The time-dependent capacitated profitable tour problem with time windows and precedence constraints

The time-dependent capacitated profitable tour problem with

time windows and precedence constraints

Citation for published version (APA):

Sun, P., Veelenturf, L. P., Dabia, S., & van Woensel, T. (2018). The time-dependent capacitated profitable tour

problem with time windows and precedence constraints. European Journal of Operational Research, 264(3),

1058-1073. https://doi.org/10.1016/j.ejor.2017.07.004

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10.1016/j.ejor.2017.07.004

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ContentslistsavailableatScienceDirect

European

Journal

of

Operational

Research

journalhomepage:www.elsevier.com/locate/ejor

Production,

Manufacturing

and

Logistics

The

time-dependent

capacitated

profitable

tour

problem

with

time

windows

and

precedence

constraints

Peng

Sun

a,∗

_,

_Lucas

_P.

_Veelenturf

a

_,

_Said

_Dabia

b

_,

_Tom

_Van

_Woensel

a

a School of Industrial Engineering, Operations, Planning, Accounting and Control (OPAC), Eindhoven University of Technology, Eindhoven 5600 MB, The

Netherlands

b Department of Information, Logistics and Innovation, Vrije Universiteit Amsterdam, Amsterdam 1081 HV, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 16 August 2016 Accepted 3 July 2017 Available online 12 July 2017 Keywords:

Transportation Profitable tour problem Pickup and delivery problem Tailored labeling algorithm Time-dependent travel times

a

b

s

t

r

a

c

t

We introducethe time-dependentcapacitatedprofitabletourproblem withtimewindows and prece-dence constraints.Thisproblemconcernsdeterminingatourand itsdeparturetimeatthe depotthat maximizesthecollectedprofitminusthetotaltravelcost(measuredbytotaltraveltime).Todealwith road congestion,traveltimesareconsideredto betime-dependent.We developatailoredlabeling al-gorithmtofindtheoptimaltour.Furthermore,weintroducedominancecriteriatodiscardunpromising labels.Ourcomputationalresultsdemonstratethatthealgorithmiscapableofsolvinginstanceswithup to 150locations(75 pickupand deliveryrequests)to optimality.Additionally,wepresent arestricted dynamicprogramingheuristictoimprovethecomputationtime.Thisheuristicdoesnotguarantee opti-mality,butisabletofindtheoptimalsolutionfor32instancesoutofthe34instances.

© 2017ElsevierB.V.Allrightsreserved.

1. Introduction

The effective usage of empty vehicles’ space is an important opportunitytoincreasetheefficiencyofurbantransportation sys-temsandtoreducetraffic congestion, fuelconsumption,and pol-lution.Companies(e.g.,Uber) cangenerateextraincomeby rent-ing out vehicles’s empty spacerising in their transportation pro-cesses.Several mobile applicationsare developed to improvethe last-miledeliveries by involvingthe city’s residents.For instance,

Roadie created an on-the-way delivery network which is an

on-linemarketwherepeopleposttheirrequiredshipmentsandwhere anyonecan offer to execute the shipment. DHL launched a plat-formcalledMyWays,enablingindividualstodeliverpackageswith productsorderedonlinedirectlytootherendconsumers.Forthose individualswithlimitedtransportationresources,itisimportantto knowwhich parcels are profitableto collect anddeliver. Onthe one hand, serving a request may be attractive because it gener-atesrevenue.Ontheother hand,thereisadditionalcostfor serv-ing the request. Consequently, it might not always be economi-cally beneficial to serve a request. Moreover, customers are not always available to receivethe ordered goods (i.e., shopsare not open24/7ore-commercecustomersarenotalwayshome). There-fore,thesecustomersproposespecifictimewindowsinwhichthey

∗ _{Corresponding author.}

E-mail addresses: P.Sun@tue.nl (P. Sun), L.p.veelenturf@tue.nl (L.P. Veelenturf),

S.dabia@vu.nl (S. Dabia), T.v.Woensel@tue.nl (T. Van Woensel).

wanttobe served.Theseextraconstraintsmakeitachallengefor theindependentcourier tobeeverywhere ontime.Hehasto se-lect a subset of the requests to be served within the requested

time windows. Furthermore, due to the limited capacity of the

roadnetworkandincreasedtraffic congestion, thetravelspeed is serious affected by thesetraffic fluctuations. This resultsin great variationsintravel timesandtherebyonthearrivaltimesat cus-tomers.Therefore,thetraveltime betweenlocationsisdependent onthetime thedriverdeparts (e.g.,inrushhourstravelingtakes more time). Not considering the time-dependenttravel time re-sults in toolate arrivalsat the customers (see e.g., Ta¸s,Dellaert, vanWoensel, & deKok,2014). All thesecomplex situations illus-tratetheneedfortheindependentcourierstohaveatooltohelp themoutinmakingdecisionsonwhichrequesttoserveandwhich timetodepart.However,scarceliteraturecanbe foundthatfocus onthisarea.

Thispaperaimsatbuildingsuchatoolbymodelingandsolving

a time-dependent capacitated profitabletour problem withtime

windowsandprecedenceconstraints.Weconsiderasinglevehicle withcapacitylimitandasetofrequestswhichhaveapick-upand adelivery node.Eachpickup node anddelivery nodehasits own time window in whichit should be served. Moreover,a delivery nodeofa requestcanonlybeserved afteritspickupnode is vis-ited.Foreachservedrequestaprofitiscollected.Tocapturetravel speed variationduringa day, atime-dependenttravel time func-tionisassignedtoeachedgelinkingtwonodes.Theobjectiveisto determinethevehicle’stourstartingandendingatthedepot,and

http://dx.doi.org/10.1016/j.ejor.2017.07.004

Fig. 1. An illustration of the time-dependent capacitated profitable tour problem with time windows and precedence constraints. P i and D i are the pickup node and delivery

node of request i , respectively.

maximizing thedifference betweenthe totalcollectedprofitsand total travel cost (see Fig.1). Following the literature inthis area, thetour’stotaltravelcostisequaltothetour’sduration.

ThedescribedproblemisNP-hardbecauseitisanextensionof thetravelingsalesmanproblemwithpickupanddelivery(TSPPD),

which itself is an extension of the traveling salesman problem

(TSP). Incontrastto theTSPPD andtheTSP, inourproblem, itis notnecessarytovisitalltherequests.

The main contributions of this paperare summarized as

fol-lows.First,weintroduceanewmodelwhichextendstheclassical

TSPPD by having time-dependenttravel timesand the option to

rejectrequests.Secondly,weproposeanexactsolutionmethodfor thisproblembydevelopingatailoredlabelingalgorithminwhich novelandstrongdominance criteriaareused.Finally,arestricted

dynamic programing heuristic is proposed with a high solution

qualityandlowercomputationtimesthanthelabelingalgorithm.

The remainder of the paper is structured as follows.

Section2 provides a brief review of related existing work.

Section3 defines the problem and introduces a mathematical formulation ofthe problem. In Section4,we presentthetailored labeling algorithm and in Section5 the restricted dynamic pro-gramingheuristic isintroduced. Finally,computational resultsare reportedinSection6,followedbyconclusionsinSection7. 2. Literaturereview

Therearethreeclassesofproblemscloselyrelatedtothe

prob-lem studied in this paper: the traveling salesman problem with

pickup and delivery (TSPPD), thetime-dependent vehicle routing problem(TDVRP)andthetravelingsalesmanproblemwithprofits (TSPwithprofits).

2.1. Thetravelingsalesmanproblemwithpickupanddelivery (TSPPD)

OurproblemextendstheTSPPDbyconsideringtime-dependent

travel times.The TSPPDisfirstlyintroduced byRuland andRodin (1997)andisproven tobe ofgreatuseinapplicationslike dial-a-ridesystemsandcourier services.Althoughthepickup and deliv-eryproblem(PDP),inwhichtheTSPPDcanappearasa subprob-lem,is extensivelystudiedinthe literature,only limitedresearch focusesontheTSPPD.

Currently, the most popular methodology for solving the

TSPPD is branch-and-cut. Ruland (1994) and Ruland and Rodin

(1997) considered the undirected case of this problem and de-veloped four classes of valid inequalities that are embedded in a branch-and-cutalgorithm. The algorithm istested oninstances withup to15pickupanddeliveryrequests.Recently, Dumitrescu, Ropke,andCordeau(2010)studiedthesameproblem, theauthors analyzeditspolyhedralstructureandproposednewvalid inequal-ities that are shown to be facets for the TSPPD polytopes. Their algorithmiscapableofsolvinginstanceswithupto35pickupand deliveryrequeststooptimality.

The TSPPD appears as pricing problem for the PDP and is

in that situation usually named as the elementary shortest path

problem with time windows, capacity and pickup and delivery

(ESPPTWCPD). Sol(1994), Sigurd and Pisinger (2004) andRopke, Cordeau, and Laporte (2009) presented labeling algorithms with severaldifferentdominancerulestosolvethisproblemto optimal-ity.

2.2.Thetime-dependentvehicleroutingproblemwithtimewindows (TDVRP)

AnotherrelatedproblemistheTDVRP.AlthoughtheTDVRPhas attractedtheattentionofmanyresearchers,literatureonthis sub-ject remains scarce. The pioneering work is done by Malandraki andDaskin(1992)andMalandrakiandDial(1996).Inthesepapers mixedintegerlinearprograms andseveralheuristicsto solve the problemareproposed.TheFirst-In-First-Out(FIFO)property,which impliesthat forevery arcalaterdeparture timeresultsinalater (or equal) arrival time, is an intuitive anddesirable property for

time dependent routing problems. Ichoua, Gendreau, and Potvin

(2003)andDabia, Ropke,vanWoensel, anddeKok(2013) consid-eredtheTDVRPwithtraveltime variabilitymodeled by“constant speed” timeperiods,whichensurestheFIFOproperty.Theideaof constantspeedtimeperiodsisadoptedinourproblemaswell.

Duetothecomplexityofthetime-dependentproblem,mostof the existing algorithms are based on heuristics. In vanWoensel, Kerbache, Peremans, and Vandaele (2008) a tabu search heuris-ticis usedto solve the capacitatedvehicle routing problemwith

time dependent travel times.An approximation based on

queue-ing theory andthe number of vehicles on a link is used to de-termine travel speeds. Donati, Montemanni, Casagrande, Rizzolo, and Gambardella (2008) developed a multi-ant colony system for the TDVRP and Ibaraki etal. (2008) proposed an iterated lo-cal search heuristic forthetime-dependentvehicle routing prob-lemwith time windows(TDVRPTW). Recently, Dabia etal.(2013)

Table 1 Parameters. Notation Definition R = { 1 , . . . , n } Set of requests N Set of nodes A Set of arcs

NP Set of pickup nodes

ND Set of delivery nodes

(i, n + i ) A transportation request R i

ri Profit of request R i

qi Demand of request R i

Q Carrying capacity of the vehicle [ e i , l i ] Time window of node i

si Service time at node i

ti Departure time from node i

τij ( t i ) Travel time from node i to node j with departure time t i at node i

developedabranch-and-pricealgorithm forthe TDVRPTW,where

a tailored labeling algorithm is presented to solve the

time-dependentshortestpathproblemwithresourceconstraint (TDSP-PRC),whichisthepricingprobleminthealgorithm.

2.3.Thetravelingsalesmanproblemwithprofits(TSPwithprofits)

Theproposed problemextendsthetravelingsalesmanproblem

withprofits(TSPwithprofits),inwhichprofitsareassociatedwith eachrequestandtheoverallgoalistofindtheshortesttourwith themaximumcollectedprofits.Thismeansthatincontrasttothe originalTSP, notall nodeshavetobe visited.Incomparisonwith ourstudy,itdoesnotincludetimedependencyandrequestswith pickupanddeliverynodes.

According to Feillet, Dejax, and Gendreau (2005), TSPs with profitscanbe categorizedintothree generic problemsdepending onthewaythetermsprofitsandtraveltimeare addressedin the objective function and constraints. They can be classified in the followingcategories:theprofitabletourproblem(PTP),the prize-collectingtravelingsalesman problem(PCTSP), andthe orienteer-ingproblem(OP).Dell’Amico,Maffioli,andVarbrand(1995) stud-iedthe profitabletourproblem(PTP) whereboth the profitsand thetraveltime arecombinedintheobjectivefunction.Ourstudy buildsupon thePTP by havingthe profitsandthe travel time in theobjective aswell. In theprize-collecting TSP (PCTSP) the ob-jectiveis similar to PTP buta constraintis added toensure that aminimumamount ofprofitsiscollectedwithin thetour. In the original definition of the PCTSP by Balas (1989) there were also penalty values forunvisited nodes within the objectivefunction. Anabundantnumberofpublicationsisdevotedtotheorienteering problem(OP),whichaimstomaximizethecollectedprofitssubject toaconstraintonthemaximumallowedtourlength.Thisproblem isalsoknownastheselectivetravelingsalesmanproblem(Laporte &Martello,1990).

A variant of the OP is the time-dependentorienteering

prob-lem (TDOP) which includes time-dependent travel times. Fomin

andLingas(2002)provideda

(

2+



)

-approximationalgorithmfor

the TDOP which runs in polynomial time if the ratio between

the minimum and maximum travel time between any two sites

is constant. Li (2011) designed a novel dynamic labeling algo-rithm forthe TDOPin whichtime ismeasured in discrete units. Therefore,theFIFO property maynotbe satisfied intheir model.

Verbeeck,Aghezzaf,andVansteenwegen(2014)providedafast

so-lution method for the TDOP based on an ant colony

optimiza-tion algorithm. Recently, Verbeeck, Vansteenwegen, and

Aghez-zaf(2016) presented an ant colony optimization basedalgorithm forthestochastic variant oftheTDOP, which isaddressedas the

stochastic time-dependent orienteering problem with time

win-dows.Formoredetails aboutthe OPandits variants,readersare

referredtoVansteenwegen,Souffriau,andOudheusden(2011)and

Gunawan,Lau,andVansteenwegen(2016).

Tothebest ofourknowledge,noliterature isfound that han-dles a combinationof precedence in pickup and delivery, profit-maximizingselectionandtime-dependenttraveltimerouting cost minimization atthesametime. Thus, inthisstudy,we introduce thetime-dependentcapacitatedprofitabletourproblemwithtime

windows and precedence constraints, which takes care of these

threechallengessimultaneously. Moreover,bothexactand heuris-ticmethodsareproposedtosolvethisproblem.

3. Problemdescriptionandmathematicalformulation

In thissection, we firstdefine the problemandintroduce the

notation used throughout the paper. Afterwards, we present a

mathematicalformulationfortheproblem.

3.1. Problemdefinition

The time-dependent capacitated profitable tour problem with

time windows and precedence constraints is defined as follows.

Weconsidera setofnrequestsR1,...,Rn,whereRi (i=1,...,n

)

is associated withthe pickup node i and the corresponding de-livery node n+i. Let G=

(

N,A

)

be a directed graph, where N=

{

0,1,...,2n+1

}

isthe set ofall nodes,and0 and2n+1 repre-senttheoriginanddestinationdepotofthevehicle.Wedefinethe subsetsNP=

{

1,...,n

}

andND=

{

n+1,...,2n

}

asthepickupand

deliverynodes,respectively.Witheachpickup nodei∈NPaprofit r_iandaloadq_iareassociated,andwitheachdeliverynodeaload

qn+i is associated.For the requests, it musthold that qi=−qn+i.

There isno inventoryatthe depotsandthereforeq0=q2n+1=0.

Toserve the requestswe have one vehicleavailable withlimited capacityQ.

A hard time window [ej, lj] is associated with each node j∈NP∪ND, whereej and lj representthe earliest andlatest time,

respectively,atwhichtheserviceatnode jmaystart.Theservice timeisdenotedbysj.Avehicleneedstowaituntiltime ej,ifitis

arriving atnode jbefore time ej; andarrivinglater thanljis not

allowed. Wedenote[e0, l0],[e2n+1,l2n+1] asthetime windows of

theoriginandthedestinationdepot, respectively.Withoutlossof generality,weassumethate0=0ands0=s2n+1=0.

Let

τ

_ij(t_i)denote thetraveltime fromnode ito node j,which dependson thedeparture time ti atnode i.Then, we can define

the set of feasible arcs as A=

{

(

i,j

)

∈N× N:i=j and ei+si+

τ

i j

(

ei+si

)

≤ lj

}

. Thismeans that an arc fromnode i to node j is

onlyincludedifitispossibletogofromnodeitojwhile respect-ingthetimewindowsofbothnodes.

ThenotationissummarizedinTable1.

Theplanninghorizonisdividedintoseveraltime periods.Each arc(i,j)∈Ahasaspeedprofileassociatedwithit,whichconsistsof

Fig. 2. Speed and travel time functions.

aconstantspeedwithineachtimeperiod.Byusingthosestepwise speedfunctions,theFIFOpropertyholdsforeveryarcinthegraph

G(i.e.,a laterdeparture alwaysleadstoa laterarrival and there-foreovertakingwillnotoccur).Thespeedprofilescanbedifferent foreacharc.

Fig.2depictsaspeedprofileandthecorrespondingtraveltime functionforsomearc(i,j).UsingtheideadescribedinIchouaetal. (2003), we denote the points a, b, c, d, ande where the speed changesasspeedbreakpoints.Therearealsotra

v

el timebreakpoints

in thetravel time function. Theseare the departure timeswhich ensure an arrival atnode j exactly atthetime of aspeed break-point(e.g.,a isthedeparturetimeatnodeitoarriveatnodejat timea).

The travel time function is piecewise linear and can be rep-resented by the breakpoints values. Note that in case of time-dependent traveltimes,thetriangleinequalitydoesnot necessar-ilyhold.Intuitively,whenthedirectlink betweennode handlis heavilycongested,wemayreachdestinationnodelearlierby tak-ingadivertedrouted(i.e.,viaoneorseveralothernodes)thanby thedirectlinkfromnodeh.

BecauseoftheFIFOpropertyofthetraveltimefunctions,alater departure atthe depot0 always resultsin a later arrival time at nodei.Therefore,ifapathisinfeasibleforacertaindeparturetime

t0 attheorigindepot(i.e.,atimewindowofanodeinthepathis

violated),itwillalsobeinfeasibleforanydeparturetimet_{≥ t}0 at

theorigindepot.Givenapath p=

(

v

0,

v

1,...,

v

k

)

with

v

0=0and

v

ibeingthenodeatpositioniinthepathp,wedefine

δ

vpi

(

t

)

asthe readytimefunction atnode

v

iinpathpgivenadeparture timet

atnode0.Thisreadytimefunctionisnondecreasing int andcan becalculatedrecursivelyforeachnodeinthepathasfollows:

δ

p vi

(

t

)

=



_{t ifi=0,} max

{

evi+svi,

δ

p vi−1

(

t

)

+

τ

vi−1,vi

(δ

p vi−1

(

t

))

+svi

}

otherwise. (1) Thereadytimefunctionispiecewiselinearandthismeansthat wecan representthereadytimefunctionbyusingthereadytime function breakpoints.These arethe breakpoints of theready time function ofthe predecessor node, breakpoints of the travel time function,andtheboundaryvaluesofthetimewindowofnode

v

i.

The duration of the path given a departure time t at node 0

can be calculated as

δ

_vp

k

(

t

)

− t,which isagain apiecewise linear

function.Inthisproblemweminimizethetotaldurationofthe se-lectedtourinsteadofthesumofthearccost.Asthedurationisa piecewiselinearfunctionofthedeparturetime,itisclearthatthe minimumdurationofatourcanbecomputedbyonlyconsidering thebreakpointsofthereadytimefunction.

3.2.Mathematicalformulation

Forevery arc (i, j)∈A, we denote Tij as the set of time

peri-ods of the corresponding travel time function

τ

ij(ti). A time

pe-riodTm∈Tij,isdefinedbytwoconsecutivetraveltimebreakpoints, Tm=[wm,wm+1].As

τ

ij(ti)islinearineachtimeperiod,usingwm, wm+1,

τ

i j

(

wm

)

and

τ

i j

(

wm+1

)

, we can easily calculate the

corre-spondingslope

θ

m andits intersection

η

mwiththey-axis.

There-fore

τ

i j

(

ti

)

=

θ

mti+

η

m.

∀

ti∈Tm (2)

Furthermore,let xm

i j be a binary variablethat takesvalue 1 if

and only if the vehicle traverses the arc (i, j)∈A with a depar-turetime intimeperiodm.Avariabletm

i j isintroduced todenote

thisdeparturetimeoftravelingfromitojintimeperiodTm.This

meansthattm i j issuchthat t_{i j}m=



ti ifxmi j =1, 0 otherwise. (3)

Consequently,whentravelingfromitoj,wehavethat:

ti=  j∈N\{0} |Ti j|  m=0 tm i j. (4)

Itmeansthatthetraveltimefunction

τ

_ij(t_i)ofarc(i,j)atnode

icanbewrittenas:

τ

i j

(

ti

)

=

|Ti j|



m=0

(θ

mti jm+

η

mxmi j

)

. (5)

Let yi be a binary variable that equals 1 if and only if node i∈NP∪ND isvisited.Furthermore,letQi,i∈Nbeanonnegative

in-tegervariablethatistheloadofthevehicleupon departurefrom nodei.Thenthemixedintegerprogramingformulationisgivenas follows: max j∈NP rjyj−

(

t2n+1− t0

)

(6) subjectto  j∈NP |T0j|  m=0 xm_0j=1 (7)  i∈ND |T_i,2n+1| m=0 xm i,2n+1=1 (8)  i∈N\{2n+1} |Ti j|  m=0 xm_{i j} =yj

∀

j∈N

\

{

0

}

(9)  i∈N\{2n+1} |Tik|  m=0 xm_ik−  j∈N\{0} |Tk j|  m=0 xm_{k j}=0

∀

k∈N

\

{

0,2n+1

}

(10)  j∈N\{0} |Ti j|  m=0 xm_{i j}−  j∈N\{0} |T_n+i, j| m=0 xm_{n+i, j}=0

∀

i∈NP (11)

tj≥

(

1+

θ

m

)

ti jm+

η

mxmi j+sjxmi j

∀

i∈N

\

{

2n+1

}

,j∈N

\

{

0

}

(12) Qi+qj≤ Qj+M

(

1− xmi j

)

∀

i,j∈N,

∀

m,m+1∈

|

Ti j

|

(13) tn+i≥ ti

∀

i∈NP (14) ti=  j∈N\{0} |Ti j|  m=0 tm i j

∀

i∈N

\

{

2n+1

}

(15) wmxmi j ≤ ti jm≤ wm+1xmi j

∀

i,j∈N,

∀

m,m+1∈

|

Ti j

|

(16) eiyi≤ ti≤ liyi

∀

i∈NP∪ND (17) max

{

0,qi

}

≤ Qi≤ min

{

Q,Q+qi

}

∀

i∈N (18) xm_{i j},yi∈

{

0,1

}

∀

i,j∈N,

∀

m∈

|

Ti j

|

(19)

The objective function (6)aims to finda tour that maximizes thecollectedprofitsminusthetotaltravelingduration.Constraints

(7)–(8) guarantee that the path starts atthe origindepot 0 and endsatthedestinationdepot2n+1.Constraints(9)guaranteethat every node, exceptthe nodesrepresenting the startand end de-pots,is visitedatmostonce. Constraints (10)keep theflow con-servation. Constraints (11) ensure that it is not possible to visit only the pickup node or only the delivery node of a certain re-quest.Constraints(12)guaranteethatthedeparturetimeatanode intherouteislargerorequalthanthesumofthedeparturetime fromthepreviousnode,thetraveltime betweenthesetwonodes andtherequiredservicetime.Constraints(13)determinethe con-sistency ofthe loadvariables. Precedenceconstraints (14)ensure thatfor eachrequest i, thepickup node is visitedbefore the de-liverynode. Constraints (15) are formulated asmentioned before in(4).Constraints (16)and(17)force thedeparture time ofeach requesttobeinthegiventimeperiodandthegiventimewindow. Finally,Constraints(18)ensurethattheloadinthevehicleisnever largerthanthecapacityofthevehicle.

Duetothecomputationalinefficiency ofsolving large-scale in-stanceswithacommercialILPsolver,wedevelopatailored label-ingalgorithmtosolvethisproblem.

4. Atailoredlabelingalgorithm

In order to solve ourproblem, we introduce a newexact

dy-namic programing algorithm, that is named as the tailored

la-beling algorithm. Ropke etal. (2009) developed a labeling algo-rithm tosolve the pickup anddelivery problems withtime win-dows(PDPTW).However,thistime-independentalgorithmisonly efficient if the triangle inequality holds. More recently, Dabia etal. (2013) proposed another labeling algorithm for the time-dependentvehicleroutingproblemwithtimewindows.Their algo-rithmhasgreatpotential forthetime-dependentroutingproblem withoutprecedenceconstraints.

Inoursituationthetriangleinequalitydoesnotnecessarilyhold

dueto the time dependent travel times and furthermore

prece-denceconstraintsarepresent.Therefore,weneedtodevelopanew algorithm. Note that the proposed algorithm can be generalized tosolve othertime-dependentrouting problemswithprecedence constraints.

Thealgorithmstartsgeneratinglabelsfromthedepot0.It pro-gressivelyextendsallfeasiblelabelsuntiltheyreachtheenddepot 2n+1.Moreover,tospeed up ourtailoredlabelingalgorithm, in-steadofstartingthelabelextension onlyfromthe origindepot0 inaforwarddirection,wesimultaneouslygeneratelabelsin back-ward directionfrom the destination depot to its predecessors as well.Wherebothforwardlabelsandbackwardlabelsareextended tosometimetm(e.g.,themiddleoftheplanninghorizon)butnot

further.At theend,completepaths aregeneratedby mergingthe partial paths offorward andbackward labels.All complete paths areevaluated andthepath withthebestobjectivefunction value is the optimalpath. Thisbidirectional approach hasshown great potential forimprovingthe runningtimeofrelatedresource con-strainedshortestpathproblems(see,e.g.,RighiniandSalani,2006;

Dabiaetal.,2013).

TheforwardlabelingalgorithmisintroducedinSection4.1, fol-lowedby thebackward labelingalgorithm inSection4.2.Iflabels are dominated by the criterion introduced in Sections4.1.3 and

4.2.3, they are removed fromthe list.Note that ifin the proce-dure none of the labels are dominated, this algorithm is equal to the complete enumeration of all feasible paths. Therefore the dominancecriterionisveryimportant.Finally,wediscusstheway

to merge the partial paths of forward and backward labels in

Section4.3.

4.1. Theforwardlabelingalgorithm

In the forward labeling algorithm we start generating labels fromthestartdepot.Thedefinitionofaforwardlabelisdiscussed inSection4.1.1.Thelabelsareextendediftheextensionisfeasible asdiscussed inSection4.1.2.The dominancecriterion forforward labelsisdiscussedinSection4.1.3.

4.1.1. Forwardlabel

ForeachforwardlabelL_f,weusethefollowingnotation:

p(Lf) ThepartialpathoflabelLf.

v

(

Lf

)

∗ Thelastnodevisitedonthepartialpathp(Lf).

L−1

(

L_f

)

∗ TheparentlabelfromwhichLforiginatesbyextending

itwith

v

(

Lf

)

.

O(Lf)∗ Thesetofincompleterequestsinp(Lf),i.e., thepickup

nodeisvisitedbutnotthedeliverynode.

U(Lf)∗ Thesetofrequestsforwhichthepickupnodesare

al-ready visited along the partial path p(Lf). It contains

boththecompleteandtheincompleterequests. There-fore,O(Lf)⊆U(Lf).

P(Lf) The setofpickup nodesnot visitedinp(Lf),i.e., j∈NP

andR_j∈U(L_f).

D(Lf) The set of delivery nodes of incomplete requests in

p(Lf),i.e.,j∈ND andRj−n∈O

(

Lf

)

.

q(Lf)∗ Theloadofthevehicleaftervisitingnode

v

(

Lf

)

.

δ

L_f

(

t

)

∗ Thepiecewiselinearfunctionthatrepresentstheready

timeat

v

(

L_f

)

ifthevehicledepartedattheorigindepot attandreached

v

(

Lf

)

throughpartialpathp(Lf).

More-over,

δ

L_f

(

0

)

istheearliestreadytimeat

v

(

Lf

)

sincethe

earliestdeparturetimeattheorigindepotis0.

r(Lf)∗ Theoverallprofitscollectedbyservingtherequests

vis-itedonthepartialpathp(Lf).

Onlytheitemsmarkedwitha∗arestoredinthelabel.Theset

D(Lf) andP(Lf) canbe deducedfromthesets O(Lf)andU(Lf).

Fur-thermore,thepartialpathcanbe deducedfromiteratively check-ingthelastnodevisitedintheparentlabelofwhichthethislabel wasanextension.

Fig. 3. Illustration of φ(L1

f , L 2f ).

4.1.2. Labelextension

We extend a label L_f along an arc

(

v

(

L_f

)

,j

)

, only when the extension isfeasibleintermsoftimewindowsandcapacity.First, thefollowingtwoconditionsshouldbemet:

δ

L_f

(

0

)

+

τ

v(L_f), j

(δ

L_f

(

0

))

+sj≤ min

{

tm,lj+sj

}

∧ j∈N

\

{

0

}

(20)

q

(

Lf

)

+qj≤ Q ∧ j∈N

\

{

0

}

(21)

Condition(20)ensuresthat anextensiontonodejcanonlybe

performed ifnode j can be reached within its time window and

guarantees that the extension is stopped before tm is exceeded.

Condition (21) ensuresthat an extension to node j is only possi-bleifthereisenoughcapacitytodealwiththeloadofnodej.

Secondly,L_f andj mustalsosatisfy oneofthefollowingthree conditions:

j∈/U

(

L_f

)

∧ j∈NP (22)

j− n∈O

(

L_f

)

∧ j∈ND (23)

O

(

L_f

)

=∅ ∧ j=2n+1 (24)

Condition (22) states that j should not have been visited be-fore,if itis a pickup node. Condition (23)indicates that ifj isa

delivery node, the corresponding pickup node should have been

visited already.The last condition, Condition (24), statesthat if j

isthe enddepotthen all visitedrequestsshould havebeen com-pleted.Inthepresenceofthoseconditions,onlyelementarypaths thatsatisfyprecedenceconstraint(11)aregenerated.

At last, we need to check that all delivery nodes of requests forwhich thepickup nodeis alreadyvisitedin p

(

L_f

)

canstill be reached. In casethetriangleinequality holds, anode is unreach-able if traversing the direct arc from j to this node is not pos-sible by capacity, time window or precedence constraints. How-ever, time-dependent travel times cannot guarantee the triangle inequality.Therefore,anode thatisunreachableviathedirectarc fromnodej bythetime windowconstraintsmightstill be reach-ableindirectlyviaadivertedroute.First,weneedtoknowthe ear-liestreadytimeatnodejafterfollowingpartialpathp

(

L_f

)

before visiting node j, which will be denoted by tr

(

L_f,j

)

. It holds that

tr

(

L_f,j

)

=max

{

ej+sj,

δ

_L

f

(

0

)

+

τ

L

f, j

(

δ

Lf

(

0

))

+sj

}

. Then, we need

forany unvisited node k the earliest arrival time given that the vehiclevisits node j andk consecutivelyafterpartial path p

(

L_f

)

_,

whichcanbecomputedbytr

(

L_f,j

)

+

τ

jk

(

tr

(

L_f,j

))

.Finally,the

ear-liest possible time the vehicle could reach a node after node j

isgivenbyte

(

L_f,j

)

=min_k∈P(L

f)∪D(L



f)

{

tr

(

L_f,j

)

+

τ

jk

(

tr

(

L_f,j

))

}

.This

meansthatanynodekwiththelatestallowedarrival timelk

ear-lierthanthistime(i.e.,l_k<te

(

L_f,j

)

)isunreachablefromj,alsoin

anindirectwayasnonode couldbe reachedbeforete

(

L_f,j

)

.Ifa

deliverynodekisunreachableafterjanditscorrespondingpickup nodeisalreadyvisited(i.e.,k− n∈O

(

L_f

)

),theextensiontojisnot feasibleasthepickedupitemcannotbedeliveredanymore. There-fore,asstatedincondition(25),all delivery nodesofrequests of whichthe pickup node isalready visitedin p

(

L_f

)

shouldstill be reachabletomakesurethatextendinglabelL_f tojisfeasible.Note thatthistestcanbedonequickly,butwemightfailtofindall un-reachabledeliverynodes.

lk≥ te

(

L 

f,j

)

∀

k∈D

(

L 

f

)

:k=j (25)

Iftheextensionalongthearc

(

v

(

L_f

)

_,j

)

isfeasibleaccordingtoall describedconditions,thena newlabelLfiscreated.The

informa-tioninlabelLfisupdatedasfollows:

L−1

(

Lf

)

=L  f (26)

v

(

Lf

)

=j (27)

δ

Lf

(

t

)

=max

{

ej+sj,

δ

Lf

(

t

)

+

τ

Lf, j

(δ

Lf

(

t

))

+sj

}

(28) q

(

Lf

)

=q

(

L  f

)

+qj (29) r

(

Lf

)

=



r

(

L_f

)

+rj if j∈NP, r

(

L_f

)

otherwise. (30) O

(

Lf

)

=



O

(

L_f

)

∪

{

j

}

if j∈NP, O

(

L_f

)

\

{

j− n

}

if j∈ND. (31)

U

(

Lf

)

=



U

(

L_f

)

∪

{

j

}

if j∈NP,

U

(

L_f

)

otherwise. (32)

Eqs.(26)–(30)setthelast visited node,theready time function, theload,andthecollectedprofitsofthenewlabel,respectively.Eq. (31)updatesthesetofincompleterequestsO(Lf)andEq.(32)

up-datesthesetofvisitedpickupnodesU(Lf). 4.1.3. Labeldominance

Letdom(Lf) andimg(Lf)be thedomainandimage oftheready

time function

δ

L_f

(

t

)

, respectively. If the partial path is feasible,

a departure at time 0 from the origin depot is always feasible. Therefore, dom(L_f) is always of the form [0, t] for some t_{≥ 0.}

When

v

(

L_f

)

=2n+1,theobjectivefunctionvalueofthepath cor-respondingtoLfis:

ob j

(

Lf

)

=r

(

Lf

)

− min t∈dom(Lf)

{

δ

Lf

(

t

)

− t

}

(33)

Inthelabelingalgorithm,allpossibleextensionsareprocessedand storedforeach label. However, thenumber oflabels thatcan be processed is typically very large and computationally expensive. Therefore,adominance testisestablishedbetweenpairsoflabels thathavethe samelast visitednode.The numberoflabelsis re-ducedbyonlystoringthenon-dominatedlabels.Beforethe domi-nancecriterionisintroducedsomedefinitionsneedtobeprovided. First,similartotheideaofFeillet,Dejax,andGendreau(2004), weintroduceforevery labelLf thesetU

(

Lf

)

whichextendsU(Lf)

byaddingrequests ofwhichthepickupnode isunreachable from

v

(

L_f

)

. Similar to the discussion in Section4.1.2 it can be derived that the earliest possible time the vehicle could reach a pickup node after

v

(

Lf

)

is given by te

(

Lf

)

=minj∈P(L_f)∪D(L_f)

{

δ

L_f

(

0

)

+

τ

v(L_f)j

(

δ

L_f

(

0

))

}

. Thismeans that any pickup node j with the

lat-estallowed arrival time l_j earlierthanthis time(i.e., l_j<te(Lf))is

unreachablefrom

v

(

Lf

)

.Therefore,thecorrespondingrequest

can-not be served anymore andcan be added tothe setU

(

Lf

)

. Note

thatthechecklj<te(Lf)doesnotguaranteetofindallunreachable

pickupnodes.

Secondly,wedefinetheinterval

I⊆

(

−∞,max

(

dom

(

L1

f

))

− max

(

dom

(

L 2

f

)))

. (34)

BasedonI,wealsodefinearealnumber

φ

(

L1 f,L2f

)

,

φ(

L1 f,L2f

)

=max

{

x∈I:

δ

L1 f

(

max

{

0,t+x

}

)

≤

δ

L 2 f

(

t

)

,

∀

t∈dom

(

L2 f

)

}

. (35) When

φ

(

L1 f,L 2

f

)

ispositive,itindicatesthatthevehiclecan

de-partatmaximum

φ

(

L1

f,L2f

)

timeunitslaterwhentraversingpartial

pathp

(

L1

f

)

insteadoftraversingp

(

L 2

f

)

,tostillreachnode

v

(

L 1 f

)

ear-lierviapathp

(

L1

f

)

thenviapartialpath p

(

L2f

)

.Whenitisnegative,

itindicates that thevehicle should departat least

φ

(

L1

f,L2f

)

time

unitsearlierwhentraversingpartialpathp

(

L1

f

)

insteadof

travers-ing p

(

L2

f

)

, to be able to reach node

v

(

L1f

)

earlier via path p

(

L1f

)

thenviapartialpathp

(

L2 f

)

.

In Fig.3, we depict several simple examples: if there is

no intersection between labels L1 f and L 2 f,

φ

(

L 1 f,L 2 f

)

is positive

when max(dom

(

L1

f

))

>max

(

dom

(

L2f

))

(see Fig.3(a)), or negative

when max(dom

(

L1

f

))

<max

(

dom

(

L2f

))

(see Fig.3(b)). Otherwise,

φ

(

L1 f,L

2

f

)

canonlybenegative(seeFig.3(c)).

Finally, the dominance test is statedin Proposition4.1 as fol-lows: Proposition4.1. LabelL2 f isdominatedbylabelL 1 f if 1.

v

(

L1 f

)

=

v

(

L2f

)

2.U

(

L1 f

)

⊆ U

(

L 2 f

)

3.O

(

L1 f

)

=O

(

L2f

)

4.

δ

_L1 f

(

0

)

≤

δ

L2f

(

0

)

5.r

(

L1 f

)

≥ r

(

L2f

)

−

φ

(

L1f,L2f

)

6.q

(

L1 f

)

≤ q

(

L2f

)

ProofofProposition4.1. Considertwo labelsL1 f andL

2

f that

sat-isfythesixconditionsinProposition4.1.Weneedtoshowthat(i) anyfeasibleextensionLthatextendsp

(

L2

f

)

to2n+1isalsoa

feasi-bleextensionforp

(

L1

f

)

to2n+1and(ii)thatforallthesefeasible

extensionsLitholdsthat ob j

(

L1 f _L

₎

≥ obj

(

L2 f _L

₎

,whereL_fL

isthelabelresultingfromextendingL_fwithL.

With regards to point (i), first, capacity will not be violated alongthepath p

(

L1

f



L

)

asitwasnot violatedonpath p

(

L2 f



L

)

andbycondition6itholdsthat q

(

L1

f

)

≤ q

(

L2f

)

.Secondly,thepath p

(

L1

f

_L

₎

_{iselementary.Byconditions2and3,allnodesvisitedin}

p

(

L1

f

)

areeithernodesvisitedinp

(

L2f

)

ornodeswhichcouldnotbe

reachedbyanyextensionoflabelL2

f (i.e.,thenodesofrequests

in-cludedinU

(

L2

f

)

\

U

(

L2f)).Afeasibleextension ofL2f cannotcontain

anynode visitedalong path p

(

L2

f

)

or node which is unreachable

fromlabelL2

f.Therefore,allnodesvisitedalongpathp

(

L1f

)

arenot

visitedin L,so path p

(

L1 f



L

)

iselementary aswell. Third,there existsa departuretime forpath p

(

L1

f

_L

₎

_{whichdoesnot violate} timewindows.AsLisafeasibleextensionofL2

f itmeansthatthere

isadeparturetimeat

v

(

L1

f

)

after

δ

L2

f

(

0

)

makingsurethatallnodes inLare visitedwithin their time windows.Ifcondition4is met, thevehicleisviapathp

(

L1

f

)

alwaysabletoreach

v

(

L 1

f

)

beforethis

time. Forexample by departing at0, the vehiclearrives at

v

(

L1 f

)

at time

δ

_L1

f

(

0

)

whichis by condition4 smaller than or equal to

δ

L2

f

(

0

)

. Therefore, a departure at0 over path p

(

L

1 f



L

)

does not violateanytimewindows.Inconclusion,anyextension Lof p

(

L2

f

)

to 2n+1 willbe a feasibleextension of p

(

L1

f

)

to 2n+1 asit

re-sultsin an elementary pathwith doesnot violate time windows andcapacityconstraints.

Then for (ii), it still has to be proven that for all feasible extensions of L of p

(

L2 f

)

to 2n+1 it holds that ob j

(

L1f  L

)

≥ ob j

(

L2 f  L

)

.LetL1∗ f =L 1 f  LandL2∗ f =L 2 f  L.Wealsodenotet2 0= argmin_t∈dom(L2∗ f )

{

δ

L2f∗

(

t

)

− t}astheoptimaldeparturetimefromthe depotforpath p

(

L2∗

f

)

andr(L)asthesumoftheprofitsassociated

withthenodesvisitedalongpathp(L).The objectivevalueofthe pathis: ob j

(

L2∗ f

)

=r

(

L2f∗

)

−

(δ

L2∗ f

(

t 2 0

)

− t02

)

=r

(

L2 f

)

+r

(

L

)

−

(δ

L2∗ f

(

t 2 0

)

− t02

)

(36)

Now consider the path p

(

L1∗

f

)

resulting fromextending L 1 f by L.Moreover, consider a departure time at the depotof thispath oft1

0=max

{

0,t02+

φ

(

L1f,L2f

)

}

. The time t01 is a feasible departure

timeforlabelL1∗

f becauseadeparturetimeof0isalwayspossible

(as the extension of L1

f by L is feasible)and by the definitionof

φ

(

L1

f,L2f

)

inEq.(35)t20+

φ

(

L1f,L2f

)

belongstodom

(

L1f

)

ifitis

non-negative.Thisdeparturetimet1

0 ensuresthatwereachnode

v

(

L1f

)

attime

δ

_L2 f

(

t 2 0

)

orearlier,meaning:

δ

L1 f

(

t 1 0

)

≤

δ

L2 f

(

t 2 0

)

. (37)

Moreover, ast1

0 is a feasibledeparture time it can be usedto

computealowerboundonob j

(

L1∗ f

)

: ob j

(

L1∗ f

)

≥ r

(

L 1∗ f

)

−

(δ

L1∗ f

(

t 1 0

)

− t01

)

=r

(

L1_f

)

+r

(

L

)

−

(δ

L1∗ f

(

t 1 0

)

− max

{

0,t02+

φ(

L1f,L 2 f

)

}

)

(38) ≥ r

(

L1 f

)

+r

(

L

)

−

(δ

L2∗ f

(

t 2 0

)

− max

{

0,t02+

φ(

L1f,L 2 f

)

}

)

(39) ≥ r

(

L1 f

)

+r

(

L

)

−

(δ

L2∗ f

(

t 2 0

)

− t02−

φ(

L1f,L 2 f

))

(40) ≥ r

(

L2 f

)

−

φ(

L1f,L2f

)

+r

(

L

)

−

(δ

L2∗ f

(

t 2 0

)

− t02−

φ(

L1f,L2f

))

(41) ≥ r

(

L2_f∗

)

+r

(

L

)

−

(δ

L2∗ f

(

t 2 0

)

− t02

)

=ob j

(

L2f∗

)

(42)

Note that in inequality (39) we use the property derived at

(37). Inequality (40) is derived by the simple fact that

∀

x∈

R:− max

{

0,x

}

≤ −x, and inequality (41) uses condition 5 of

Proposition4.1. 

4.2. Thebackwardlabelingalgorithm

Inthebackwardlabelingalgorithmwestartintheopposite di-rectionandstartgeneratinglabelsfromtheenddepot.The defini-tionofabackwardlabelisdiscussedinSection4.2.1.Thelabelsare extendedasdiscussedinSection4.2.2.Thedominancecriterionfor backwardlabelsisdiscussedinSection4.2.3.

4.2.1. Backwardlabel

In the backward labeling algorithm, labels are extended from theenddepot2n+1toits predecessors. Fora labelLb,we

asso-ciatethefollowingcomponents:

p(Lf) ThepartialpathoflabelLb.

v

(

Lb

)

∗ Thefirstnodevisitedonthepartialpathp(Lb). L−1

(

L_b

)

∗ TheparentlabelfromwhichL_boriginatesbyextending

itwith

v

(

L_b

)

.

O(Lb)∗ The setofincompleterequests,i.e.,thedelivery is

vis-itedbutnotthepickupnode.

U(Lb)∗ The set of requests for which the delivery nodes are

alreadyvisitedalong thepartialpathp(Lb).It contains

both thecompleteandincompleterequests.Therefore,

O(L_b)_⊆U(L_b).

P(Lb) Thesetofpickupnodesofincompleterequestinp(Lb),

i.e.,j∈NPandRj+n∈O

(

Lb

)

.

D(L_b) Thesetofdeliverynodesnotvisitedinp(L_b),.i.e.,j_∈ND

andRj−n∈/U

(

Lb

)

.

q(Lb)∗ Theloadofthetouraftervisitingnode

v

(

Lb

)

.

δ

L_b

(

t

)

∗ The arrival time at the end node 2n+1 through the

partialpathrepresentedbyLbwhenleavingnode

v

(

Lb

)

attimet.

r(Lb)∗ The overall profits collected with the requests

com-pletedonthepartialpathp(Lb).

Again, onlythe itemsmarkedwitha ∗ are storedinthe label andthe sets D(Lb) andP(Lb) can be deducedfrom thesets O(Lb)

andU(L_b).Furthermore,thepartialpathcanbededucedfrom iter-ativelycheckingthefirstnodevisitedintheparentlabelofwhich thethislabelwasanextension.

4.2.2. Labelextension

Let dom(Lb) be the domain of the function

δ

L_b

(

t

)

and let tl(Lb) denote the latest possible ready time at

v

(

Lb

)

: tl

(

Lb

)

=

max

(

dom(Lb)).WeextendalabelLbalonganarc

(

j,

v

(

L 

b

))

to

cre-ateanewlabelL_b.Tobeafeasibleextensionsatleastthefollowing twoconditionsshouldbemet:

tl

(

Lb

)

≥ max

{

tm,ej+sj

}

∧ j∈N

\

{

2n+1

}

(43)

q

(

Lb

)

+qj≤ Q ∧ j∈N

\

{

2n+1

}

(44)

Condition(43)ensuresthat nodejcanbereachedwithinitstime window andthat the extension willbe stopped before tm is

ex-ceeded,while condition(44)ensures capacity feasibility. Further-more,L_bandjmustsatisfyoneofthefollowingthreeconditions:

j+n∈O

(

L_b

)

∧ j∈NP (45)

j∈/U

(

L_b

)

∧ j∈ND (46)

O

(

L_f

)

=∅ ∧ j=0 (47)

Condition(45)indicatesthatifjisapickupnode,the correspond-ing delivery node should havebeen visitedalready. Furthermore, condition(46)statesthatifjisadeliverynode,itshouldnothave beenvisitedbefore.Finally,condition(47),statesthatifjisthe be-gindepotthenallvisitedrequestsshouldhavebeencompleted.In thepresenceofthose conditions,onlyelementarypaths that sat-isfyprecedenceconstraint(11)aregenerated.

Atlast,itneedstobe checkedthatall pickupnodesof incom-pleterequestsforwhichthedeliverynodeisincludedinp

(

L_b

)

can be visitedbeforenode j.Todo so,first thelatest possiblearrival timeatnodejwhichensuresareadytimeoft_l(L_b)at

v

(

L_b

)

should be determined. Let this be denoted by tr

(

j,L_b

)

, then it holds

thattr

(

j,L_b

)

=min

{

lj,max

{

t:t+sj+

τ

_jv(L

b)

(

t

)

+s

v(L

b)≤ tl

(

L_b

)

}

.As showntr

(

j,L_b

)

isdeterminedbythelatestpossiblearrivaltimeat j(i.e.,l_j)orbythelatestpossibilitydeparture timetoreach

v

(

L_b

)

ontime.

Then, all nodes from which the vehicle cannot depart before

time tr

(

j,L_b

)

due to time window constraints, cannot be visited

before node j andare defined asunreachable. This can be made

stronger by considering the travel time to node j as well. How-ever,again dueto theabsence ofthe triangleinequality and the timedependenttraveltimeswecannotsimplyconsiderthetravel time of thedirect connection. Therefore,we need for any unvis-itednodekthelatestpossibledeparturetimetoarriveatjattime

tr

(

j,L_b

)

,whichcanbecomputedbymax

{

t:t+

τ

k j

(

t

)

≤ tr

(

j,L_b

))

}

.

Finally,thelatest possibletimethevehiclecoulddepart fromany nodebefore node jis givenbytd

(

j,Lf

)

=maxk∈P(L

f)∪D(Lf)

{

max

{

t:

t+

τ

k j

(

t

)

≤ tr

(

j,L_b

)

}}

.

This means that any node k with the earliest allowed arrival timeeklater thanthistime (i.e.,lk>td

(

j,Lf

)

)cannot bea

prede-cessorofnode j,alsonot inanindirectwayasnonode could be leftaftertd

(

j,Lf

)

andstillreachingnodejontime.

Ifa pickup node k cannot be a predecessor of node j andits correspondingdelivery node isalreadyvisited(i.e.,k+n∈O

(

L_f

)

), theextensiontojisnot feasibleastheitemwhichshouldbe de-liveredcannot bepicked upanymore.Therefore,asstatedin con-dition(48),allpickupnodesofrequestsofwhichthedeliverynode isalreadyvisitedin p

(

L_b

)

shouldbeapossiblepredecessorofjto makesurethatextendinglabelL_btojisfeasible.

lk≤ td

(

j,L 

b

)

∀

k∈P

(

L 

Fig. 4. Illustration of φ(L1

f , L 2f ).

Iftheextensionalongthearc

(

j_,L_b

)

isfeasibleaccordingtothe providedconditions,theinformationinlabelLbissetasfollows:

L−1

(

Lb

)

=L  b (49)

v

(

Lb

)

=j (50)

δ

Lb

(

t

)

=

δ

Lb

(

max

{

ev(Lb),t+

τ

jv(Lb)

(

t

)

}

+sv(Lb)

)

(51) q

(

Lb

)

=q

(

L  b

)

+qj (52) r

(

Lb

)

=



r

(

L_b

)

+rj if j∈NP, r

(

L_b

)

otherwise. (53) O

(

Lb

)

=



O

(

L_b

)

\

{

j

}

if j∈NP, O

(

L_b

)

∪

{

j− n

}

if j∈ND. (54) U

(

Lb

)

=



U

(

L_b

)

∪

{

j− n

}

if j∈ND, U

(

L_b

)

otherwise. (55) 4.2.3. Labeldominance

Dominance of the backward algorithm can be constructed in

the same way as in the case of the forward algorithm, because

thearrival time functionsarenon-decreasing andstepwise linear asbefore.

Similar totheforwardalgorithm,inProposition4.2,we extend the set U(Lb) to U

(

Lb

)

by includingrequest of which the pickup

ordelivery node cannot be a predecessor of

v

(

Lb

)

.Via the same

reasoningasinSection4.2.2 itcanbederived thatthelatest pos-sible time a vehicle could depart from a node to reach

v

(

Lb

)

on time is td

(

Lb

)

=maxj∈P(L_b)∪D(L_b)

{

max

{

t: t+

τ

jv(L_b)

(

t

)

≤ tl

(

Lb

)

− s_v(L

b)

}}

.Thismeansthat anynode j withthe earliestpossible de-parturetime ej+sj laterthanthistime(i.e.,ej+sj>td

(

Lb

)

)

can-notbeapredecessorof

v

(

Lb

)

.Therefore,thecorrespondingrequest

cannotbeservedanymoreandcanbeaddedtothesetU

(

Lb

)

.

Furthermore,wedefine

φ

(

L1 b,L

2

b

)

(seeFig.4)as:

φ(

L1 b,L 2 b

)

=max

{

x∈R:

δ

L1 b

(

t

)

+x≤

δ

Lb2

(

t

)

,

∀

t∈dom

(

L 2 b

)

}

(56)

Thenthedominancecriterionwillbecome: Proposition4.2. LabelL2 bisdominatedbylabelL1b if 1.

v

(

L1 b

)

=

v

(

L2b

)

2. U

(

L1 b

)

⊆ U

(

L2b

)

3. O

(

L1 b

)

=O

(

L2b

)

4.t

(

L1 b

)

≥ t

(

L2b

)

5. r

(

L1 b

)

≥ r

(

L2b

)

−

φ

(

L1b,L2b

)

6. q

(

L1 b

)

≤ q

(

L2b

)

FortheproofofProposition4.2thesamereasoningastheproof ofProposition4.1couldbefollowed.

4.3. Mergingforwardandbackwardlabels

Whenall forwardandbackward labelsaregenerated,they are merged to construct feasible profitable tours. A forward label Lf

andabackwardlabelL_bcanbemergedifthefollowingconditions aresatisfied: 1.

v

(

L_f

)

=

v

(

L_b

)

2. O

(

Lf

)

∩O

(

Lb

)

=

{

v

(

Lf

)

}

3.

(

U

(

L_f

)

\

O

(

L_f

))

_∩

(

U

(

L_b

)

\

O

(

L_b

))

_=∅ 4. q

(

L_f

)

+q

(

L_b

)

=q_v(L f) 5. Img(Lf)∩dom(Lb)=∅

Theresultingpath p

(

L

)

=

(

p

(

Lf

)

p

(

Lb

))

hasthefollowing

at-tributes: 1.

v

(

L

)

₌2n₊1 2. r

(

L

)

=r

(

Lf

)

+r

(

Lb

)

− rv(L_f) 3. O

(

L

)

_=∅ 4.U

(

L

)

=U

(

Lf

)

∪U

(

Lb

)

5. q

(

L

)

=0

6.

δ

L

(

t

)

=

δ

L_b

(

δ

L_f

(

t

))

,

∀

t∈dom

(

Lf

)

suchthat

δ

L_f

(

t

)

∈dom

(

Lb

)

However,thisbidirectionallabelingalgorithmcangenerate du-plicatesolutions.Consider afeasiblesolutionp∗includingnodesi,

jandkinthisorder.Eachnodex∈p∗isassociatedwithaforward labelLf(x)andabackwardlabelLb(x)(i.e.,

v

(

Lf

(

x

))

=

v

(

Lb

(

x

))

=x).

Therefore,thepathp∗canbeobtainedbymerging L_f(i)withL_b(i) aswell asmergingby Lf(j)withLb(j).Toovercomethisdrawback,

we devised an additionaltest: we accept asolution onlywhen a furtherextensionoftheforwardlabelisimpossible.Inour exam-ple(seeFig.5)theextensionfromLf(i)tonodejisfeasibleandthe

extensionfromLf(j)tonodekisinfeasiblebythepredefinedfixed

timetm.Wegeneratesolutionp∗bymergingLf(j)andLb(j)instead

ofLf(i) andLb(i).The testisperformedforeachcandidatepair of