Pareto-Based Optimization for Multi-objective Nurse Scheduling

School of Computer Science and Information Technology University of Nottingham Jubilee Campus NOTTINGHAM NG8 1BB, UK

Computer Science Technical Report No. NOTTCS-TR-2007-5

Pareto-Based Optimization for Multi-objective Nurse Scheduling Edmund K. Burke, Jingpeng Li and Rong Qu

First released: June 2007

© Copyright 2007 Edmund K. Burke, Jingpeng Li and Rong Qu

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Pareto-Based O ptim ization for M ulti-objective N urse Scheduling Edm und K .Burke,Jingpeng Li* and Rong Q u {ekb,jpl,rxq}@ 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 * Corresponding author A bstract:The developm entof robustnurse scheduling system s thatcan handle a w ide range of constraints and requirem ents w ould provide significantbenefitfor hospitaladm inistrators and nurses. In this paper, w e propose a Pareto-based optim ization technique to the m ultiobjective hospitalnurse scheduling problem .Foreach nurse,w e firstrandom ly generate a set oflegalshiftpatterns w hich satisfy allshift-related hard constraints.W e then em ploy an adaptive heuristic to quickly find a solution w ith the leastnum ber of violations on the coveragerelated hard constraint,by assigning one ofthe available shiftpatterns to each nurse.N ext,w e apply a coverage repairing procedure to m ake the resulting solution feasible,by adding /rem oving any under-covered / over-covered shifts. Finally, to satisfy the soft constraints (or preferences),w e suggesta sim ulated annealing based m ulti-objective approach w ith the follow ing tw o optional acceptance criteria: one w ith a w eighted-sum objective function w hich encourages m oves tow ards users’ predefined preferences,and another one w ith dom inationbased fitness function w hich encourages m oves tow ards a m ore diversified approxim ated Pareto set.To introduce m ore w ell-spread non-dom inated solutions in the Pareto set,w e restart the w hole process from different sets of random ly-produced shift patterns. Com putationalresults dem onstrate thatthe proposed technique is applicable to m odern hospitalenvironm ents. 1 Introduction Recent years have seen an increased research interest in personnel scheduling in general (Ernstetal,2004) and nurse scheduling in particular (Burke etal,2004b).The developm ent of robust nurse scheduling system s that can handle a w ide range of requirem ents and constraints w ould be ofgreatbenefitto hospitaladm inistrators and nurses.They w ould help to alleviate the chronic shortage of nurses thatthe U K and other European countries are currently facing.They can notonly lead to a m ore efficientuse of resources in hospitals,butalso lead to a happier w orkforce by generating personalschedules thatreflectthe nurses’ow n requirem ents. In addition, they can lead to safer w orking environm ents as tired and overw orked nurses are m ore likely to m ake m istakes. Research approaches to nurse scheduling have been investigated over the pastfour decades, and this paper only m akes a very brief overview .For a com prehensive review of various approaches during different periods, see survey papers by Sitom pul and Randhaw a (1990), Cheang (2003) and Burke et al (2004b). Basically, the approaches range from traditional m athem aticalprogram m ing m ethods (W arnerand Praw da,1972;Beaum ont,1997;Jaum ard et al,1998;Bard and Purnom o,2005)to specialpurpose heuristic m ethods (Isken and H ancock, 1990;Randhaw a and Sitom pul,1993).O ne ofthe m ajorresearch directions ofnurse scheduling in recent years is the study of m eta-heuristic m ethods,in w hich genetic algorithm s has form ed an im portant class (Easton and M ansour, 1999; A ickelin and D ow sland, 2000; K aw anaka et al, 2001; A ickelin and D ow sland, 2004). O ther m eta-heuristics have also been used, including sim ulated annealing (Brusco and Jacobs, 1993), tabu search (Burke et al, 1999),m em etic algorithm s (Burke etal,2001),variable neighbourhood search (Burke et al, 2004a and 2007)and Bayesian optim ization algorithm (Liand A ickelin,2006).M any ofthese m eta-heuristic approaches are attem pting to solve m odels w hich capture the increasing com -

plexity and w ide range ofdem ands required in m odern hospitalenvironm ents. N urse scheduling can be regarded as a type of resource allocation problem , in w hich the w orkload needs to be assigned to nurses periodically,taking into account a num ber of constraints and requirem ents.H ard constraints are those thatm ustbe satisfied in order to have a feasible schedule.They are often generated by physicalresource restrictions and legislation. W hen requirem ents are desirable but not obligatory they are referred to as soft constraints, w hich are often used to evaluate the quality offeasible schedules.In nurse rostering,there are a large num ber of variations on legal regulations and individual preferences, depending on different countries and institutions. Typical issues concern coverage dem and, day-off requirem ents,w eekend-off requirem ents,m inim um and m axim um w orkforce,etc (Burke etal, 2004b). H ence,the nurse scheduling problem is inherently a m ulti-objective com binatorial problem , w ith each objective, possibly in conflict each other, corresponding to a soft constraint (or preference).H ow ever,untilnow there has been very lim ited w ork on the application ofm ultiobjective techniques to this problem due to the com plex nature ofits real-w orld applications. G oalprogram m ing is the m ostcom m only used m ethod,w hich defines a targetlevelfor each criterion and relative priorities to achieve these goals,w ith the aim offinding a solution thatis as close as possible to each of the objectives in the order of the priorities given (A rthur and Ravindran, 1981; M usa and Saxena, 1984; O zkarahan and Bailey, 1988; O zkarahan 1991; Chen and Y eung,1993;A zaieza and A iSharif,2005).Berrada etal(1996) proposed a tabu search,w hich considers only the m ostprom ising m ove to im prove the objective function having the w orstvalue ateach iteration.Burke etal(2002)also presented a tabu search,butusing the m ethod ofcom prom ising program m ing to take allthe objectives into account.Jaszkiew icz (1997) introduced a Pareto sim ulated annealing based on a w eighted-sum objective function w ith adaptively changing w eights,w hich is probably the firstattem ptto address the problem undera conceptofPareto-based optim ization. In this paper,w e presenta Pareto-based optim ization technique,tow ards the targetof developing m ore flexible system s thatare capable to address nurse scheduling problem s in the real w orld.By applying an iterative heuristic w hich takes only the satisfaction ofshift-related hard constraints into account,w e random ly generate a setoflegalshiftpatterns foreach nurse.W e then em ploy an adaptive heuristic to find a quick solution by assigning one of the available shiftpatterns to each nurse.H ow ever,solutions obtained atthis stage are rarely feasible,let along in good quality,because the satisfaction ofcoverage dem ands (a hard constraint)has no w ay to guarantee based on th lim ited num ber of shiftpatterns generated and none of the soft constraints has been addressed yet.To dealw ith the coverage dem ands,w e design a repairing heuristic w hich is capable of elim inating allthe under-covers and over-covers w ithin several iterations of its run.To satisfy the softconstraints associated w ith objectives,w e proposed a sim ulated annealing based m ulti-objective approach w ith the follow ing tw o optional acceptance criteria: one w ith a w eighted-sum objective function encouraging m oves tow ards the users’ predefined preferences,and another one w ith a dom ination-based fitness function encouraging m oves tow ards a m ore diversified approxim ated Pareto set containing nondom inated solutions.To constantly update and extend the Pareto set,w e repeatedly execute the above processes from differentshiftpattern sets generated atrandom . The paperis organized as follow s.In Section 2,w e introduce the nurse scheduling problem to be addressed. In Section 3, w e form ulate its integer program m ing m odel. In Section 4, w e presentan adaptive heuristic together w ith a coverage repairing procedure to obtain feasible solutions quickly.In Section 5,w e describe a sim ulated annealing based m ulti-objective approach to dealw ith the preferences.In Section 6,w e carry outthe experim ents on realw orld instances.W e finally give ourconclusions in Section 7. 2 The N urse Scheduling Problem

2

The nurse rostering problem tackled here is based on the situation ofintensive care units in a D utch hospital,w hich involves the assignm entoffourtypes ofshifts (i.e.shifts ofearly,day, late and night) w ithin a planning period of 4-5 w eeks to 16 nurses of differentw orking contracts in a w ard (Burke etal,2007). In brief,the problem has the follow ing hard constraints (abbreviated as H C): • H C-1:D aily coverage dem and ofeach shifttype; • H C-2:Foreach day,a nurse m ay startatm ostone shift; • H C-3:M axim um num berofw orking days ofeach nurse; • H C-4:M axim um three on-duty w eekends; • H C-5:M axim um three nightshifts; • H C-6:N o nightshiftbetw een tw o non-nightshifts; • H C-7:M inim um tw o free days aftera series ofnightshifts; • H C-8:M axim um num berofconsecutive nightshifts; • H C-9:M axim um num berofconsecutive w orking days; • H C-10:N o late shifts forone particularnurse. In addition,the problem has the follow ing softconstraints (abbreviated as SC): • SC-1:Eitherno shifts ortw o shifts in w eekends; • SC-2:A voiding a single day betw een tw o days off; • SC-3:M inim um num beroffree days aftera series ofshifts; • SC-4:M axim um num berofconsecutive assignm ents ofa specific shifttype; • SC-5:M inim um num berofconsecutive assignm ents ofa specific shifttype; • SC-6:M axim um num berofw eekly w orking days; • SC-7:M inim um num berofw eekly w orking days; • SC-8:M axim um num berofconsecutive w orking days forpart-tim e nurses; • SC-9:A voiding certain shifttype successions (e.g.day shiftfollow ed by early shift). 3 A n Integer Program m ing M odel The above problem could be solved by a com m on approach of G eneration and Allocation, w hich has been used to solve m any personnelscheduling problem s successfully (Fores etal, 2002; Li and A ickelin,2004).The G eneration phase first generates a large num ber of legal shiftpatterns (i.e.possible w ork patterns during the planning period)foreach person,and the Allocation phase then allocates one of the shift patterns to each person to create a practical schedule.To our nurse scheduling problem ,w e use the follow ing G eneration steps to generate shiftpatterns fornurse i, i˛ I w here Iis the setofnurses. Step 1

Step 2 Step 3

Step 4

LetJ the setof days during the planning period,W the setof w eeks contained in the planning period, and K the set of shift types including {1(early), 2(day), 3(late)},4(night)}; Setindex g = 1 and A(i)= f ,w here A(i)is the setofshiftpatterns ofnurse i; Let Aig be the g-th shift pattern generated for nurse i, represented as Aig = (agjk |for j= 1,...,|J |and k = 1,...,|K |) w here agjk is 1 if the g-th pattern covers shifttype k on day jand 0 otherw ise.Setagjk to be 0 or1 atrandom ,w here 0 represents a free shiftand 1 a w orking shift; Satisfy H C-2: if (∑ agjk > 1|j˛ J), random ly locate a k¢˛ K having agjk¢ = 1 . k˛ K

Set agjk = 0 |" k ˛ K - {k¢} ; Step 5

Satisfy H C-4: if (∑ ∑ ag(7w )k > 3) , random ly locate a w ˛ W

having

w˛W k˛ K

3

ag(7w )k = 1.Set ag(7w )k = 0 ;

Step 6

Satisfy H C-6: if (ag(j-1)4 - agj4 + ag (i+1)4 < 0 |j˛ {2,...,|J |-1}) , random ly set either ag(j-1)4 = 0 or ag(j+1)4 = 0 ; r+n1

Step 7

Satisfy H C-8:if (∑ agj4 > n1 |r˛ {1,...,|J |-n1}) w here n1 is the m axim um num j=r

berofconsecutive nightshifts,random ly seteither agr4 = 0 or ag(r+n1)4 = 0 ; Step 8

Satisfy H C-5: if (∑ agj4 > 3) , random ly locate a j˛ J having agj4 = 1 . Set j˛ J

agj4 = 0 ;

Step 9

Satisfy H C-7: if (agj4 + ∑ ag(j+1)k + ∑ ag(j+2)k ‡ 2 |j˛ {2,...,|J |-1}) , set k˛ K

k˛ K

ag(j+1)k = 0 |" k ˛ K and ag(j+2)k = 0 |" k ˛ K ; r+n2

Step 10 Satisfy H C-9: if (∑ ∑ agjk > n2 |r˛ {1,...,|J |-n2}) w here n2 is the m axim um j=r k˛ K

num ber of consecutive w orking days, random ly locate a j˛ [r,r + n2 ] . Set agjk = 0 |" k ˛ K ; Step 11 Satisfy H C-3:if (∑ ∑ agjk > m i) w here m i is the m axim um num ber of w orking j˛ J k˛ K

days for nurse i, random ly locate a j˛ J having

∑ agjk = 1

. Set

k˛K

agjk = 0 |" k ˛ K ;

Step 12 Satisfy H C-10:if(i= 16),set agj3 = 1|" j˛ J ; Step 13 Check the legality ofAig.Ifillegal,go to step 3; Step 14 A dd Aig to A(i).If (Aig ˇ A(i)),setg = g +1; Step 15 Ifg ≤ N(i)w here N(i)isthe num berofshiftpatterns to be generated fornurse i,go to step 3. The Allocation phase can be m odelled as the follow ing Integer Program m ing (IP) problem . D ecision variable xig is 1 ifnurse iw orks on shiftpattern g and 0 otherw ise.Param eters J,K, A(i)and agjk are defined the sam e as above.D jk is the dem and ofnurses ofshifttype k on day j and cig is the preference costofnurse iw orking on shiftpattern g. |A (i)|

M inim ize ∑ ∑ cig xig

(1)

i˛ I g=1

Subjectto: |A(i)|

∑ xig = 1, " i˛ I

(2)

g=1

n |A (i)|

∑ ∑ agjk xig = D jk , " j˛ J,k˛ K

(3)

i=1 g=1

xig ˛ {0,1}, " i˛ I,g ˛ {1,...,|A(i)|}

(4)

O bjective function (1) m inim izes total cost of all nurses. Constraint (2) ensures that every nurse w orks exactly on one of his/her available shiftpatterns.Constraint(3) ensures thatthe dem and for nurses of each shifttype is fulfilled on every day,and constraint(4) ensures the integrality ofvariable xjg.

4

A ssigning every shiftpattern a sam e costvalue (e.g. cig = 1|" i,g),w e have tried to solve the above IP problem by CPLEX 10.0,the latestversion ofa com m ercialIP solver.H ow ever,not a single integer solution could be found even if w e generate severalm illion shiftpatterns for each nurse and allow several days’ runtim e. Considering that the equality constraint in (3) m ightbe too tightand over-covered shifts in rem ained in a solution could be rem oved heuristically later,w e relax (3)to be |A (i)|

∑ ∑ agjk xig ‡ D jk , " j˛ J,k˛ K

(5)

i˛ I g=1

A gain,no integersolution can be found on ourm achine regardless ofthe num berofshiftpatterns generated and the m axim um runtim e allow ed.This show sthe com plexity ofthe problem and proves thatthe traditionalapproach ofG eneration and Allocation is notsufficientto solve our problem alone.W e are therefore seeking for a m ore sophisticated technique to solve the problem effectively. 4 A n A daptive H euristic w ith a C overage R epairing Procedure

In this section,w e presenta heuristic search m ethod to address the problem atthe Allocation phase.Itfirstcarries outan Im proved Squeaky W heelO ptim ization (ISW O )search tow ards a solution w ith the leastnum ber of violations on coverage dem ands,and then applies a coverage repairing procedure to m ake the resulting solution feasible. 4.1 A n A daptive H euristic Search Process by ISW O

The ISW O is based on the observation thatthe solutions ofm any realw orld problem s consist ofcom ponents w hich are intricately w oven togetherin a non-linear,non-additive fashion.To dealw ith these com ponents,Joslin and Clem ents (1999)proposed a technique called Squeaky W heelO ptim isation (SW O ),and (A ickelin etal(2006)suggested an im proved version called the ISW O w hich incorporates som e evolutionary features,i.e.tw o additionalsteps of Selection and M utation,into the searching.In this section,w e adaptthe ISW O forournurse scheduling problem .Starting from an initialsolution created by random ly assigning a shiftpattern to each nurse,the steps of Analysis,Selection,M utation,Prioritization and Construction are executed in a loop until a user specified param eter is reached or no im provem ent has been achieved fora certain num berofiterations. The firstAnalysis step evaluates the fitness ofeach com ponent,i.e.a shiftpattern assigned to each nurse,by taking a currentschedule into account.The evaluating function used should be able to evaluate the contribution ofthis assignm enttow ards the solution feasibility.Let Si be the shiftpattern assigned to nurse i,its evaluating function can be form ulated as m ax(f1g1 ,...,fngn )- figi (6) F (Si)= , " i˛ I , m ax(f1g1 ,...,fngn )- m in(f1g1 ,...,fngn ) w here figi denotes the contribution of nurse i w orking on the gi-th shift pattern tow ards reduction in nurse shortfall(as solving the problem of nurse shortfallis surely m ore im portant than thatof nurse surplus). figi can be calculated as the num ber of shifts thatw ould becom e uncovered ifnurse idoesnotw ork his/hergi-th shiftpattern,form ulated as figi = ∑ ∑ agijkd jk ," i˛ I ,

(7)

j˛ J k˛ K

and n  1, if (∑ agi¢jk - agijk £ D jk ) , d jk =  i¢=1 0, otherw ise 

(8)

5

w here agjk uses the sam e definitions in form ula (3),djk is 1 ifthere are stillnurses ofshifttype k needed on day jbefore nurse iw orks on his/herassigned shiftpattern and 0 otherw ise. The second Selection step determ ines w hether a shift pattern Si should be retained or discarded.The decision is m ade by com paring its fitness value F(Si)to a random num bergenerated foreach iteration in the range [0,1].IfF(Si)is larger,then Si w illrem ain in its presentallocation,otherw ise Si w illbe rem oved from the currentschedule and the shifts itcovers are then released w ith the coverage dem ands updated accordingly.By using the Selection,a shift pattern w ith largerfitness value hasa higherprobability to survive in the currentschedule. The third M utation step follow s to m utate the shiftpatterns ofthe rem aining nurses,i.e.itrandom ly discards them from the partialschedule ata sm allgiven rate pm .The days and nights thata m utated Si coversare then released and coverage dem ands are updated. The fourth Prioritization step generates a new sequence for the nurses thatare w aiting to be rescheduled (i.e.the ones that have been rem oved by the steps of Selection and M utation), w ith poor-scheduled nurses being earlier in the sequence.U sing the results of Analysis,this step firstsorts the problem (orrem oved)shiftpatterns in ascending orderoftheirfitness values.A s each shiftpattern in the sequence is associated w ith a nurse,w e can then obtain a sequence ofproblem nurses. The fifth Construction step repairs a schedule by assigning one of the shiftpatterns to each unscheduled nurse,in the orderthe nurses appearin the priority sequence.A new schedule is form ed after each unscheduled nurse has been assigned a new shiftpattern.W e use the follow ing constructing heuristic to schedule one nurse ata tim e to coverthe m ostnum berofuncovered shifts.Foreach shiftpattern in a nurse’s feasible set,itcalculates the totalnum berof uncovered shifts thatw ould be covered ifthe nurse w orked on thatshiftpattern.Forinstance, considering a shortone-w eek planning period,w e assum e thata shiftpattern covers M onday to Friday day shifts.W e further assum e thatthe currentrequirem ents for the day shifts from M onday to Sunday are as follow s:(-4,0,+1,-3,-1,-2,0),w here a negative num ber m eans undercoverand a positive num berm eans over-cover.The M onday to Friday day-shiftpattern hence has a covervalue of8 as the sum ofundercoveris -8.Ifthere is m ore than one shiftpattern w ith the sam e highestundercovervalue,w e choose the firstfitted one. A ftereach run ofthe above five steps,w e need to calculate the fitness ofthe obtained solution so thatthe ISW O can alw ays search from a best-im proved solution.The chosen encoding of ISW O autom atically satisfies constraints (2) and (4) of the IP form ulation,and the targetof ISW O is to achieve a solution thatis as closer to a feasible schedule as possible.H ence,in designing our fitness function,w e can ignore the objective (1) and justevaluates the num ber of violations on coverage dem ands,i.e.try to satisfy constraint(3) as m uch as possible.W e use the follow ing function to calculate the fitness ofan obtained solution: n |A (i)|

M inim ize

∑ ∑ D jk - ∑ ∑ agjk xig j˛ J k˛ K

(9)

i=1 g=1

4.3 A C overage R epairing Procedure

D ue to the highly-constrained nature of the problem , solutions obtained by the ISW O are rarely feasible (i.e.there exists atleastone under-covered orover-covered shift).To m ake the resulting solutions feasible, w e em ploy a coverage repairing procedure to elim inate all the over-covers and under-covers as follow s. Step 1

Setinitialsolution S ={S1,… ,Sn),w here Si denotes the shiftpattern thatnurse i w orks;

6

n

Step 2

Rem ove a single over-covered shift: if (∑ agijk > D jk ) and (agijk = 1) , set i=1

agijk = 0 .Shiftpattern Si is thus revised.If Si is infeasible w hich m eans atleast

one ofthe hard constraintshas been violated,set agijk = 1 back; n

Step 3

Rem ove tw o consecutive over-covered shifts: if (∑ (agijk + agi(j+1)k )> D jk ) and i=1

(agijk = agi(j+1)k = 1), set (agijk = agi(j+1)k = 0). If the revised Si is infeasible, set (agijk = agi(j+1)k = 1) back; n

Step 4

A dd a single under-covered shift:if (∑ agijk < D jk ) and (agijk = 0),set agijk = 1 . i=1

Ifthe revised Aigi isinfeasible,set agijk = 0 back; n

Step 5

A dd tw o consecutive under-covered shifts: if (∑ (agijk + agi(j+1)k )< D jk ) i=1

and (agijk = agi(j+1)k = 0),set (agijk = agi(j+1)k = 1).Ifthe revised Aigi is infeasible, set (agijk = agi(j+1)k = 0) back; n

Step 6

Sw ap an under-covered shift w ith an over-covered shift: if (∑ agijk < D jk ) and i=1 n

(agijk = 0) and (∑ agi¢j¢k¢ > D j¢k¢) and (agi¢j¢k¢ = 1), set agijk = 1 and agi¢j¢k¢ = 0 . If i¢=1

one ofthe revised Aigi and Aigi¢ is infeasible,set agijk = 0 and agi¢j¢k¢ = 1 back; n

Step 7

Check the feasibility of the revised schedule S:if (∑ agijk „ D jk |j˛ J,k ˛ K ),go

Step 8

to step 2; Stop and outputthe finalschedule S.

i=1

Providing a given schedule is infeasible butsatisfies m ostofthe coverage dem ands,the above coverage repairing procedure can transform this infeasible schedule to a feasible one quickly. H ow ever,underthe rare circum stance w here the procedure fails to m ake the repair,w e should consider another initialschedule.This can be obtained by sim ply rerunning the ISW O for a differentnum berofiterations,orfrom differentsets oflegalshiftpatterns. 5 A Sim ulated-based M ulti-objective A pproach for N urse Scheduling

A fterobtaining a feasible solution w hich satisfies allthe hard constraints by the above heuristic search m ethod,w e then need to edititforpracticaluse by satisfying the softconstraints as m uch as possible.Regarding the softconstraints,a hospitaladm inistratornorm ally has a general priority ordering in m ind beforehand,but in m aking actual schedules such an ordering m ightnotbe im plem ented strictly.H ence,the constrainthandling in nurse becom es a m ultiobjective problem ,w ith each softconstraintassociated w ith an individualgoal.In this section, w e presenta sim ulated annealing based m ulti-objective approach to dealw ith the problem . 5.1 O bjective Functions

W e firstdefine the decision variable xijk to be 1 ifnurse iw orks on shiftpattern k on day j,0 otherw ise.The definition ofeach param eter,ifnotspecified separately,is the sam e as before. W e use the follow ing nine objective functions to form ulate the corresponding nine goals. 1) G oal1

7

This goalisto achieve com plete w eekends during the planning period,form ulated as ∑ (xi(7w -1)k - xi(7w )k )= 0," i˛ I,w ˛ W .

(10)

k˛ K

Thus,w e can define the objective function f1(x)ofgoal1 as M inim ize f1(x)= ∑

∑ ∑ (xi(7w -1)k - xi(7w )k ) .

(11)

i˛ I w˛W k˛ K

2) G oal2 This goalisto avoid any stand-alone shiftduring the planning period,form ulated as ∑ (xi(j-1)k - xijk + xi(j+1)k )‡ 0," i˛ I,j˛ {2,...,J - 1} .

(12)

k˛ K

Thus,w e can define the objective function f2(x)ofgoal2 as |J|-1   M inim ize f2 (x)= ∑ ∑ m ax0,∑ (- xi(j-1)k + xijk - xi(j+1)k ) . i˛ I j=2  k˛ K 

(13)

3) G oal3 This goalis to allocate atleasttw o free days aftera series ofshifts during the planning period, form ulated as (14) ∑ (xi(j-1)k - xijk + xi(j+1)k )£ 1," i˛ I,j˛ {2,...,J - 1}. k˛ K

Thus,w e can define the objective function f3(x)ofgoal3 as |J|-1   M inim ize f3 (x)= ∑ ∑ m ax0,∑ (xi(j-1)k - xijk + xi(j+1)k )- 1 . i˛ I j=2  k˛ K 

(15)

4) G oal4 This goalis to allocate atm osta certain num berofconsecutive shifts ofa particularshifttype during the planning period,form ulated as r+3

∑ xijk £ ck ," i˛ I,r˛ {1,...,J - 3},k ˛ {1,3},

(16)

j=r

w here ck is the m axim um num berofconsecutive shifts oftype k.Thus,w e can define the objective function f4(x)ofgoal4 as |J|-3  r+3  M inim ize f4 (x)= ∑ ∑ ∑ m ax 0,∑ xijk - ck  . (17)  j=r  i˛ I r=1 k˛{1,3} 5) G oal5 This goalis to m inim ize the num berofconsecutive assignm ents ofa specific shifttype during the planning period,form ulated as xi(j-1)k - xijk + xi(j+1)k ‡ 0," i˛ I,j˛ {2,..., J - 1},k ˛ {1,3} . (18) Thus,w e can define the objective function f5(x)ofgoal5 as |J|-1

M inim ize f5 (x)= ∑ ∑

∑ m ax{0,- xi(j-1)k + xijk - xi(j+1)k }.

(19)

i˛ I j=2 k˛{1,3}

6) G oal6 This goalis to allocate atm osta certain num ber of w eekly w orking days to each nurse w ith differentw orking contract,form ulated as 7w

∑ ∑ xijk £ gt," t˛ {1,2,3},i˛ It,w ˛ W

,

(20)

j=7w -6 k˛ K

w here It is the subsetof nurses w orking on the t-th contractsatisfying I = {I1 (fulltim e),I2 (shortparttim e),I3 (long parttim e)},and gt is the m axim um num berofw eekly w orking days

8

ofnursesin subsetIt.Thus,w e can define the objective function f6(x)ofgoal6 as |W |  3 7w    M inim ize f6 (x)= ∑ ∑ ∑  m ax 0, ∑ ∑ xijk - gt  .  j=7w -6 k˛ K   t=1 i˛ It w =1  

(21)

7) G oal7 This goalis to allocate atleasta certain num ber of w eekly w orking days to each nurse w ith differentw orking contract,form ulated as 7w

∑ ∑ xijk ‡ ht," t˛ {1,2,3},i˛ It,w ˛ W

,

(22)

j=7w -6 k˛ K

w here ht is the m inim um num berofw eekly w orking days ofnurses in subsetIt.Thus,w e can define the objective function f7(x)ofgoal7 as |W |  3 7w    (23) M inim ize f7 (x)= ∑ ∑ ∑  m ax 0,ht - ∑ ∑ xijk   .    t=1 i˛ It w =1  j=7w -6 k˛ K  8) G oal8 This goalis to m axim ize the num berofconsecutive w orking days forpart-tim e nurses during the planning period,form ulated as r+3

∑ ∑ xijk £ 3," i˛ I1,r˛ {1,...,J - 3}.

(24)

j=r k˛ K

Thus,w e can define the objective function f8(x)ofgoal8 as |J|-3  r+3  M inim ize f8 (x)= ∑ ∑ m ax0,∑ ∑ xijk - 3 .  j=r k˛ K  i˛ I1 r=1

(25)

9) G oal9 This goalisto avoid certain shifttype successions during the planning period,form ulated as xijk1 + xi(j+1)k2 £ 2," i˛ I,j˛ {1,..., J - 1},(k1,k2 )˛ K ¢, (26) w here K ¢ is the setofundesirable shifttype successions including {(2,1),(3,1),(3,2),(1,4)}. Thus,w e can define the objective function f9(x)ofgoal9 as |J|-1

M inim ize f9 (x)= ∑ ∑

∑ m ax{0,xijk

1

}

+ xi(j+1)k2 - 2 .

(27)

i˛ I j=1 (k1,k2)˛ K ¢

5.2 G eneration ofN on-dom inated Solutions by M ulti-objectSim ulated A nnealing

W ith the above nine goals,the nurse scheduling problem can be regarded as a m ulti-objective optim ization problem expressed as (28) M inim ize f(x)= (f1(x),...,f9 (x)). Conceptofdom inance has been used to m ake com parison betw een tw o solutions.A solution x is said to dom inate another solution y if and only if fi(x)£ fi(y) for i=1,… ,9 and fi(x)< fi(y) for atleast one i.A solution is said to be globally non-dom inated (or Paretooptim al)ifno othersolution can dom inate it.The setofallPareto-optim alsolutions is called the Pareto-optim alfront(orthe Pareto set),and solutions in the Pareto setrepresentthe possible optim altrade-offs betw een conflicting objectives.A usercan then selecta preferred solution from the Pareto setonce itis revealed.W hen using (m eta-)heuristic approaches,the nondom inated setproduced w illonly be an approxim ation to the true Pareto front,thus in this paper w e refer the set generated by our approach as the archive of the approxim ated Pareto front.

9

The traditional approach to the m ulti-objective problem is to use a w eighted-sum objective function w hich com bines the m ultiple objectives into one scalar objective (Chankong and H aim es,1983;Ehrgott,2000).O ver the recentyears,researchers have proposed a num ber of m eta-heuristic approaches,w hich are m ostly evolutionary algorithm based (see survey paper (Coello Coello (2006)). Sim ulated annealing (SA )is a stochastic search algorithm firstintroduced by K irkpatrick etal (1983)to a spin glass m odel.SA has been used to solve a w ide variety ofsingle optim ization problem s, w ith a substantial reduction in com putational tim e, for m ore than tw enty years. H ow ever,the applications ofSA to m ulti-objective problem s are very lim ited (see survey paper(Sum an and K um ar,2006)),m ostofw hich stilluse the traditionalw eighted-sum objective functions.This type of functions w ould either dam age the proof of SA ’s convergence under m ulti-objective circum stances,or restrictSA ’s ability in fully exploring the trade-off surface of the Pareto front.In this paper,w e presenta SA -based m ulti-objective approach w ith tw o optional acceptance criteria to address user’s preferences in different w ays: one w ith a w eighted-sum objective function encouraging m oves tow ards users’ predefined preferences, and another one w ith a dom ination-based fitness function encouraging m oves tow ards m ore non-dom inated solutions w hich are w ell-spread in the approxim ated Pareto set. W e firstdefine the neighbourhoods in w hich new solutions are generated.In this paper,w e apply the neighbourhoods of sw apping blocks of consecutive shifts, w hich are inspired by hum an scheduling process of re-allocating sections of schedules.Consecutive shifts w ithin a period from one day to the w hole planning period can be sw itched betw een any pair of tw o nurses in the schedule.To avoid violating the coverage dem ands again,sw aps w illonly m ade vertically. For a better illustration, w e use Figure 1 to show the m oves allow ed in these neighbourhoods w ithin a short3-day planning period,w ith an arrow representing a possible m ove.Each day a nurse can w ork atm ostone ofthe fourshifttypes:Early (E),D ay (D ),Late (L)and Night(N ).

N urse 1 N urse 2 N urse 3

M on D E

Tue L

W ed E

E

L

L

N

N eighbourhood N t,t= 1

N urse 1 N urse 2 N urse 3

M on D E

Tue L

W ed E

E

L

L

N

N eighbourhood N t,t= 2

N urse 1 N urse 2 N urse 3

M on D E

Tue L E

L

W ed E L N

N eighbourhood N t,t= 3 Figure 1.Possible m oves in neighbourhoods N t betw een nurse 1 and nurse 3 W e then describe ourproposed SA -based m ulti-objective approach in detailsas follow s: Step 1 Provide tw o options as the solution acceptance criteria of SA : option ‘1’ for a w eighted-sum objective function and option ‘2’ for a dom ination-based fitness function; Step 2 Random ly generate a setoflegalshiftpatterns foreach nurse (see steps described

10

in Section 3); A pply the ISW O to create a quick solution tow ards the leastnum ber of coverage violations (see Section 4.1), and then apply the coverage repairing procedure to obtain a feasible solution x (see Section 4.2); Step 4 Letr be the num berofruns.Set r = 1; Step 5 Letk be the num berofiterations.Set k = 0 ; Step 6 LetP(r)be the setofpotentially non-dom inated solutions.SetP(r)= {x}; Step 7 Setcurrenttem perature T(k)to an initialtem perature T0; Step 8 Construct a new solution y by a random m ove w ithin a random ly selected neighbourhood N t ofx; Step 9 Ify is infeasible,go to step 8; Step 10 Replace x w ith y w ith acceptance probability p1 = m in (1,exp(−∆E(y,x) /T )):if Step 3

9

the option is ‘1’, DE (y,x)= ∑ wi(fi(y)- fi(x)),w here w i is the priority w eight i=1

Step 11

Step 12 Step 13 Step 14 Step 15 Step 16

of the i-th objective; if the option is ‘2’, DE (y,x)= y - x ,w here y and x denote the num berofsolutions in P(r)dom inating y and x respectively; If y is accepted and y is notdom inated by x,update the setP(r) w ith y in the follow ing w ay:check y for Pareto dom inance am ong allthe solutions in P(r),add y to P(r)ifitis non-dom inated,and rem ove the solutions originally in P(r)thatare dom inated by y. W ith a predefined sm allrate p2,replace x w ith a random ly selected solution from setP(r); Setr = r+1 and k = k+1.D ecease T(k)by using a proportionaltem perature cooling schedule:T(k)= αT(k-1),w here cooling rate α ˛ [0.80,0.99]; Repeatsteps 5-13 untila predefined num berofiterations w ithin SA iscarried out; Repeatsteps 2-14 untila predefined num berofrunsofSA is carried out. Set P = P (1)+ ...+ P (r).Rem ove any solutions in P thatare redundantor dom inated by the otherones.Thus,P is ourfinalsetofPareto-dom inated solutions.

6 C om putationalR esults

The proposed approach has been tested on a real-w orld problem w ith tw elve data instances provided by O RTEC,an internationalconsultancy com pany specializing in planning,optim ization and decision supportsolutions.The hospitalhas a generalpreference ordering regarding the softconstraints listed in Section 2,w hich is {SC-1,SC-2} f {SC-3} f {SC-4,SC-5, SC-6,SC-7,SC-8} f {SC-9},w here ‘f ’denotes “be m ore preferred than”.H ow ever,due to the “soft” nature of these constraints,the above ordering is notnecessarily the one thatm ust be com plied w ith strictly.W hile choosing a schedule foractualuse,the hospitalm ay consider candidate schedules w ith differenttrade-offs betw een the softconstraints,e.g.accepta schedule w hich just violates one or tw o constraints deem ed as “highly preferred” in general,but satisfies allof the restconstraints.H ence,there is stilla need for us to provide such a setof candidate schedules. Three earlierapproaches have been proposed on the sam e testinstances.The firstone is a hybrid genetic algorithm (Postand V eltm an,2004) w hich carries outa localsearch after each generation of the genetic algorithm to m ake im provem ent.The second is a hybrid V ariable N eighbourhood Search (V N S) (Burke etal,2007) w hich starts from an initialschedule created by an adaptive ordering technique,and sequentially runs the steps of V N S,feasibility correction,schedule disruption and schedule reconstruction in a loop until stopping criteria m et.The third is an IP-based V N S (Burke etal,2006)w hich uses a non-shift-pattern-based IP to firstsolve a sm allproblem including the fullsetofhard constraints and a subsetofthe soft constrains,and then executes a basic V N S to satisfy allthe rem aining constraints.The above three approaches solved the problem under a fram ew ork of single objective optim ization.

11

They allused the sam e w eighted-sum objective function to com bine allthe objectives fi as 9

M inim ize f(x)= ∑ wi fi(x),

(29)

i=1

w here w eights w i w ere setto be (1000,1000,100,10,10,10,10,10,5). Table 1 lists the results of these three approaches after 1 hour’s runtim e.In general,the IPbased V N S has produced the bestresults.The hybrid genetic algorithm and the hybrid V N S w ere coded in D elphi5 and im plem ented on a Pentium 1.7 G H z PC underW indow 2000 operating system .The IP-based V N S w as im plem ented on a 2.0 G H z PC underW indow s X P,of w hich the IP partw as solved by CPLEX 10.0 and the V N S partw as coded in Java 2. D ata H ybrid G A JA N 775 FEB 1791 M AR 2030 A PR 612 M AY 2296 JU N 9466 JU L 781 AUG 4850 SEP 615 O CT 736 NOV 2126 D EC 625 A V E. 2225

H ybrid V N S 735 1866 2010 457 2161 9291 481 4880 647 665 2030 520 2145

IP-based V N S 460 1526 1713 391 2090 8826 425 3488 330 445 1613 405 1809

Table 1.Results ofearlierthree approaches after1-hourruntim e The SA -based m ulti-objective approach is also coded in Java 2 and im plem ented on a 2.0 G H z PC underW indow s X P.Foreach data instance,w e allow the sam e m axim um runtim e of 1 hour.In addition,w e setthe num ber of shiftpatterns generated for each nurse to be 1000, the initialtem perature ofSA to be 100,replacem entrate p2 ofSA to be 0.02,the cooling rate ofSA to be 0.99 and the num berofiterations w ithin SA to be 1,000,000.Table 2 lists the results of using differentevaluation functions on the obtained solutions.For the w eighted-sum objective function,w e use the sam e setofw eightvalues as in form ula (29),and listthe num ber of archived non-dom inated solutions (see colum n 2) and the best solution under this evaluation function (see colum n 3).For com parison,w e also listthe relative percentage deviations of this best solution over the best solutions by the hybrid genetic algorithm (i.e. ∆% 1),the hybrid V N S (i.e.∆% 2)and the IP-based V N S (i.e.∆% 3). For the dom ination-based fitness function,the com parison in the m ulti-objective testbeds is difficultdue to the lack of a system atic criterion to m easure the perform ance of our Paretobased approach.In m ulti-objective optim ization,the objective value itselfdoes nothave a significantm eaning.Rather,the configuration ofobjective values is m ore im portant.H ence,the com m only-used m easure is only the plotting ofthe Pareto set.Fora bi-objective problem ,itis easy to draw a 2-D graph to show this m easure.W hen the dim ension of objectives increases to three,itbecom es harder to determ ine from a 3-D graph w hether the Pareto setis a good one.W hen the dim ension of objectives is larger than three,it is im possible to draw such a graph.Even if itis possible to plotthe graph for m ore than three objectives,itis nota good m easure as no quantitative inform ation exists.Therefore,for our nurse scheduling problem w ith a dim ension ofobjectives being nine,w e can only listthe num berofnon-dom inated solutionsin the Pareto setforreader’s inform ation (see the lastcolum n).

12

W eighted-sum objective function N um berof Bestsolution ∆% 1 ∆% 2 solutions JA N 640 44 17.4 12.9 FEB 1645 58 8.15 11.8 M AR 1780 39 12.3 11.4 A PR 465 133 24.1 -1.8 M AY 1590 48 30.7 26.4 JU N 9026 27 4.6 2.9 JU L 446 56 42.9 7.3 AUG 1735 76 64.2 64.4 SEP 339 90 44.9 47.6 O CT 540 85 26.6 18.8 NOV 1780 39 16.2 12.3 D EC 295 37 52.8 43.3 A V E. 1690 61 25.8% 21.5% D ata

∆% 3

-39.1 -7.8 -3.9 -18.9 23.9 -2.3 -4.9 50.3 -2.7 -21.3 -10.4 27.2 -0.8%

D om ination-based fitness function (num berofsolutions) 1431 1744 976 2235 1154 876 1362 1865 2033 2287 1006 1192 1513

Table 2.Results ofthe ourproposed m ulti-objective approach after1-hourruntim e A ccording to the results in Table 2,w e can see thatourproposed approach is very prom ising in solving the m ulti-objective nurse scheduling problem . In term s of the solution quality evaluated by the sam e objective function,ourapproach perform s sim ilarto the IP-based V N S, and significantly im prove the bestresults ofthe hybrid genetic algorithm and the hybrid V N S by 25.8% and 21.5% on average,respectively.In term s ofthe size ofits approxim ated Pareto set,our approach can generate as m any as tw o thousand non-dom inated solutions,thus leaving users plenty ofchoicesin m aking theirdecisions. 7 C onclusions

In this paper,w e propose a Pareto-based optim ization technique to solve the m ulti-objective nurse scheduling problem .W e firstdesign a generating heuristic w hich random ly generates a set of legal shift patterns for each nurse.W e then em ploy an adaptive heuristic,called im proved squeaky w heel optim ization, to quickly find a solution w ith the least violations on coverage dem ands.N ext,w e apply a coverage repairing heuristic to m ake the resulting solution feasible.Finally,w e propose a m ulti-objective version ofsim ulated annealing to create a setof approxim ated Pareto-optim alsolutions.The above processes are executed in a loop to update and extend the Pareto setrepeatedly. A pplying the proposed Pareto-based optim ization to nurse scheduling has the follow ing advantages.The firstis its optim ality:based on a benchm ark com parison by the sam e w eightedsum objective function, our approach perform s sim ilar to the IP-based m eta-heuristic approach and significantly betterthan the otherhybrid m eta-heuristic approaches.The second is its generality: it can be em ployed in a w ide range of hospital environm ents by altering its form ulations ofconstraints and requirem ents easily.The third is its flexibility:itcan generate notonly a group ofhigh quality solutions tow ards the users’predefined preferences,butalso a broaderrange ofcandidate solutions ifusers do nothave such specific preferences,thus allow ing usersto interactw ith the system by expressing theirw ide range ofpreferences. A cknow ledgem ents

The w ork w as funded by the U K ’s Engineering and Physical Sciences Research Council (EPSRC),undergrantG R/S31150/01. R eferences

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