Solving School Bus Routing Problems through Integer Programming
Author(s): T. Bektaş and Seda Elmastaş
Source: The Journal of the Operational Research Society, Vol. 58, No. 12 (Dec., 2007), pp. 15991604
Published by: Palgrave Macmillan Journals on behalf of the Operational Research Society
Stable URL: http://www.jstor.org/stable/4622856 .
Accessed: 22/01/2014 23:27
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
.
Palgrave Macmillan Journals and Operational Research Society are collaborating with JSTOR to digitize,
preserve and extend access to The Journal of the Operational Research Society.
http://www.jstor.org
This content downloaded from 131.247.152.60 on Wed, 22 Jan 2014 23:27:42 PM
All use subject to JSTOR Terms and Conditions
Journal of the Operational Research Society (2007) 58, 1599-1604 @ 2007 Operational Research Society Ltd. All rights reserved. 0160-5682/07 $30.00
www.palgravejournals.com/jors
Solving
integer
school
bus
routing
programming
problems
through
T Bektag1,2* and Seda Elmasta?2'3
1Universite"de Montreal, HEC Montreal, Quebec, Canada; 2Batkent University, Ankara, Turkey;and
3Bilkent University, Ankara, Turkey
In this paper,an exact solutionapproachis describedfor solving a real-lifeschool bus routingproblem(SBRP)
for transportingthe studentsof an elementaryschool throughoutcentral Ankara,Turkey.The problem is
modelled as a capacitatedand distanceconstrainedopen vehicle routingproblemand an associatedinteger
linear programis presented.The integer programborrowssome well-known inequalitiesfrom the vehicle
routingproblem,which are also shown to be valid for the SBRP underconsideration.The optimalsolution
of the problemis computedusing the proposedformulation,resultingin a saving of up to 28.6% in total
travellingcost as comparedto the currentimplementation.
Journal of the Operational Research Society (2007) 58, 1599-1604. doi:10.1057/palgrave.jors.2602305
Publishedonline6 December2006
exactsolution;openvehicleroutingproblem;
Keywords:schoolbus routing;integerprogramming;
distanceconstraints
capacityconstraints;
1. Introduction
The school bus routingproblem(SBRP) is simply concerned
with transportingstudentsto schools via publictransportation
systems(such as buses).This is a problemwhichhas received
attentionfrom the scientific communityfor over 30 years.
The generalityof the problemmakes it very importantand
scientificways shouldbe employedto deal with it. However,
it is often the case that the routingsare plannedratherintuitively in real life, which may resultin excessive cost for the
transportation.
In some ways, the SBRP resembles the vehicle routing
problem(VRP),whichis extensivelystudiedin the operations
researchliterature(see, eg, thebookby TothandVigo (2002)).
However,as also pointedout by Mandl(1979), therearesome
specific characteristicsof the problemand these are outlined
below:
1. The buses in general do not have to returnto the school
aftercompletingtheirtours.In specific,they may end their
tours at any point otherthanthe depot.Consequently,the
routes of the buses in an SBRP will be paths as opposed
to the tours of the VRP.
2. The total number of students each bus carries cannot
exceed the capacityof the bus.
3. The length (or time) of each touris restrictedby a certain
amount, since the students must be transportedto the
school before a specific time.
*
Correspondence: T Bektay, Centre for Research on Transportation,
University de Montreal, HEC Montreal, 3000 chemin de la C6teSainte-Catherine,Montreal, Canada H3T 2A7.
E-mail: [email protected]
In fact, the problemdescribedabovefalls into anotherversion
of theVRP,namelythe openvehicleroutingproblem(OVRP),
which has recently attractedattentionof severalresearchers
(see, eg, Fu et al (2005), Tarantilis et al (2005), Li et al (2006),
and Repoussiset al (2006)).
In this study,we are concernedwith an SBRP that arises
in transportingthe studentsof an elementaryschool that is
locatedin centralAnkara,Turkey.The school has a contractor
firm, which takes care of transportingthe students.It is of
interest for the contractorfirm to minimize the total cost
associated with transportingthe students to and from the
school to theirhomes. The firm also has to obey the capacity
and distanceconstraintsstatedabove.
The problemconsideredhere correspondsto a capacitated
and distance constrainedOVRP. In this paper, we present
an integerlinear programmingformulationfor this problem
which is solved to optimality using a commercial integer
programmingoptimizer.
The paperis organizedas follows: We briefly review the
relatedwork on the SBRP and the OVRPin the next section.
Section 3 formally defines the problem considered in this
paperandpresentsthe associatedintegerlinearprogramming
formulation,along with some valid inequalities.Input data
and the solution of the integer linear programmingformulation are discussed in Section 4. Finally, the last section
presentsour conclusions.
2. Related work
In this section, we present some previous work related to
school bus routing.Ourattemptis by no meansto providean
This content downloaded from 131.247.152.60 on Wed, 22 Jan 2014 23:27:42 PM
All use subject to JSTOR Terms and Conditions
1600
Research
of the Operational
Journal
SocietyVol.58, No.12
exhaustivereview on the subject,but ratherto focus on the
type of problemandthe solutionapproachconsideredin each
study (For a detailedliteraturereview,the readeris referred
to the relevantsections of these studies).
One of the earliest studies on the subjectis due to Angel
et al (1972). In this study,the authorspresentedan algorithm
for a school bus schedulingproblem,which minimizes the
numberof buses and the total distance covered, such that
capacity and time constraintsare respected.The case study
for which the algorithmis developedconsists of transporting
1500 studentslocatedon a region of about 150 squaremiles,
in Indiana,USA.
Bracaet al (1997) offereda computerizedapproachfor the
of studentsto multipleschools locatedin New
transportation
York, USA. Their problem includes capacity, distance and
time windowconstraints.In addition,theyrequirethata lower
bound on the numberof studentsthat form a route should
also be respected. The problem consists of 4619 students
to be picked up from 838 bus stops and transportedto 73
schools. The authorsproposeda routingalgorithmbased on
the location-basedheuristic for the capacitatedVRP. One
interestingaspect of this study is the estimationof distances
andtraveltimes, which areperformedvia a geographicinformation systems-basedprogram(MapInfo)and a regression
analysis,respectively.The authorsalso presentedtwo integer
programmingformulations,namely a set partitionmodel and
an assignment-basedmodel, although these do not explicitly include the capacityand distanceconstraintsand are not
utilized in the proposedroutingalgorithm.
A recent study relatedto the subjectis due to Li and Fu
(2002). These authors provided planning techniques for a
single SBRP in Hong Kong, China,which consists of transporting86 studentslocatedat 54 pick-uppoints.The problem
is of a multi-objectivenature,includingthe minimizationof
the total numberof buses used, the total traveltime of all the
students,the totalbus traveltime andbalancingthe loads and
traveltimes betweenbuses. A heuristicalgorithmis proposed
to solve the problem. The authorsalso presented a threeindex flow-basedinteger programmingformulation,which,
however,is not utilized in the solutionalgorithm.
The OVRPhas recentlyattractedattentionfrom the operations researchcommunity.One of the first studies on the
OVRP is due to Sariklis and Powell (2000), who proposed
a heuristic to solve the capacity constrainedversion. Later
on, Tarantiliset al (2005) describeda single parametermetaheuristicalgorithmfor the problem.To the best of our knowledge, the only exact methodto solve the capacitatedOVRP
is due to Letchfordet al (2006).
The OVRP may have additionalconstraintsbesides the
capacity restrictions. In a recent study, Repoussis et al
(2006) studied the OVRP with capacity and time window
constraintsanddescribeda greedylook-aheadrouteconstruction heuristicalgorithmto solve the model that is proposed
for the problem.For the OVRP with capacity and distance
constraints,Brandao(2004) and Fu et al (2005) presented
tabu search algorithms, and Li et al (2006) described a
record-to-recordtravelheuristic.To our knowledge,there is
no exact solution algorithmoffered for the solution of the
OVRPwith distanceand/ortime window constraints.
In what follows, we breakawayfrom the previousstudies'
problem-specificsolution approaches for the SBRP. Our
approachand contributionlies in presentingan integerlinear
programfor the problem under consideration,which also
has a potentialto be used for other similarproblems.As the
case study is of moderatesize, our focus is on well-model
buildingratherthandevisinga specializedsolutionprocedure.
The formal definitionof the problemand the corresponding
integer programmingformulationare presentedin the next
section.
3. Problem definition and integer programming
formulation
The SBRP we consider in this paper is defined as follows:
A numberof buses are to be used to transportthe students
between an elementaryschool and their houses. Each bus is
dedicatedto a single path and this bus utilizes the same path
for both picking-upor dropping-offthe students.In addition,
each bus has a limited capacitywith an upperbound on the
total amount of distance it may traverse.Each pick-up or
drop-offpoint is visited only once by a single bus, that is,
no partialpick-upsare allowed. The problemlies in finding
the minimumnumberof buses requiredfor the transportation
of all the studentsand their correspondingroutes, so as to
minimize the total cost of transportation.
We formulatethe problemon a completegraphG = (V, A)
with IV Inodes and IAIarcs.The distancesbetweeneach node
pair is characterizedby a symmetricdistance matrix D :=
[dij], where dij representsthe distance requiredto traverse
from node i to node j (and also from node j to node i). The
node set is partitionedas V = {0} U I, where node 0 is the
depot(the school) and I is the set of intermediatenodes.Each
intermediatenode i E I has a numberof students(denotedby
qi) to be picked up. Then, the SBRP consists of determining
k node disjoint paths connected to the depot such that the
total capacityon each pathdoes not exceed a pre-determined
capacity limit (denotedby Q) and the total length of each
pathdoes not exceed some amount(denotedby T). A feasible
solution to an SBRP resembles a star-liketopology on the
graphG, where thereare k paths all startingfrom the depot,
thatis, node 0. Hence, we have a numberof pathsoriginating
from a single originpointratherthana collection of tours(as
in the case of the VRP).Note thatthis problemcorrespondsto
the OVRPwith capacityand distanceconstraints.Therefore,
the following formulationis also valid for this problem.
In orderto model the SBRP,we introducea 'dummy'node
d, to which all the last nodes of each path will be connected
to. In this case, the problemreducesto findingk node disjoint
pathsbetweentwo pointson an expandedgraphG'=(V', A'),
This content downloaded from 131.247.152.60 on Wed, 22 Jan 2014 23:27:42 PM
All use subject to JSTOR Terms and Conditions
andS Elmasta--Solving
schoolbusrouting
T Bekta?
problems1601
such that all intermediatenodes are visited exactly once. In
the expandedgraph,V'= VU{d) and the new distancesin A'
are definedas follows:
=
0
ifi E I and j = d
M
if i E 0 and j =d
dij
otherwise
di
=
in thesolution
S=
1 if arc(i, j) is traversed
0
otherwise
Then, the integerprogramis constructedas follows:
minimize
ui - uj + Qxij + (Q - qi - qj)xji ?
Vi #j E I
ui - qi
Before presentingthe integer linear program,we define the
following binaryvariable:
xi
capacitatedvehicle routing problem. These constraintsare
given as follows:
cijxij + f.k
(1)
ieV' jEV'
ui - qixoi + Qxoi
Vi E I
Q vi E I
Q - qj
(9)
(10)
(11)
Note that under these constraints,variablesxij and xji are
only definedif qi +?qj < Q. Constraints(9) and (10) arethose
presentedby Kara et al (2004). Constraint(11) is derived
specially for the SBRP. In these constraints,the variableui
representsthe total amount of students picked up by the
vehicle just afterleaving node i. The following propositions
show the validityof these constraints.
Proposition 1 The constraints(9), (10) and (11) are valid
capacity constraintsfor the SBRP.
subjectto
Exoi
k
(2)
k
(3)
iEI
EXid
ielI
xij =1
jEIU{d}
Z
iEIU{o}xij-=1
Vi
E
Vj EI
(4)
(5)
+ capacityconstraints
(6)
+ distanceconstraints
(7)
xij E {0, 1}
'i, j E V'
(8)
In this formulation,the objectivefunctionrepresentsthe total
cost of the travel and the fixed cost of the numberof buses
used. Here, cij is the cost of traversingarc (i, j) and is typically a function of the distance as cij = adij, with a being
the unit distancecost. In the second term, f is the fixed unit
cost of dispatchinga bus. Constraints(2) and (3) allow at
most k vehicles to departfrom node 0 and arriveto node d.
Constraints(4) and(5) arethe degreeconstraints,which force
each intermediatenode to be visited exactly once. Capacity
and distancerestrictionsimposedon each bus are denotedby
(6) and (7). In what follows, we will presentthe associated
capacityand distanceconstraintsthat are polynomialin size
and which also ensurethat valid pathswill be formed.
Proof We firstobservethat constraints(10) and (11) imply
that if i E I is the first node on a path (ie, xoi = 1), then
ui = qi. Also observe that if xij = 1 in a solution to SBRP,
writing constraints(9) for pairs (i, j) and (j, i) results in
i1, ik} in
uj = ui + qj. Now, considera path := {il, i2,
a solutionto the SBRP.Writingconstraints(9) ....
for each pair
of arc in path 0 results in Uik= qik qi,
4- .. + qi2 + qil,
that is, uik representsthe total capacityon path3?. Since ik is
the last node on the path, constraints(10) implying
uik, Q
restrictthe total capacityof the pathby Q. D
We now show that the capacity constraints presented
above also preventthe formationof illegal tours that are not
connectedto the depot, which are named as subtoursin the
routingliterature.
Proposition 2 Constraints(9) preventtheformationof
subtourswithinthe intermediatenodes.
Proof It is shown in the proof of Proposition 1 that
constraints(9) 'link' all the nodes in a single path via the ui
variables.Now, assumean illegal subtouras (k, 1, m, k) where
k, 1, m E I. Then, constraints(9) imply uk = Um+ qk =
i
+4qm + qk = Uk
+- qk, which is impossible since
ql1 qm
qi > O, Vi e I (any i with qi = 0 can be removed from
the graph without loss of generality).By raising a similar
argumentfor all the subtoursdisconnectedfrom the depot,
we can conclude that no subtourwill be formed among the
intermediatenodes. O
3.1. Capacity restrictions
By definition,the SBRPrequiresthatthe capacityof each bus
is respected,that is, no bus can carrymore than its capacity.
To impose such a restriction,we use the Miller, Tucker,and
Zemlin (MTZ)-basedconstraints(Miller et al, 1960) for the
3.2. Distance restrictions
Similarto the capacityrestrictions,we now presentdistance
constraintsthat restrictthe total length of each path to some
pre-determinedamount T. The constraintsare given in the
This content downloaded from 131.247.152.60 on Wed, 22 Jan 2014 23:27:42 PM
All use subject to JSTOR Terms and Conditions
1602
Research
Journal
of the Operational
SocietyVol.58, No.12
following proposition:
Proposition 3
\-r7
Constraints
vi -vj +(T -did -doj +dij)xij +(T -did -doj -dji)xji
< T - did - doj Vi 0 j E
(12)
I
vi - doixoi>0
Vi E I
vi - doixoi + Txoi < T
Vi E I
Jr
-Th
(13)
-
•
(14)
are valid for the SBRP for doi + dij + djo< T for every pair
(i, j), where the variable vi denotes the total length travelled
from the depot to node i.
i '2
Proof Similarto that of Proposition1. O
In these constraints,variablesvi denote the distance that
the vehicle travelled up until point i. Constraints(12) are
liftings of those proposedby Naddef (1994) (see Desrochers
and Laporte (1991) and Kara and Bekta? (2005) for the
lifting results),whereasconstraints(13) and (14) are specifically derivedfor the SBRP.The formerconstraintis used to
'connect' the nodes in each tour and the lattertwo are used
to set initialize the value of vi to doi if i is the first node
on the tour.We would like to note that these constraintscan
also be used to restrictthe total travellingtime of each bus
in a similarfashion, since time is typically a function of the
travelledamountof distance.
As a resultof the precedingdiscussion,the integerprogramming formulationof the SBRP may now be given in full as
Minimize
Eiv'~jev,
cijxij : s.t. (2)-(5), (8), (9)-(14). In
the next section, we will describethe solutionof the problem
underconsiderationusing the proposedformulation.
4. Solution and proposed implementation
This section describesthe currentimplementationof the case
study,explainshow the inputdatawas processedandpresents
the results obtainedby solving the integerprogram.
4.1. Current implementation
In the current implementation,the transportationof the
studentsis handledby a contractorfirm, which has k = 26
identicalvehicles with a commoncapacityof Q= 33. There
are 519 studentsto be picked up from differentlocations in
Ankara.The routingis plannedratherintuitively,and the one
currentlyimplementedis presentedin Figure 1. As demonstratedin the figure, most of the buses (23 out of 26) are
dedicatedto a single destinationonly. Totaldistancetravelled
by all buses in the currentimplementationis calculatedto
be 246.736 km. The school requiresthat each studentshould
not travelmore than a total of T = 25 km by bus.
4.2. Input data processing
Since the locations of the students are points scattered
throughoutAnkara, we have grouped all of these points
Figure 1 The currentroutingplan.
Table1 Thenumberof studentslocatedin eachsubregion
Subregion
Number of students
1
2
3
25
24
26
4
19
5
6
7
8
9
18
22
24
19
22
10
11
22
9
12
13
14
15
8
24
19
24
16
23
17
32
18
19
12
13
20
21
22
23
24
25
26
27
28
29
24
12
12
22
21
11
11
5
9
7
into approximatelyequal-sized clusters, which resulted in
29 different subregions. These sub-regions along with the
numberof studentslocated in each are given in Table 1. The
centroid of each subregionis consideredto be the pick-up
point of all the studentslocated in this subregion.Since the
size of each sub-regionis quite small as comparedto the
entire routing area, the inter-travelwalking distances from
the homes of each studentto the centralpoint in each region
can be neglected.
This content downloaded from 131.247.152.60 on Wed, 22 Jan 2014 23:27:42 PM
All use subject to JSTOR Terms and Conditions
T Bekta?
andS Elmasta--Solving
schoolbusrouting
problems1603
Table 2
Bus No.
-
,)
:
-
"
~.~-n
i
"IN
r
/
/ \
Figure 2
The proposed routing plan.
The distancesbetween each pair of centres are calculated
via MapInfo,by takinginto accountthe inostoften used paths
in the currentimplementationand the paths on which the
buses are allowedto travelaccordingto the trafficregulations.
The resultingdistance matrixis symmetricbut clearly nonEuclidean.
The unit distance cost for each bus is calculated to be
a = 300 TurkishLiras (TLs) per metre and the fixed cost
of each bus is calculated to be f = 18568 181 TLs. The
total cost of transportingthe studentsto school in the current
implementationis calculated as 556 793 506 TLs, using 26
buses in total.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Capacity utilization and distance figures for the buses
in the currentimplementation
Used Cap.
Cap (%)
25
24
26
19
18
22
24
19
22
22
9
8
24
19
24
23
32
25
24
24
22
21
22
5
9
7
75.76
72.73
78.79
57.58
54.55
66.67
72.73
57.58
66.67
66.67
27.27
24.24
72.73
57.58
72.73
69.7
96.97
75.76
72.73
72.73
66.67
63.64
66.67
15.15
27.27
21.21
7450
6920
6512
13 820
18020
3290
5420
850
4430
3550
13220
16980
7960
3810
5814
4310
8620
10940
7990
9850
8350
13980
8950
26580
8100
21 020
60.49
15.15
96.97
9489.84
850
26580
Avg
Min
Max
Distance
4.3. Solution of the model
Using CPLEX 9.0 as the commercialoptimizer,the integer
linearprogrampresentedwas solved to optimalityin 202.31
CPU seconds on a Sun UltraSPARC12 x 400 MHz with
3 GB RAM. The optimal solution came up with a total cost
of 397 151058 TLs, using 18 buses in total. Comparedto
the currentimplementation,the reductionin the total cost is
28.6%. The correspondingroutingplan of the optimal solution is given in Figure 2.
We also present some data in Tables 2 and 3, relevantto
the capacity utilization of the 26 buses used in the current
implementationandthe 18 buses used in the optimalsolution,
respectively.
In Tables 2 and 3, the first column denotes the bus
number,the second column indicateshow many studentsare
carriedby this bus, the third column shows the respective
capacityutilization(in percent)and the last columnindicates
the amountof distancetraversedby this bus. These tablesalso
presentthe maximum,minimumandaveragecapacityutilization rates of all the buses. Table2 indicatesthat the capacity
utilizationin the currentimplementationvarieshighly (about
82% between the maximum and minimum values) and the
averageutilizationrateof 60.49%,is ratherlow. On the other
hand,the optimalsolutionhas an averagecapacityutilization
Table 3
Bus No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Avg
Min
Max
Capacity utilization and distance figures for the buses
in the optimal solution
Used Cap.
25
29
26
25
33
32
31
33
22
24
31
24
32
32
32
24
33
31
Cap (%)
Distance
75.76
87.88
78.79
75.76
100
96.97
93.94
100
66.67
72.73
93.94
72.73
96.97
96.97
96.97
72.73
100
93.94
7450
24500
6512
21020
10355
17600
10890
6770
3550
7960
12960
5814
13220
8620
15590
7990
16975
11970
87.37
66.67
100
11652.55
3550
24500
This content downloaded from 131.247.152.60 on Wed, 22 Jan 2014 23:27:42 PM
All use subject to JSTOR Terms and Conditions
1604
Journal
of the Operational
Research
SocietyVol.58, No.12
of 87.37%,as presentedin Table3, with the variationin the
capacityusage being decreased(to about33%).
Comparingthe currentimplementationwith the optimal
solution terms of distance, the total distance that the buses
travelhas decreasedfrom 246 736 m to 209 746 m. However,
the averagedistancethateach bus travelshas increasedfrom
9489.84 m to 11 652.55 m.
References
5. Conclusionand furtherremarks
DesrochersM and LaporteG (1991). Improvementsand extensions
to the Miller-Tucker-Zemlin
subtoureliminationconstraints.Oper
Res Lett 10: 27-36.
Fu Z, Eglese R and Li LYO (2005). A new tabu searchheuristic
for the open vehicle routing problem. J Opl Res Soc 56:
267-274.
Angel RD, CaudleWL,NoonanR andWhinstonA (1972). Computerassistedschool bus scheduling.MngtSci 18: B279-B288.
Braca J, Bramel J, Posner B and Simchi-Levi D (1997). A
computerizedapproachto the New YorkCity school bus routing
problem.IIE Trans29: 693-702.
BrandioJ (2004).A tabusearchalgorithmfor theopenvehiclerouting
problem. Eur J Opl Res 157: 552-564.
In this paper,we have presentedan integer linear programming formulationto solve a real-life SBRP. The integer
programwas solved to optimalityeasily, due to the moderate
size of the case study.The optimalsolutionobtainedfor the
SBRP consideredhere resultedin a 28.6% savings in total
cost as comparedto that of the currentroutingscheme. As
the school bus routing is most often intuitivelyplanned in
real life, we see thatone can surelybenefitfrom a betterplan
offeredthroughintegerprogramming.
It goes without argument that large-size problems
do need specialized algorithms, as is the case in other
studies mentioned previously, since such problems cannot
in general be directly solved using commercial packas
the
However,
ages.
hardware
and
software
is
computer
technology
rapidly
improving,we believe thatthe focus shouldbe on developing
better formulationsfor problems of moderate size, rather
than devising solution algorithmsthat are problem-specific.
This is exactly the approachtakenin this paper.In fact, it is
quite interestingto note that the integer programpresented
here could not be solved by CPLEX 8.0 on a PentiumIII
1400Mhz PC runningLinux (it was stoppedafter 900 CPU
seconds without reachingan optimal solution), whereas the
same integerprogramwas easily solved on a fastercomputer
using CPLEX9.0.
We finally note that the formulationpresentedhere is also
capableof accommodatingseveraladditionalconstraints,such
as the upper and lower bounds on the numberof students
each bus carriesor time-windowconstraints.In specific, the
time windowconstraintspresentedby DesrochersandLaporte
(1991) can be directlyincludedin the formulationto restrict
each node to be visited in certain time intervals. Further
researchmight thereforeconsidertesting the applicabilityof
the proposed formulationin solving OVRP with capacity,
distance and time window constraintsusing test problems
takenfrom the literature.
Kara I and Bekta? T (2005). Integer linear programmingformulations
of distance constrained vehicle routing problems. Research
Report,Departmentof IndustrialEngineering,BagkentUniversity
(availableon requestfromthe authors).
Kara I, Laporte G and Bektag A (2004). A note on the lifted
Miller-Tucker-Zemlinsubtour elimination constraints for the
capacitated vehicle routing problem. Eur J Opl Res 158:
793-795.
Letchford AN, Lysgaard J and Eglese RW (2006). A
branch-and-cut algorithm for
the capacitated open vehicle
routing problem. Unpublished research report available at
http://www.lancs.ac.uk/staff/letchfoa/ovrp.pdf.
Li F, GoldenB andWasilE (2006).The openvehicleroutingproblem:
Algorithms,large-scaletest problems,and computationalresults.
ComputOperRes, forthcoming.
Li L andFu Z (2002). The schoolbus routingproblem:A case study.
J Opl Res Soc 53: 552-558.
Mandl C (1979). Applied Network Optimization. Academic Press:
London.
MillerCE, TuckerAW andZemlinRA (1960). Integerprogramming
formulationsand travelingsalesmanproblems.J Assoc Comput
Mach 7: 326-329.
Naddef D (1994). A remark on 'Integer linear programming
formulationfor a vehicle routingproblem'by N.R. Achutanand
L. Caccetta,or how to use the Clark& Wrightsavings to write
such integerlinearprogramming
formulations.Eur J Opl Res 75:
238-241.
Repoussis PP, TarantilisCD and Ioannou G (2006). The open
vehicle routingproblemwith time windows.J Opl Res Soc, doi:
10.1057/palgrave.jors.2602143.
SariklisD and Powell S (2000). A heuristicmethod for the open
vehicle routingproblem.J Opl Res Soc 51: 564-573.
TarantilisCD, IoannouG, KiranoudisCT and PrastacosGP (2005).
Solving the open vehicle routingproblemvia a single parameter
metaheuristicalgorithm.J Opl Res Soc 56: 588-596.
Toth P and Vigo D (2002). The Vehicle Routing Problem, SIAM
Monographs on Discrete Mathematics and Applications. SIAM:
Philadelphia.
We aregratefulto an anonymous
reviewer,whose
Acknowledgementscommentsand carefulreadingof the manuscript
haveprovedto be of
greatuse in improvingthe paper.
This content downloaded from 131.247.152.60 on Wed, 22 Jan 2014 23:27:42 PM
All use subject to JSTOR Terms and Conditions
Received July 2005;
accepted June 2006