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CIENACorporationInternalReportCoreSwitchingDivision,Cupertino,CAReportingManagerDerekSandersbyEswarSivaraman(seswar@okstate.edu)

TheMulti-CommodityNetworkFlowProblem

ProblemStatement:Suppose we have a network of nodes connected by lines. Each line has a particularcapacity(aka "bandwidth"), aswellas a cost associated with it. There issome trafficflowing along each line (where the flow is bi-directional) consuming some of thebandwidth. Thetraffic flow ismade upof node-to-node demands that maygo throughmultiplehopsalongitsroute inorder togetfromoriginto destination. We can assumethateachpossiblepairofnodeshassometrafficflowbetweenthem.Thegoalis tofindtheoptimal(ie,minimumcost)setof routes throughthenetworkforeachofthosedemands.Theconstraintsarethattotalflowalongeachlinkmustbelessthanthelink'scapacity,andthatthetrafficdemandscannotbesplitontodifferentroutes,i.e.,thecompleteconnectionisroutedthroughasinglepath.

AnExample:

Considerthefollowingnetworkof5nodes,and8links.(Numbersshowncorrespondthenodelabels,andthelinklabels.)

BasicNotation: N=#ofnodes;Nsetofallnodes

L=#oflinks/lines;Lsetofalllines/links

i,jNdij=demandfromnodeitonodej iNdefineLiLtobethesetofalllinksincidentatnodei

kLdefineYkLtobethesetofalllinksadjacenttolinkk(i.e.,linksthatsproutfromthenodesoneitherendsoflinkk)

kLCk=Capacityoflinkk

kLMk=Costofusinglinkktoprovisionaflowofunitsize

kLiNwki=0ifkLi,=1otherwise

Alittleexplanationhere:Asource(sink)lineisapotentiallinkthroughwhichademandeitherstarts(ends)itsjourneyacrossthenetwork.Fore.g.,ifapacketneedstobetransferredfromnode1tonode5,ithastostartitsjourney(i.e.,leavenode1)ononeoflinks1/2/3,andsimilarlyhastoenditsjourney(i.e.,enternode5)ononeoflinks6/7/8.Thus,links1/2/3arepotentialsourcelinksford15,andlikewise,links6/7/8arepotentialsinksford15.Consequently,alllinesincidentatanodearepotentialsources(sinks)for

demandsstarting(ending)therefromthismotivatestheconstructionoftheparameterswki,forkL,andiN.

Choice Variables: The question we are trying to answer is this: consider any link is it(capableof)carryingapackageofsizedijthatneedstoberoutedfromnodeitonodejwhilesatisfyingallcapacityconstraints?Consequently,thechoicevariablesare:

otherwise0

jnodetoinodefromddemandcarriesklinkif1Xj,ik ijkij

=

= NL

1

2

3

4

5

1

2

3

6

7

8

4

5

7/28/2019 CIENA_MCNFP_2

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CIENACorporationInternalReportCoreSwitchingDivision,Cupertino,CAReportingManagerDerekSandersbyEswarSivaraman(seswar@okstate.edu)

Constraints:

(1) CapacityConstraint: = =

N

1i

N

1j

kkijij CXdk L

(2) Demandcompletionconstraint:

=

=

i

i

Lk

ijijkij

Lk

jijikji

ddXj,i2(b)Eqn

ddXj,i2(a)Eqn

N

N

(tomakesurethatallpackagesarereceived,andthatallpackagesareforcedoutoftheirsourcenodes)

(3) FlowBalanceEquations:

++

p

k

Lk

kij

Yr

rij

kijkjkikjki

2Xp,j,i3(b)Eqn

0XX)1ww()ww(kj,i3(a)Eqn

N

LN

(tomakesurethatwhatgoesincomesoutandtoavoidanyloops);actually,eqn3(b)isredundantsincethefinalsolution,beingminimalincost,willobviouslynotincludeanywastefulloops.

IntegerProgrammingFormulation:

3(b)3(a),2(b),2(a),1,-sintConstra

MdXk j,i

kijkij

TOSUBJECT

MINIMIZE

L N

(otherobviousconstraintschoicevariablesarenon-negativeandtakeonvaluesof0/1)

AdditionalNotes:

Thisso-to-speakmiserlinessinmynotationistopurposelyavoidthestandardapproachofflowin=flowout,andalsotoavoidexplicitlydocumentingthedirectionofflow,which,Ifeel only adds tothe complexity of the formulation. This IPformulationwillidentifytheshortestpathforeachdemandpacketacrossthenetwork.

Also,equations 1,2(a),and 2(b) will automatically identify ifit impossible tosatisfy alldemands,i.e.,iftheIPhasaninfeasiblesolution.IftheIPisfeasible,theLagrangian,i.e.,themarginalpricesforequations2(a)and(b)willbezero.IftheIPisinfeasible,some/allofthemarginalpricesforconstraints2(a)and(b) will be non-zero; consequently elimination of the corresponding demand pairs willrendertheIPfeasible,allowingfortheremainingflowstoberouted.

Fortheexamplenetwork,theNodeadjacenciesare:

L1={1,2,3},L2={1,4,6},L3={2,4,5,7},L4={3,5,8},L5={6,7,8}and,theLinkadjacenciesare: Y1={2,3,4,6},Y2={1,3,4,5,7},Y3={1,2,5,8},Y4={1,2,5,6,7},

Y5={2,3,4,7,8},Y6={1,4,7,8},Y7={2,4,5,6,8},Y8={3,5,6,7}