tsp in excel.pdf

7/25/2019 TSP IN EXCEL.pdf

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TSPinSpreadsheetsaGuided

Tour

RasmusRasmussen

Abstract

Thetravellingsalesmanproblem(TSP)isawellknownbusinessproblem,andvariantslikethemaximum

benefitTSPorthepricecollectingTSPmayhavenumerouseconomicapplications.Wearelookingatseveral

differentvariantsofTSP;allsolvedinspreadsheets,notusingtailoredsolversforTSP.Astheseproblemsare

NPhard,solvingthoseusingstandardLP/MIPsolvershasbeenregardedfeasibleonlyforverysmallsized

problems.However,acarefulconsiderationofthespreadsheetlayoutmayfacilitateefficientsoftware

utilisation.Forrealworldproblemsthiscanhaveconsiderableeffects,andwiththerecentadvancementsin

solverengines,problemspreviouslyregardedasbigarenoweasilysolvableinspreadsheets.Thispaper

showsyouhow;andhowtheflexibilityofspreadsheetsmakesitaconvenienttoolsolvingmanyvariantsof

TSP,wheretailoredsolverssimplywouldnotfit.

JELclassification: C61,Z00

1. Introduction

Afteraformalstatementoftheproblem,threedifferentspreadsheetmodelswillbeillustrated.The

flexibilityofspreadsheetswillalsobedemonstrated,aswillhowspreadsheetlayoutmayhelpinmaking

anefficientproblemformulation,inadditiontohelpingtoclearlycommunicateanddisplaythe

solution.Thedirectpermutationapproachispresentedfirst,applyingintegervariablestodescribethe

sequenceofthevisits.Thedirectpermutationapproachfitssmallproblemswell,andrequiresverylittle

workafterdatahasbeenobtained.Noconstraintstoeliminatesubtoursareneeded,buttheproblemis

nonlinearandnonsmooth,requiringheuristicsolvers.Second,anetworkformulationispresented,

wherebinaryvariablesareusedtomakealinearformulationoftheproblem.Anefficientspreadsheet

layoutispresentedfornoncompletegraphs.Thirdlyanassignmentformulationispresented,applying

aspreadsheetlayoutmoresuitableforcompletegraphs.

VariantsofTSPnotfittingtailoredTSPsoftwarearealsosolved.Inadditionsomeconfiningsideeffects

ofcommonsubtoureliminatingconstraintsarediscussed,particularlywhenmultiplevisitsarerequired.

2. ThestandardTSP

Travellingsalesmanproblems(TSP)areeasytodescribe:asalesmanneedstovisitallhiscustomers

locatedindifferentcitiesinhisregion,andhewouldliketofindthecheapesttourthatwillassurethat

allcitieshavebeenvisited.UnfortunatelyTSPisnotsoeasytoformulate,andrelativelyhardtosolve.

Whenmakingamathematicalformulationoftheseproblemswewillforthemostpartuseanetwork

framework.Thecitiesarethencallednodes,andtheroadsconnectingthecitiesarecalledarcs.See

GutinandPunnen(2007)forafulltreatmentofTSPanditsvariants.

ThesetofnodestobevisitedaredefinedasN={1,2,...,n}wherenisthetotalnumberofnodes

(referredtoasthesizeofaTSP),andthesetofarcsconnectingthenodesisdefinedasA={(i,j):i,jN,


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ij},wherethepair(i,j)indicatesthearcbetweennodeiandj.AstandardassumptioninTSPisto

assumedirectlinksbetweeneverypairofnodes,usuallyreferredtoasacompletegraph.Thegraph

consistingofthenodesNandarcsAisthenconnected;thereisaconnectionorpathfromanynodeto

anyothernodeinthegraph.Thebasicstandardassumptionistorestrictthenumberofvisitstoexactly

oneforeachnode.Whythesalesmanisnotallowedtovisitanodemorethanonceisnotobvious.Onecanspeculatethatsucharequirementmakesiteasiertodevelopsolutionprocedures,therebyfitting

theproblemtothetoolsathand.AcommondefinitionofthesetofdecisionvariablesisX{xij:i,jN,

ij}wherexij=1ifthesalesmantravelsfromnodeitoj(nodeiisvisitedimmediatelybeforenodej),

and0otherwise.ThecostmatrixisdefinedasC={cij:i,jN,i j}andusuallyassumedtobepositive,

wherecijrepresentsthecostoftraversingfromnodeitonodej.InstandardTSPacommonassumption

isthatthesquarecostmatrixissymmetric,cij=cji,thecostisthesameinbothdirections.Another

standardassumptionistoassumethetriangleinequality;cij+cjkciki,j,kN,thedirectconnection

betweentwonodesisalwaysthecheapest.

OnebasicassumptioninTSPistoassumethatthesalesmanhastoreturntothenodewherehestarts

thetour;thisnodeisusuallyreferredtoasthebasecityordepot.Thisassumptioniscalledaclosed

tour.Foraclosedtouranynodecanbeselectedasthestartingnode,butforpracticalreasonsnode1is

settobethestartingnode.Node1isthenthebasecityordepot.

ForastandardTSPthereisalwaysafeasiblesolution(asacompletegraphisalwaysconnected),andwe

canchooseanynodetostart(asthetourisclosedandallnodesarevisited).Therearealways

alternativeoptimalsolutions;thetourcangoineitherdirection(asthecostsaresymmetric).Andinthe

optimaltour(s)everynodeisvisitedonlyonce(becauseofthetriangleinequality,andtheobjectiveis

alwaysminimisation).

3. VariantsofTSP

Quite

a

lot

of

real

life

problems

do

not

fit

these

assumptions.

Often

we

must

allow

for

the

set

A

not

beingcomplete,incaseswheresomenodesdonothavedirectlinkstoallothernodes.Graphsthatare

notcompletearenolongerguaranteedtobeconnected,andfordisconnectedgraphsthereisno

feasiblesolution.Inreallifewealsohavetoallowforcijcji,thecostoftravellingfromnodeitojmay

notbethesameastravellingfromjtonodei.Thisrepresentstheasymmetrictravellingsalesman

problem(ATSP),andimpliesdirectedarcs.Similarlyitisnotalwayscheapesttotravelthedirectlink

fromnodeitonodek,sometimesitmaybecheapertotravelvianodej.Thuswemustallowforthe

triangleinequalitynottoapply.Thebasicstandardassumptiontorestrictthevisitstoexactlyonefor

eachnodemayalsobeskipped;TSPwithmultiplevisitsisreferredtoasTSPM,asinGutinandPunnen

(2007).

Ofcoursethereasonforthesalesmantomakethetouristoderivesomebenefitfromvisitingthe

nodes.ThenletB={bj:jN},wherebjisthebenefitfromvisitingnodej.Forsuchproblemswehave

themaximumbenefittravellingsalesmanproblem(MBTSP);seeMalandrakiandDaskin(1993).Another

variantisthepricecollectingTSP(orPCTSP),seeGutin(2007).

Sometimesthesalesmandoesnothavetoreturntothebase,andrelaxingsucharequirementiscalled

anopentour.Foranopentouritmaybeadvantageoustobeabletoselecttheendingnodeaspartof

theproblemsolution,butthismayincreasetheproblemsizeforsometypesofformulations,exceptfor

thedirectpermutationapproach.

Thereisawideselectionofliteratureontheseproblems,andseveralvariantsofproblemformulations.

Wewillgrouptheformulationsintwoclasses:theassignmentformulationsandtheflowformulations.

Further,

in

each

group

the

models

vary

according

to

which

assumptions

are

made,

most

notably

whetheracompletegraphisassumed.


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4. AssignmentformulationofTSP

Fortheclosedtouranassignmentformulationcouldbeofthefollowingform:

1 1

n n

ij ij

i j

Minimize c x= =

(1)

1

1 ,n

ij

i

x j N=

= (2)

1

1 ,n

ij

j

x i N=

= (3)

{ }0,1 , ,ijx i j N (4)

Inadditionsubtoureliminationconstraints(SECs)areneeded.Constraints(2)and(3)arethestandard

assignmentconstraints.Theobjectivein(1)willminimisethetotalcostalongallthearcsusedto

completethetour.However,aswrittenthisformulationassumesacompletegraph,andifthedataare

beingarrangedinasquarematrixwillalsoincludethediagonal.Foracompletegraphtheonlyarcsthat

donotexistarerelatedtotheselfloopvariablexi,i(alongthediagonal).Thereforeitusuallyismore

convenienttoexcludethesevariablesbyanewconstraint(5),insteadofexcludingtheminthe

definitionofthesetX.Thisconveniencecomesatthecostofincreasedproblemsize(bothintermsof

variablesandconstraints).Foracompletegraphthefollowingconstraintwillfixthediagonalinasquare

nmatrixofthebinaryvariablesxijequaltozero:

, 0i ix i N=

(5)

Adifferentapproachtorectifythis,andallowforinstancesofnoncompletegraphs;istosetthecostcij

sufficientlylargefornonexistingarcs,therebypreventingthemfromenteringthefinalsolution.

Howeverthisisnotafoolprooftrick.Inaconnectedgraphthereisapathfromanynodetoanyother

nodeinthegraph,andacompletegraphisalwaysconnected,andthushasafeasiblesolution.Non

completegraphsmaynotbeconnected(disconnected),andwillassuchhavenofeasiblesolution.A

highcostfornonexistingarcsisthennoguaranteeforthesesarcstobeexcludedinthefinalsolution.

Thereforeanotherstrategyistosetanewparameter:eij=1ifnodeiisdirectlyconnectedtonodej,

otherwise0;andreplaceconstraint(5)with(6):

, ,ij ijx e i j N (6)

Thisformulationdoesnotrequireacompletegraphandallowsforasymmetriccostsandalsoforthe

triangleinequalitynottoapply,butunfortunatelyithassomeflaws.Iftheassumptionofacomplete

graphisnotsatisfied(andthereforeconstraint(6)isrequired),thenafeasiblesolutionforaclosedtour

mayrequiresomenodestobevisitedtwice,breakingthe=requirementinconstraint(2).

Thelimitationofvisitingeachnodeexactlyoncemayalsocausedifficultiesevenforproblemswitha

completegraph,ifthetriangleinequalityisnotsatisfied.Thisrequirementwilleffectivelyprohibithub

likesolutions,evenwhensuchsolutionsarethemostcosteffective.Aproblemformulationthat

excludessuchpossibleoptimalsolutionsisgenerallynotrecommended.


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5. FlowformulationofTSP

Aflowformulationoftheclosedtour,thatexplicitlyconsidersvalidconnectionsonly,canbemadeafter

redefiningC={ci,j:(i,j)A}andX{xi,j:(i,j)A}.Thisformulationwillthusworkevenwhenthe

graph

is

not

complete:

, ,

( , )

i j i ji j A

Minimize c x

(7)

,

: ( , )

1i ji i j A

x j N

(8)

, ,

: ( , ) : ( , )

i k k ji i k A j k j A

x x k N

= (9)

{ } ( ), 0,1 ,i jx i j A (10)

OfcourseSECsarealsorequired.Theobjective(7)willminimisethetotalcostofthetour,only

consideringvalidarcs.Constraint(8)statesthatthesalesmanhastoarriveeachnodeatleastonce.

Constraint(9)statesthatthesalesmanhastoleaveeachnodeasmanytimesashearrivethenode.By

usinginsteadof=in(8),weavoidthepossibilityofmakingtheprobleminfeasiblefornoncomplete

graphswheresomenodesneedtobevisitedtwice,andwedonotexcludehublikeoptimalsolutions

ifthetriangleinequalitydoesnotapply.

6. FlowformulationofopentourTSP

FortheopentourformulationweaddtheparametersD={di:iN}wherediisthenetdemandin

nodei;anddi=1forthestartnode(thebasecityisnumberednode1);di=+1fortheendnode,forthetransitorintermediatenodesdi=0.Theopentourformulationcanthenbestatedas:

, ,

( , )

i j i ji j A

Minimize c x

(11)

,

: ( , )

1 1i ji i j A

x j N

> (12)

, ,

: ( , ) : ( , )

i k k j k i i k A j k j A

x x d k N

=

(13)

{ } ( ), 0,1 ,i jx i j A (14)

WemustaddSECstocompletetheTSPformulation.Theobjectivein(11)isidenticalto(7).Constraint

(12)issimilarto(8),exceptthatwedonotrequirethesalesmantoarrive/returntothestartingnode1.

Constraints(13)requirethesalesmantoleavethestartingnodeonemoretimethanentering,enterthe

stoppingnodeonemoretimethanleaving,andleaveanyintermediatenodeasoftenasarrivingthe

node.Byremovingconstraint(12)wehavethecommonshortestpathproblem.Iftheendnodeisnot

specified,thediparametersmaybeconvertedtobinaryvariables(exceptforthestartnode),requiring

theirsumtoequal1.


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7. Subtoureliminatingconstraints(SECs)

AkeypartofaTSPistomakesurethetouriscontinuous,thatthearcsarelinkedfromthebasecityall

thewaytoeverycityvisited.Withoutsuchconstraintswequiteoftenwillgetsolutionscontaining

degenerate

tours

between

intermediate

nodes

and

not

connected

to

the

base

city.

The

originally

SECs

wasformedin1954byDantzigFulkersonJohnson(DFJ)(seeDantzig, FulkersonandJohnson,1954):

, 1, \{1},i ji S j S

x S S N S

(15)

Unfortunatelythisintroducesanexponentialnumberofconstraints,andbecomesimpracticalevenfor

smallsizedproblems.AdifferentSECproposedin1960byMillerTuckerZemlin(MTZ)(seeMiller,

TuckerandZemlin,1960)introducesonlyamaximum1of(n2)

2constraints,atthedisadvantageofa

weakLPrelaxation:

( )( ) ( ),1 1 1 , , : , 1

i j i j

u u n x i j A i j +

(16)

In(16)anewsetofvariablesU={ui:iN,i1}isrequired.Theuiarearbitraryrealnumbers,butcan

berankedtononnegativeintegers,representingthesequencethenodesarebeingvisited.For

conveniencewemayaddu11(node1isthebasecity),andlimittherangeofui,thushelpingthe

optimisationsoftware(seealsoPataki(2003)):

2 1iu n i N > (17)

TheMTZSECswillbeusedinthispaper,andhavethefollowingproperties:

node1isrequiredtobethebasecity;

theymakesurethateverycityvisitedbelongstoatourconnectedtothebasecity,thereby

eliminatingsubtours;

theyallownodestobevisitedmorethanonce(unlessotherconstraintspreventsuchasolution);

theydonotrequireallnodesbeingvisited(unlessotherconstraintsmakesuchrequirements);

theyallowunidirectionalarcstobeutilisedinbothdirectionsonthesametour.

Foraclosedtourvisitingallnodesthebasenodecanalwaysbechosenarbitrarily.Afundamental

weaknessofMTZSECsisthatfeasibilityandfinalsolutionmaydependonwhichnodeisselectedasthe

basecity.TheMTZSECsmayfailtofindafeasiblesolutionevenifsuchexists,andtheymayfailtofind

theglobaloptimalsolution.Problemswithfeasibilitymayoccurinnoncompletegraphs,whereall

feasiblesolutionsrequiresomenodestobevisitedtwice.Problemsfindingtheglobaloptimalsolution

mayoccurincompletegraphswherethetriangleinequalitydoesnotapply,andwheretheglobal

optimalsolutionrequiressomenodestobevisitedmorethanonce.Itisthereforeimportanttobe

awareofthesetwosituationswhereapplyingtheMTZSECsmaymakethefinalsolutionsensitiveasto

whichnodeisselectedasthestartingnode.Theywillneverfailiftheglobaloptimalsolutionvisitseach

nodeonlyonce.

8. ClosedTSPinacompletegraph

AsanexampleofaTSPinacompletegraphweshallusethefollowingexample.Asupplyshipisserving

10oilrigsatsea.Thebaseislocatedatcoordinates(0,0),andtherigsarelocatedasdisplayedinFigure

1Inasquarennmatrix;thefirstrow,firstcolumnandthediagonalareexcluded.


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1.Assumingopenseathedistancesbetweenanypairofnodes(oilrigs)canbecalculatedasstraight

lines(ignoringthefactthatthesealevelisnotflat).ThisisastandardsymmetricTSPwithacomplete

graphwherethetriangleinequalityapplies.DataistakenfromRagsdale(2001).

Figure1Locationsoftheoilrigstovisit

9. Adirectpermutationapproach

Inthissimpleformtheproblemistofindtheorderforeachnodeinthesequenceofthetourthat

minimisesthetotaldistance(cost).Ifthesupplyshiptakesthetourbasedontherignumbers:012...

9100;thetotaldistanceis205.67.Weseektheorderorpermutationthatminimisesthetotal

distance.Thisdirectapproachisveryeasytoimplementinspreadsheets,asdisplayedinFigure2.

Figure2SpreadsheetfordirectpermutationofTSP


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Table1:FormulasforspreadsheetinFigure2

Figure3SolversettingsforthespreadsheetinFigure2

Thespreadsheetisorganisedintwoparts.Theupperpartholdsatableofthecoordinatesforthe

nodes,andacorrespondingtablecalculatingthedistances.Thelowerpartholdsatableofthetour

sequenceandthecostofeachleg,andacorrespondingtablewiththecoordinatesofeachleg,to

facilitateaplotofthetour.Thetableofthetoursequencestartsatthebase.Notethatnodenumber0

isusedforthedepotinthisexample,tofacilitateuseoftheAlldifferentconstraintinSolver.(Atrial

versionofSolverisavailableatwww.solver.com.)Theproblemistoselectwhichnodetogotonextin

thesequence(headingSequenceinFigure2).Thelastleghastoreturntothebase.Theminimum

totaldistance/costof122.77isachievedbythetoursequence0946572810130(orreverse).To

modelanopentoursimplydeleterow27inthesheet.Figure4displaystheoptimalopentour,which

hasacostof103.58.

Thescatterplotconsistsoftwoseries.Oneseriesisaplotofthenodes(C3:D13inFigure2),withmarkersbutnoline.Thesecondseriesisthetour(G13:H27inFigure2),withnomarkersandaline.

Cell Formula Copiedto Name/Task

G3

=SQRT((INDEX($C$3:$D$13;$F3+1;1)

INDEX($C$3:$D$13;G$2+1;1))^2

+(INDEX($C$3:$D$13;$F3+1;2)

INDEX($C$3:$D$13;G$2+1;2))^2)

G3:Q13CalculateEuclediandistances

betweenanypairofnodes

D17 =INDEX($G$3:$Q$13;C16+1;C17+1) D18:D27 Costonaleg

D28 =SUM(D17:D27) Totalcost

C17:C26 Sequence

G16 =INDEX($B$3:$D$13;$C16+1;2) G17:G27 Avisitednodesxcoordinate

H16 =INDEX($B$3:$D$13;$F16+1;3) H17:H27 Avisitednodesycoordinate


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Figure4TheopenTSPsolution

BoththespreadsheetandtheSolversettingsareverysimple.Wehave10decisionvariables(numberof

nodeslessthedepot),namedSequenceinthespreadsheet.TheobjectiveintheSolversettingsistominimisethevalueinthecellnamedTotal_cost,andtheonlyconstraintisthatthevariablesmustbe

alldifferent.Thealldifferentconstraintsetsthevariablestointegersrangingfrom1tothenumberof

variables,andallaredifferent.(ThistypeofconstraintisnotavailableintheStandardSolverthatships

withExcelpriortoExcel2010,butisintroducedintheeducationalversionofSolver,includedinmany

textbooks.)

TheuseoftheIndexfunctioninExceltolookupthecostateachlegmakestheobjectivefunctionnon

smooth,becausethedecisionvariablesareusedasargumentsintheIndexfunction.Anintegernon

smoothproblemisnoteasytosolve,andisdefinitelynotthepreferredformforlargeproblems.Inthis

casethePremiumSolverPlatform(PSP)selectstheOptQuestsolverengine,andSolverspendslessthan

twosecondsinfindingtheoptimaltour(theAutoStopoptionforOptQuestwasincreasedfrom100to

1000iterationstoavoidaprematureending).Asthissolverengineappliesheuristics,itcannot

guaranteethataglobaloptimalsolutionhasbeenfound.Whensuchproblemsbecomelarge,thisnon

linearapproachisnolongerefficient.Wewillthereforeintroducethelinearformulation,whichwillbe

appliedintherestofthepaper.Alsonotethatthedirectpermutationapproachdoesnotallowfor

multiplevisits.

10.TSPinanoncompletegraph,flowformulation

AsanintroductoryexampleforanoncompletegraphtheGridspeedpuzzlewillbeused,takenfrom

Chlond(2008).Figure5presentsthepuzzle,basedonarectangulargridstreetplan,wherethedistance

betweenanytwointersectionsis10kilometres.(Ihavetakenthelibertytotransformthedatatothe

metricsystem.)Thespeedalongallnorthsouthstreetsandalleastwestavenuesisconstant.However

thespeedonthenorthsouthstreetsishighestontheeastendofthegrid,andfortheavenueseast

westthespeedishighestinthesouthendofthegrid.Thefastestareaisthereforeatthesoutheast

edgesofthegrid,andslowestinnorthwest.


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Figure5Streetplan

OnepuzzlerelatedtoFigure5istofindthefastestroutefromintersection(6,1)(northwest)to

intersection(1,1)(southwest),butvisitingeachintersectionatleastonce.Theoriginalproblemisto

visiteachintersectiononceandonlyonce.Howeverthisismorerestrictedthanrequired.Sinceit

obviouslywilltakemoretimetovisitanintersectionmorethanonce,andwewanttospendasshort

timeaspossibleonthetour,itissufficienttousetherequirementtovisiteachintersectionatleast

once.

Itisnecessarytotransformtheproblembynumberingtheintersectionsandcalculatethetravelling

timebetweeneach(directlyconnectednode),tofacilitateamathematicalformulation.Thenumbered

intersectionsarethenodes,andthelinesconnectingthenodesarethearcs.Thetravellingtime(in

minutes)alongeacharciscalculatedasshowninFigure6.

Figure6Relabelledstreetplan

10 km/h

20 km/h

30 km/h

40 km/h

50 km/h

60 km/h10 km

10 km/h 20 km/h 30 km/h 40 km/h 50 km/h 60 km/h(6, 6)(6, 1)

(1, 1) (1, 6)

60 60 60 60 60

60

60

60

60

60

30

30

30

30

30

30 30 30 30 30

20

20

20

20

20

20 20 20 20 20

15

15

15

15

15

15 15 15 15 15

12

12

12

12

12

12 12 12 12 12

10

10

10

10

10

10 10 10 10 10

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 27 28 29 30

31 32 33 34 35 36


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Figure7SpreadsheetofopenTSP,noncompletegraph(rows66121arehidden)


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Table2FormulasforthespreadsheetinFigure7

11.Theopentour

Wewillfirsthavealookattheopentourvariantofthepuzzle.Anefficientlayoutofthisnetworkina

spreadsheetwouldbetoorganisetheproblemintwotables,onetableforthearcsandthebinary

decisionvariables,andanothertableforthenodesandthecontinuousvariables(asinRagsdale,2001).

Thiswillfacilitatetheentryoftheequations(11)(14),(16)and(17),andalsomakeasolutioneasyto

understand.Oncethedatahasbeenenteredinthespreadsheet,themodelcaneasilybebuiltaround

thedata.Noticethatfornondirectedarcsitissufficienttoentertheminonedirection,andusesimple

formulastomirrortheotherdirection.Athirdtablehasbeenaddedtothespreadsheettofacilitatea

plotofthetour,whichofcourseisnotneededforsolvingtheproblem,buthandyfordisplayingthe

solution.

Anewconstrainthasbeenadded,tospeedupthesolutionprocess:

( ), , 1 ,i j j ix x i j A+ (18)

Constraint(18)simplystatesthatnoarcwillbeusedinbothdirections,whichisquiteobviousforthis

problem.Suchboundsonthevariablesareveryhelpfulfortheoptimisationprocess,particularlysofor

binaryvariables.Howevertheyshouldbeusedwithcare,sinceaddingthemcanmakesomeproblems

infeasible.

InFigure7thetableforthearcsislistedfirst;thenthetableforthenodesandfinallythetablefor

facilitatingaplotofthetour.Thefirsthalfofthearcsarelisted(allarcsinonedirection)andafewof

therest,togetherwiththeobjective.(Therestofthearcsareinthehiddenrows66121.)Weseethat


B63 =C3 B64:B122 Reversearcs:stopstart

C63 =B3 C64:C122 Reversearcs:startstop

D63 =D3 D64:D122 Reversearcs:copycosts

D123 =SUMPRODUCT(D3:D122;E3:E122) Eq11

F63 =E3+E63 F64:F122 Eq18(LHS)

G3=IF(OR(B3=1;C3=1);0;INDEX($J$3:$J$38;B3)

INDEX($J$3:$J$38;C3) +($I$37*E3))G4:G122 Eq16(LHS)

K4 =SUMIF($C$3:$C$122;$I$3:$I$38;$E$3:$E$122) K5:K38 Eq12(LHS)

L3 =SUMIF($B$3:$B$122;$I$3:$I$38;$E$3:$E$122) L4:L38 FirstpartofEq13

M3 =K3L3 M4:M38 Eq13(LHS)

E3:E122 Var_x

I36 Param_n_2

I38 Param_n

J4:J38 Var_u

N3:N38 Param_d

P3 =RANK(J3;$J$3:$J$38;1) P4:P38 Therankofanode

Q3 =MATCH(U3;$P$3:$P$38;0) Q4:Q38 Visitingsequence

R3 =INDEX($U$3:$W$38;$Q3;2) R4:R38 Avisitednodesxcoordinate

S3 =INDEX($U$3:$W$38;$Q3;3) S4:S38 Avisitednodesycoordinate


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theoptimalvalueoftheobjective(11)is726;thefastestopentourfromnode1tonode31takesa

minimumof726minutes.TheformulasinthespreadsheetaredisplayedinTable2,andtheSolver

settingsarelistedinFigure8.

ThefirstthreerowsinTable2arepurelyforeasydataentry.Theoptimisationmodelconsistsofthe

nextsixrows.Thefollowingfiverowsareusedfornamingsomekeycells,makingthemodeleasierto

read.Thelastfourrowsfacilitateaplotofthesolution,assumingonlynlegsinthetour(eachnodeis

visitedonlyonce).ThescatterplotinFigure7consistsoftwodataseries.Oneseriesisthexy

coordinatesofthenodes(incolumnVandW),withnoline,andacircle(size20)asmarker.2The

secondseriesisthexycoordinatesofthetour(incolumnRandS),withnomarkerandaline.Fora

closedtourafinallegisaddedattheend(byreferringtothefirstlegincolumnQ).

Figure8SolversettingsforFigure7

TheStandardSolverParameterDialogBoxdisplayedinFigure8hasascrollbartodisplaythe

constraintsnotfittingthefixedsizeofthebox.Hereconstraint(14)isnotdisplayed;thisisthedeclarationofthexijvariablebeingbinary.Observethatconstraint(14)and(17)isentereddirectlyin

Solver,involvingnoformulasinthespreadsheet.Constraints(17)arethelasttwovisibleconstraintsin

theSolverParameterDialogBox.

Thismodelhas120binaryvariablesand35continuousvariables,155boundsonthevariablesand191

constraints.Thenumberofconstraintscanbereducedby4ifwegroupthefourarcsconnectedtonode

1andlistthemfirst,thennotincludetheminconstraints(16).Inthespreadsheettheformulafor(16)

includethesearcs,butfixtheirvalueto0,therebysatisfyingtheconstraint.

ExcelandtheStandardSolvertakelessthanfivesecondstofindtheoptimalsolutionfortheopentour.

(ThesolutiontimewillofcoursedependontheversionofExcel,theoperatingsystem,andthe

computer.)Thisspreadsheetdesignisquiteversatileformanytypesofnetworkproblems.Ifwedrop

(12),(16),(17)and(18)wehavetheshortestpathproblem.(WemaythendeletecolumnF,G,JandP

W.)

12.AclosedTSPinanoncompletegraph,assignmentformulation

Wewillnowrephrasetheproblemtoaclosedtour,requiringthesalesmantoreturntothebase.We

willimplementtheassignmentformulationandcompareitwithaflowformulation(notshown).We

willalsodemonstrateanefficientlayoutforthespreadsheetofaTSPinacompletegraph,eventhough

thisparticularexampleisnoncomplete.ForaTSPinacompletegraph,itismoreefficienttogroupthe

probleminthreetables;onetableforthecostmatrixandtheobjective(1),asecondtableforthexij

2ThelabelsofthescatterplotweremadebytheXYChartLabeleraddinforExcel,freeatwww.appspro.com.


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106

binarydecisionvariablesandconstraint(2)and(3),andathirdtablefortheSECsandtherelatedui

variables.Forconvenienceatableofthecoordinatesofthenodescanbeaddedtofacilitateaplotof

thetour.Forlargeproblemsthesematrixesmaybeenteredindifferentsheetsintheworkbook.(Itis

moreeffectivethoughforSolvertohavetheobjective,constraintsandvariablesinonesheet.Thecost

matrixandtheplotdatacanbestoredinaseparatesheet.)

Amatrixlayoutofthecostsisveryefficientforacompletegraph,whentherearedirectlinksbetween

anynodetoeveryothernode.However,itisalsoverycommontousethesameapproachfornon

completegraphs,maybebecauseitisconsideredhandierfordataentry,atleastinthepreferred

softwaretoolsmostcommonlyused.

Unfortunatelythisconveniencehasatradeoff.Addingenormousamountofnonexistingvariablesand

correctingthisbyaddinganequalamountofnonexistingconstraints,makesasubstantialburdenon

thesoftware.Incontrast,enteringthedatainaspreadsheetmayactuallybeeasierfornoncomplete

graphsituations,usingonlytwotablesinsteadofthree.

Figure9MatrixofcostsforTSPnoncompletegraph,assignmentformulation


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107

Figure10MatrixofbinaryvariablesforTSPnoncompletegraph,assignmentformulation


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108

Figure11MatrixofSECsandplotdataforTSPnoncompletegraph,assignmentformulation

Figure12TheplotoftheclosedTSPinanoncompletegraph,assignmentformulation


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109

Figure13SolversettingsforFigures9to11

Table3:

FormulasforspreadsheetinFigures9to11


AM39 =SUMPRODUCT(C3:AL38;C42:AL77) Eq1

AM42 =SUM(C42:AL42) AM43:AM77 Eq3(LHS)

C78 =SUM(C42:C77) D78:AL78 Eq2(LHS)

D83

=INDEX($AM$82:$AM$117;$B83)

INDEX($AM$82:$AM$117;D$81)

+INDEX($C$42:$AL$77;$B83;D$81)*$B$37

D83:AL117 Eq15(LHS)

C3:AL38 Eq6(RHS)

C42:AL77 Var_x

B36 Param_n_2

B38 Param_n

AM83:AM117 Var_u

AO82 =RANK(AM82;$AM$82:$AM$117;1) AO83:AO117 Therankofanode

AP82 =MATCH(AT82;$AO$82:$AO$117;0) AP83:AP117 Visitingsequence

AP118 =AP82 Lastleg,returntobase

AQ82 =INDEX($AT$82:$AV$117;$AP82;2) AQ83:AQ118Avisitednodesx

coordinate

AR82 =INDEX($AT$82:$AV$117;$AP82;3) AR83:AR118 Avisitednodesycoordinate


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110

DuetoitssizethisformulationcannotbesolvedbytheStandardSolverinExcel,whichhasalimitof

200variables.WeseefromFigure9thattheminimumtimetocompleteaclosedtouris834,thevalue

fortheobjective(1).

InTable4themodelinFigure7,aclosednetworkversion(notshown),andtheclosedassignment

versioninFigures9to11arecompared.Thesolutiontimefortheassignmentformulationoftheclosed

tourismorethan6.2timesthesolutiontimefortheflowformulation(usingtheGurobiSolverEnginein

RSPV9.04forExcel).

Table4:Keyfeaturesofthemodels

Lessonslearnedfromthissmallexamplearethatformulationmatters.Avoidusingnonexisting

variablesrectifiedbynonexistingconstraints.However,addingconstraintsmayhaveagreatimpact,

evenwhennotchangingtheoptimalsolution.Constraintsthataretighteningthefeasiblespacemay

speedupthesolutiontime(orthecontrary),butsuchconstraintsmustnoteliminateanotherwise

optimalsolution.

13.AclosedMBTSPinanoncompletegraph,flowformulation

Theultimateminimumcostforanyproblemiszero;simplydonothing.Sotheremustbeareasonfor

doingsomething(presumablythereisanullalternative).Minimisingcostscanoftenturnouttobe

solvingthewrongproblem.Unlessanyrevenuesarecompletelyunaffectedbythedecisionsathandwe

cannotbesurethatminimisingcostsisavalidmodelthataccuratelyrepresentstherelevant

characteristicsoftheproblem.

Let

us

assume

there

are

some

revenues

or

benefits

bj

by

visiting

node

j;

where

j

N.

Also

introduce

the

setY={yj:jN,j1,yj{0,1}}wherethebinarydecisionvariableyj=1ifthesalesmanvisitsnodej,

else0.Definetheparametery1=1astherequirementtovisit/returntothedepot(node1),a

consequenceofourSECs.

Theobjectivenowistomaximisethetotalnetbenefit(totalbenefitsminustotalcosts):

, ,

( , )

maximizej j i j i j

j N i j A

b y c x

(19)

Thesalesmanhastoarrive(atleast)onceeachnodehedecidestovisit:

,

:( , )

i j ji i j A

x y j N

(20)

ModelsofTSP Figure7 Notshown Figure911

Type,

model

Open,

network

Closed,

network

Closed,

assignment

Integervariables 155 155 1296Continuousvariables 35 35 35

Constraints 248 249 1297

Boundsonvariables 70 70 1331

Solutiontime(seconds) 0.44 3.84 24.01

TimeStandardSolver 5 Overnight Notsolvable


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111

Inadditionthebalanceconstraint(9)andbinaryconditions(10)apply,aswellastheSECs(16)andthe

correspondingbounds(17).Here(19)and(20)replacetheoriginalobjective(7)andconstraint(8),

whencomparedtotheclosedTSPflowformulation.

TherevisedproblemispresentedinFigure14.Therevenuesareenteredwith$symbolsundereachnodenumber,indicatingtherevenueifthenodeisbeingvisited.Thecostisassumedtobe1$foreach

minuteoftraveltime,sowehaveacommonunitofmeasurementintheobjective.Observethatnode

10andnode28havenorevenues,andarethereforeobviouscandidatesfornovisits.

Figure14RelabelledstreetplanforMBTSP

ThisisavariationofthePriceCollectingTSP.Thecommonfeatureoftheseproblemsinthiscategoryis

thecombinationoftwokindsofdecisions,theselectionofsomenodes,andtheorderingofthenodes

selectedforthetour(seeGutin,2007).

-$60 -$60 -$60 -$60 -$60

-$60

-$60

-$60

-$60

-$60

-$30

-$30

-$30

-$30

-$30

-$30 -$30 -$30 -$30 -$30

-$20

-$20

-$20

-$20

-$20

-$20 -$20 -$20 -$20 -$20

-$15

-$15

-$15

-$15

-$15

-$15 -$15 -$15 -$15 -$15

-$12

-$12

-$12

-$12

-$12

-$12 -$12 -$12 -$12 -$12

-$10

-$10

-$10

-$10

-$10

-$10 -$10 -$10 -$10 -$10

1$90

2$80

3$70

4$60

5$50

6$40

7$30

8$20

9$10

10$0

11$10

12$20

13$30

14$40

15$50

16$60

17$70

18$80

19$90

20$80

21$70

22$60

23$50

24$40

25$30 26$20 27$10 28$0 29$10 30$20

31$30

32$40

33$50

34$60

35$70

36$80


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112

Figure15ClosedMBTSPflowformulationinaspreadsheet,rows66119arehidden


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113

Table5:FormulasforspreadsheetinFigure15


B63 =C3 B64:B122 Reversearcs:stopstart

C63 =B3 C64:C122 Reversearcs:startstop

D63 =D3 D64:D122 Reversearcs:copycosts

D123 =SUMPRODUCT(D3:D122;E3:E122) Computingcosts

F3=IF(OR(B3=1;C3=1);0;INDEX($K$3:$K$38;B3)

INDEX($K$3:$K$38;C3) +$H$37*E3)F4:F122 Eq16(LHS)

I39 =SUMPRODUCT(I3:I38;J3:J38) Computingrevenues

I40 =I39D123 Eq19

L3 =SUMIF($C$3:$C$122;$H$3:$H$38;$E$3:$E$122) L4:L38 Eq20(LHS)

M3 =SUMIF($B$3:$B$122;$H$3:$H$38;$E$3:$E$122) M4:M38 Eq9(RHS)

E3:E122 Var_x

H36 Param_n_2

H38 Param_n

K4:K38 Var_u

J4:J38 Var_y

O3 =IF(J3=0;"";RANK(K3;$K$3:$K$38;1)) O4:O38 Apreliminaryrankofnodes

P3 =IF(J3=0;"";RANK(O3;$O$3:$O$38;1)) P4:P38 Rankofanodeinthetour

Q3 =MATCH(U3;$P$3:$P$38;0) Q4:Q38 Visitingsequence

Q39 =Q3 Lastleg,returntodepot

R3 =INDEX($U$3:$W$38;$Q3;2) R4:R39 Avisitednodesxcoordinate

S3 =INDEX($U$3:$W$38;$Q3;3) S4:S39 Avisitednodesycoordinate

Figure16SolversettingsforspreadsheetinFigure15

FromFigure15weseethatthesalesmanvisitsonly30ofthe36nodes,withanetprofitof$812.Notice

thatthetouractuallyincludesnode10with$0benefits,butskipnode13with$30inbenefits.Using

theStandardSolverittakesmorethanafullweekendtofindtheoptimalsolution.Anearlytestversion

ofExcel2010(TechnicalPreview)withthenewStandardSolverfinishedduringanovernightrun.Again,

anassignmentformulationapplyingafullnnmatrixofthexvariableswouldincludetoomany

variablesfortheStandardSolver.UsingthecommercialPSPV9.04,thesolutiontimeinExcelis9.06

secondsfortheflowformulation.Thisalsorevealsanalternativeoptimalsolution,visiting32nodes.

Thesolutiontimewhentheassignmentformulationisimplementedis11.45seconds,onthesame


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114

computer.Intheassignmentformulationacompletegraphisassumed,andtheextravariablesare

eliminatedbyextraconstraints,therebyincreasingthesolutiontime.

UnfortunatelytheMTZSEQsused(16)(17)assumesnode1tobethedepot.Thereforewecannot

applythistypeofSECifourgoalistofindtheoptimallocationforthedepot.

14.PitfallsusingMTZSECs

Foraclosedtourthestartingnodecanalwaysbechosenarbitrarily,ifallnodeshavetobevisited.

Unfortunately,whenapplyingtheMTZSECs,thesolutionmaydependuponwhichnodehasbeen

selectedasthestartnode.Whendoweneedtostayalert?

15.Noncompletegraphsrequiringmultiplevisits

TakealookatFigure17.Twoidenticalgraphsaredisplayed,exceptthatnode1and2havebeen

renumbered.Youcaneasilyfindtheoptimalsolutionbyvisualinspection.Tryingtosolveoneofthem

fails,

whereas

the

other

succeeds,

if

the

MTZ

SECs

are

applied

in

a

closed

TSP.

There

are

two

other

nodesthatcouldberenumberedasnumber1.Noneofthemwillsucceedifwetrytosolveusingthe

MTZSECs.

Figure17TSPinanoncompletegraph,requiringmultiplevisitsinaclosedtour

Sincethisisanoncompletegraph,aspreadsheetlayoutsimilartoFigure7isprobablymostefficient,

skippingequation(18),asitishardlyneeded.

16.Thetriangleinequalityisnotalwayssatisfied

HavealookatthecostmatrixesinTable6.Theonlydifferenceisthatoncemorethenodes1and2

havebeenrenumbered.Thisisacompletegraph,asthereisadirectlinkbetweenanynodetoevery

othernode.

3 2 1

45

4

3

2

1

1 2 3 4 5 6

32

4

4.47

3 1 2

45

4

3

2

1

1 2 3 4 5 6

32

44.47


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115

Table6:Identicalcostmatrixesofacompletegraph,thetriangleinequalitydoesnotapply

c ij 1 2 3 4 5 cij 1 2 3 4 5

1 - 2 3 4 5 1 - 5 130 140 150

2 5 - 130 140 150 2 2 - 3 4 5

3 4 123 - 40 50 3 123 4 - 40 50

4 3 124 34 - 45 4 124 3 34 - 45

5 2 135 35 54 - 5 135 2 35 54 -

Solvingoneofthemusingtheflowformulation(7)(10)andtheMTZSECs(16)(17)returnsa

solutionwithaminimalcostof209,visitingeverynodeonce.Solvingtheotherreturnsaminimumcost

of28,visitingfourofthenodesonceandvisitingonenodefourtimes.Thenumberingofthenodeshas

thusagreatimpactonwhichsolutionisobtained.Theassignmentformulationwouldreturna

minimumcostof209insteadof28.SeealsoLeeandRaffensperger(2006)forusingAMPLteachingTSP,

wheretheDFJSECsarebeingimplemented.

ForacompletegraphaspreadsheetlayoutsimilartoFigures9to11isprobablymostefficient,skipping

columnsAOARandtheplot,sincenocoordinatesaregiveninthisexample.

17.Conclusion

SolvingTSPusinggeneralpurposeoptimisationtoolslikeMIPsolversinspreadsheetshasbeen

regardedpracticalonlyforproblemsofasmallsize.Recentadvancementsinthesetypesofsoftware

haveincreasedthislimit;problemsofsize358ofnoncompletegraphshavebeensolvedinlessthan10

minutes.

SuchgeneralpurposeoptimisationtoolsalsoallowforagreatervarietyoftypesofTSP,whereas

proceduresdesignedspecificallyforTSPoftenrestricttheproblemtoalimitednumberofvariants.In

fact,wemayoverlookthebestsolutionbyapplyingthestandardapproachusingtheassignment

formulationorthesespecifictoolsforsolvingTSP.Wemustbeabsolutelysureourproblemformulation

isvalidallrelevantcostsandrevenueshavetobeconsidered,andtheconstraintsmustnotbetoo

limiting.Otherwisewemayendupsolvingthewrongproblem.Anticipatingaspecifictypeofsolution

whenformulatingtheproblemislikestartingatthewrongend,andmayleadtoapoorresult.

However,theSECsneededinaTSPformulationmayhaveunfortunateconsequencesandlimitations.

WhenusingtheMTZSECs,theselectionofthebasenodecanbecritical,evenforaclosedtourina

completegraph.Itmayalsoprohibitasolution,eveninaconnectedgraph.

Wehavefurtherseenthatformulationmattersincludingnonexistingvariablesandeliminatingthem

byaddingnonexistingconstraintscanbothincreasesolutiontimeandcauseproblemsinfindingthe

optimalsolution(thenumberofvariablesorconstraintsmayevenbecometoobigforthesolver).A

wideformulationwillalwaysincludetheoptimalsolution.Atightformulationmayhelpfindingthe

optimalsolution,butmayalsoexcludetheoptimalsolution.Atightformulationmayreduceor

increasethesolutiontime,thisdependsonthetypeofsolverusedandtheproblemathand.

Toplaysafeawideformulationseemslikeagoodstrategy.Ifthegraphisnotcompleteorthetriangle

inequalitydoesnotapply,bothimplysituationswheremultiplevisitsmayberequired,thena

replacementofthecostmatrixmaybeadvisable.Thenthecostsshouldbereplacedbyaminimumcost

matrix(youcanusetheshortestpathformulation(n1)2times),whichformanylinksmayinvolvea

lengthytourvisitingmanynodesfromnodeitonodej.TheTSPmodelcanthenbetight,whichmayor

maynotbehelpfulforsomesolvers.UnfortunatelytheTSPmodelwillthenalsobeblindeyed;ithasno


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realtrackofthetourorhowmanyvisitsareactuallybeingmadeateachnode.TheTSPsolutionthen

onlyindicatesthesequenceofthevisits,andignoresanyrevisits.

References

Chlond,M.J.(2008).Shashasgridspeedpuzzle,INFORMSTransactionsofEducation,Vol.9(1),pp.4652.

Availableonlineathttp://ite.pubs.informs.org/.

Dantzig,G.,Fulkerson,D.andJohnson,S.(1954).Solutionofalargescaletravelingsalesmanproblem,

OperationsResearch,Vol.2,pp.393410.

Gutin,G.andPunnen,A.P.(eds)(2007).Thetravelingsalesmanproblemanditsvariations, NewYork:

Springer.

LeeJ.,andJ.F.Raffensperger:(2006).UsingAMPLforteachingtheTSP,INFORMSTransactionson

Education,Vol.7(1),pp.3769.http://archive.ite.journal.informs.org/Vol7No1/LeeRaffensperger/

Malandraki,C.andDaskin,M.S.(1993).ThemaximumbenefitChinesepostmanproblemandthemaximum

benefittravelingsalesmanproblem,EuropeanJournalofOperationalResearch,Vol.65,pp.21834.

Miller,C.E.,Tucker,A..W.andZemlin,R.A.(1960).Integerprogrammingformulationoftravelingsalesman

problems,JournalofACM,Vol.7,pp.3269.

Pataki,G(2003).Teachingintegerprogrammingformulationsusingthetravelingsalesmanproblem,SIAM

Review,Vol.45(1),pp.11623.

Ragsdale,C.T.(2001).SpreadsheetModelingandDecisionAnalysis, Cincinatti:SouthWesternCollege

Publishing.

AuthorBiography

RasmusRasmussensteachingexperienceismostlyinthefieldsofbusinesseconomics,finance and

managementscience.Hisresearchinterestsareinthefieldsofappliedmanagementsciencerelatedto

problemsinbusinesseconomics,quiteoftenusingspreadsheets,thetoolfrequentlyusedalsoin

teaching.

Contactdetails

RasmusRasmussen

MoldeUniversityCollege

P.O.Box2110

6402Molde,Norway

Tel:+4771214242

Fax:+4771214100

[email protected]

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