demands The capacitated vehicle routing problem with evidential International Journal of Approximate Reasoning

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International Journal of Approximate Reasoning

www.elsevier.com/locate/ijar

The capacitated vehicle routing problem with evidential demands ^✩

Nathalie Helal

^a

^,∗ , Frédéric Pichon

^a

, Daniel Porumbel

^b

, David Mercier

^a

, Éric Lefèvre

^a

aUniv.Artois,EA3926,LaboratoiredeGénieInformatiqueetd’Automatiquedel’Artois(LGI2A),F-62400Béthune,France bConservatoireNationaldesArtsetMétiers,EA4629,Cedric,75003Paris,France

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received12September2017

Receivedinrevisedform8February2018 Accepted8February2018

Availableonline13February2018 Keywords:

Vehicleroutingproblem Chanceconstrainedprogramming Stochasticprogrammingwithrecourse Belieffunction

We propose to represent uncertainty on customer demands in the Capacitated Vehicle Routing Problem (CVRP) using the theory of evidence. To tackle this problem, we extend classicalstochastic programming modelling approaches. Specifically,we propose twomodels for thisproblem. The firstmodel is an extension ofthe chance-constrained programming approach, which imposes certain minimum bounds on the belief and plausibilitythatthesumofthedemandsoneachrouterespectsthevehiclecapacity.The secondmodel extendsthestochasticprogrammingwithrecourseapproach:foreachroute, itrepresentsbyabelieffunctiontheuncertainty onitsrecourses,i.e.,corrective actions performedwhenthe vehiclecapacityis exceeded,and definesthecost ofaroute asits classicalcost (withoutrecourse) plusthe worstexpectedcostofitsrecourses.We solve theproposedmodelsusingametaheuristicalgorithmandpresentexperimentalresultson instancesadaptedfromawell-knownCVRPdataset.

©^{2018ElsevierInc.Allrightsreserved.}

1. Introduction

IntheCapacitatedVehicleRoutingProblem(CVRP),wearegivenafleet ofvehicles withidenticalcapacitylocatedata depotandasetofcustomerswithknowndemandslocatedontheverticesofagraph.Thegoalofthisproblemistodeter- minearouteforeachvehicle,suchthatthesetofroutesforallthevehicleshastheleasttotalcost,allcustomerdemands are fullyserviced,the capacityofeach vehicleisalwaysrespected andeach customerisvisitedby exactlyoneroute. The CVRPisNP-hardsinceitcontains thetravelingsalesmanproblemasaparticularcase(onerouteandunboundedcapacity).

It can bewritten asan integerlinearprogram. The CVRPhasgenerated alarge body ofresearch, sinceit belongsto the classoflocaltransportationordeliveryproblemsaffectingthemostexpensivecomponentinthedistributionnetwork [8].

Yet,manyindustrialapplicationsareconfrontedwithuncertaintyoncustomerdemandsintheirdistributionproblemsin- volvingtheCVRP,andtheexactcustomerdemandsaremostlyrevealedwhentheservicingvehiclesarriveatthecustomers.

Accordingly, several authors (see, e.g., [28,29] and the references therein) tackled this issue by assuming that customer demands are random variables andthe associatedproblemis the well-knownCapacitated VehicleRoutingProblem with Stochastic Demands (CVRPSD). Twoof the mostwidely-used frameworksfor modellingstochastic problems, such as the

✩ Thispaperisanextendedandrevisedversionof[33] and[35].

*

Correspondingauthor.

E-mailaddresses:nathalie_helal@ens.univ-artois.fr(N. Helal),frederic.pichon@univ-artois.fr(F. Pichon),daniel.porumbel@cnam.fr(D. Porumbel), david.mercier@univ-artois.fr(D. Mercier),eric.lefevre@univ-artois.fr(É. Lefèvre).

https://doi.org/10.1016/j.ijar.2018.02.003

0888-613X/©^{2018ElsevierInc.Allrightsreserved.}

CVRPSD,aretheChance-ConstrainedProgramming(CCP)approachandtheStochasticProgrammingwithRecourse(SPR)ap- proach [7].ModellingtheCVRPSDviaCCPamountstousingaprobabilisticcapacityconstraintthatrequirestheprobability ofrespecting thecapacityconstrainttobeabove acertain threshold.TheCCP modellingtechniquedoesnot considerthe additionalcostofrecourse(orcorrective)actionsnecessaryifcapacityconstraintsfailtobesatisfied.TheSPRapproachdoes considersituationsneedingrecoursesanditaimsatminimizingtheinitially-plannedtravelcostplustheexpectedcostof the recoursesexecuted along routes, e.g.,returning to thedepotand unloadingin orderto bring tofeasibility a violated capacityconstraint.

Theprobabilisticapproachtomodellinguncertaintyisnotnecessarilywell-suitedtoallreal-lifesituations.Inparticular, itsabilitytohandleepistemicuncertainty(uncertaintyarisingfromlackofknowledge)hasbeencriticised [4,1].Thetypical approach to representing basic epistemic uncertainty is the set-valued approach [26]. It is sensible when, e.g., all that is known about the customer demands is that they belong to some intervals. This kindof uncertainty in the CVRP is generallyaddressedusingrobust optimisation,whereoneoptimises againsttheworst-casescenario,that is,one wantsto obtainsolutionsthatarerobusttoallrealisationsofcustomerdemandsthataredeemedpossible(see,e.g., [49]).However, the set-valuedapproach to uncertainty representation may be too coarse and may thus lead to solutions that are too conservative,hencenotuseful.

In the last fortyyears, the necessityto account forall facets ofuncertainty has beenrecognized and alternative un- certainty frameworksextendingboth theprobabilistic andset-valuedoneshaveappeared [4]. Inparticular,the theory of evidenceintroducedbyShafer [47],basedonsomepreviousworkfromDempster [15],hasemergedasatheoryofferinga compromisebetweenexpressivityandcomplexity,whichseemsinterestinginpracticeasitssuccessfulapplicationinseveral domainstestifies(see [18] forarecentsurvey ofevidencetheoryapplications).Thistheory,alsoknownasbelieffunction theory,maybeusedtomodelvariousformsofinformation,suchasexpertjudgementsandstatisticalevidence,anditalso offerstoolstocombineandpropagateuncertainty [1].

InthecontextoftheCVRP,thetheoryofevidencemaybeusedtorepresentuncertaintyoncustomer demandsleading toanoptimisationproblem,whichwillbereferredtoastheCVRPwithEvidentialDemands(CVRPED).Usingthetheoryof evidenceinthisproblemseemsparticularlyinterestingasitallowsonetoaccountforimperfectknowledgeaboutcustomer demands,such asknowingthateach customerdemand belongsto oneormoresets witha givenprobability allocatedto eachset–an intermediarysituationbetweenprobabilisticandset-valuedknowledge.Inthispaper,weproposetoaddress theCVRPEDbyextendingtheCCPandSPRmodellingapproachesintotheformalismofevidencetheory.Althoughthefocus willbetoextendstochasticprogrammingapproaches,wewillalsoconnectourformulationswithrobustoptimisation.

Toourknowledge,evidencetheoryhasnotyetbeenconsideredtomodeluncertaintyinlarge-scaleinstancesofanNP- hardoptimisationproblemliketheCVRP.Indeed,itseemsthatsofar,onlyother non-classical uncertaintytheories,andin particularfuzzyset theory [50,9,42,11],havebeen usedinsuch problems.Besides, modellinguncertaintyinoptimisation problemsusingevidencetheoryhasconcernedonlycontinuousdesignoptimisationproblems¹ [41,48] andcontinuouslin- ear programs [40].Specificallyin [41], thereliabilityofthe systemis optimized,whileuncertaintyishandledby limiting the plausibilityof constraints violationinto a small degree; while in [48] the problem was handled differently, andthe plausibilityofaconstraintfailure wasconvertedintoasecondobjectivetotheproblemthatshouldbeminimized.Ofpar- ticularinterestistheworkofMasriandBenAbdelaziz [40],whoextendedtheCCPandSPRmodellingapproaches,inorder to modelcontinuous linear programs embedding belieffunctions, which they calledthe BeliefConstrained Programming (BCP)andtherecourseapproaches,respectively.Incomparison,inthiswork,wegeneraliseCCPandSPRtoanintegerlinear program involving uncertaintymodelled by evidence theory.Borrowing from [40], we propose to model the CVRPEDby methodsthatmaybecalledtheBCPmodellingoftheCVRPEDandtherecoursemodellingoftheCVRPED.Forbothmodels, theresolutionalgorithmisasimulatedannealingalgorithm;weuseametaheuristic,astheCVRPEDderivesfromtheCVRP, whichisNP-hard.

The paper is structured as follows. Section 2 summarises the basic preliminaries on the CVRP and on the CVRPSD modelling via CCP andSPR, along withthe necessary background on evidence theory.In Section 3, the BCP model and therecoursemodelfortheCVRPEDare presentedandsomeoftheirpropertiesare studied.InSection4wesolvetheBCP modelandtherecoursemodeloftheCVRPEDusingasimulatedannealingalgorithmandperformexperimentsoninstances generatedfromCVRPbenchmarks.InSection5,weconcludeandstatetheperspectivesofthepresentwork.

2. Background

ThissectionrecallsnecessarybackgroundontheCVRP,theCVRPSDanditsstochasticprogrammingformulations,aswell assomeconceptsofbelieffunctiontheoryneededinthispaper.

1 Designingphysicalsystemsintheengineeringfieldusingoptimisationtechniques,sodesigncostsareminimized,whilethesystemperformanceis fulfilled [2].

2.1. TheCVRP

IntheCVRP,afleetofmidenticalvehicleswithagivencapacitylimit Q,initiallylocatedatadepot,mustcollect² goods fromncustomers,withdi such that0

<

di≤^{Q the}deterministiccollectdemandofclienti, i=¹

, . . . ,

n.Theobjectivein theCVRP istofindasetofmrouteswithminimumcosttoserveallthecustomerssuchthattotalcustomerdemands on anyroutemustnotexceedQ,eachroutestartsandendsatthedepot,andeachcustomerisservicedonlyonce.

Formally,itisconvenienttorepresentthedepotbyanartificialclienti=^{0,whosedemandalwaysequals0,i.e.,d}0=^0.

TheCVRPmaybedefinedonagraphG=

(

V

,

E

)

suchthat V = {⁰

, . . . ,

n}isthevertexsetandE= {

(

i

,

j

)

|ⁱ=^j;ⁱ

,

j∈^V}is thearcset.V representsthecustomersandthedepotthatcorrespondstovertex0.Atravelcost(ortraveltimeordistance –thesetermsareinterchangeable) ci,j isassociatedwitheveryedgein E.Travelcostsaresuchthat ci,j=^cj,i

,

∀

(

i

,

j

)

∈^E and they satisfy the triangleinequality: c_i,j≤^ci,l+^cl,j, ∀ⁱ

,

l

,

j∈^{V.Besides, c}i,i= +∞^,∀ⁱ∈^{V [51]. Let R}k be the route associatedtovehiclek andw^k_i,j abinaryvariablethatequals 1 ifvehiclek travelsfromi to j andserves j (exceptif j is thedepot),and0 ifitdoesnot.AproperformulationfortheCVRP [8,38] is:

min

m k=1

C

(

R_k

),

(1)

where C

(

R_k

) =

n i=⁰

n j=⁰

c_i,jw^k_i,j

,

(2)

subjectto

n i=⁰

m k=¹

w^k_i,j

=

¹

,

j

=

¹

, . . . ,

n

,

(3)

n i=⁰

w^k_i,

=

n

j=⁰

w^k_,j

,

k

=

¹

, . . . ,

m

, =

⁰

, . . . ,

n

,

(4)

n j=¹

w^k_0,j

≤

¹

,

k

=

¹

, . . . ,

m

,

(5)

i,j∈L i=^j

m k=1

w^k_i,j

≤ |

^L

] −

¹

,

L

⊆

^V

\ {

⁰

},

⁽⁶⁾

n i=1

di

n j=0

w^k_i,j

≤

Q

,

k

=

1

, . . . ,

m

.

(7)

Constraints (3) makesurethatexactlyonevehiclearrivesatclient j, j=¹

, . . . ,

n.Constraints (4) ensurethecontinuityof theroutes(flow):ifvehiclekleavesvertex

,vehiclekmustalsoentervertex

,ensuringthattherouteisaproperunbroken cycleinthegraph.Constraints (5) obligevehiclek,k=¹

, . . . ,

m,toleaveatmostonetimethedepot.Thechoicesofthearcs that are representedby w^k_i,j isalso restrictedbyconstraints (6), that forbidssubtourssolutions [8]. Withouttheselatter constraints,wecanhaveavehicleperformingthepath

(

i1

,

i2

, . . . ,

it

)

with0∈ {

/

ⁱ1

,

i2

, . . . ,

it}^.Constraints (7) statethatevery vehicle cannotcarrymorethanits capacitylimit. Wenotethat constraints (3) and(4) imply

n i=⁰

m k=¹

w^k_j,i=^{1, j}=¹

, . . . ,

n, i.e.,exactlyone vehicleleavesclient j.Constraints (5) and (4) imply

n i=¹

w^k_i,0≤^1,k=¹

, . . . ,

m,i.e.,vehiclek isobligedto returnatmostonetimetothedepot.Finally,weremarkthatthismodelrequiresusingatmost mvehicles,sinceforsomek, wemighthavew^k_i,j=^{0, i}

,

j=¹

, . . . ,

n.

Example1.Supposem=^{2 vehicleswithcapacitylimit Q} =^{10,whichmustcollectthedemandsofn}=4 customerswith demands d1=³

,

d2=⁴

,

d3=⁵

,

d4=^{6.Thesecustomersare}illustratedinFig.1a(thedepotisdenotedby “0”),alongwith their associatedtravelcostmatrixinFig.1b.Acandidatesolution,i.e.,asetofroutessatisfyingconstraints (3)–(7),tothis problem isshownin Fig.1c; thetotal travelcost (the value ofthe objectivefunction inEquation (1))ofthissolution is

2 Theproblemcanalsobepresentedintermsofdelivery,ratherthancollection,ofgoods.

Fig. 1.A simple CVRP.

24

.

1.Anoptimalsolution,i.e.,a setofroutessatisfyingconstraints (3)–(7) andwithminimumcost amongthecandidate solutions,isprovidedinFig.1d;itstotaltravelcostis17

.

8.

2.2. TheCVRPSD

TheCVRPSD isavariation oftheCVRP,whichintroduces stochastic demandsintheCVRP, i.e.,d_i

,

i=¹

, . . . ,

n,arenow randomvariables,such that P

(

d_i≤^Q

)

=^{1 (theserandom variablesareusually assumedtobe}independent). TheCVRPSD istypicallyhandledusingthe frameworkofstochasticprogramming, whichmodelsstochastic programsintwo stages:an

“apriori”solutionisestablishedinthefirststage,andtheninthesecondstage therealisations oftherandomvariables– theactualdemandsinthecaseoftheCVRPSD–arerevealedandcorrectiveactionsarecarriedoutifnecessaryonthefirst stagesolution [28].Moreprecisely,theCVRPSDiseithermodelledasa so-calledchance-constrainedprogram [10] orasa stochasticprogramwithrecourse [7];thesetwomodelsaredetailedinthenexttwosections.

2.2.1. TheCVRPSDmodelledbyCCP

Chanceconstrainedprogrammingconsistsinfindingafirststagesolutionforwhichtheprobabilitythatthetotaldemand onanyrouteexceedsthecapacityisconstrainedtobebelowagiventhreshold.Formally,aCCPformulationfortheCVRPSD corresponds tothesameoptimisationproblemdescribedfortheCVRPinSection 2.1exceptthatdeterministicconstraints representedby (7) arereplacedbythefollowingsocalledchance-constraints:

P

⎛

⎝

ⁿ

i=0

d_i

n

j=0

w^k_i,j

≤

^Q

⎞

⎠ ≥

¹

− β,

k

=

¹

, . . . ,

m

,

(8)

where1−

β

istheminimum allowableprobabilitythat anyroute respectsvehicle capacityandthussucceeds.Note that thismodel representsa so-calledindividual chance-constrained model,since theinequality mustbe satisfied forevery k separately;see [45,7] formoredetails.

This model does not consider the cost of corrective actions that may be necessary when the first stage solution is implemented.Indeed,whenimplementingthissolution,itisunlikelyyetpossiblethatthevehiclecapacityisexceeded,i.e., routefailuresoccur, whentheactualdemands arerevealedandthus correctiveactions mayhavetobe carriedoutinthe secondstage.

2.2.2. TheCVRPSDmodelledbySPR

Stochasticprogrammingwithrecoursedealsexplicitlywiththepossibilityofafirststagesolutionfailure,byincorporat- ingintotheobjectiveoftheproblemthepenaltycostofcorrective,orrecourse,actionssuchasallowingvehiclestoreturnto thedepottounload.Morespecifically,intheSPRmodellingoftheCVRPSD,theexpectedpenaltycostoftherecourseactions happeninginthesecondstageisconsideredandtheproblemistofindasetofrouteswhichhastheminimalexpectedcost definedasthecostofthefirststagesolutionifnofailuresoccur,plustheexpectedpenalty costoftherecourseactionsof thesecondstage.Formally,letCE

(

R_k

)

denotetheexpectedcostofaroute R_kdefinedby

C_E

(

R_k

) =

^C

(

R_k

) +

^CP

(

R_k

),

(9)

withC

(

R_k

)

thecostdefinedby (2) representingthecostoftravelingalongR_kifnorecourseactionisperformed,andC_P

(

R_k

)

theexpectedpenaltycost on R_k – CP

(

R_k

)

maybe definedinmanydifferentwaysdependingontherecourse policyused

(see,e.g., [12,27,39,23]).Then,aSPRmodelfortheCVRPSDconsistsinmodifyingtheCVRPmodelpresentedinSection2.1 asfollows.TheobjectiveistofindasetofroutesminimizingthesumoftheexpectedcostsofroutesR_k,i.e.,

min

m k=1

CE

(

Rk

),

(10)

subjecttoconstraints (3)–(6) excludingconstraints (5),whichisreplacedby

n j=¹

w^k_0,j

=

¹

,

k

=

¹

, . . . ,

m

,

(11)

that isexactly mvehiclesmustbeused.Constraints (5) maybeconsidered insteadof (11), butthentheproblembecomes evenmoredifficulttosolve.Inaddition,notethatthebinaryvariables w^k_i,j donotencoderecourseactions:theyrepresent onlytheinitiallyplannedsolutionroutes,i.e.,thefirststagesolution.

2.3. Evidencetheory

Inthissection,basicconcepts aswell assome moreadvancednotionsofevidencetheory [47],whicharenecessaryin ourstudyontheCVRPED,arerecalled.

2.3.1. Basicsofevidencetheory

LetxbeavariabletakingitsvaluesinafinitedomainX= {^x1

, . . . ,

xK}^{.Inthistheory,uncertainknowledgeaboutxmay} berepresentedbyaMassFunction(MF)definedasamappingm^X:^2X→^[0

,

1] suchthatm^X

(∅)

=^{0 and}

A⊆^Xm^X

(

A

)

=^1.

Thesuperscript^X canbeomittedwhenthereisnoriskofconfusion.Eachmassm^X

(

A

)

representstheprobabilityofknowing onlythatx∈^A.SubsetsA⊆^{X suchthatmX}

(

A

) >

0 arecalledthefocalsetsofm^X.Tobeconsistentwiththestochasticcase terminology,avariablexwhosetruevalueisknownintheformofaMFwillbecalledanevidentialvariable.

Massfunctionsgeneralisebothprobabilisticandsetvaluedrepresentationsofuncertaintysince:

•^{aMFwhosefocalsetsare}singletons,i.e.,m^X

(

A

) >

0 iff|^A|=^1,correspondstoaprobabilitymassfunctionandiscalled aBayesianMF;

•^{aMFhavingonlyonefocalset,i.e.,mX}

(

A

)

=^{1 forsome A}⊆^X,correspondstoasetandiscalledacategoricalMF.

Anotherspecialcaseofmassfunctionsarethosewhosefocalsetsarenested,inwhichcasetheyarecalledconsonant.

EquivalentrepresentationsofaMFm^X arethebelief andplausibilityfunctionsdefined,respectively,as Bel^X

(

x

∈

^A

) =

C⊆^A

m^X

(

C

), ∀

^A

⊆

^X

,

Pl^X

(

x

∈

^A

) =

C∩^A=∅

m^X

(

C

), ∀

^A

⊆

^X

.

The degreeofbelief Bel^X

(

x∈^A

)

can be interpreted asthe probability that the evidenceabout x andrepresented bym^X, supports(implies) x∈^{A,whereasthedegreeof}plausibility Pl^X

(

x∈^A

)

istheprobabilitythattheevidenceisconsistentwith x∈^{A.Forall A}⊆^X,wehaveBelX

(

x∈^A

)

≤^PlX

(

x∈^A

)

and

Pl^X

(

x

∈

A

) =

1

−

Bel^X

(

x

∈

A

),

(12)

where A denotesthecomplementof A.Besides,ifm^X isBayesian,then Bel^X

(

x∈^A

)

=^PlX

(

x∈^A

)

,forall A⊆^X,andthis function isaprobabilitymeasure.Ifm^X iscategorical,then Bel^X

(

x∈^A

)

∈ {⁰

,

1}^andPlX

(

x∈^A

)

∈ {⁰

,

1}^{,forall A}⊆^X,and theplausibilityfunctionrestrictedtothesingletonscorrespondstotheindicatorfunctionofthesetassociatedtom^X.Ifm^X isconsonant,thenitsassociatedplausibilityfunctionisapossibilitymeasure [54]:itverifies Pl^X

(

x∈^A∪^B

)

=^max

(

Pl^X

(

x∈

A

),

Pl^X

(

x∈^B

))

,forall A

,

B⊆^X.

Given a MF m^X anda function h:^X→R^{+, it is possible to compute the lowerexpectedvalue E}∗

(

h

,

m^X

)

and upper expectedvalue E^∗

(

h

,

m^X

)

ofhrelativetom^X defined,respectively,as [16]

E_∗

(

h

,

m^X

) =

A⊆^X

m^X

(

A

)

min

x∈^Ah

(

x

),

(13)

E^∗

(

h

,

m^X

) =

A⊆^X

m^X

(

A

)

max

x∈Ah

(

x

).

(14)

Ifm^X isBayesian,then E_∗

(

h

,

m^X

)

and E^∗

(

h

,

m^X

)

reduceto theclassical(probabilistic) expectedvalue ofh relative tothe probabilitymassfunctionm^X.

2.3.2. Comparisonsofbelieffunctions

Theinformativecontentoftwoset-valuedpiecesofinformationx∈^Aandx∈^{B, A}

,

B⊆^{X,aboutxisnaturallycompared} by sayingthat x∈^{A ismore}informativethan x∈^{B if A}⊂^{B.Anextension ofthistocomparethe}informativecontent of mass functionsin terms ofspecificity is the notion ofspecialisation [24]:a MF m₁^X definedon X is said to be at least asinformative (orspecific) asanotherMF m₂^X definedon X,which isdenoted by m₁^{X m}2^X,ifandonly ifthereexists a non-negativesquarematrixS= [^S

(

A

,

B

)]

^{, A}

,

B∈^2X,verifying

A⊆^X

S

(

A

,

B

) =

¹

, ∀

^B

⊆

^X

,

(15)

S

(

A

,

B

) >

0

⇒

^A

⊆

^B

,

A

,

B

⊆

^X

,

(16)

m₁^X

(

A

) =

B⊆^X

S

(

A

,

B

)

m₂^X

(

B

), ∀

^A

⊆

^X

.

(17)

Theterm S

(

A

,

B

)

maybeseen astheproportionofthemassm₂^X

(

B

)

which“flows down”to A.Letusalsorecallthat we have [24]

m₁^X

m₂^X

⇒ [

Bel₁^X

(

x

∈

A

),

Pl₁^X

(

x

∈

A

)] ⊆ [

Bel₂^X

(

x

∈

A

),

Pl₂^X

(

x

∈

A

)], ∀

A

⊆

X

.

(18) Assumenowthatanorderinghasbeendefinedamongtheelementsof X.Byconvention,assumethat x₁

< . . . <

x_K.Let A_a,a denotethesubset{^xa

, . . . ,

x_a}^,for1≤^a≤^a≤^{K andlet}I ^{denotethesetofintervalsof X:}I= {^Aa,a

,

1≤^a≤^a≤^K}^. DecidingwhetheranintervalA_a,a,i.e.aninterval-valuedpieceofinformationx∈^Aa,aaboutx,issmallerorequaltoanother interval B_b,b can bedone inseveralways, andinparticularthe so-calledlatticeordering denoted ≤lo isdefinedas [20]:

A_a,a≤loB_b,b ifa≤^{b anda}≤^{b.ThisorderingcanbeextendedtoarbitrarysubsetsAandBof X asfollows:A}≤loBifa≤^b anda≤^{b,whereaandb(resp.aandb)denotetheindicesofthelowest(resp.highest)valuesin AandB.Moregenerally,} followingtheextensionaboveofinclusionbetweensetstomassfunctions,theordering≤lo ofsubsetscanbeextendedto comparemassfunctionsintermsofrankingasfollows:aMFm^X₁ issaidtobeatleastassmallasanotherMFm₂^X,whichis denotedbym₁^Xm₂^{X,ifandonlyifthereexistsa}non-negativesquarematrix R= [^R

(

A

,

B

)

]^{, A}

,

B∈^2X,verifying

A⊆X

R

(

A

,

B

) =

¹

, ∀

^B

⊆

^X

,

(19)

R

(

A

,

B

) >

0

⇒

^A

≤

loB

,

A

,

B

⊆

^X

,

(20)

m₁^X

(

A

) =

B⊆X

R

(

A

,

B

)

m₂^X

(

B

), ∀

^A

⊆

^X

.

(21)

Inotherwords,themassm₂^X

(

B

)

canbesharedamongsmaller(accordingto≤lo)subsetsthanB.Wenotethatextensionsof intervalrankingstobelieffunctionswerealreadyproposedin [17],butinthecontextofbelieffunctionsonthereallineand theranking≤lo wasnotconsidered.Toourknowledge,thedefinitionof^{appearsthustobe new;itwillbe} particularly usefulinconjunctionwithProposition1,whichissomewhatofacounterparttoEq. (18) for^{,toexhibitapropertyofthe} BCPmodelfortheCVPRED:

Proposition1.ForanyQ ,1≤^Q ≤^{K ,wehave}

m₁^X

^m₂^X

⇒

Bel₁^X

(

x

∈

^A1,Q

) ≥

^Bel₂^X

(

x

∈

^A1,Q

),

Pl₁^X

(

x

∈

^A1,Q

) ≥

^Pl₂^X

(

x

∈

^A1,Q

).

⁽²²⁾

Theconversedoesnothold,i.e.,theimplicationin(22)isstrict.

Proof. SeeAppendixA. 2

Proposition1basically saysthatifaMFm₁^X isatleastassmallasaMFm₂^X,thenthebelief,accordingtothepiece of evidencem₁^X,that thevalue ofx issmaller orequalthana value x_Q isatleastasgreat asthe beliefofthesame event accordingtothepieceofevidencem₂^X (andthesamegoesfortheplausibility),asmaybeexpectedfromthemeaningof^. Tosumupthissection,m₁^Xm₂^{X basicallymeansthatm}₂^{X representsalessprecisepieceofuncertainknowledgeabout} xthanm₁^X,whereasm₁^Xm2^X meansthatm₂^X representsapieceofuncertainknowledgetellingthatxtakesahighervalue thanwhatm^X₁ tells.

2.3.3. Uncertaintypropagation

Let x¹

, . . . ,

x^N be N variables definedon thefinite domains X₁

, . . . ,

X_N, respectively. AMF m^{X1×···×XN} definedon the Cartesianproduct X1× · · · ×^XN representjointknowledgeaboutthevaluesofthesevariables.

Similarlyasinprobabilitytheory,onecanobtainjointknowledgeaboutasubsetoftheevidentialvariablesx¹

, . . . ,

x^N by marginalising MFm^{X1×···×XN} onthedomainsofthesevariables.Forinstance,andwithoutlackofgenerality,themarginali- sationofm^{X1×···×XN} on X1×^X2 istheMFm^{X1×···×XN↓X1×X2} on X1×^X2 definedas,∀^A⊆^X1×^X2,

m^{X1×···×XN↓X1×X2}

(

A

) =

B⊆^X1×···×^XN,B^↓X1×X2=^Am^{X1×···×XN}

(

B

),

(23) where B^↓X1×X2 denotestheprojectionofBonto X1×^X2.

IfMFm^{X1×···×XN} satisfies,forall A⊆^X1× · · · ×^XN, m^{X1×···×XN}

(

A

) =

_N

i=1m^{X1×···×XN↓Xi}

(

A^↓Xi

)

ifA

= ×

_i^N₌₁^A↓Xi

,

0 otherwise, (24)

then variables x¹

, . . . ,

x^N are said to beevidentiallyindependent(or independentfor short) [47].In practice,thishappens when joint knowledge about variables x¹

, . . . ,

x^N,is builtfrom marginal knowledge m^Xi, i=¹

, . . . ,

N, on each of these variables,suppliedbysourcesassumedtobeindependent [43],asillustratedbyExample2(otherreasonsforthistohappen canalsobefoundin [14,13]).

Example2.Let X1= {^x1₁

,

x¹₂

,

x¹₃}^{and X}2= {^x2₁

,

x²₂}^.Furthermore,assume twosourcesprovidingthepiecesofevidencem^X1 andm^X2,respectively,aboutx1 andx2,definedasm^X1

(

{^x11

,

x¹₂}

)

=⁰

.

8,m^X1

(

{^x12

,

x¹₃}

)

=⁰

.

2 andm^X2

(

{^x21}

)

=⁰

.

7,m^X2

(

X2

)

= 0

.

3.Assumingthatthesourcesareindependent,weobtain

m^X1×X2

({

x¹₁

,

x¹₂

} × {

x²₁

}) :=

m^X1

({

x¹₁

,

x¹₂

}) ·

m^X2

({

x²₁

}) =

0

.

56

,

m^X1×X2

({

x¹₁

,

x¹₂

} ×

X2

) :=

m^X1

({

x¹₁

,

x¹₂

}) ·

m^X2

(

X2

) =

0

.

24

,

m^X1×X2

({

^x1₂

,

x¹₃

} × {

^x2₁

}) :=

^mX1

({

^x1₂

,

x¹₃

}) ·

^mX2

({

^x2₁

}) =

⁰

.

14

,

m^X1×X2

({

^x1₂

,

x¹₃

} ×

^X2

) :=

^mX1

({

^x1₂

,

x¹₃

}) ·

^mX2

(

X₂

) =

⁰

.

06

.

m^X1×X2 clearlysatisfies(24).

Furthermore, let y be a variable withfinite domain Y, such that y= ^f

(

x¹

, . . . ,

x^N

)

for some mapping f:^X1× · · · × X_N→^{Y.Asshownin [25],uncertainknowledgemX1×···×XN aboutvariablesx1}

, . . . ,

x^N,inducesMFm^Y aboutthevalueof ydefinedas

m^Y

(

B

) =

f(A)=B

m^{X1×···×XN}

(

A

), ∀

^B

⊆

^Y

,

(25)

with f

(

A

)

= {^f

(

x¹_k1

, . . . ,

x^N_kN

)

|

(

x¹_k1

, . . . ,

x_k^NN

)

∈^A}^{forall A}⊆^X1× · · · ×^XN. 3. ModellingtheCVRPED

This section formalisesandstudies theCVRPED, whichis an integerlinearprogram involvinguncertainty represented by belieffunctions.We obtainthisproblemwhencustomer demandsintheCVRP arenolonger deterministicorrandom, butevidential,i.e.,thevariablesdi,i=¹

, ...,

n,areevidential.Followingwhathasbeendoneforthecaseoflinearprograms involvingevidentialuncertainty [40],wemayextendstochasticprogrammingapproachestothisintegerlinearprogramem- beddingbelieffunctions:theCCPmodellingoftheCVRPSDisgeneralisedintoaBCPmodellingoftheCVRPEDinSection3.1, andtherecoursemodellingoftheCVRPSDisgeneralisedintoarecoursemodellingoftheCVRPEDinSection3.2.

Notethat,tosimplifytheexposition,weassumeactualcustomerdemandstobepositiveintegers,hencethevalueofthe demandofanycustomer belongstotheset

= {¹

,

2

, . . . ,

Q}^{.Inaddition,sincetheCVPREDinvolvesnevidentialvariables} di, i=¹

, ...,

n, with respective domains

i:=

, i=¹

, ...,

n, then formally this means that knowledge about customer demands inthisproblem isrepresented by a MF mⁿ on

ⁿ:= ×ⁿ_i=1

i. Inpractical situations, it maybe the casethat only marginal knowledge in the formof a MF mⁱ may be available aboutthe individual demand of each customer i, i=¹

, . . . ,

n.In such case,asexplained inSection 2.3.3,mⁿ canbe derived by assuming thatthesepieces ofknowledge aboutindividualcustomerdemandshavebeenprovidedbyindependentsources.Inotherwords,ifnecessaryandjustified, evidentialvariablesdi,i=¹

, ...,

n,maybeassumedtobe independent,similarly asitmaybedone inthestochasticcase.

However, let us underline that the BCP and recourse modellings of the CVRPEDproposed in the next two sections, are generalinthatthey donot relyonsuchindependenceassumption,i.e.,theydonotneedmⁿ to satisfyapropertyofthe form (24).

3.1. TheCVRPEDmodelledbyBCP

AgeneralisationoftheCCPmodellingoftheCVRPSDtothecaseofevidentialdemandsisproposedinthissection.The model is provided in Section 3.1.1.Important particular cases of thismodel are discussed inSection 3.1.2. Influences of modelparameters andof customer demandranking on theoptimalsolution cost, arestudied inSections 3.1.3 and3.1.4, respectively.

3.1.1. Formalisation

ABCPmodellingoftheCVRPEDamountstokeepingthesameoptimisationproblemdescribedfortheCVRPinSection2.1 exceptthatcapacityconstraints (7) arereplacedbythefollowingbelief-constraints:

Bel

⎛

⎝

ⁿ

i=1

di

n j=0

w^k_i,j

≤

^Q

⎞

⎠ ≥

¹

− β,

k

=

¹

, . . . ,

m

,

(26)

Pl

⎛

⎝

ⁿ

i=¹ d_i

n j=⁰

w^k_i,j

≤

^Q

⎞

⎠ ≥

¹

− β,

k

=

¹

, . . . ,

m

,

(27)

with

β

≥

β

andwhere 1−

β

(resp. 1−

β

) is the minimumallowable degree of belief(resp. plausibility) that a vehicle capacityisrespectedonanyroute.

Remark1.From (12),constraints (27) areequivalentto

Bel

⎛

⎝

ⁿ

i=¹ d_i

n j=⁰

w^k_i,j

>

Q

⎞

⎠ ≤ β,

k

=

¹

, . . . ,

m

.

(28)

Hence,constraints (26) and (27) amounttorequiringthatforanyroutethereisalot(atleast1−

β

)ofsupport(belief)of respectingvehiclecapacityandnotalot(atmost

β

)ofsupportofviolatingvehiclecapacity.

Notethatinordertoevaluatethebelief-constraints (26) and (27),thetotaldemandoneveryroutemustbedetermined bysummingallcustomerdemandsonthatroute.Supposearoute R having N clients,thenthesumofcustomerdemands on R isobtainedusing (25),where f isthe additionofintegers andwherem^{X1×···×XN} is themarginalisation ofmⁿ on thedomainsoftheevidentialvariablesdr1

, . . . ,

drN associatedwiththe N clientsontheroute,with Xi thedomainofthe evidentialvariabledriassociatedwiththei-thclienton R.Thecomputationofthetotaldemandonarouteaswellasthe evaluationofconstraints (26) and (27) forthatrouteareillustratedbyExample3.

Example3. Suppose that

β

=⁰

.

1 and

β

=⁰

.

05 and that we have n=5 customers and m=^{2 vehicles with capacity} limit Q =^{15. Moreover,suppose knowledge aboutcustomer demands is}represented by MFmⁿ definedon

ⁿ=

⁵=

1×

2×

3×

4×

5by:

m⁵

({(

²

,

3

,

8

,

4

,

5

), (

3

,

5

,

6

,

7

,

4

), (

3

,

4

,

7

,

6

,

2

)}) =

⁰

.

5

,

m⁵

({(

⁵

,

5

,

6

,

4

,

7

), (

7

,

6

,

5

,

3

,

4

)}) =

⁰

.

3

,

m⁵

({(

⁴

,

6

,

7

,

4

,

6

), (

5

,

5

,

6

,

5

,

7

)}) =

⁰

.

2

.

(29) Consider the two routes represented in Fig. 2. Let us compute the sum of the customer demands on the route

(

0

,

4

,

1

,

2

,

0

)

,i.e.,the route that collects the demand ofcustomer 4, then the demandof customer 1 and finally the de- mandofcustomer2.Callthisroute R.Onthisroute,thereare N=^{3 clients.Thefirstclientisclient4,henceaccordingto} theabovenotation,wehavedr1=^d4 andX1=

4.Similarly,wehave

dr2

=

d1

,

X2

=

1

,

dr3

=

d2

,

X3

=

2

.

Themarginalisationofm⁵ on X1×^X2×^X3 istheMFm^{5↓X1×X2×X3} definedas:

m^{5↓X1×X2×X3}

({(

4

,

2

,

3

), (

7

,

3

,

5

), (

6

,

3

,

4

)}) =

0

.

5

,

m^{5↓X1×X2×X3}

({(

4

,

5

,

5

), (

3

,

7

,

6

)}) =

0

.

3

,

m^{5↓X1×X2×X3}

({(

⁴

,

4

,

6

), (

5

,

5

,

5

)}) =

⁰

.

2

.

(30)

Fig. 2.Evaluation of constraints (26) and (27) on the route(0,4,1,2,0)(route circled in red).

Now givenm^{5↓X1×X2×X3} andusingEquation (25) suchthat f istheadditionofintegers,uncertaintyonthesumofclient demandsonroute R isrepresentedbyaMFdenotedm^R anddefinedonthedomain

R:= {¹

,

2

, . . . ,

N·^Q}= {¹

, . . . ,

45} by:

m^R

( {

⁹

,

15

,

13

} ) =

⁰

.

5

,

m^R

({

¹⁴

,

16

}) =

⁰

.

3

,

m^R

({

¹⁴

,

15

}) =

⁰

.

2

.

(31)

ThebeliefandplausibilitythatthesumofcustomerdemandsonR issmallerorequalthanthevehiclecapacityQ arethen respectively:

Bel

(

d4

+

^d1

+

^d2

≤

¹⁵

) =

^mR

( {

⁹

,

15

,

13

} ) +

^mR

( {

¹⁴

,

15

} )

=

0

.

7

,

Pl

(

d4

+

d1

+

d2

≤

15

) =

1

.

Hence,wehave

Bel

(

dr1

+

^dr2

+

^dr3

≤

^Q

) <

1

− β =

⁰

.

9

,

Pl

(

dr1

+

^dr2

+

^dr3

≤

^Q

) >

1

− β =

⁰

.

95

.

Inotherwords,constraint (27) issatisfiedon R butconstraint (26) isnot,andthusanysetofroutescontainingthisroute, suchastheoneshowninFig.2,isnotacandidatesolution.

SupposefurtherthatthenumberoffocalsetsofMFm^{X1×···×XN} isatmostc,thentheworstcasecomplexityofevaluating eachofthebeliefconstraints (26) and (27) onthisrouteisO

(

N·^QN·^c

)

.Thislattercomplexityemergesfromthefollowing:

the Q^N factoristhemaximal numberofelementsofafocalsetofMFm^{X1×···×XN}.Aswehave N clientsonaroute, then foreachelementofafocalsetofMFm^{X1×···×XN},theadditionof N integersmustbeperformed,thisexplainsN·^QN.The last factorinthecomplexitywhichisc,isrelatedtoperformingtheproduct N·^{QN forthec focalsetsofMFmX1×···×XN.} Nonetheless,inaparticularcase,theworstcasecomplexitydropsdownsignificantly:

Remark2.When thefocalsetsofm^{X1×···×XN} areall Cartesianproducts ofN intervals,i.e.,forall A⊆^X1× · · · ×^XN such thatm^{X1×···×XN}

(

A

) >

0,wehaveA=^A↓X1× · · · ×^{A↓XN with,fori}=¹

, . . . ,

N, A^↓Xi=J^Ai;^AiK^{forsomeintegers A}i

,

Ai∈^Xi suchthat A_i≤^Ai,theworstcasecomplexityisO(^N·^c

)

.Thiscomplexitytoevaluateconstraint (26) forroute Rcomesfrom thefactthat(withdritheevidentialvariableassociatedwiththei-thclientonR):

Bel

(

N

i=1

dri

≤

Q

) =

{

m^{X1×···×XN}

(

A

)|

A

:

max

a∈^A f

(

a

) ≤

Q

}

=

{

^{mX1×···×XN}

(

A

)|

^A

:

^max

(a1,···,aN)∈A

N i=¹

a_i

≤

^Q

}

⁽³²⁾

=

{

^{mX1×···×XN}

(

A

) |

^A

:

N i=¹

A_i

≤

^Q

} ,