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Long-term home care scheduling
Gamst, Mette; Jensen, Thomas Sejr
Publication date:
2011
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Gamst, M., & Jensen, T. S. (2011). Long-term home care scheduling. Kgs. Lyngby: DTU Management. (DTU
Management 2011; No. 12).
Mette Gamst
Thomas Sejr Jensen
September 2011
Report 12.2011
DTU Management Engineering
M. Gamst
⋆
and T. Sejr Jensen
⋆
DTU Management Engineering,gamstman.dtu.dk
University of Southern Denmark, thomassejrgmail.om
September 28,2011
Abstrat
Inseveralountries,home areis provided forertainitizens livingat home. Home areoersleaning,groeryshopping,helpingwithpersonalhygieneandmediine,helping itizens togetinand outofbed,et. Thelong-termhomeareshedulingproblemisto generateworkplans suh thatahigh qualityof servieismaintained,theworkhoursof theemployeesarerespeted,andtheoverallostiskeptaslowaspossible. Theproblem oversseveraldaysofhomearesheduling. Asolutionprovidesdetailedinformationon visitsandvisittimes,foreahemployeeoneahday.
We propose a branh-and-prie algorithm for the long-term home are sheduling problem. Thepriingproblemgeneratesaone-dayplan foranemployee,andthemaster problemmergestheplanswithrespettoregularityonstraints. Themethodisapableof generatingplanswithupto99visitsduringoneweek. Thistrulyillustratestheomplexity oftheproblem.
Inmany ountries a large number of itizens wholive intheir homes but are not able to do so without help, reeive regularservies from so-alled home are entres. The itizens may reeive a substantial numberof visits during the week. There aremany typesof suh visits: some are simple tasks like leaning, bringing out food, doing the laundry et., while others involve personalhygiene, mediation, getting out ofand into bedet.
Thegoalofthe homeare enter istoplanall thesevisitssuhthatthepereived quality of servie is high without overloading the individual home are personnel. The overall ost of performing this servie must be kept as low as possible. This is highly nontrivial as the pereivedqualityof servieisoftenindiretonit withtheost ofperformingthe servie.
In pratie, a two-level approah is used for planning home health are. The rst phase is a master-plan whih is a long-term plan. The seond phase is daily planning whih uses the master-plan as a starting point but inorporates last minute hanges suh as employees allinginsik, adho visits, andotherunforeseenevents. Thispaperfouses ononstrution ofthemaster-plan. Themaster-planspeieswhenitizensarevisitedandtheemployeesthat onduteah visit.
Everyvisitinthemaster-planisrepeatedregularly,asspeiedbyitsperiod. Avisitwith period
p
is repeated afterp
time, e.g.,a period of one day means that thevisit is onduted every day. Sineeveryvisit inthe master-planis repeated indenitely,the samegoesfor the entire master-plan. In this sense the master-plan isperiodi, whih means thateah weekin the master-plan is rolledout to an innite number of atual alendar weeks. If for example the lengthofthe master-planisfour weeks,everyfourthalendar weekwillbeidential(until the master-planisupdateddue to additionor removalof visits).Quality ofserviepartlyonsistsofregularity, i.e.,that avisit isalways onduted atthe same timeof the dayand that a itizenis visitedbythe same (small group of) employee(s). Espeiallythelatterisofhighimportanetomanyitizenswhofeelsaferwhenbeingservied bypersonsthey arefamiliar with.
The other aspet of quality of servie is skill set requirements, i.e., assigning suitable employees to a visit. How suitable an employee is for a visit depends on the employee's professional skillsand personality featuresofthe employee and itizen.
Eieny isquantied bythe total time spent traveling between visits for allemployees. Clearly,thetotal travel time shouldbe aslowaspossible.
Thelong-termhomeareshedulingproblemis
N P
-hardanddiersfrompreviousworkon homeareshedulingbyalulatingdetailedworkplansforalongerperiodwithouthaving pre-xedvisitsto speidays,and withtakingqualityof servieinto aount. Thelatteraspet auses inter-dependent onstraints between the daily work plans. It is thus not possible to alulate thedailyplansindependentlyfromoneanother, even ifallvisitsarexedtospei days. We propose at branh-and-prie algorithm for solving the problemto optimality. The priing problem generates a plan for an employee on a given day, and the master problem mergestheplansinto an overallsolution.Theproposedsolution methodhasbeen tested onreal-life dataprovidedbyPapirgården, whih is a home are provider loated on Funen, Denmark. It has about 25 employees and servesitizens spread overadiameter ofabout 3.5kilometers. Thismeans thatallemployees
itizensinlessthan15minutes. ComputationalresultsshowthatwhileoutperformingCPLEX usedfor solving abasi formulationof theproblem, thebranh-and-prie algorithm isunable to alulate large plans due to time and spaeusage. Instead thealgorithm is well-suited for benhmarking heuristis. Thistrulyillustratesthe omplexityof theproblem.
This paper is organized asfollows. First, a review of related work from the literature is presented inSetion 2. In Setion 3, theproblem is formallydened and formulated mathe-matially. Setion 4ontains the proposedbranh-and-prie solutionmethod. Themethodis omputationally evaluated inSetion 5,and nalonlusionsaregiven inSetion 6.
2 Related work
In thissetion, anoverviewisgivenon workfromthe literatureonhome aresheduling and related problems.
Anearlyappliation ofoperationsresearhmethodsto homehealtharewasdesribedby Begur et al. in [1℄. The daily planning problem was modeled as a site dependent VRPTW, i.e., aVRPTWwhereaustomeranonlybevisitedbyasubset ofthevehiles. Theproblem wassolved usingthesequential savings heuristi developed byClarke and Wright[4 ℄.
Evebornetal. developedadeisionsupportsystemforsolvingdailyplanning. Thesystem is alledLapsCare, andthe reent paper[8℄ doumentsitssuessinreal-lifeuse. LapsCare modeled daily planning as a VRP whih wassolved heuristially. The heuristi onsisted of iteratively ombining pathsusing generalized mathing, and splittingpaths into singlevisits whih are then ombined withthe remaining paths. In another paper about Laps Care [7 ℄, the authors assumed theexistene ofa base master-plan on whih thedailyplan was based, but they didnot onsiderhowto onstrut andmaintain themaster-plan.
The master'stheses ofThomsen [17 ℄, Lessel[11℄, and Godskesen[9℄ also dealt withdaily planning,andtheyallusedheuristisolutionmethods. ThomsenandGodskesenmodeledthe problem asa so alled rih VRP, i.e., a VRP with many real-life onstraints and objetives whih leads to ompliatedmodels,see e.g. [18 ℄.
Nikolajsen [13℄ onsidered a routing problem spanning several days, but eah visit was ondutedonlyoneandthereforedependeniesbetweenindividualdayswerenotintrodued. The work is related to the Periodi VRP (PVRP), where the planning horizon spans several days and eah ustomer must be visited on a speied number of these days. The speied number of days for eah ustomer is denoted its shedule: ifevery ustomer must be visited every day,PVRP redues to VRP sine thesame path an be usedevery day. Theshedules make PVRPdierfrom thelong-term homearesheduling problem; inthelattervisitsat a itizen an be ondutedon any(unspeied) day. PVRP wasintrodued byRusselland Igo [15 ℄ anddened formallybyChristodesand Beasley[3℄. A reent surveyfousingon PVRP mentioned manyreal-lifeappliations ofPVRP,none of whih washome health are[16℄.
Dohn etal. [10℄ modeled daily planning and solved theproblem using branh-and-prie. Themasterproblemwasasetpartitioningmodel,where eaholumnorrespondedtoapath for an employee. The priing problem was the
N P
-hard elementary shortest path problem with resoureonstraints andwassolved usinga labeling algorithm.BredströmandRönnqvist[2℄introduedamixed-integermathematialformulation forthe ombinedvehileroutingandshedulingproblemwithtimewindowsandtemporalonstraints. They showed how the formulationis appliableon the dailyplanning problem.
This setionintroduesnotation for planning homehealth are. Employeesareintroduedin Setion 3.1, theplanning horizoninSetion 3.2,visitsinSetion 3.3, ativitiesinSetion 3.4, and nallythe objetivesoftheprobleminSetion3.5. Theentities aregathered ina mathe-matial formulation ofthe long-term homearesheduling probleminSetion 3.6. Notethat the long-term home are sheduling problem and thus all entities desribed in the following operate inadisrete timespae.
3.1 Employees
Let
E
denote the set of employees. LetH
j
denote the work hours for employeej
in a given periodofk
days. Aninterval[
a
jh
, b
jh
]
∈
H
j
meansthat employeej
ison dutyfromtimea
jh
totimeb
jh
ondayh
∈ {
1
, . . . , k
}
.3.2 Planning horizon
Themaster-planoversaperiodoftimewithlength
L
∈
N
,givenasanumberofdisretetime steps. Inorderto inlude all visitsproperly,L
should besetto theleastommon multiple of thevisit periods. For example, if theplan inludes visits, whih are repeatedevery 2nd and 7th day,thenL
is 14 days. Ifthe planinludes visits, whih arerepeated every 3rd and7th day,thenL
is21 days.Some visits, for example those thatare repeatedinfrequently,ause a large
L
and onse-quently alargerproblem instane. Thisan be remediedintwo ways. Onewayisto exlude suh visits from the master-plan and only handle them during daily planning. Another way is to redue the visit's period by dividing it by an integerk
whih divides evenly into the original period. Thisimplies that the visit is repeated too often in themaster-plan, so only everyk
'threpetition is inluded indaily planning. If, for example,a visit hasa period of 21 days,dividing by3 redues the period to 7 days and only every 3rd repetitionis inluded in daily planning.3.3 Visits
The setofvisitsisdenoted
V
. Eah visiti
∈
V
is repeatedatregular intervals,e.g.,oneper dayoroneperweek,asindiatedbyitsperiodwhihisdenotedp
i
. Thetraveltimebetween two visitsi, j
∈
V
isdenotedc
ij
∈
N
andismeasuredintimesteps. The durationof visiti
isd
i
≥
0
,and itstimewindow offeasible visitingtimes is[
a
i
, b
i
]
Notallemployeesareequallysuitedforondutingaertainvisit
i
,andsomearenoteven allowed to do so. Therefore,pr
i
is a vetor of non-negative osts of letting eah employee onduti
. Ifan employee isunsuited for onduting thevisit, thentheorresponding ostis innite.Finally, let
C
denote thesetof allitizens reeivingservieandV
c
thesetof allvisits for itizenc
∈
C
.3.4 Ativities
Visit
j
withperiodp
j
is repeatedL
/
p
j
times in the planning horizon of lengthL
, and these repetitions aresheduled independently of eah other. Therefore, visitj
is rolledout toL
/
p
j
ativities, and these ativities are sheduled instead of
j
itself. LetA
denote the set of all ativities. Theset of ativitiesfor visitj
∈
V
isdenoted byA
j
. Two onseutive ativitiesi
andk
aredenoted(
i, k
) :
i, k
∈
A
j
.Every ativity
i
∈
A
j
inherits most datafrom visitj
∈
V
,so theduration ofi
isd
i
=
d
j
, the time window isW
i
=
W
j
, i.e.,[
a
i
, b
i
] = [
a
j
, b
j
]
, and the ost vetor of letting employees ondut the ativity ispr
i
. The travel time between two ativitiesi
∈
A
j
andk
∈
A
j
′
isc
ik
=
c
jj
′
.Inadditionto theativitiesin
A
thereisaspeialativity0
representingthedepot, whih is the loation where employees start and end their work days. The duration of the depot ativity iszero, and itan be performed atall times.Asolutionspeiesstart time andassignedemployees foreahativity. Let
s
i
∈
N
denote the start time of ativityi
∈
A
, lete
i
denote the employee assigned to ativityi
∈
A
, and letE
c
⊆
E
denote the set of employees whih are assigned to itizenc
∈
C
. In the formal problemdenition,wedenotethedailyshedule for anemployee apath. Apath onsistsofa listof ativitieswith orrespondingstart visit times. The master-plan thus onsistsof paths forall employees onall days overing allativities.3.5 Objetive funtions
Thissetion denes the objetive funtions, i.e., the aspets of quality of servie, how busy the home are employees are, and travel time. We hoose to aggregate the objetives even though this is not trivial due to the relatively large number of objetives and the dierent units of measurement. How to weigh the objetives is not disussed any further here but is insteadonsideredinSetion 5.
3.5.1 Travel time
Let
A
(
h
)
⊆
A
denote the setof ativitiesonduted inworkshifth
∈
H
j
ofemployeej
∈
E
. Letσ
i
∈
A
(
h
)
denotethe suessor of ativityi
. The total traveltimeis omputed as:f
T T
(
S
) =
X
j
∈
E
X
h
∈
H
j
X
i
∈
A
(
h
)
c
iσ
i
(1)Sinethestartand enddepotsareonsideredasativitiesin
A
(
h
)
,we denethesuessor of the enddepot to alsobe the enddepot and we lettheorresponding traveltimebezero.3.5.2 Employee priority
An ativity
i
∈
A
should be onduted by the most suitable employees, as speied by the non-negative ost vetorpr
i
. Letpr
i
(
j
)
≥
0
denote the ost of employeej
∈
E
onduting ativityi
∈
A
. If employeej
∈
E
does not have the required skill set for onduting the ativity,thenpr
i
(
j
) =
∞
. Theemployee priorityobjetiveis dened as:f
EP
(
S
) =
X
i
∈
A
X
j
∈
E
pr
i
(
j
)
.
(2)Anemployee issaidto be busy ifthe timeperiodinthe sheduleviolatesonstraintsontime windows and/ortraveltime. Speially,busyness appearsinthefollowing ases:
1. Thestart timesattwoativitiesaretoolose,i. e.,
s
σ
i
−
(
s
i
+
d
i
+
c
iσ
i
)
<
0
⇔
s
i
+
d
i
>
s
σ
i
−
c
iσ
i
2. Thestarttime atanativityistoolate,i.e.,let
b
jh
betheendtimewindowofemployeej
performing ativityi
on dayh
thens
i
+
d
i
> b
jh
.In both ases, busyness is penalized in the objetive funtion. There are two reasons for treating busyness as an objetive rather than a hard onstraint inthe master-plan. First of all,itreetshowthe planis onstruted manually. Seond, itintroduesslakinthemodel suh that onstraints onerning time windows for visits and overtime for employees an be enforedwithoutonstrainingthesolutionspaetothepointwherenofeasiblesolutionexists. The busynessobjetive isomputed as:
f
B
(
S
) =
X
j
∈
E
X
h
∈
H
j
X
i
∈
A
(
h
)
max(0
, s
i
+
d
i
−
min
{
s
σ
i
−
c
iσ
i
, b
jh
}}
Ifaworkshiftisempty,itsbusynessiszerobydenition. Anybusyness inthemaster-planis taken are of during dailyplanning, whih spreads out ativities or assign some ativities to anotheremployee.
3.5.4 Employee regularity
The employee regularity objetive ounts the number of dierent employees visiting eah itizen:
f
ER
(
S
) =
X
c
∈
C
|
E
c
|
.
(3) 3.5.5 Visit periodsReallthatvisitsarerolledout toativitiessuh thatthelatteraresheduledindependently, allowingmoreexibilityinthemaster-plan. Let
(
i, k
) :
i, k
∈
A
j
betwoonseutiveativities for visitj
∈
V
. Ifthetimeperiod betweens
i
ands
k
diersfrom thegiven timeperiodofthe visit,p
j
,thenthe objetive ispenalized:f
V P
(
S
) =
X
j
∈
V
X
(
i,k
):
i,k
∈
A
j
|
s
i
+
p
j
−
s
k
|
(4) 3.6 Mathematial FormulationThelong-term home aresheduling probleman nowbedened mathematially. Reallthe notation introdued previously. Furthermore,dene:
[
a
h
ij
, b
h
ij
] = [max
{
a
i
, a
jh
}
,
min
{
b
i
−
d
i
, b
jh
}
]
AsmentionedinSetion3.5weoptimizeanaggregated objetivefuntion,whereeahpart is weighed appropriately. Let
w
be a non-negative vetor of suh weights, wherew
T T
≥
0
theweightfortraveltimes,
w
EP
≥
0
isforemployeepriority,
w
B
≥
0
isforbusyness,
w
ER
≥
0
isfor employee regularity,and
w
V P
≥
0
istheweight vetor forvisit periods. Also,let
M
be somelarge number.In addition to start time variables
s
i
≥
0
, the variables are as follows. Letx
jh
ik
∈ {
0
,
1
}
equal one i employee
j
∈
E
travels from ativityi
tok
on dayh
. Letu
ik
≥
0
denote the dierene in the start times between two onseutive ativities(
i, k
) :
i, k
∈
A
j
for visitj
∈
V
. Letz
ij
≥
0
denote the busyness of employeej
∈
E
aused by ativityi
∈
A
. Finally, lety
j
c
∈ {
0
,
1
}
equal onei employeej
∈
E
visitsitizenc
∈
C
. Thelong-term homearesheduling problem isformulated as:min
X
j
∈
E
X
h
∈
H
j
X
i
∈
A
X
k
∈
A
w
T T
c
ik
+
w
EP
pr
i
(
j
)
x
jh
ik
+
X
i
∈
A
X
j
∈
E
w
B
z
ij
+
X
c
∈
C
X
j
∈
E
w
ER
y
j
c
+
X
j
∈
V
X
(
i,k
):
i,k
∈
A
j
w
V P
u
ik
(5) s. t.X
k
∈
A
X
j
∈
E
X
h
∈
H
j
x
jh
ik
= 1
∀
i
∈
A
\{
0
}
(6)X
k
∈
A
x
jh
ik
−
X
k
∈
A
x
jh
ki
= 0
∀
i
∈
A
\{
0
}
,
∀
j
∈
E,
∀
h
∈
H
j
(7)X
k
∈
A
x
jh
0
k
=
X
k
∈
A
x
jh
k
0
= 1
∀
j
∈
E,
∀
j
∈
H
j
(8)s
i
+
d
i
+
c
ik
−
M
(1
−
x
jh
ik
)
≤
s
k
+
z
kj
∀
i, k
∈
A,
∀
j
∈
E,
∀
h
∈
H
j
(9)s
i
≥
a
h
ij
−
M
(1
−
X
k
∈
A
x
jh
ik
)
∀
i
∈
A,
∀
j
∈
E,
∀
h
∈
H
j
(10)s
i
≤
b
h
ij
+
M
(1
−
X
k
∈
A
x
jh
ik
)
∀
i
∈
A,
∀
j
∈
E,
∀
h
∈
H
j
(11)s
i
+
p
j
−
s
k
≤
u
ik
∀
(
i, k
) :
i, k
∈
A
j
,
∀
j
∈
V
(12)s
k
−
(
s
i
+
p
j
)
≤
u
ik
∀
(
i, k
) :
i, k
∈
A
j
,
∀
j
∈
V
(13)X
h
∈
H
j
X
k
∈
A
x
jh
ik
≤
y
j
c
∀
c
∈
C,
∀
v
∈
V
c
,
∀
i
∈
A
v
,
∀
j
∈
E
(14)s
i
+
M
(1
−
X
j
∈
E
X
h
∈
H
j
x
jh
ik
)
≤
s
k
∀
i, k
∈
A
(15)x
jh
ik
∈ {
0
,
1
}
∀
i, k
∈
A,
∀
j
∈
E,
∀
h
∈
H
j
(16)s
i
≥
0
∀
i
∈
A
(17)z
ij
≥
0
∀
i
∈
A,
∀
j
∈
E
(18)u
ik
≥
0
∀
(
i, k
) :
i, k
∈
A
j
,
∀
j
∈
V
(19)y
c
j
∈ {
0
,
1
}
∀
c
∈
C,
∀
j
∈
E
(20)The objetive funtion (5)minimizes a weighted sum of theobjetives desribed in Se-tion 3.5. First partoftheobjetive funtiononsistsoftraveltimesand employee priority. If
x
jh
ik
= 1
for ativitiesi, k
∈
A
,employeej
∈
E
anddayh
∈
H
j
,thenthe objetive must pay the travel time fromi
tok
and the ost of letting employeej
ondut ativityi
. Next, the objetive funtiononsistsof threeparts: busyness, employee regularity andvisit periods.of an employee isonneted, andonstraints(8)fore pathsto start andend inthedepot. Constraints(9) measurebusyness: ifa visit at ativity
k
startstoo late,thenbusyness is added to the variablez
kj
≥
0
. Furthermore, onstraints (9) eliminate subtours. Constraints (10) and (11) say that any variables
k
must satisfythe time windows of ativityk
∈
A
and employeej
∈
E
.Constraints(12)and(13)measuretheamount oftimedierene between two onseutive visits
(
i, k
) :
i, k
∈
A
j
, j
∈
V
. Ifthe gap between the start times ati
andk
diers fromp
j
, thenthevariableu
ik
is setto the absolutetimedeviation.Constraints(14)ountthenumberofemployeesvisitingaitizen: ifemployee
j
∈
E
visits ativityi
∈
A
v
, v
∈
V
c
, c
∈
C
, then the variabley
j
c
is set to one. Constraints (15) limit the amount of allowed busyness suh that visit times on a path are non-dereasing. Finally, bounds(16)- (20) forevariables totake onfeasible values.4 Exat Solution Approah
Abranh-and-prie algorithm for solvingthelong-term homeare sheduling problemis pre-sentedinthis setion. Themathematial formulation (5)-(20)isDantzig-Wolfe deomposed [5℄. Thepriingproblemgeneratesapathfor agivenemployeeonagivenday,andthemaster problemmergesthe paths into anoverallfeasible solution.
Let
p
be a path andP
theset ofall generated paths. Themaster problemontains three types ofvariables. Variablex
p
∈ {
0
,
1
}
denotes whetheror not pathp
is partofthesolution. Variablesu
ik
≥
0
andy
j
c
∈ {
0
,
1
}
areasdened for theoriginal formulation (5)- (20) . Eahpathp
hasanumberof onstantsattahed. Constantδ
i
p
is setto one,ifpathp
∈
P
visitsativityi
∈
A
,otherwiseδ
i
p
iszero. Letonstantδ
ij
bp
≥
0
denotetheamount ofbusyness inpathp
∈
P
for employeej
∈
E
and ativityi
∈
A
. Letδ
i
sp
denote thestart timeof pathp
∈
P
at ativityi
∈
A
,andletδ
i
sp
beundenedifδ
i
p
= 0
. Letδ
ij
p
besettoone,ifpathp
∈
P
isgenerated for employeej
∈
E
and visitsativityi
∈
A
,otherwiseδ
ij
p
is zero. Constantδ
jh
p
isset to one, if path
p
∈
P
is generated for employeej
∈
E
on dayh
∈
H
j
, otherwiseδ
jh
p
is zero. Finally,letc
p
≥
0
denotethe total traveltimeinplanp
.min
X
p
∈
P
w
T T
c
p
x
p
+
X
i
∈
A
X
j
∈
E
X
p
∈
P
w
EP
pr
i
(
j
)
δ
ij
p
x
p
+
X
i
∈
A
X
j
∈
E
X
p
∈
P
w
B
δ
ij
bp
x
p
+
X
c
∈
C
X
j
∈
E
w
ER
y
c
j
+
X
j
∈
V
X
(
i,k
):
i,k
∈
A
j
w
V P
u
ik
(21) s. t.X
p
∈
P
δ
p
i
x
p
= 1
∀
i
∈
A
(22)X
p
∈
P
δ
sp
i
x
p
+
p
j
−
X
p
∈
P
δ
sp
k
x
p
≤
u
ik
∀
(
i, k
) :
i, k
∈
A
j
,
∀
j
∈
V
(23)X
p
∈
P
δ
sp
k
x
p
−
(
X
p
∈
P
δ
i
sp
x
p
+
p
j
)
≤
u
ik
∀
(
i, k
) :
i, k
∈
A
j
,
∀
j
∈
V
(24)X
p
∈
P
δ
ij
p
x
p
≤
y
j
c
∀
c
∈
C,
∀
v
∈
V
c
,
∀
i
∈
A
v
,
∀
j
∈
E
(25)X
p
∈
P
δ
jh
p
x
p
≤
1
∀
j
∈
E,
∀
h
∈
H
j
(26)x
p
∈ {
0
,
1
}
∀
p
∈
P
(27)u
ik
≥
0
∀
(
i, k
) :
i, k
∈
A
j
,
∀
j
∈
V
(28)y
c
j
∈ {
0
,
1
}
∀
c
∈
C,
∀
j
∈
E
(29)The objetive funtion (21) orresponds to the objetive in theoriginal formulation (5) , only withvariables
x
p
insteadofx
jh
ik
. The objetive funtion onsistsoftheweightedsum of traveltimes, employee priorities, busyness,employee regularityand visitperiods.Constraints (22)ensure thatevery ativityis visited. Constraints (23)and (24) measure time deviation similar to onstraints (12) and (13) in the original formulation. Constraints (25)measureemployeeregularity. Constraints(26)ensurethatatmostonepathperemployee perdayispartofasolution(notethattheoriginal formulationforeseveryemployee toleave the depot every day,but it isfeasible for an employee to travelfrom thedepot ativitybak to thedepot ativity,seeonstraints(7)and(8) . Thisorrespondsto notassigning apath to an employee inonstraint (26) ).
The number of olumns in the master problem is redued by xing ertain ativities to ertaindays. Speially,ifavisit mustbe repeatedeveryday,thentheorrespondingseven ativities arexed to Monday, Tuesday, Wednesday, et., respetively. The priing problem only allows suhativities to be partof pathsonappropriate days.
4.1 Priing problem
Assoiate the dualvariables
π
(22)i
∈
R
withonstraints(22) ,π
(23)ik
≤
0
withonstraints (23) ,π
(24)ik
≤
0
withonstraints(24) ,π
(25)icj
≤
0
withonstraints(25) , andπ
(26)jh
≤
0
withonstraints (26) .em-ployee
j
∈
E
on agiven dayh
∈
H
j
. Theredued ostis dened as:¯
c
jh
=
X
i,k
∈
A
w
T T
c
ik
+
X
i
∈
A
w
EP
pr
i
(
j
) +
X
i
∈
A
w
B
z
ij
−
X
i
∈
A
π
(22)i
−
X
v
∈
V
X
(
i,k
):
i,k
∈
A
v
(
s
i
−
s
k
)
π
(23)ik
+ (
s
k
−
s
i
)
π
(24)ik
−
X
c
∈
C
X
v
∈
V
c
X
i
∈
A
v
π
(25)icj
−
π
(26)jh
≤
0
(30) wherez
ij
≥
0
denotes the amount of busyness for employeej
∈
E
at ativityi
∈
A
, ands
i
≥
0
denotes the start time at ativityi
. The following notation is introdued to simplify Inequality (30) . Let:¯
c
i
0
=
(
s
i
(
π
(23)ik
−
π
(24)ik
)
∃
k
: (
i, k
) :
i, k
∈
A
v
, v
∈
V
0
otherwise and:¯
c
i
1
=
(
s
i
(
π
(24)ki
−
π
(23)ki
)
∃
k
: (
k, i
) :
i, k
∈
A
v
, v
∈
V
0
otherwiseThereduedost for visitingan ativity
i
∈
A
v
, v
∈
V
c
, c
∈
C
annowbe expressedas:¯
c
i
jh
=
w
EP
pr
i
(
j
)
−
π
(22)i
−
π
(25)icj
−
¯
c
i
0
−
c
¯
i
1
Amaster-plan makessure thatan ativity
i
∈
A
is visitedexatlyone, henewe knowthat exatly one employee leaves ativityi
∈
A
exatlyone onexatly one day. For thatreason, itisfeasible todene the reduedost between anytwo ativitiesi, k
∈
A
as:¯
c
ik
jh
=
w
T T
c
ik
+ ¯
c
i
jh
Now, theredued ost(30)is rewrittenas:
¯
c
jh
=
X
i
∈
A
X
k
∈
A
¯
c
ik
jh
+
w
B
X
i
∈
A
z
ij
≤
π
(26)jh
(31) Thepriingproblemissolvedforeahemployeej
∈
E
ondayh
∈
H
j
,heneπ
(26)
jh
isaonstant andisolatedontheright-handside. Ifthepriingproblemgeneratesapathwherec
¯
jh
< π
(26)
jh
, thenthepath hasnegative redued ost and the orrespondingolumn maybe added to the master problem.Now, using onstraints similar to the original formulation (5)- (20), the priingproblem for employee
j
∈
E
ondayh
∈
H
j
isformulated as:min
X
i
∈
A
X
k
∈
A
¯
c
ik
jh
x
ik
+
w
B
X
i
∈
A
z
ij
(32) s. t.X
k
∈
A
x
ik
−
X
k
∈
A
x
ki
= 0
∀
i
∈
A
\{
0
}
(33)X
k
∈
A
x
0
k
=
X
k
∈
A
x
k
0
= 1
(34)s
i
+
d
i
+
c
ik
−
M
(1
−
x
ik
)
≤
s
k
+
z
kj
∀
i, k
∈
A
(35)s
i
≥
a
h
ij
−
M
(1
−
X
k
∈
A
x
ik
)
∀
i
∈
A
(36)s
i
≤
b
h
ij
+
M
(1
−
X
k
∈
A
x
ik
)
∀
i
∈
A
(37)s
i
+
M
(1
−
x
ik
)
≤
s
k
∀
i, k
∈
A
(38)x
ik
∈ {
0
,
1
}
∀
i, k
∈
A
(39)s
i
≥
0
∀
i
∈
A
(40)z
ij
≥
0
∀
i
∈
A
(41)The objetive funtion (32) minimizes the redued ost as dened in (31) . Constraints (33) and(34)ensurepath onnetivityandthatthe pathstartsandendsinthedepot. Constraints (35) - (37) make sure that the start time
s
i
at ativityi
∈
A
is set appropriately, that any busyness isaddedto thevariablez
ij
andthatsubtoursareeliminated. Constraints (38)limit the amount ofallowed busyness byensuringthat starttimes onthe pathare non-dereasing. Finally,thebounds(39)- (41)fore variables to take on feasiblevalues.The priing problem is reognized as a shortest path problem withtime onstraints and potentiallynegativeedgeweightsdenedby(32). ThisisalsodenotedtheElementaryShortest Path Problem with ResoureConstrained (ESPPRC).An instane of ESPPRC onsistsof a number of resoures and a weighted graph whose edges and verties onsume resoures and have a lower and upperbound on total onsumption of eah resoure. The taskis to nd a shortest simple path from node
s
to nodet
. The resoure onsumption at every node and everyedgeonthispathmustbewithinthespeiedbounds. InthisappliationofESPPRC,a nodeorrespondstoanativity,anedgetravelsbetweentwoativities,edgeshavenoresoure bounds,andthe resoureboundsonvertiesaretimewindowsoftheativitiesandemployees. Theweight of an edgeisdetermined by(32) .Beause ESPPRC is
N P
-hard, see Dror [6℄, we rst try to solve the priing problem heuristially. Iftheheuristiannotndanypathwithnegativereduedostforanyemployee on anyday,thenthe priingproblemis solved tooptimality.Labeling algorithm
Both the heuristi and exat solution approahes for the priing problem use a labeling al-gorithm. The approah assoiates a set of labels with eah ativity. A label for ativity
i
represents a path from the (soure) depot ativitys
toi
. Assoiated with a labelℓ
are the following attributes:•
The start timet
(
ℓ
)
ofthe last visitedativityv
(
ℓ
)
•
The setof ativities, whihℓ
an beextended to,denotedextendables(
ℓ
)
•
The reduedost of the labelredued_ost(
ℓ
)
A label an be extended to ativities, whih have not yet been onduted, and whose time window is still open. The redued ost of a label isdened in(32) and is basedon thepath leadingup to
v
(
ℓ
)
.Algorithm1 is ageneri label-setting algorithm that nds ashortest paths withnegative redued ost from node
s
tot
. A buket is in this ontext a set of labels, and we use one buket foreah ativity.Algorithm1 Generilabelingalgorithm whihomputesaset
P
ontaininguptok
resoure onstrainedshortestpathswithnegativereduedostfromnodes
tot
. Thesetofallbukets isdenotedbyB
.1:
P ← ∅
2:
ℓ
init
←
initialize_label(
s
)
3:B
(
s
)
← B
(
s
)
∪ {
ℓ
init
}
4: whilea non-emptybuketin
B
existsand|P|
< k
do 5:L
←
dequeued non-emptybuketfromB
6: for all
ℓ
∈
L
do7: for all ativity
i
∈
extendables(
ℓ
)
do 8: for all feasiblestart timess
i
ofi
do9:
ℓ
′
←
reate_label(
i, s
i
)
10: ifi
=
t
then 11: if redued_ost(
ℓ
′
)
< π
(26)jh
then 12:p
←
get_path(
ℓ
)
13: if keep_path(
p
)
then 14:P ← P ∪ {
p
}
15: end if 16: endif 17: else 18:B
(
i
)
← B
(
i
)
∪ {
ℓ
}
19: remove_dominated(
B
(
i
)
, ℓ
)
20: end if 21: end for 22: endfor 23: end for 24: end whileIntherstthreelinesoftheAlgorithm, anemptyset ofsolutionsisinitialized alongwith alabelinthe (soure)depot ativity. Thelatteris addedto thebuket oflabelsinthedepot ativity. Thealgorithm thenextratssome non-emptybuket inLine (5). InLine(6) -(9)the algorithm extendsthepath ofeahlabelinthebuketwithfeasibleativitiesandstarttimes, whih results innew labels. If a path hasreahed the (target) depot ativity
t
,the redued ostisnegative,andwewishto savethepath,thenitisaddedtothesetofsolutions,seeLine (10) -(14) .We wishto save apath
p
if:•
No more than 100 paths are already saved. The number 100 is reahed through pa-rameter tuning, or•
Morethan100 pathsaresaved andthenewpath hassmallerreduedostthan another savedpathIfthe (target) depot ativity
t
hasnot been reahed in Line(10) , thenthe labelis added to theappropriatebuket. Finally,inLine(19)thealgorithm heksifthenewlabelℓ
dominates anylabels inthe buketor ifitisdominated byany labelsinthebuket.A label
ℓ
dominates anotherlabelℓ
′
ifthefollowing onditions hold:
•
The labelsℓ
andℓ
′
end at the sameativity:
v
(
ℓ
) =
v
(
ℓ
′
)
•
The reduedost ofℓ
isno greaterthan thatofℓ
′
: redued_ost
(
ℓ
)
≤
redued_ost(
ℓ
′
)
•
The path ofℓ
endsnolater than that ofℓ
′
:
t
(
ℓ
)
≤
t
(
ℓ
′
)
•
Labelℓ
anat leastextend to thesame ativitiesasℓ
′
These riteria ensure that if at least one path with negative redued ost exists, then the labeling algorithm isguaranteed to ndit.
When using Algorithm 1 as a heuristi, only the rst label added to every buket is proessed. Keepingonlyonelabelineahbuketspeedsupthepriingalgorithmsigniantly and it is still apable of nding olumns with negative redued ost inthe rst iterations of the branh-and-prie algorithm.
4.2 Branhing strategy
Branhing is neessary when the optimal solution ina branh node is frational. Frational solutions ourinthefollowing situations:
Frational itizen visits: Thatis
0
< y
j
c
<
1
for some itizenc
∈
C
andemployeej
∈
E
. In this asetwo branhinghildrenaregenerated withadded ut:y
c
j
= 0
resp.y
j
c
= 1
.
This doesnot hange thepriingproblem, beausetheutis not onthe
x
p
-variables. An ativity is visited by several employees or on several days: Thatis,0
< x
p
, x
p
′
<
1
for pathsp, p
′
∈
P
, onstantsδ
i
p
=
δ
p
i
′
= 1
for some ativityi
∈
A
, and onstantsδ
jh
p
=
δ
j
′
h
′
p
′
= 1
for some employee(s)j, j
′
∈
J
and day(s)h, h
′
∈
H
j
witheitherj
6
=
j
′
orh
6
=
h
′
.In this asetwo branhinghildrenaregenerated withthefollowing rules:
X
p
∈
P
δ
jh
p
x
p
= 0
resp.X
p
∈
P
δ
j
p
′
h
′
x
p
= 0
The branhing rule is maintained inthe priing problem, whih ensures thatemployee
j
(resp.j
′
) never visits ativity
i
on dayh
(resp.h
′
). The branhing rule does not ompliate the strutureof thepriingproblem.
is,
0
< x
p
, x
p
′
<
1
for pathsp, p
′
∈
P
,and onstantsδ
jh
p
=
δ
p
jh
′
= 1
for some employeej
∈
J
and dayh
∈
H
j
. Leti
be the rst ativityfrom whih thepathsp
andp
′
dier. Let
i, k, k
′
∈
A
,suh thatp
travelsfromi
tok
at times
ik
andsuhthatp
′
travelsfrom
i
tok
′
at time
s
ik
′
. Furthermore,letk
6
=
k
′
,or
s
ik
6
=
s
ik
′
. Two branhinghildrenaregenerated withthefollowing rules:X
p
∈
P
δ
jh
p
δ
p
s
ik
x
p
= 0
resp.X
p
∈
P
δ
jh
p
δ
s
ik
′
p
x
p
= 0
where onstantδ
s
ik
p
(resp.δ
s
ik
′
p
) is set to one, if pathp
travels fromi
tok
at times
ik
(resp. fromi
tok
′
at time
s
ik
′
), otherwise it is set to zero. The branhing rule is maintained bythe priingproblem, whih ensuresthat employeej
never travels fromi
tok
(resp.k
′
)attime
s
ik
(resp.s
ik
′
)ondayh
. Thebranhingruledoesnotompliate thestruture of thepriingproblem.Togetherthethreebranhingstrategiesareniteandeventuallyensureanintegersolution. Thestrategy generates branhinghildren intheorder given above, and bestrst is usedas searhstrategyinthebranh-and-boundtree. Strongbranhingisapplied: foreahbranhing andidate,alowerboundonitsLPrelaxationofeahofthehildrenisobtained. Theandidate thatleadsto hildrenwiththe lowestboundsisseleted.
4.3 Inumbent
Beforethebranh-and-prie proessan begin, an initialsolution tothelong-term homeare sheduling problemmustbegenerated. Thesolution, also denotedtheinumbent, isused for ndinginitial valuesfor the dualvariables.
Algorithm 2 tries to assign ativities to the rst employee on the rst day, with respet to time windows and xed days. If unable to assign an ativity to this employee and day, the algorithm tries to assign it to the employee on the next day. Eventually, the algorithm triesto assignthe ativityto thenext employee on therst day,et. An ativity isassigned to an employee ina feasible way and suh that busyness is avoided when possible, see Line (8)and (14), i.e., the algorithm only allows busyness to our when having reahed the last employee onthe last dayof the instane. Lines (13) - (15)measuretraveltimeand busyness forreturning to the depot.
The algorithm always nds a feasible solution if one suh exists. The reasons for this arethat ativities are sorted aordingto their end times in non-dereasing order, and that busynessis allowed.
5 Computational Results
The exat solution method is tested on a number of real-life benhmark instanes. In this setion, the benhmark instanes are rst introdued. This is followed by omputational resultsfor the branh-and-prie algorithm.
5.1 Real-life test instane
The proposed solution method is tested on real-life data provided by Papirgården, a home are enter in Funen, Denmark. The real-life instanes onsist of up to 99 ativities to be
initialsolution.
1: for all employees
j
∈
E
do 2: for all daysh
∈
H
j
do3:
t
←
a
jh
(startof work shift) 4:i
←
0
(the depot)5: sortativities innon-desending orderof
b
i
(endof timewindow) 6: for all ativitiesk
∈
A
do7: if
k
is unsheduled andan besheduled on dayh
then 8: Updatet
aordingtoc
ik
andb
h
kj
9: start
k
at timet
and assignemployeej
10:
i
←
k
11: end if 12: endfor
13:
k
←
0
(thedepot)14: Update
t
aordingtoc
ik
andb
h
kj
15: start
k
at timet
and assignemployeej
16: end for17: end for
ondutedin 7daysbyat most 2 employees. The timewindowof an ativityis setto either 7.309.00, 9.0011.00, 11.3013.00, or13.0015.00. Employees workfrom 7.00 13.00, 7.00 14.00 or 7.00 15.00. Timeisdisretized into either5 or 10minutetimesteps.
The distanes between itizens are found with the Google MapsAPI, whih means that the atual distane is used in the master-plan rather than straight line distanes. When omputing travel times, we assume that an employee travels at 15 kilometers per hour (all employees travel by biyle). Unless two ativities areat the same itizen, two minutes are added to the travel time between them to aount for thetime it takes to enter and leave a residene. All traveltimesareeiled to nearestinteger.
Weightsmustbesetfortheaggregatedobjetivefuntion. Papirgårdenhasreommended the following priorities: highest priority is given to minimizing busyness and employee reg-ularity, followed by minimizing travel times. Visit regularity has fourth priority, and em-ployee skill requirements are of no importane for Papirgården, beause its employees have similar skill sets. Visit regularity is given low priority, beause the time window of an a-tivity is relatively small and beause high priority is given to redue busyness. Inluding the priorities intheobjetivefuntion isdonebyassigninglarge numbersto theweights, i.e.,
w
T T
= 500
, w
B
= 750
, w
EP
= 0
·
5
/τ, w
ER
= 750
·
5
/τ
and
w
V R
= 50
,where
τ
isthenumber ofminutespertimestep. Thetraveltime, busyness,andvisitregularityobjetivesdependon thetimedisretization,whilethisisnottheaseforemployeepriorityandemployeeregularity. Theweights ofthe latter twoare thus multiplied withatimestep dependent fator.5.2 Results
The branh-and-prie algorithm has been implemented using theCOIN Bp framework, see Lougee-Heimer [12 ℄, and is tested on an Intel 2.13GHz Xeon CPU with 4 ores and 8 GB RAM.Notethatallresultsstemfromusingoneore. CPLEX12.1wasusedasstandardMIP
the branh-and-prie algorithm.
CPLEX is unable to solve the original formulation (5) -(20) for instanes withmore than 11ativitiesandisthusnotevaluatedfurther. Foraninstanewith12ativities,CPLEXwas stopped after 75000 seonds. It had generated a searh tree with 4057600 nodes, but only reduedthegapbetween theupperandlowerboundfromaninitial14.8%to8.25%. Applying CPLEX on the originalformulationis nota suessfulstrategy,henetherest ofthis setion onentrateson the branh-and-prie algorithm.
TestresultsaresummarizedinTable1and2. Thersttabledisplaysresultsfortheoptimal branh-and-prie algorithm, while the seond table displays results for the branh-and-prie algorithm withonlyheuristially generated olumns.
An instane is named
|
E
| − |
A
| −
τ
, whereτ
is the number of minutes per time step, i.e., either5 or 10 minutes. All instanes have a planning horizon of 7 days. A time limit of 30 minutes has been imposed on the runs, and an * in the last olumn indiates that an provably optimal solution was not found within these 30 minutes (some runs exeeded this limit slightly beause elapsed timeisnot heked everywhereintheprogram).As an be seen in Table 1, only four instanes with 5 minute time steps an be solved to optimality within half an hour. A oarser disretization helps, but the branh-and-prie algorithm stillsuers froma large timeusage.
An interesting observation for instanes with 10 minutes timesteps isthat the instanes with 30 ativities time out, whereas the instanes with 33 and 40 ativities are solved to optimality. This is due to the fat that the instanes with 30 ativities ontain many visits withaperiodofoneweek,whihanbesheduledonanyday. Theotherinstaneshavemore visits withaperiodofone daywhose ativitiesare thus xed tospei days.
Thenumberofolumnsislargefor severalinstanes,whihisaused partlybylargetime windowsandpartlybybusyness,i.e.,thattimewindowsmaybeviolated. Thetreegrowslarge for many instanesnot solvedto optimality,henebranhing alsoonstitutes abottlenek.
As an be seen in Table 2, the branh-and-prie algorithm is generally faster when only generating olumns heuristially. Note that the gap in this table denotes the gap between theheuristi upperandlowerbound,and not between theheuristi upperandoptimal lower bound. Someinstanesstillsuerfromlargetreesizesandmanyolumns,butthefarmajority of instanes are solved in seonds. The objetive values generally suer from the heuristi approah. Comparing the two tablesshows thatthe heuristi solutions are between 4% and 85%solution. Theaveragegapbetweenprovablyoptimalandheuristisolutionsis47%. Even thoughsolvingthepriingproblemheuristiallyredues theoverallrunningtimesigniantly, thebranh-and-prieapproah isstillunabletosolvetheinstanewitha10minutetimestep, 2 employeesand 99 instanes.
Considering the omplexity of the master-plan problem, it is no surprise that the exat branh-and-priealgorithm an solveonly limitedsizedinstanes. Theresults trulyillustrate the omplexityof the problem.
6 Conlusion
Inthispaper,wepresentedabranh-and-prie algorithmforthelong-termhomeare shedul-ing problem. The priing problem onsisted of alulating a work planon a given dayfor a
1-20-5 6327 199 1 0 0.00 27000.0 3.97 1-25-5 8671 250 4687 59 0.03 29750.0 1801.33* 1-30-5 12883 295 2079 71 0.04 38000.0 1801.97* 1-33-5 11447 328 1615 75 0.00 38800.0 1655.35 1-40-5 28365 397 171 51 0.00 43000.0 812.94 1-44-5 40445 433 179 89 0.01 97100.0 1802.57* 1-50-5 31480 493 27 13 0.07 87000.0 1803.35* 1-55-5 37944 544 159 28 0.00 57000.0 1804.30* 1-58-5 39997 571 9 4 0.08 156200.0 1807.19* 1-80-5 8737 787 1 0 NoLB 259250.0 1906.50* 2-20-5 8917 346 5 2 0.00 27000.0 8.58 2-25-5 10089 432 2405 69 0.03 29750.0 1803.39* 2-30-5 16383 512 695 66 0.05 38400.0 1803.29* 2-33-5 18316 566 701 90 0.03 39300.0 1803.90* 2-40-5 23921 684 455 70 0.05 42100.0 1801.46* 2-44-5 34527 748 125 62 0.08 84600.0 1835.07* 2-50-5 30642 850 71 35 0.07 85000.0 1806.21* 2-55-5 34414 936 89 32 0.03 51500.0 1817.40* 2-58-5 30582 984 17 8 0.02 143700.0 2174.36* 2-80-5 8261 1354 1 0 NoLB 237500.0 1908.73* 1-20-10 4877 199 59 28 0.00 17250.0 6.37 1-25-10 5507 250 277 51 0.00 18125.0 27.36 1-30-10 10934 295 2449 101 0.01 25300.0 1801.07* 1-33-10 10688 328 43 21 0.00 23875.0 42.04 1-40-10 14037 397 63 31 0.00 27750.0 90.16 1-44-10 27925 433 567 98 0.03 33750.0 1803.50* 1-50-10 19907 493 3 1 0.00 36500.0 337.04 1-55-10 19912 544 3 1 0.00 60375.0 123.64 1-58-10 40434 571 111 55 0.00 64450.0 1803.82* 1-80-10 19711 787 1 0 NoLB 161100.0 1847.78* 2-20-10 5913 346 39 19 0.00 17250.0 9.19 2-25-10 6594 432 81 40 0.00 18125.0 16.30 2-30-10 10109 512 1245 56 0.00 25200.0 1802.52* 2-33-10 14485 566 89 44 0.00 23750.0 146.03 2-40-10 15937 684 77 36 0.00 27250.0 162.93 2-44-10 27208 748 271 101 0.01 33400.0 1805.10* 2-50-10 25457 850 21 10 0.00 35750.0 654.53 2-55-10 29331 936 49 24 0.01 80625.0 1841.08* 2-58-10 21593 984 31 15 0.01 96750.0 1826.54* 2-80-10 22136 1354 3 1 0.02 149750.0 2475.74*
Table1: Resultsforinstaneswitheither1or2employeesandeither5or10minutespertime step when using the optimal branh-and-prie algorithm. The Table shows total number of generated olumnsandtotalnumberofonstraintsinthemasterproblem,numberofnodesin the branh-and-bound tree,depthofthe tree,gap between lowerandupperbound,objetive valueofbestfoundsolution,andrunningtimeinseonds. An* intherunningtimeindiates that thealgorithm timedout.
1-20-5 1794 199 1 0 0.00 38750.0 0.75 1-25-5 608 250 1 0 0.00 44500.0 0.18 1-30-5 2852 295 1 0 0.00 86000.0 1.45 1-33-5 848 328 1 0 0.00 59750.0 0.45 1-40-5 903 397 1 0 0.00 67500.0 0.55 1-44-5 1710 433 1 0 0.00 97100.0 1.95 1-50-5 1235 493 1 0 0.00 97500.0 1.45 1-55-5 1599 544 1 0 0.00 112000.0 1.20 1-58-5 2919 571 1 0 0.00 156200.0 4.76 1-80-5 2954 787 1 0 0.00 259250.0 10.96 1-99-5 2449 970 1 0 0.00 328800.0 22.10 2-20-5 1917 346 1 0 0.00 31750.0 0.66 2-25-5 1416 432 3 1 0.00 34750.0 0.27 2-30-5 5637 512 481 25 0.00 73500.0 117.37 2-33-5 4171 566 7 3 0.00 41750.0 1.49 2-40-5 4252 684 7 3 0.00 49500.0 1.98 2-44-5 7236 748 481 28 0.00 84600.0 242.35 2-50-5 8629 850 101 23 0.00 85000.0 48.85 2-55-5 7660 936 35 17 0.00 99500.0 18.33 2-58-5 8770 984 123 29 0.00 143700.0 116.47 2-80-5 12677 1354 185 35 0.00 237500.0 388.09 2-99-5 15601 1670 159 30 0.00 285750.0 1232.10 1-20-10 1286 199 1 0 0.00 24750.0 0.29 1-25-10 324 250 1 0 0.00 29625.0 0.11 1-30-10 1845 295 1 0 0.00 54250.0 0.71 1-33-10 433 328 1 0 0.00 36125.0 0.27 1-40-10 733 397 1 0 0.00 48750.0 0.32 1-44-10 1473 433 1 0 0.00 61400.0 1.61 1-50-10 1002 493 1 0 0.00 67500.0 0.71 1-55-10 1409 544 1 0 0.00 86875.0 0.69 1-58-10 2270 571 1 0 0.00 106650.0 1.89 1-80-10 2255 787 1 0 0.00 161100.0 4.44 1-99-10 3076 970 1 0 0.00 215425.0 16.13 2-20-10 1770 346 1 0 0.00 20750.0 0.43 2-25-10 714 432 3 1 0.00 23125.0 0.15 2-30-10 4585 512 751 25 0.00 48000.0 115.49 2-33-10 2717 566 7 3 0.00 24875.0 0.66 2-40-10 3267 684 7 3 0.00 37500.0 1.00 2-44-10 5338 748 713 30 0.00 55150.0 220.10 2-50-10 6914 850 191 26 0.00 61250.0 58.38 2-55-10 6090 936 67 23 0.00 80625.0 21.94 2-58-10 7242 984 143 25 0.00 96750.0 76.81 2-80-10 8552 1354 221 36 0.00 149500.0 265.50 2-99-10 11410 1670 557 58 0.01 195250.0 1802.64*
Table 2: Results for instanes with either 1 or 2 employees and either 5 or 10 minutes per timestepwhenonly generatingolumnsheuristiallyinthebranh-and-prie algorithm. The tableshowstotalnumberofgenerated olumnsandtotal numberofonstraintsinthemaster problem, numberof nodesinthe branh-and-bound tree,depth ofthe tree,gap between the heuristi lowerand upper bound,objetive value ofbestfound solution, and runningtime in seonds. An* inthe running timeindiates thatthealgorithm timedout.
priing problem was
N P
-hard and solved through a labeling algorithm. Initially, the pri-ingproblem wassolved heuristially by only onsidering a small subset of labels. When this approah wasunsuessful,thelabelingalgorithm solved thepriing problemto optimality.The branh-and-prie algorithm was implemented and tested on a number of real-life in-stanesprovided bythePapirgården homeareservieinFunen, Denmark. The branh-and-prie algorithm outperformed applying CPLEX to the original formulation. The algorithm, however,showed performanediultiesfor larger instanesdue tothelarge numberof om-binationsof visits,visit times andemployees.
Improvingthe branh-and-prie approahwouldrequiremethodsforreduing thenumber of olumns and limiting the searh tree size. The authors attempted stabilizing the value of dual variables using the interior point method of Rousseau et al. [14℄, but with no avail. Otherstabilizationmethodsouldbeinvestigated,asbettervaluesforthedualvariablesould reduethenumberofgeneratedolumns. Dierentprimalandinumbentheuristishavebeen implemented and tested without improving theboundsor pruning larger parts of thesearh tree. Future work should fous on nding better bounds, through primal and inumbent heuristisand throughthebranhing strategy.
AdierentapproahouldalsobetakentotheDantzig-Wolfedeomposition. Ifthemaster problem was to deide the time of visits, then the number of olumns would be redued signiantly. This, however, would ome at a prie, beause the omplexity of the master problemwouldbeaetednegatively.
Aknowledgements
We wouldlike to thankPapirgåden for sharing knowledge onhome are sheduling and pro-vidingreal-lifedata. Furthermore,wethanktheVillum-Kann-Rasmussenfoundationfortheir supportofthis work.
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In several countries, home care is provided for certain citizens living
at home. The long-term home care scheduling problem is to generate work
plans spanning several days such that a high quality of service is
maintained and the overall cost is kept as low as possible. A solution
to the problem provides detailed information on visits and visit times
for each employee on each of the covered days.
We propose a branch-and-price algorithm for the long-term home care
scheduling problem. The pricing problem generates one-day plans for an
employee, and the master problem merges the plans with respect to
regularity constraints. The method solves instances with up to 99 visits
during one week. This truly illustrates the complexity of the problem.
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