totalthere are 56 infeasible solutions for6 data instances.

20 15 10 5 0 -5 -10 -15 -20 N o.infeasible

N o.optim al

N o.w ithin 3

Figure 3:Results ofthe bestindirectgenetic algorithm (i.e.G A -5).

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Figure 4 show s a sum m ary of Table 1 in graphical form at and gives an overall com parison of perform ance of the other approaches w ith our proposed m ethodology. The bestresults for these instances are obtained by the IP softw are,and in general,our approach perform s better than the previous best-perform ing approach. The basic genetic algorithm (i.e.G A -1),the adaptive genetic algorithm (i.e. G A -2), the m ulti-population genetic algorithm (i.e. G A -3) and even the hillclim bing genetic algorithm (i.e.G A -4)w hich includes m ultiple populations and an elaborate local search are allsignificantly outperform ed in term s offeasibility,bestand average results. The other three approaches (i.e.the G A -5,the ED A and the LCS) belong to the class ofindirect approaches,in w hich a setof heuristic rules,including the ‘Cover’ rule and the ‘Com bined’ rule used in our approach,is used for schedule building. Com pared w ith the ED A and the LCS,our new approach perform s m uch betterin term s ofthe bestand average results,and slightly w orse in term s of feasibility. Com pared w ith the G A -5 w hich perform s best am ong all the heuristic algorithm s,our approach perform s better in allaspects of feasibility (99% vs.95% ),bestresults (21.7 versus 22.0)and average results (28.6 vs.35.6). In addition,itis w orth m entioning thatthe G A -5 uses the bestpossible orderofthe nurses (w hich,ofcourse,has to be found)forthe greedy heuristic to build a schedule,w hile ouralgorithm only uses a fixed indexing ordering given in the originaldata sets. Feasi bi l i ty

A verage

B est

Fe a s i bi l i t y/ So l ut i o nCo s t

100 90 80 70 60 50 40 30 20 10 0 IP

G A -1

G A -2

G A -3

G A -4

G A -5

ED A

LC S

CH A P

Figure 4:Sum m ary results ofvarious search algorithm s.

5 C onclusions This paper presents a new approach to address the hospitalpersonnelscheduling problem . The m ajor idea behind this m ethod is to decom pose a solution into com ponents,and then to m im ic a naturalevolutionary process on these com ponents to m ake iterative im provem ents in each single schedule. In each iteration,an unfitportion of the solution is rem oved. A ny broken solution is repaired by a refined greedy building process. Taken as a w hole, the proposed approach has a num ber of distinct advantages. Firstly, it is sim ple and easy to im plem entbecause ituses greedy algorithm s and localheuristics. Secondly, due to its features ofm aintaining only a single solution ateach iteration and elim inating inferior parts from this solution,itcan quickly converge to localoptim a. Thirdly,the technique has the ability to jum p outof localoptim a in an effective m anner. Finally,this approach can be easily

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com bined w ith other m eta-heuristics to achieve its peak perform ance on solution quality ifCPU tim e is notthe m ajorconcern. Forexam ple,tabu search can be used in the Reconstruction step to explore the neighbouring solutions in an aggressive w ay and avoid cycles by declaring attributes of visited solutions as tabu. In addition, sim ulated annealing could be used as the acceptance criteria forthe resulting solutions afterReconstruction to acceptnotonly im proved solutions as in the currentform ,butalso w orse ones w ith a certain levelofprobability.

A cknow ledgem ents The w ork w as funded by the U K G overnm ent’s m ajor funding agency, the Engineering and PhysicalSciences Research Council(EPSRC),undergrants G R/R92899/02 and G R/S70197/1.

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