2.5.1 A Lagrangian heuristi for the Prize Colleting Travelling Salesman
Problem [8℄
An artile inspetinghowto solve PCTSP witha Lagrangian heuristi byM. Dell'Amio, F.
Maoli and A. Siomahen. A good introdution to the PCTSP. The underlying
unstru-tion of the bus route problem, presented in the report, is based in partially on the PCTSP
modelinthisartile.Theproblempresentedintheartileisminimized. Thereforetoassistin
determining the valitityof alulatedsolutions a lowerbound wasalso alulated.Thislower
bound is found in [9 ℄. A feasible solution is found by using Adding-Nodes Proedure where
two rules, R1 and R2,are ompared. From these omparisons R2 was shown to be better in
this instane. Thisfeasible solution isthendened asan upperbound asnofeasible soultion
with alower value objetive valueisknow.
To improve upon feasible solutions two methods are ombined. The rst was the so alled
Extensionphasetriestoimprovetheoverallprotoftheurrentyle.Theseondmethodwas
alledCollapsephaseand ittriestoremovethemost expensive node eahtime.Togetherthe
methodwasalledExtensionandCollapse.LastlyaLagragianheuristiwasdevelopedsothat
Extension and Collapse was applied ineah omputation of the Lagrangian multiplier. This
methodwasthenusedonafewomputationalexperiments.Theonlusionoftheexperiments
was that with inreased prot, that needs to be olleted, the omputational time required
inreased while the quality of thesolutions dereases. Thisquality of solutions was mesured
asthe ratiobetween upperbound and lowerbound.
2.5.2 Prie Colleting Travelling Salesman Problem [1℄
This is an artile by E. Balas onerning theprie olleting travelling salesman problem. It
wasBalas who, along with Martin, rst introdued thePCTSP. There is an introdution to
PCTSP initsrstsetion.Afterthisthe artilebeomesverymathematial andompliated.
Themain fousofthis artile is to disussthe strutural propertiesof thePCTSP polytope,
theonvex hull ofthe solutions tothePCTSP.
2.5.3 On Prize-ColletingTours andThe AsymmetriTravellingSalesman
Problem [9℄
An artile by M. Dell'Amio, F. Maoli and P. V
¨ a
rbrand. The artile ontains a shortin-trodutionto PCTSP anda modelispresented.There isalso adenitionfor PTP,protable
setionontestswhihproved tobehelpfulinondutingtestsforthemodelinspetedinthis
report.Testwere randomlygenerated.
TheartiledenesPTPbyremoving ertainonstraintsfromPCTSPandallowing theempty
solution. Asimple heuristi is dened to solve PTP.It isalso disussed how thePTPan be
polynomialy redued to Asymmetri TSP on a large diagraph. Three previously disovered
lowerboundsfor PCTSP are presented andalso a new lowerbound for PCTSP is put forth.
ForasymmetriPTPtwolowerboundsarepresentedbyremovingonstraints.Theartileends
with a setion on omputational experiments both for PTP and PCTSP. Were all instanes
weresolved inlessthan one minuteof CPUtime. Itwasalsoonluded, by inspeting ratios
between lowerbounds, thatsolutions to large asymmetriPTPproblems were good.
2.5.4 Hybrid algorithms with detetion of promising areas for the prize
olleting travelling salesman problem [4 ℄
This artilebyAgusto and Lorenaon PCTSP presents some ideasof lustering, using
evolu-tionarylustersearhandahybridapproahalledCS*.Thishybridapproahwasonstruted
from Greedy Randomized Adaptive Searh Proedure, or GRASP, and Variable
neighbour-hood searh. The methods are given a short desription and how they an solve PCTSP is
explained.Theseideas ouldbeuseful infurtherdevelopment of insertmovesor bus moves.
Theartile startswithanintrodution wherePCTSP isintrodued andashorthistoryofthe
problemis given.The nextsetionputsfortha mathematial modelofPCTSP,this modelis
alittledierent fromtheonein [8℄.InthethirdsetionECS,evolutionarylustersearh,and
its omponents, evolutionary algorithm, interative lustering, analyzer module and a loal
searh; are explained. Then a setion desribes how ECS is applied for PCTSP. The hybrid
approah alled CS* is then applied to PCTSP. In this setion a few interesting moves are
dened.These6movesweredierent fromthe onesusedinthisprojet.Onemove alled
m 4,
isomparablewithinsertmove13 3
.Other movesweresimilarbut oftenusedmore nodes, for
example
m 1 inserted2 nodesinsteadof one.Thelast setionison omputationalresults and show solution from ECS and CS*. The results from these two are also ompared to results
from a CPLEX 7.5 solver. In onlusion the authors nd that CS* returns better solutions
and useof thesemethods isvalidated.
2.5.5 A tabu searh algorithm for the open vehile routing problem [3℄
This artile by Brandao ontains a good introdution to OVRP and ompares it to VRP.
Mostoftheinformation inthesetionon OVRPame fromthis soure.There isalsoa short
introdution on the history of OVRP and relatively few, ompared to VRP, have studied
it. The meta-heuristi used in the artile is tabu searh. The importane of a good initial
solutionisdisussed andhowto attain suha solution, themethods usedfor thisarenearest
neighbourheuristi,orNNH,andasolutionbasedonapseudolowerbound.Thepseudolower
boundisamethodbasedonminimumostspanningtreewithdegreeksubjettorelaxations.
Initialsolutions givenwithaninsertion heuristi andalower bound wereexperimentedupon.
Beforeapplyingthetabu searh tothisinitial solutionthesolutionis submittedtoone oftwo
methods: nearest neighbour or unstringing and stringing method. This was done to improve
thesolution.Inthetabusearhswapandinsertmovesareused.Thegoalofthealgorithmwas
3
to minimize the number of routes and therefore new routes ould not be reated. A method
was inluded that tried to join the two routes with the lowest demand. This is lever and
ould be implemented to the algorithm used in the report in the future. In onlusion it is
statedthatthe algorithmgavegoodsolutions foraveryshortomputingtime,outperforming
former algorithms suhas theone proposedbySariklis and Powell. For example themethod
of using psuedo lower bound gave an average travel time of 416.1 while Sariklis and Powell
algorithmhadanaveragetraveltime of488.2.Thesearefromalulations with50pointdata
setsand thedierane inrunning timeswas 88.6seonds, Sariklisand Powell methodsolved
theproblemin0.22seonds.
2.5.6 OpenVehileRoutingProblemwithTimeDeadlines:Solutions
Meth-ods and Appliation [17℄
Thisartile,byAksen,ArasandÖzyurt;fousedontheOVRPwithtimedeadlines,or
OVRP-TD.Clarke-Wrightparallel savingalgorithmmodiedforOVRPwasimplementedalongwith
greedy nearest neighbour algorithm and a tabu searh heuristi. The artile also ontains a
short desription for most of these methods. The artile explained how Clark-Wright, CW,
ismodied for OVRP-TD, mostly bysetting ertain distanes to innity. ThenCW and the
nearestneighbour algorithm were usedto nd an initial solution. There neighbourhood
on-sistedofthreemoves,whihwere1-0move,1-1exhange and2-Optmove.Thesethreemoves
are the same as the swap moves desribed in this report. Loal searh with these moves is
inorporatedintoTSasatoolofloalpostoptimization,LPO.Thehapteronomputational
resultssolvingverandomresultsandonerealproblem,ashoolbusprobleminIstanbul.In
onlusionitwasapparentthatCW initial solutionperformedbetterthan lassialheuristis
withLPO.Overallthis isa very shortartile thatdoesnot go muh into details.
2.5.7 A general heuristi for vehile routing problems [11 ℄
Thisartile,byPisinger andRopke,is alarge,extensiveand takeson variousvehile routing
problems.VRPwithtime windows,apaitated VRP,multi-depot VRP,site dependant VRP
and OVRP are all disussed and solved by transforming eah instane into a single type
of model. The model is alled Rih Pik up and Delivery Problem with Time windows, or
RPDPTW. There is a mathematial presentation of this model that is a little onfusing, on
aountofthe numberofsets involved.Allthe modelsRPDPTWsolvesareVRP modelsand
therfore have to visit all nodespresentedinthesystem, whih means theRPDPTW annot
be appplied to the bus route problem. Next there is a setion on how one transforms these
vedierent VRPproblems into aRPDPTW. Thisartile andthemodelpresentedaregood
reading materialwhen presentedwith aproblem asdisussed in this report. The artile also
explains dierent objetives of its model. The rst objetive is to minimize the number of
vehiles while the seond objetive is to minimize the travel distane. This is in aordane
withtheproblem presentedin this reportwhere therst objetive is to visit asmany nodes
aspossible,withgiven travel onstraints, whileusingasfewbuses aspossible andtheseond
objetive is to minimize the travel distane/time. The heuristi used to solve RPDPTW is
adaptive large neighbourhood searh, ALNS, a method that uses two, a onstrutive and a
destrutive, neighbourhoods to nd an optimal solution. It is explained how one applies the
itisstatedthattheALNSshouldbeonsideredasoneofthestandard frameworksfor solving
large-sizedoptimization problems,asthemethod isvery general andgave good results.
2.5.8 Open vehile routing problem with driver nodes and time
dead-lines [16℄
Thisartile looksat apartiular variant oftheOVRPwhere thevehiles,routes,start atthe
depot andvisitanumberofnodesbutallroutes arerequiredtoend atertaintypesof nodes
alleddrivernodes,thisproblemalsohastimedeadlinesthathavetobekept.Amathematial
modelispresentedforthispartiular typeofproblem. Theproblemisquitedierent fromthe
one presented inthis reportbut as with artiles on similar subjets it iswortha lookto get
abetterunderstanding on OVRP.
The introdution setion in this artile, by Aksen, Aras and Özyurt, ontains an exellent
historial overview of OVRP. Instrumental artiles and methods used are mentioned. The
authors also state that they know of no other artile where a similar problem, OVRP using
drivernodes,istakled.To solvethe problemanewheuristi alledopentabu searhisused.
Itmakesuseofthree move operatorsingeneratingthe solutions intheneighbourhood ofthe
urrent solution. These moves are the same as dened in [17℄. The initial solution is found
withanearestinsertion heuristi and aClark-Wright parallelsaving algorithm. Theproblem
alledOVRP-dismathematially presentedasamixedintegerproblemintheseondsetion.
Thisislearlypresentedandnotompliated.Thenextsetionisonthetabusearhalgorithm
previouslydesribed.Theforthsetionisonomputationalresultswheretheopentabusearh,
OTS,isompared tovarious lassialheuristis. Theninonlusionitis determinedthatthe
newheuristi, OTS,giveshigher qualitysolutions thenthelassial heuristis.
2.5.9 A TABU Searh Heuristi for the Team Orienteering Problem [13 ℄
Thisartile,byTandand Miller-Hooks,ontheteamorienteeringproblemwasveryuseful for
theprojet. The team orienteering problem, TOP, is very similar to the model presentedin
thisreport.Also theauthors supplieddatasoomparison tests,betweentheir resultsandthe
algorithm inthisprojet, ouldbe performed.
The artile starts out with a good introdution to TOP. The onnetion between TOP and
several other problems is disussed. Also the method that have be inspeted when solving
TOP are listed, simulated annealing is not one of them. The next setion puts forth the
mathematial model ina very straight forward manner. The artile explains how theinitial
solutionisalulated witha method knownasadaptive memory proedure, AMP.This isan
exellent method for alulating an initial solution, although might in some ases be
prob-lemati if the best solution is using no routes 4
.Interestingly thetabu searh algorithm uses
intermediate infeasiblesolutions to aidinthe searh proess,bymoving solutionsout ofloal
optimums. Other methods like small and large neighbourhood searh and methods used for
tourimprovement arealsodisussed.Thesetiononomputational resultsshows omparison
between TABUsearh,5-stepheuristi anda versionof theTsiligiridesheuristi extendedfor
TOP by Chao. In onlusion it is noted that AMP and its mehanism, alternating between
smallandlargeneighbourhoodsstagesandusingbothrandominsertionandgreedyproedures
led to aneetive tabu searhalgorithm.
4
Simulated Annealing for the BRP