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An exponential (matching based) neighborhood for the vehicle routing problem
Eric Angel, Evripidis Bampis, Fanny Pascual
To cite this version:
Eric Angel, Evripidis Bampis, Fanny Pascual. An exponential (matching based) neighborhood for
the vehicle routing problem. Journal of Combinatorial Optimization, Springer Verlag, 2008, 15 (2),
pp.179–190. �10.1007/s10878-007-9075-3�. �hal-00341352�
Routing Problem
Eri Angel 1
, Evripidis Bampis 1
, and Fanny Pasual 1
Abstrat
WeintrodueanexponentialneighborhoodfortheVehileRoutingProblem(vrp)withunit
ustomers'demands,and weshowthat itanbeexploredeÆientlyin polynomialtimeby
reduing its exploration to a partiular ase of the Restrited Complete Mathing (rm)
problem that weprovetobepolynomialtime solvableusing owtehniques. Furthermore,
weshow thatin the generalasewith non-unitustomers'demands theexploration ofthe
neighborhoodbeomesanNP-hardproblem.
Keywords: loalsearh,exponentialneighborhood,vehile routingproblem, mathing
1 Introdution
The intratabilityof manyombinatorial optimizationproblems motivated theuse of heuristi
algorithms that aim to nda nearly optimal solution in a reasonable amount of omputation
time. An important lass of heuristis is the lass of loal searh algorithms [AL97 ℄. A loal
searh algorithm starts witha feasiblesolutionand iteratively tries to improve it by searhing
at eah iteration the \neighborhood" of the urrent solution, i.e. a set of feasible solutions
that are lose to the urrent solution. The qualityof the returned solution(loal optimum) as
wellasthe omputationtime of suh an algorithm are very losely related to thehoie of the
neighborhoodstruture. Animportantparameterof thestrutureofaneighborhoodisits size:
intuitively, the larger the neighborhood is, the better the quality of the returned solution is,
but at the same time the longer it takes to searh the neighborhood at eah iteration. Thus,
a larger neighborhooddoesnotneessarily impliesa more eÆient heuristi algorithm. This is
the reason why many reent worksare foused on exponentialneighborhoodsand try to iden-
tify the borderline between neighborhoods that an / that annot be explored eÆiently, i.e.,
in polynomial time with respet to the size of the input [AEOP02℄. Studies in this vein have
been onduted for dierent ombinatorial optimization problems, inluding sheduling prob-
lems [CPvdV02, Hur99℄, the traveling salesman problem [Gut99 , GYZ02 ℄ and generalizations
suh as the quadrati assignment problem [DW00℄. For some of these problems there exists
exponentialneighborhoodsthat an be searhedinpolynomialtime(TSP,sheduling)whereas
forothers,likethequadratiassignmentproblem,thesituationismorediÆultsinethesearh
of varioustypes ofexponentialneighborhoodsleadsto NP-hardoptimization problems.
In thispaper,we fous on theVehile Routing Problem(vrp) [TV01 ℄. The designof expo-
nential neighborhoodsforthis problemhasbeenpreviously addressed. Ergunet al. [EOAF02℄
haveproposedseveralexponentialneighborhoods,andtheyshowedhowtondagoodneighbor-
ing solution by proposing a heuristi forsearhing a onstrained shortest path on an auxiliary
graph(a similarapproahwasusedin[TP93 ℄). Xu andKelly[XK96 ℄proposedatabusearhin
1
IBISC{Universited'
EvryVald'Essonne,CNRSFRE2873{,523PlaedesTerrasses,91000
Evry,Frane.
Tel: 33160873906;Fax: 33160873789;E-mail: fe.angel,bampis,f.pasualgibis.univ-evry.fr
several deletions/insertionsof lientsourinasingleroute simultaneously,theostoftheow
mayonlyapproximatethe atualmove ost.
Inthispaper,weonsideranewneighborhoodofexponentialsizedenedbymathings that
adapts the exponentialneighborhood introdued by Gutin for the traveling salesman problem
[Gut99℄to thevrpproblem. Gutinproved that, intheaseof thetravelingsalesmanproblem,
the neighborhood is polynomial time searhable using a redution to the minimum weighted
perfet mathing problem in bipartite graphs. We prove in what follows that the exponential
neighborhood that we introdue here for the vrp an also be (fully) explored in polynomial
time inthe ase where eah ustomerasks for exatly one unit of goods. To do so, we give a
redutionto apartiularase oftheRestrited Complete Mathing(rm)problem[IRT78 ℄,the
min wprm problem dened below, that we solve by using ow tehniques. Furthermore, we
showthatthe problembeomesNP-hard fornon-unitdemands.
Overview. Intheremainingpartofthissetion,we introdueformallythevariantsofthevrp
that we onsiderthroughout thispaper, the min wprmproblem and some notations that we
use inthe sequel. InSetion2,we denetheexponentialneighborhoodforthevrp. InSetion
3,wegive aredutionof theproblemofexploringtheproposedneighborhoodforunit demands
to themin wprm problem, and inSetion 4, we solve this problem usingowtehniques. In
Setion 5 we show thattheonsidered neighborhood annotbe exploredinpolynomialtimein
the general ase of the vrp where theustomers areallowed to ask forany arbitrary quantity
of goods. Setion6 onludes thepaper.
Notation and denitions. The Vehile Routing Problem (vrp) is dened as follows: n
ustomers must be served from a single depot. Eah ustomer asksfor some amount of goods
andavehileofapaityCisavailabletodelivergoods. Sinethevehileapaityislimited,the
vehilehasto periodiallyreturnto thedepotforreloading. It isnotpossibleto splitustomer
delivery. Therefore, a vrp solutionis a olletion of tours where eah ustomer is visitedonly
one, and thequantityofgoodsdeliveredalong atour doesnotexeedthevehile apaityC.
In the rst part of the paper, we will onsidera restritedversionof thevrp in whih the
demand of eah ustomer is exatly one, and thusthe numberof ustomers served per tour is
at most C. We all this variant the vrp with unit demands. In the last part we onsiderthe
general vrpproblemasdened above.
From a graph theory point of view the vrp with unit demands may be stated as follows:
Let G(V;E) be aompletegraph withnode set V =f0;1;2;:::;ng and arset E. Inthisgraph
model, 0 is the depot and the other nodes are the ustomers to be served. Eah ar (i;j) is
assoiated withvalue d
i;j
representing thedistane (or travel time) between iand j. The goal
is to nd a set of tours of minimum total distane (travel time). Eah tour starts from and
terminates at the depot 0, eah node must bevisited exatly oneand thelength of eah tour
is at mostC.
Example 1. Figure 1 shows an instane of the vrp with unit demands, where there are 7
ustomers. The apaityof thevehileis 4 and the distane matrixis shown on Figure 1 Left.
A solutionof ost 16 is shown on Figure 1 Right (the ost is the total length of the tours, i.e.
0 0 2 1 4 4 1 4 2
1 4 0 3 4 1 4 4 4
2 4 4 0 2 4 4 1 4
3 1 4 4 0 1 4 4 4
4 1 4 4 2 0 4 4 4
5 4 1 4 4 4 0 3 4
6 4 4 4 4 4 4 0 1
7 2 4 4 4 4 4 4 0
1 2
1 3
1 2 2
1
6
3
7
4 0 2
1
5 3
Figure 1: Left: Distanematrix. Right: Asolutionof thevrp withunit demands.
the sum of the lengthsof the ars of thetours). There are 2 tours: inthe rst one the vehile
visits the ustomers 1, 2, 3, 4 and go bak to the depot, and in the seond tour it visits the
lients5,6,7 and go bakto thedepot.
The Restrited Complete Mathing (rm) probleman be denedas follows: in a bipartite
graphG(V;E),i.e. inagraphwhereV anbepartitionedintotwodisjointsets, V
1 andV
2 with
alledgeshavingoneendpointinV
1
andtheotherendpointinV
2
,asetM Eisamathingifno
vertexofV isinidentwithmorethanoneedgeofM. ThesizeofamathingM isthenumberof
itsedges. IfjV
1 jjV
2
jandeveryvertexx2V
1
isinidentwithanedgeofM,thenthemathing
is said to be omplete. Let E
1
;E
2
;:::;E
k
be subsets of E, and r
1
;r
2
;:::;r
k
positive integers.
The restrited omplete mathing problem(rm)onsistsindeterminingwhether thereexistsa
omplete mathing M forGwhihalso satisesthe restritions:
jM \E
j jr
j
8j 2f1;:::;kg:
Ithasbeenshownin[IRT78 ℄thatthisproblemisNP-omplete(redutionfromthesatisability
problem of Boolean expressions). For the ase of a single restrition, the authors presented a
polynomialtimealgorithm.
We are interested inthefollowingsubproblem of rm: Theset of subsets E
1
;E
2
;:::;E
k is
not arbitrary but it orrespondsto a partition of V
2 : T
1
;T
2
;:::;T
k
. For every j 2f1;:::;kg,
let E
j
be thesetof all theedges of E whihhave an endpoint inT
j
. The Partiular Restrited
Complete Mathing problem(prm) isto determine whether there existsa ompletemathing
M forGwhih also satisestherestritions:
jM \E
j jr
j
8j 2f1;:::;kg:
In the weighted version of prm, every edge [v;w℄ has a weight
vw
. Our aim is to nd a
minimum-weightpartiularrestritedompletemathing. We willallthisversionmin wprm
in thesequel. Asusually,m willdenote thenumberof edges ofG.
Example 2. Figure2 shows an exampleof an instane of min wprm: we have a (omplete)
bipartitegraphwithV
1
=f2;4;5;7g,V
2
=T
1 [T
2 ,T
1
=f0;6gandT
2
=f0 0
;1;3g. E
1
isthesetof
alltheedgeswhihhave0or6asendpoint: E
1
=f[2;0℄;[2;6℄;[4;0 ℄;[ 4;6 ℄;[5;0℄;[5 ;6℄ ;[7 ;0 ℄;[7 ;6 ℄g ,
and E
2
isthesetofedgeswhihhave0,1,or6asendpoint,thatistheremainingedges. Figure
2 Right represents thedistane matrix (or the ost matrix), i.e. it indiates theosts between
verties of V
1
and verties of V
2
. We x r
1
= r
2
= 2. Figure 2 Left represents a solution of
the min wprm problem: edges f[2;0℄;[4;1℄;[5;0 0
℄;[7;6℄g form a minimum-weight partiular
restritedompletemathing ofost -4.
0’
3 1 0
6 2
4
5
7
T1
T2
V1 V2
−2
0
−1
−1
0 6 0' 1 3
2 -2 4 3 1 7
4 4 1 6 -1 1
5 0 4 0 4 7
7 1 -1 3 4 4
Figure 2: An instane of the min wprm problem, with V
1
= f2;4;5;7g; V
2
=
f0;0 0
;1;3;6g; T
1
= f0;6g; T
2
= f0 0
;1;3g and r
1
= r
2
= 2. Left: An optimal solution with
ost 4. Right: Distanematrix.
2 An exponential neighborhood for the vrp
Loalsearhremainsthemainpratialtoolforndingnearoptimalsolutionsforlargeinstanes
of thevrp. Letus onsiderthefollowingloalsearh algorithm:
We startwithan arbitrary feasiblesolutionS of thevrp. Then,inthissolution,wehoose
a subset of ustomers,whihwill be onsidered asmobile. The other ustomers and thedepot
will be xed. Let us all verties the set of lients and the depot. A neighbor solution of S is
a solution S 0
in whih eah mobile vertex has been inserted between two xed verties of the
initial solution(these two verties may possiblybe both the depot in the ase where the only
xed vertex of a tour is the depot). Note that we an insert at most one vertex between two
xedvertiesandsothenumberofmobilevertiesmustbesmallerthanorequaltothenumber
ofars inthegraphofxedverties,whihissmallerthanorequaltotwiethenumberofxed
verties. Note also thatthenumberoftoursannot inrease: it isonstant, ordereasesinthe
ase where the onlyxed vertex of atour is the depot and no mobilevertex is insertedinthis
tour.
We do notfousin thispaperon thebestway to determine theinitialsolutionand the set
of mobileverties. The set of mobileverties maybe forexample randomlyhosen among the
quantityofdeliverygoodsinthistour (numberofvertiesintheaseof unitdemands)isequal
to the apaity of thevehile. Indeedno mobilevertex ould be insertedin thistour. In suh
a tour,moving at leastafew vertiesfrom theset ofxed vertiesto theset ofmobileverties
ould onlyimprove thequalityof thebestneighborsolution, sineitwouldjust add some new
possible solutions inthe neighborhood. It is easy to hek whenever a ouple (initialsolution,
set of mobile verties) has a non-empty neighborhood. This is the ase if and onlyif the two
following onditions are satised. First, it must be possible to insert all the mobile verties
between two xedverties, i.e. thenumberof ars inthegraph wherethe mobilevertieshave
beenremoved hasto belargerthanorequaltothenumberofmobileverties. Moreoveritmust
be possible to obtain a neighbor solution where every tour has a quantity of delivered goods
(length, in the ase of unit demands) smaller than or equal to the vehile apaity C. In the
ase ofthe vrp withunit demands, if, step by step,we add a mobilevertex to the tourwhih
hasthesmallestlength,thenthelengthof thelongesttourmustbesmallerthanorequaltoC,
when all themobilevertieshave beeninserted.
Example 3. Figure3showsan initialsolutionof ost16 (Left). The xedvertiesforma new
graph (Right), in whih we an at most insert one mobilevertex between two (xed) verties.
Figure 4 shows a neighbor solution of Figure 3 Left. The ost of this new solution is 11 (the
distane matrixis theone given inFigure 1). The orientation of eah touris preserved during
the onstrution oftheneighborsolution.
5 7
2
4 6
3 1
0
3
1 2 2
1
3 1 2
1
6
3 1
0 4
4 2
4 1
Figure3: Left: asolutionofthevrp. Vertiesirledwiththiklinesaremobileverties. Right:
the orresponding graphwhere mobileverties have beenremoved.
1
4
3 6
7
2
0
1
1 2 1 1
1 1
2 1
5
Figure4: A solutionof ost11 forthevrp.
Letusdenotetheabovedenedneighborhoodthemulti-insertionneighborhood. Weonsider
of theurrentsolution. This solution(tiesare broken arbitrarily)is theurrent solutionof the
next step. Toobtainan eÆientalgorithm, thetimeneeded to nda bestneighborsolutionat
eah step mustbe polynomial.
Theorem 2.1 In the multi-insertion neighborhood, the number of neighbor solutions of a solu-
tion may beexponential, evenfor the vrp withunit demands.
Proof: Let us onsider the following Vehile Routing Problem: we have n lients to supply
(where n is an even number) and theapaity of thevehile is at least n. Let ussupposethat
the initial solution is a solution in whih there is only one tour and that we have n
2
mobile
verties. The numberof neighborsolutions isthen( n
2
+1)! 2
Sinethenumberof neighborsolutionsispossiblyexponential,weannotenumerate allthe
solutions and look for the best one, but we have to nd a method to get the best neighbor
solutioninapolynomialtime. Inorder todothat,wewillnowshowthatitispossibletoredue
the searh of the best solutionof the multi-insertion neighborhood into a partiular restrited
weighted mathing problem.
3 Redution of the neighborhood exploration to the min wprm
in the ase of unit demands
In thissetion, we will saythat, given a solution S and a set of mobile verties, S 0
is a neigh-
bor of S if and only if S 0
an be obtained by a step of the following loal searh algorithm:
S 0
is obtained from S byremoving the mobileverties and by inserting eah of them between
two xed verties. S 0
is a best neighborof S if and onlyif S 0
is a neighbor of S and there is
nootherneighborofSwhihhasaost(atotaldistaneofthetours)smallerthantheostofS 0
.
Let ussupposethat we have an instane of thevrp with unit demands: a distane matrix
D (where d
i;j
is the distane from i to j), a vehile apaity C, an initial feasible solution S
(representedas a graph G(V;E)) and a set of mobileustomers. Let k denotes the number of
toursinS. InS,eahxedvertexuhasasuessors(u)whihisthexedvertexwhihfollows
it inthe tour(we have s(u)=u ifu is thedepot and itis theonlyxed vertex ofa tour). For
example, ifS isthesolutionrepresentedinFigure 3Left, thesuessors(1) of vertex1,is 3. It
is thediret suessorof1 onewe removedthe mobileedges (see Figure 3Right).
LetusmodelthesearhofthebestsolutionintheneighborhoodofSbyamathingproblem.
Let G 0
(V 0
;E 0
) be a omplete bipartite graph suh that V 0
=V
1 [V
2
, where the verties of V
1
are the mobile ustomers and the verties of V
2
are the xed ustomers plus k opies of the
depot. We have then E 0
= f[u;v℄ j u 2 V
1
;v 2 V
2
g. The weights of the edges are dened as
follows: let u 2 V
1
and v 2 V
2
, the weight of the edge [u;v℄ is equal to d
v;u +d
u;s(v) d
v;s(v) .
If an edge [u;v℄ belongs to the mathing we obtained, it means that the vertex u is inserted
between the verties v and s(v). Let fT
1
;T
2
;:::;T
k
g be a partition of V
2
suh that for eah
i2f1;:::;kg,thevertiesofT
i
arethexedustomersof thei th
tourofS andone opyofthe
depot. Let r
i
=C jT
i
j+1. Our aim is to nd in G 0
a minimum-weight omplete mathing
