BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of Informaton Technologes, 1113 Sofa Abstract: The Pareto Optmal set for the multcrtera network flow problem s descrbed by the theorem for ParetoOptmal flow augmentng paths An approach for determnng of the unsupported nteger PO solutons of the nvestgated problem s proposed Keywords: multcrtera network flow problem, Pareto optmal solutons, multcrtera shortest path 1 Introducton A flow n a network s defned as the functon: X, X X(G,v,c, where X(G,v,c s defned by the lnear constrants (1-(2 on the graph G = {N, U}; N = {1, 2,, n} s a set of nodes and U = {(, j:, j N}, U = m a set of arcs: (1 1 4 X ( G, v, c : x j x (2 0 x j c j, (, j U j v f s, 0 f s,t, v f t; The nodes s = 1 N and t = j 1 N, are called a source and a snk resp, c j denotes the capactes of the arcs (, j U and v the value of the feasble network flow We denote X = {x j : (, j U} If X X(G, v, c, we also denote X = X v The sngle-crteron problems for an optmal flow n a network are solved wth the help of effcent polynomal algorthms that fnd an nteger soluton at nteger arcs capactes Ths property determnes the separaton of the flow problems n an ndependent class of lnear programmng problems
Both the theoretcal nvestgatons and the practcal applcatons requre generalzaton of the problem consdered As such generalzatons may be consdered the problem for an optmal flow wth surrogate constrants (FSGP, when the set X s reduced by settng addtonal lnear constrants on the values of the flow along the problem arcs, and also the multcrtera flow problem (MCF, n whch mprovement of the values of more than one lnear crteron s requred Each Pareto optmal (PO soluton of MCF s a soluton of the ε-constrant problem, whch s a problem for mnmzng only one of the crtera put on the set, whch conssts of the flow constrants and constrants on the values of the remanng crtera, e FSGP It has been proved for FSGP that t s equvalent to the general problem of lnear programmng The nteger propertes are lost n t and also the possblty to solve t by flow methods Some adaptatons of the smplex methods are desgned, whch use the embedded flow structure n the constrants The essence of MCF problem s n the search for such a flow, usually wth a fxed or maxmal value, the crtera values of whch satsfy the DM If at a gven stage of ts soluton t turns out that some crtera should not be stll deterorated, then the determnaton of a flow on the network wth a value lower than the desred one, could become necessary MCF problem s a problem of lnear multcrtera programmng The methods solvng lnear multcrtera problems can be appled to t and approprately adapted wth respect to the flow structure of the constrants In ths case MCF wll be solved as a lnear problem, the set of the basc PO nteger solutons and ts lnear combnatons wll be defned, but ths set wll not comprse all nteger solutons of MCF The problem whch determnes all nteger solutons of the multcrtera flow problem wll be denoted as MCIF Thorough analyss and revew of the methods solvng MCF and MCIF problems s proposed n [1] The present paper examnes the structure of the set of PO solutons of MCIF and a method for determnnteger PO solutons of the bcrtera flow problem s proposed 2 Condtons for Pareto optmalty of the multcrtera flow Let us defne the multcrtera problem MCF for a flow wth a value v on graph G: MCF: mn* { (X, I k }, s t X X(G, v, c, where (X = a j x j, (, j U Lemma 1 Let p be a PO path n graph G Let X p be a feasble flow along the path p wth a value of 1 Then X p s a PO flow n G P r o o f: Assumng the opposte, a contradcton s obtaned wth the Pareto optmalty along the path p Theorem 1 Every nteger PO soluton of MCF wth a value of the flow v may be presented as a sum of a PO soluton of MCF wth a value of the flow v 1 and a flow wth a value 1 along a PO augmentng the flow path and vce versa 1 5
P r o o f: 1Let X v be a PO soluton of the problem MCF Then there exst numbers λ e 0, Σ λ = 1, such that X v s an optmal soluton of the problem I k MF(X: mn F(X = Σ λ g (X, st X X(G, v, c The problem MF(X s a sngle-crteron mnmum flow problem If we fnd the flow X v by a mnmal path method, then X v =X v 1 + X p 1 6 I k Here X v 1 = {x j v 1 : (, j U} s an optmal soluton of the followng problem: MF(X: mn F(X = Σ λ (X, I k st X X(G, v 1, c, e t s a PO soluton wth a value of the flow v 1 Hereby X p s an optmal soluton of the problem MF(X MF(X: mn F(X = Σ λ (X, st I k 1, f s, X ( GI ( X v 1,1,1 : xj x j 0, f s, t, 1, f t; v 1 v 1 x j x j c j x j, (, j U Actually MF(X 1 s a problem of the shortest augmentng path p n the ncremental graph G I for the flow X v 1 Hence, p s a PO soluton (PO path for the graph G I gven the crtera (X, I k e, p s a PO augmentng path for the flow X v 1 2Let X v 1 be a PO soluton of the problem MCF and p a PO soluton (path n the ncremental graph G I Let X p be a flow wth a value of 1 on the path p We need to prove that the flow X v = X v 1 + X p s a PO flow for the problem MCF The reverse s assumed X v s not a PO soluton for MCF, e there exsts a flow Y v for whch the followns satsfed: (3 (Y v (X v, 1 (Y v < 1 (X v ;, 1 I k In conformance wth the theorem for flow decomposton on paths and cycles, t can be wrtten: (4 Y v = X v = X v 1 + X p In 1 In 1 It s desgned wth X, In 1 flows on the cycles σ, In 1 n graph G It follows from (3 and (4 that
Σ (X j 0 and Σ 1 (X j < 0,, 1 I k j In 1 j In 1 It turns out, that at least for one of the flows X σ = X v 1 and X p = X p the condtons In 1 In 1 (X σ, 1 (X σ < 1, (X pσ (X p, 1 (X pσ < 1 (X p are kept, whch contradcts to the condton for Pareto optmalty of both flows X v 1 and X p The theorem proved descrbes the whole set of PO solutons of MCF problem 3 Fndng unsupported Pareto optmal solutons of the bcrtera network flow problem For the bcrtera problem of a flow n a network (BCF, the mage of the set of all PO solutons s a pecewse lnear and convex set called effcent fronter (Ef Its breakponts are the mages of ts PO basc solutons and are called extreme ponts For the bcrtera nteger problem of a flow n a network (BCIF, the queston for determnng the set of supported nteger PO solutons (supported effcent flows s solved [3,4] It ncludes all PO basc solutons of the problem and the nteger solutons along the segments, connectng each two neghbourng basc PO flows The mages n the crtera space of the unsupported nteger PO solutons of BCIF problem are found n the trangles, determned by each segment of Ef and the lnes drawn across ts breakponts parallel to the axes n the crtera space (Fg 1 The queston of defnng all unsupported nteger PO solutons s not completely closed Let (X 1, X 2 and (Y 1, Y 2, where X = (X = p and Y = (Y = r, =1, 2, be two neghbourng extreme ponts Ther correspondng ponts on the dagram n Fg 1 are denoted by A and B The flows X and Y, correspondng to these ponts are two neghbourng basc solutons n the space of MCF problem solutons Let T X and T Y are ther correspondng trees Let T Y be obtaned by addng an approprate arc ( 1, 2 to T X, sendng an nteger flow wth a value of ε along the cycle σ, formed by T X and ( 1, 2 and excludng an approprate arc (j 1, j 2 from ths cycle Let p 1 < r 1 and p 2 > r 2 Let q = p 2 r 2 1 Let ponts A, =1,, q, be ponts from segment AB, for whch: A = (g 1, g 2, Z a PO flow for whch g 2 = p 2, e g 2 s nteger valued Some of the ponts A are nteger ponts that are receved by sendng flows wth values 1, 2,, ε 1 along the cycle σ, e they are supported PO solutons [3] For the remanng ponts A, =1,,q 1, q 1 =q+1 ε (and all ponts from AB segment, ther correspondng flows are obtaned as convex lnear combnatons of the flows X and Y; 2 1 7
1 8 Fg1 Z = λ 1 X+ λ 2 Y = Y + Z (σ, g 2 = g 2 (Y + g 2 (σ Here Z (σ denotes a flow along the cycle σ, λ 1 + λ 2 = 1 Let lnes L, = 1,, q 1, be drawn across each pont A, parallel to the axs g 1 (X Every unsupported РО soluton of MCIF les on any of L lnes There s no more than one unsupported PO soluton on each lne L If the ponts P 1 and P 2 on the lnes L and L j, <j, are correspondng to the two unsupported PO solutons, then the pont P 2 les n the rght of P 1 We desgn the flow X, X T =X as the frst unsupported PO soluton An approach s offered for defnng unsupported nteger PO solutons Its essence s that for each lne L, =1,, q 1, ponts T are successvely found from ths lne, for whch the correspondng flows can be defned The flow n pont T s defned, addng to the flow Х T flow Z j along a crculaton σ j, so that the costs g 2 and g (X +Z are equal 2 T j and g 1 (Y >g 1 (X T +Z j >g 1 A pont T forms an nterval wth the pont A, n whch there s at least one nteger pont Every such nteger pont s checked defnng ts correspondng flow X I lke convex lnear combnaton of the flows on the ponts A and T In case t s nteger or the nteger ponts of L are already consdered, X T = X I The ponts along the next lne are examned and so on In the opposte case we enlarge the nterval (А,T searchnn smlar way another pont T on L n the rght of the already nvestgated The formal content of the procedure suggested s as follows: 1 =1, j=1, flow X =X The flow Z s determned for pont А as a convex lnear combnaton of the flows X and Y 2 If Z s nteger and = q 1, end
Otherwse f Z s nteger, :=+1, go to 2 In case Z s not nteger,, go to 4 3 Let G I (X be the ncremental graph of G for the flow X 4 The followng problem s stated (5 mn ( g 1 (X, (6 X X(G I (X,0, g 2 (X 0 The problem has got feasble solutons, among them beng the ponts from lne L The search for an optmal soluton s not necessary An algorthm s appled, smlar to the algorthm of the negatve cycles for seekng a mnmal flow for the problem defned by (5 and (6, the flow beng altered only along these cycles for whch g 2 (X 0 6 Let at teraton j a crculaton σ j be found wth costs g 1 and g 2 respectvely, and wth maxmal feasble flow alont Z j wth a value w j The value v j s determned as v j = mn (w j,(g 2 (X g 2 / g 2 If (g 2 (X g 2 / g 2 w j, the flow X j s created, X j = X+ v j /w j Z j Passng to 8 Otherwse the flow X s formed: X := X j + Z j 7 The ncremental graph G I (X j s defned and Step 5 s executed 8 If n the nterval [g 2, g 2 (X j ], an nteger number t s found, then the number λ s defned: λ = (t g 1 (Z j / (g 1 (X j g 1 The flow X t = (1 λ Z + X j s determned In case the flow X t s nteger and < q 1, the flow X t s unsupported PO soluton It s assgned := +1, X I = X t and Step 2 s completed If the flow X t s not nteger, Step 7 s executed 4 Concluson Theorem 1 descrbes the whole set of PO solutons of MCF problem, but the applcaton of an algorthm an analogue to the algorthm for the mnmal path n the snglecrteron case, s practcally mpossble The problem for a multcrtera PO path s NP-hard [2]Of course, some algorthms could be developed determnng a part of the set of PO flows wth the actve partcpaton of the decson maker The eventual case when for the pont P, the convex lnear combnaton of the flows n the ponts A and T, s not nteger, but an nteger flow exsts, s stll not nvestgated It may be supposed that there s a crculaton σ whch conssts of the arcs wth non nteger flow and g 1 (σ = g 2 (σ = 0 1 9
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