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A planning and scheduling problem for an operating theatre using an open scheduling strategy

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A planning and scheduling problem for an operating theatre using an openscheduling strategy
H. Fei
a,
*
, N. Meskens
a
, C. Chu
b
a
Louvain School of Management and Catholic University of Mons, 151 Chaussée de Binche, 7000 Mons, Belgium
b
Laboratory of Industrial Engineering, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Châtenay-Malabry cedex, France
a r t i c l e i n f o
Article history:
Available online 28 February 2009
Keywords:
Operating theatre planning and schedulingproblemOpen schedulingColumn generationHeuristic procedureHybrid genetic algorithm
a b s t r a c t
The objective of this paper is to design a weekly surgery schedule in an operating theatre where timeblocksarereservedfor surgeons ratherthanspecialities. Bothoperatingroomsandplacesintherecoveryroom are assumed to be multifunctional, and the objectives are to maximise the utilisation of the oper-atingrooms, tominimisetheovertimecost intheoperatingtheatre, andtominimisetheunexpectedidletimebetween surgical cases. This weeklyoperating theatre planningandscheduling problemis solvedintwophases. First, theplanningproblemis solved togive thedate of surgery for eachpatient, allowing forthe availability of operating rooms and surgeons. Then a daily scheduling problem is devised to deter-mine the sequence of operations in each operating roomin each day, taking into account the availabilityof recovery beds. The planning problem is described as a set-partitioning integer-programming modeland is solved by a column-generation-based heuristic (CGBH) procedure. The daily scheduling problem,based on the results obtained in the planning phase, is treated as a two-stage hybrid ﬂow-shop problemand solved by a hybrid genetic algorithm (HGA). Our results are compared with several actual surgeryschedules in a Belgian university hospital, where time blocks have been assigned to either speciﬁc sur-geons or specialities several months in advance. According to the comparison results, surgery schedulesobtained by the proposed method have less idle time between surgical cases, much higher utilisation of operating rooms and produce less overtime.
2009 Elsevier Ltd. All rights reserved.
1. Introduction
Over the past decade, in response to multiple challenges (suchas the increase in the elderly population, the occurrence of newdiseases, and stricter budgetary constraints), health care organisa-tionshaveundergoneincreasingpressureofprovidinghighqualitysurgery at as low as possible costs. Therefore, hospitals are alwayslooking for the ways to not only improve patient care but also re-duce operating costs. Since the operating theatre is a unit withhighest cost and highest revenue as well and hence is of particularinterest for hospitals (Health Care Financial Management Associa-tion (HCFMA), 2005; Macario, Vitez, Dunn, & McDonald, 1995),hospital managers are always interested in ﬁnding effective waysof running the operating theatre so as to improve the efﬁciencyand quality of its services.What can hospital managers do to improve the performance of their operating theatres? Good-quality patient care usually meansexcellent service and patient satisfaction, highly ensured patientsafety, and ﬁrst-class care and outcomes. Hospital managersshould therefore take steps to improve quality in these three re-spects. With regard to costs, an efﬁcient surgery schedule shouldnot involve too much overtime, because the cost (especially thestafﬁng cost) of each additional hour in the operating theatre ismuch greater than the cost of a regular working hour.Obviously, it is difﬁcult, if not impossible, to target at all theseobjectivesinjustonemodel.Oftentimesitisbettertoconstructasol-uble model with only the most important objectives considered.According to a recent review made by Cardoen, Demeulemeester,and Belien (2008), many researchers have tried to develop an efﬁ-cient model for assigning surgical cases to the operating theatre in2000 or later. This is considered as a solvable problembecause pa-tientsaretheonly‘clients’oftheoperatingtheatre,anditseemsrea-sonabletoassignsurgicalcasestothesurgicalsuiteandthenadjusthospital resources to minimise the wait in this sector. Among allthose studies, two major classes of patients are involved: patientswith elective cases and those with urgent cases. The surgery datesof the former are normally well planned in advance while those of the latter are usually unexpected and must be arranged urgently.Considering that the elective surgical cases compose an importantpart of the operating theatre’s capacity, in this study, we restrictthe focus to the construction of an efﬁcient surgery schedule fortheﬁrstkindofpatientsduringaselectedperiod.
0360-8352/$ - see front matter
2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.cie.2009.02.012
*
Corresponding author. Tel.: +32 65323250; fax: +32 65323363.
E-mail addresses:
fei@fucam.ac.be (H. Fei), meskens@fucam.ac.be (N. Meskens),
chengbin.chu@ecp.fr (C. Chu).Computers & Industrial Engineering 58 (2010) 221–230
Contents lists available at ScienceDirect
Computers & Industrial Engineering
journal homepage: www.elsevier.com/locate/caie
Before the surgery schedule is constructed, a decision has to bemade about the operating theatre planning strategy. Three mainplanning strategies are used in hospitals:
Open scheduling strategy.
Thiswascalledthe‘anyworkday’strat-egy by Dexter, Traub, and Macario (2003), where it meant thatthe surgeons could choose any workday for a case. Patterson(1996) simpliﬁed it as a ‘the ﬁrst-come ﬁrst-served’ strategy.
Block scheduling strategy.
Surgeons or groups of surgeons areassigned to a set of time blocks in which they can arrange theirsurgicalcases.Intheory,thesurgeonsorgroups‘own’thesetimeblocks, which are reserved in advance and cannot be released inthe planning period even if some of them remain unused.
Modiﬁed block scheduling strategy.
The block scheduling strategycan be modiﬁed in two ways to provide more ﬂexibility: eithersome of the operating rooms’ opening hours are reserved whileothers are left open, or unused time blocks are released at someagreed time (e.g. 72h) before the surgery.In practice, the block scheduling or modiﬁed block schedulingstrategiesarewidelyusedinhospitals.Ingeneral,thedecisionpro-cess for the surgery schedule consists of four steps: (1) forecast of the total demand of operating time for each department based onthepastexperiencesduringthepastperiod(e.g.onetrimester);(2)allocation of OR blocks and staff scheduling for the next trimester;(3) construction of the case schedule and optimization of the caseschedule, given a speciﬁc scenario (normally for 1 or 2weeks); (4)execution of the surgical schedule as well as scheduling of emer-gency and add-on elective cases (scheduled upon arrival).In some Belgian hospitals, such as CHU Ambroise Paré, a varia-tion of modiﬁed block scheduling strategy is implemented, wheresome time blocks are reserved for speciﬁc surgeons rather thanspecialties everytrimester andsurgeonarefreetoassign hissurgi-cal casesintoblocksreservedtohisspecialtyexceptthat theprior-ity of assigning surgical cases into the reserved blocks in the nextweek will be removed on upcoming Friday, i.e. either some emptyblocks will be closed or some blocks will be assigned to anotherspeciality with further demand. Furthermore, we ﬁnd that withinthis modiﬁed block scheduling strategy, most of surgeons still pre-fer assigning as many as possible surgical cases into one block.Unfortunately, such kind of arrangement may cause inefﬁciencyof the operating theatre. On the one hand, it is quite hard for onesurgeon to focus on his operations without taking a break duringall the working day! According to a further analysis about the resttime of surgeons between two successive surgical cases in CHUAmbroiseParé, surgeonstakearest for about15minafter onesur-gerythoughsomeofthemcanstartthenextonewithinafewmin-utes. In general, the longer one surgical case is, the more time isneeded by the surgeon for a rest. It can account for the phenome-nonthat if asurgeonis assignedwithtoomanycasesduring1day,either some surgical case must be cancelled due to lack of time ormuch unexpected overtime will be needed. Although it is also truethat some unexpected idle time could occur while patients arewaiting for another surgeon in the operating room, we ﬁnd if everything is well arranged, no time is needed for changing fromone surgeon to another in one operating room; On the other hand,unexpected idle time may occur with block scheduling as long asone surgeon didn’t ﬁll his time blocks because the next surgeoncould not begin his operations before the start of his time block.Therefore we try to implement some ideas of the open schedul-ing strategy to surgery planning and scheduling in order to im-prove the performance in the operating theatre. In this study, wesupposed that surgeons could assign their surgical cases into timeblocksreservedwiththeblockschedulingstrategyasusualbuttheﬁnal surgery schedule of the coming week will be decided by anoperating theatre management committee on Friday by applyingopenschedulingstrategytargetedatoptimizingsomeperformancecriteria of the involved operating theatre.According to the literature, studies about the surgery planningandschedulingvarybetweenthetechniquesandtheobjectives. Asforthetechniques,amongawiderangeofmethodologiesintroducedfrom the domains of industrial operations research, mathematicprogramming models and discrete-event simulation tool are thetwomostcommonlyusedtechniques;theformerareusednotonlytoconstructthemastersurgeryplan,i.e.allocationofORtimeblocksfor eachsurgeonor specialty(e.g. Belien&Demeulemeester, 2007;Blake, Dexter, & Donald, 2002) but also to specify a surgery datefor each patient, i.e. assigning surgical cases into operating rooms(e.g.Cardoen,Demeulemeester,&Belien,2006;Fei,Chu,&Meskens,2009; Fei, Chu, Meskens, &Artiba, 2008; Guinet &Chaabane, 2003;Hans, Wullink, Van Houdenhoven, & Kazemier, 2008; Jebali, HadjAlouane, & Ladet, 2006; Kuo, Schroeder, Mahaffey, & Bollinger,2003; Lamiri, Xie, Dolgui, & Grimaud, 2008a, 2008b; Mulholland,Abrahamse, & Bahl, 2005; Ogulata & Erol, 2003; Perez, Arenas, Bil-bao, & Rodriguez, 2005; Pham & Klinkert, 2008) while the latter isnormallyusedfor improving thesurgeryscheduling(e.g. Bowers&Mould, 2005; Dexter, Macario, & Lubarsky, 2001; Dexter & Traub,2002; Sciomachen, Tanfani, & Testi, 2005; Testi, Tanfani, & Torre,2007). In addition, some researchers treat the surgery schedulingproblemastheworkshopschedulingproblemsandthereforesomemeta-heuristics used to solve the workshop problems are adaptedto the healthcare system(Fei, Meskens, & Chu, 2006; Fei, Meskens,Combes,&Chu,2006;Hansetal.,2008).Asfortheobjectives,weno-ticedinthesomestudieshaveattemptedtooptimizeasingleperfor-mancecriterionsuchasmaximisationofoperatingroomutilisationandminimisationoftherelatedcost(e.g.Belien&Demeulemeester,2008;Chaabane,Meskens,Guinet,&Laurent,2008;Kuoetal.,2003;Testietal.,2007;VanHoudenhoven,VanOostrum,Hans,Wullink,&Kazemier, 2007), while many others have included several perfor-mancecriteriaintheirstudy(e.g.Cardoenetal.,2006;Fei,Chu,Mes-kens, & Artiba, 2008; Fei, Chu, & Meskens, 2009; Fei, Meskens,Combes, & Chu, 2006; Guinet & Chaabane 2003; Hans et al., 2008; Jebali et al., 2006; Lamiri et al., 2008a, 2008b; Mulholland et al.,2005; Ogulata & Erol, 2003; Pham & Klinkert, 2008). This study isaimedatschedulingsurgicalcasestotheinvolvedoperatingtheatrewith the intent of both maximising the operating room utilisationandminimisingtheovertimecostoftheoperatingtheatre.As many other researchers did (such as Guinet & Chaabane,2003; Jebali et al., 2006; Van Houdenhoven et al., 2007), the con-sidered problem is divided into two-stages. At the ﬁrst stage, aweekly surgery planning problem is solved by assigning a surgerydate to each surgical case. At the second stage, the surgery sche-dule on each day is ﬁnally obtained by solving a daily surgeryscheduling problem.The rest of this paper is organised as follows. Firstly, a mathe-matic programming model is constructed for the weekly operatingroom planning problem, and it is solved by a column-generation-basedheuristic(CGBH)procedure.Secondly,adailysurgerysched-uling problem is transformed to a hybrid ﬂow-shop schedulingproblem and solved by a hybrid genetic algorithm (HGA) to deter-mine the ﬁnal sequence of the surgical cases that have been as-signed to that day in the planning phase. Thirdly, theexperimental results with data collected from CHU AmbroiseParé, one university hospital in Belgium, are used to evaluate theperformance of the proposed method. The paper is ended up withsome conclusions and perspectives.
2. Operating theatre weekly planning problem
Given that the ﬁnal surgery schedule is normally decided onFridaybeforethecomingweekinmanyBelgianhospitals,wefocus
222
H. Fei et al./Computers & Industrial Engineering 58 (2010) 221–230
on the assignment of surgical cases within 1week in this study aswell. Asmentionedintheprevioussection,thisconsideredsurgeryassignment problemis solvedintwosteps: ﬁrst, eachsurgical caseis assigned with a date for surgery; second, the start time of eachsurgical case is determined, and the objective is both to maximisethe operating room utilisation and to minimise the overtime costof the operating theatre.Whentryingtosolve anassignmentproblemintwophases, theoperating theatre planner normally assigns a surgery date to eachpatient (the planning phase) and decides in which operating room(OR) the patient will be operated on, and when he or she will betransferred to that OR (the scheduling phase). In this section, wedeal with the problem at the ﬁrst phase, i.e. the operating theatreweekly planning problem.Before going further, we would like to ﬁrst introduce somebackgroundinformation:inpractice, thepatientchoosesasurgeonattheconsultationstage,whichwilloftenmakethesurgeryforthepatient later. Therefore, we assume that the surgeon for each sur-gical case is determined in advance and cannot be changed. Nor-mally, when a surgical case is assigned with a date, the othersurgical team members will be speciﬁed by the operating theatreplanners. Thus, in this paper, we assume that the human andinstrumental resources, except for the surgeons, are always avail-able whenever they are needed.The other hypotheses, adopted to deﬁne the weekly planningproblem, are as follows:
all operating rooms are multifunctional, i.e. a patient can beoperated on in any available operating roomby the speciﬁc sur-geon before the given deadline;
an open scheduling strategy is used, i.e. no surgeon can decidethe ﬁnal order of surgical cases in the coming week;
emergency cases are not taken into consideration becausepatients admitted from the emergency department are usuallyoperatedonimmediately, andhenceonlyplannedsurgicalcasesare involved in this study;
once a surgical case gets started in an operating room, it cannotbe interrupted, i.e. there is no pre-emption.With such hypotheses, the planning phase problem can be re-garded as a resource-constrained bin-packing problem (Van Hou-denhoven et al., 2007) and can be formulated as a binary-integerproblem. Binary-integer-programming, whose decision versionwas one of Karp’s 21 NP-complete problems (Karp, 1972), is clas-siﬁed as NP-hard and so that the planning problem under consid-eration is NP-hard as well. Considering that no polynomialalgorithm has been found yet able to systematically obtain itsoptimal solution and hospital managers often prefer a good-qual-ity solution that requires a reasonable running time to an opti-mum one that needs several hours or even days for execution,we are interested in devising a heuristic procedure to ﬁnd anapproximate but good-quality solution with reasonable runningtime.Also employed in this study is Column Generation (CG) proce-dure, known as an efﬁcient technique to deal with this kind of problem (Barnhart, Johnson, Nemhauser, Savelsbergh, & Vance,1998;Belien&Demeulemeester,2008;Fei,Chu,Meskens,&Artiba,2008; Fei, Chu, & Meskens, 2009; Fei, Meskens, Combes, & Chu,2006; Lamiri et al., 2008b) and a column-generation-based heuris-tic (CGBH) procedure is developed to obtain an efﬁcient assign-ment of surgical cases in the planning phase. Since the CGprocedure is often used to solve problems involving set-partition-ing constraints. Inthe remainder of this section, we will ﬁrst intro-duce the constructed set-partitioning integer-programmingformulation for the planning problem under consideration andthen present the CGBH procedure.
2.1. Model of the weekly operating theatre planning problem withopen scheduling
Thebinaryset-partitioningmodelconsistsoftwoparts:oneisamaster problem describing the main constraints with the desiredobjective and the other is an auxiliary problem used to determinethe values of parameters of the master problem. Inthis section, wewill ﬁrst introduce the master problem and then the auxiliaryproblem. Sincetheexistenceof abasic feasiblesolutionispre-con-ditionoftheauxiliaryproblem,theconstructionoftheinitialsetof feasible plans will be introduced at the end of this section.
2.1.1. Set-partitioning model for the master problem
In the binary set-partitioning model of the master problem,each column corresponds to a feasible case plan for one operatingroomin 1day, i.e. a feasible sub-surgery schedule, named as a fea-sibleplanintherestofthispaper,generatedbyoneoftheauxiliaryproblems (details are given in Section 2.1.2).Parameters used in the set-partitioning model for the masterproblem are
N
D
numberofdaysovertheplanningperiod:inourmodel
N
D
=5,i.e. the planning period is 1week
N
dS
number of surgeons available on day
d
X
set of all surgical cases awaiting assignment
t
i
predicted duration of surgical case
iD
i
daysleftbeforethedeadlineforsurgicalcase
i
,i.e.thenumberof days within which surgical case
i
must be performed
M
number of operating rooms in the involved operating theatre
R
dk
number of regular opening hours of operating room
k
on day
d
. If operating room
k
is unavailable on day
d
,
R
dk
is set as 0
S
dk
maximumnumber of overtime hours for operating room
k
onday
d A
dl
maximumworkinghoursforsurgeon
l
onday
d
. If surgeon
l
isunavailable on day
d
,
A
dl
is set as 0
X
l
set of surgical cases to be treated by surgeon
l
b
cost ratio of a regular working hour to an overtime hour, i.e.the penalty cost of the overtime
C
j
operating cost of either unused opening hours or overtimehours for the operating room if the feasible plan
j
is adopted
N
set of all feasible plans for the planning period
a
ij
1 if surgical case
i
is assigned to feasible plan
j
; otherwise=0
b
d j
1 if feasible plan
j
is scheduled on day
d
; otherwise=0
e
kj
1 if operating room
k
is used by feasible plan
j
; otherwise=0Decision variables
x
j
1 if feasible plan
j
is accepted; otherwise=0
The set-partitioning model for the considered master problem is asfollows:
Min
X
j
2
N
C
j
x
j
ð
1
Þ
Subject to
X
j
2
N
a
ij
x
j
¼
1
;
i
2
X
;
D
i
N
D
;
ð
2
Þ
X
j
2
N
a
ij
x
j
1
;
i
2
X
;
D
i
>
N
D
;
ð
3
Þ
X
j
2
N
b
d j
e
kj
x
j
1
;
k
2 f
1
;
;
M
g
;
d
2 f
1
;
;
N
D
g
;
ð
4
Þ
X
j
2
N
b
d j
X
i
2
X
l
a
ij
t
i
!
x
j
A
dl
;
l
2 f
1
;
;
N
dS
g
;
d
2 f
1
;
;
N
D
g
;
ð
5
Þ
x
j
2 f
0
;
1
g
j
2
N
:
ð
6
Þ
The operating cost of feasible plan
j
is calculated as
H. Fei et al./Computers & Industrial Engineering 58 (2010) 221–230
223
C
j
¼
max
X
N
D
d
¼
1
X
M k
¼
1
b
d j
e
kj
R
dk
X
i
2
X
a
ij
t
i
!
;
b
X
i
2
X
a
ij
t
i
X
N
D
d
¼
1
X
M k
¼
1
b
d j
e
kj
R
dk
!()
;
j
2
N
ð
7
Þ
The objective function seeks to minimise the cost of the totalunexploited opening hours and overtime (calculated by Formula(7)). Sincethecost of aregular openinghour of theoperatingroomcan be treated as constant, it is omitted from this formula of theobjective function.Constraints (2) and (3) ensure that, during the planning period,eachsurgical case, the deadlineof whichisno laterthantheendof the planning period (1week), is treated exactly once before theend of the planning period, while the other cases are treated atmost once. Constraint (4) shows that each operating room can beoccupiedbyatmostoneacceptedfeasibleplanin1day. Constraint(5) ensures that the total operating time assigned to each surgeonper day cannot exceed his or her maximum working hours in thatday. This is necessary because one surgeon cannot work on twosurgical cases simultaneously. Although it seems that a surgeoncan be assigned to two or more surgical cases in parallel in thisplanning model, a feasible operating programme can be alwaysguaranteed if the patient sequence is well determined at the dailyscheduling stage as long as this constraint is respected at the plan-ning phase.Asmentionedabove,eachcolumninthisset-partitioningmodelcorrespondstoafeasibleplanconstructedwhileanauxiliaryprob-lemisbeingsolved. Onceaset of feasibleplans for assigningall in-volved surgical cases have been generated, i.e. values of parameters
a
ij
;
b
d j
and
e
kj
having been determined for each col-umn, this model is apparently a binary-integer linear program-ming, the linear relaxation of which can be easily solved by thelinear programming solver.
2.1.2. Generation of feasible plans with auxiliary problem
Incolumn
j
of theset-partitioning model of the master problemdescribed in Section 2.1.1, parameters
a
ij
;
b
d j
and
e
kj
must respectthefollowingconstraintsinordertoensurethatthiscolumncorre-sponds to a feasible plan:
X
i
2
X
a
ij
t
j
X
N
D
d
¼
1
X
M k
¼
1
b
d j
e
kj
ð
R
dk
þ
S
dk
Þ
;
j
2
N
;
ð
8
Þ
X
N
D
d
¼
1
X
M k
¼
1
b
d j
e
kj
¼
1
;
j
2
N
;
ð
9
Þ
X
N
D
d
¼
D
X
M k
¼
1
b
d j
e
kj
¼
0
;
if
N
D
>
D
¼
min
f
a
ij
D
i
i
2
X
g
;
j
2
N
:
ð
10
Þ
Constraint (8) implies that the total operating timeof each planwould not exceed the maximum opening hours of the operatingroom where the plan is carried out; Constraint (9) ensures thateach plan corresponds to just one available operating room duringthe planning period; Constraint (10) implies that each plan con-taining surgical cases with a deadline no later than the end of the planning period is implemented before its deadline.Set
X
B
asabasicfeasiblesolution(BFS)tothelinearrelaxationof the master problem (LMP) with corresponding basic matrix
B
andobjectivevalue
z
.Accordingtothetheoryofsimplexmethod(Dant-zig&Wolfe,1960),ifthereexistsacolumn
A
j
correspondingtoafea-sible plan but not in
B
whose reduced cost
r
j
<0 and at least oneelementin
B
1
A
j
>0,thenitispossibletoobtainanewbasicfeasiblesolutionbyreplacingonecolumnin
B
withthenewcolumn,andthenewvalueof the objective functionis no smaller thanthe previousone.Therefore,givenanexistingBFStotheLMP,theauxiliaryprob-lem can determine values of parameters
a
ij
;
b
d j
and
e
kj
for one col-umn, with an objective of minimising the corresponding reducedcost,sothatitcanbeeitherinsertedintothecurrentBFStoimprovethecurrentsolutionorusedastheindicatorofobtainingtheoptimalsolution of the LMP when the minimumreduced cost is non-nega-tive. This is the basic the Column Generation (CG) procedure. Inour study, the reduced cost corresponding to column
A
j
¼ ð
a
1
j
;
;
a
j
X
j
j
;
b
1
j
e
1
j
;
;
b
N
D
j
e
Mj
;
b
1
j
P
i
2
X
N
1
S
a
ij
;
;
b
N
D
j
P
i
2
X
N N DS
a
ij
Þ
T
is
r
j
¼
C
j
X
i
2
X
a
ij
p
I
i
X
N
D
d
¼
1
X
M
d
k
¼
1
b
d j
e
kj
p
II
dk
X
N
D
d
¼
1
X
N
S
l
¼
1
b
d j
e
kj
X
i
2
X
l
a
ij
t
i
!
p
III
dl
ð
11
Þ
where
p
I
i
ð
i
2
X
;
D
i
N
D
Þ
represent the dual variables correspond-ing to the part
ð
a
1
j
;
;
a
j
X
j
j
Þ
;
p
II
dk
ð
d
¼
1
;
. . .
;
N
D
;
k
¼
1
;
. . .
;
M
Þ
corre-sponds to the part
ð
b
1
j
e
1
j
;
;
b
N
D
j
e
Mj
Þ
and
p
III
dl
ð
l
¼
1
;
. . .
;
N
dS
;
d
¼
1
;
. . .
;
N
D
Þ
corresponds to the rest part of column
A
j
. Thisauxiliary problem can be regarded as one kind of knapsack prob-lemsandsolvedbydynamicprogrammingprocedures.Sincedetailsof those dynamic procedures can be found in a previous study thatdeals with the similar planning problem (Fei, Chu, & Meskens,2009), they are not introduced in this paper.
2.1.3. Construction of the initial set of feasible plans
Asmentionedabove,anexistingbasicfeasiblesolutionisneces-sary for determining the value of parameters in the objectivefunc-tionoftheauxiliaryproblem.Inouralgorithm,aheuristicbasedonthe Best Fit Descending with Fuzzy constraint (BFDFC) (Dexter,Macario,&Traub,1999)isusedtogenerateaninitialsetoffeasibleplans, with which a restricted master problem (RMP) can be con-structed. This BFDFC procedure works as follows.
Step 1:
Surgical cases waiting for assignment are sorted bydeadlines in ascending order. Surgical cases with the samedeadline are sorted by operation duration from the longest tothe shortest. In addition, surgical cases are considered in suchan order that the longest case with the nearest deadline isassigned to an operating room at ﬁrst.
Step 2:
Eachsurgical caseis assignedto the operatingroomthat(1) is available on 1day before the deadlineof this case; (2) hassufﬁcient regular opentime availablefor insertingthis case; (3)has the lowest amount of available regular open time.
Step 3:
If no operating room has sufﬁcient regular open timeavailable for the current case, but sufﬁcient open time is avail-able in the operating room with the most remaining time pro-vided that the surgical case duration is shortened by
6
min{15min,itsmaximalovertime},thecaseisassignedtotheoper-ating room with the most remaining time.
Step 4:
If no operating room has sufﬁcient regular open timeavailable even with the fuzzy constraint, a dummy plan willbe constructed for each case not yet arranged, i.e. a single caseplanwillbeconstructedwherethecaseisassignedtooneoper-ating room that is available before its deadline and has sufﬁ-cient regular open time. When all cases have been assigned toan operating room, an initial set of feasible plans have beenobtained. Since eachsurgical case has been included in one fea-sible plan, i.e. one column of the initial RMP, a feasible operat-ing programme can always be guaranteed with the solutionobtained from the current RMP.
2.2. Column-generation-based heuristic (CGBH) procedure
Sofarwehavedescribedtheset-partitioningmodelfortheplan-ning problemand the generation of the initial set of feasible plans.Started with an initial set of feasible plans generated by the BFDFCprocedure, a column-generated-based heuristic procedure (CGBH)proposed in previous study (Fei, Chu, & Meskens, 2009) is imple-mentedforobtainingafeasibleweeklysurgeryplanwithgoodqual-ity.ThegeneralstepsoftheCGBHprocedureareasfollows:
224
H. Fei et al./Computers & Industrial Engineering 58 (2010) 221–230
Step 0:
Make the surgical-cases-assignment problem describedin Section 2.1.1 the current problem.
Step 1:
Solve the linear relaxation of the current problem (LMP)by an explicit CG procedure.
Step 2:
If no feasible solution of the LMP is obtained, the CGBHprocedure is ended because no feasible weekly operating pro-gramme can be obtained by the CGBH procedure; otherwise,an optimal solution of this LMP is obtained; If the solutionobtained in Step 1 respects all the integer constraints, meaningthatthedecisionsvariables
x
iseitherzeroorone,theCGBHpro-cedure will be stopped. If this is the ﬁrst iteration, the optimalweekly surgery plan is obtained; otherwise, a feasible solutionwith acceptable quality is found.
Step 3:
If the solution obtained in Step 1 does not respect all theinteger constraints (because the integer constraints wererelaxed in the LMP), only a lower boundary of the current prob-lem has been identiﬁed. In that case, one plan is selected by theMaxXMinC criterion which is shown to have the largest robust-nessinFei,Chu,andMeskens(2009). Thiscriterionworksasfol-lows:ifseveraldecisionvariablesareequaltoone,theplanwiththe smallest operating cost is selected; if all the decision vari-ables are fractional, the plan with the largest decision variable,i.e. closest to one, is selected. Ties are broken by selecting theplan with the lowest operating cost.
Step 4:
Add the plan selected in Step 3 to the list of ﬁnal plans,which is empty at the beginning of this procedure, and removethe surgical cases assigned and the operating room used bythe selected plan from the current problem. A reduced planningproblem is thus obtained.
Step 5:
Whenall the surgical cases have been assigned, or all theoperating rooms have been planned, the CGBH procedure iscomplete. If all those surgical cases whose deadlines are earlierthan the end of the planning duration are assigned, a feasiblesolution is then obtained; if not, no feasible solution is obtainedby the CGBH procedure. If all the surgical cases have beenassigned but some rooms still remain free, a set of empty plansareconstructedfortheunusedrooms,i.e.thoseoperatingroomsare supposed to be closed in the next week. If there are stillsome unassigned cases and unallocated rooms, the reducedplanning problem becomes the current problem, and the proce-dure is repeated, starting from Step 1.
3. Daily operating theatre scheduling problem
Whenthesurgeryscheduleintheoperatingtheatreforthecom-ing week has beendecided, the patient shouldbe informed of boththesurgerydateandthestartingtime, i.e. hisorherpositioninthesequenceofoperations,ofthegivenday. Howevertheorderofsur-gicalcasesisnotyetoptimizedbytakingintoaccounttheconstraintthatasurgeoncannotoperateontwosurgicalcasesatthesametime.Whereastheoperatingtheatreconsistsoftwoparts:asetofoperat-ing rooms and a recovery roomcontaining several recovery places,theefﬁciencyintheoperatingtheatredependsnotonlyontheefﬁ-ciencyofoperatingroomsbutalsoontheefﬁciencyoftherecoveryroom. Therefore, anefﬁcient surgeryscheduleshouldalso considertheavailabilityofplacesintherecoveryroom.Inthissection,adailyoperating theatre scheduling model is constructed to build a dailysurgeryschedulebytakingintoaccountthesetwoconstraintsmen-tionedabovewiththeaimofminimisingthedailyoperatingcost.
3.1. Description of the scheduling model
In order to construct a soluble model of the scheduling phase,some hypotheses are made:
Surgical cases treated by one speciﬁc surgeon can be insertedinto the sequence of the cases that will be made by other sur-geons in an operating room.
Human resources and all material resources but recovery bedsand surgeons are always available whenever needed. We allowfor the facts that no surgeon can operate on more than onepatientatthesametime;similarly,norecoverybedcanbeoccu-pied by more than one patient at the same time.
As in practice, all the operatingrooms opensimultaneously, andall recovery beds are empty at the beginning.
Allthescheduledpatientsarereadyfortheirsurgeryonthegivenday,i.e.theirarrivaltimeisnottakenintoaccountinthemodel.
Once started, an operation cannot be interrupted until it is ﬁn-ished. Moreover, once transferred to a recovery bed, a patientwill stay in that recovery bed until the pre-deﬁned recoverytime elapses.
The induction time for each operation and the clean-up timebefore leaving the operating room are included in the operatingtime, operation duration.According to the literature, some researchers have treated theoperating theatre scheduling problemas ‘‘hybrid ﬂow-shop” prob-lems (e.g. Guinet & Chaabane, 2003; Jebali et al., 2006) since ananalogycanbe drawnbetweenthesetwokinds of problems. Manystudies of hybrid ﬂow-shop situations have been carried out inindustrial ﬁelds. However, to the best of our knowledge, no previ-ous studies have allowed for the fact that the recovery time afteran operation can be shared between the operating room and therecovery room although it has been in practice in most hospitals.Inthissection,weregardthedailyschedulingproblemasatwo-stagedhybridﬂow-shopproblem(withtheoperatingrooms as theﬁrst stage and the recovery roomas the second stage) and yet takeaccountof thefact that therecoverytimeafter anoperationcanbeshared between the operating room and the recovery room. Theobjective of this scheduling phase is to determine an operation se-quence that minimises the daily operating cost including the costofboththeoperatingroomsandtherecoveryroom.Sincethissched-ulingproblemisalsoanNP-hardone,weareinterestedindevelop-inganefﬁcient heuristicproceduretosolvetheproblemduetothesameconsiderationintheplanningphase. Thehybridgeneticalgo-rithm,proposedbyFei, Meskens, &Chu(2006)forsolvingthedailyscheduling problem with block scheduling, performs quite well,andasimilarhybridgeneticalgorithm(HGA),therefore,isproposedforsolvingthedailyopenschedulingproblemunderconsideration.The notation used in the scheduling phase is
N
number of surgical cases (patients) awaiting scheduling onthe given day
C
ð
s
Þ
i
completion time for operation
i
ð
i
2 f
1
;
;
N
g
at stage
s
. Inthe scheduling model, the starting time of the operatingtheatre is set as 0, so the completion time is the momentwhen the patient is leaving the operatingroom(for
s
= 1) orthe recovery room (for
s
=2)
E
k
thetimeat whichthelast patientleavestheoperatingroom
k
ð
k
2 f
1
;
;
M
1
gÞ
where
M
1
represents the number of operating rooms available on the given day
p
a feasible daily surgery schedule, namely a sequence of patients passing through the operating theatre
C
ð
1
Þ
max
the time at which the last patient leaves the ﬁrst stage(operating rooms),
C
ð
1
Þ
max
¼
max
f
C
ð
1
Þ
i
j
i
2 f
1
;
;
N
g
. Inaddition,thisindicatorcanalsobecalculatedbytheformula
C
ð
1
Þ
max
¼
max
f
E
k
j
k
2 f
1
;
;
M
1
gg
C
ð
2
Þ
max
the time at which the last patient leaves the second stage(the recovery room). This also represents the time at whichthe last patient leaves the operating theatre.
C
ð
2
Þ
max
¼
max
f
C
ð
2
Þ
i
j
i
2 f
1
;
;
N
gg
H. Fei et al./Computers & Industrial Engineering 58 (2010) 221–230
225

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