Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR) ISSN 307-453 (Prt & Ole) http://gssrr.org/dex.php?oural=jouralofbascadappled --------------------------------------------------------------------------------------------------------------------------- Dfferece of Covex Fuctos Optmzato Methods: Algorthm of Mmum Maxmal Networ Flow Problem wth Tme-Wdows Nasser A. El-Sherbey Mathematcs Departmet, Faculty of Scece, Al-Azhar Uversty, Nasr Cty(884), Caro, Egypt Mathematcs Departmet, Faculty of Appled Medcal Scece, Taf Uversty, Turabah, KSA Emal: asserelsherbey@yahoo.com Abstract I ths paper, we cosder the mmum maxmal etwor flow problem,.e., mmzg the flow value, mmzg the total tme amog maxmal flow wth tme-wdows, whch s a combatoral optmzato ad a NP-hard problem. After a mathematcal modelg problem, we troduce some formulatos of the problem ad oe of them s a mmzato of a cocave fucto over a covex set. The problem ca also be cast to a dfferece of covex fuctos programmg (ocovex optmzato). We propose ths wor a ew algorthm for solvg the Mmum Maxmal Networ Flow Problem wth Tme-Wdows (MMNFPTW). Keywords: Optmzato etwor; Mmum flow problem; Dfferece of covex fuctos optmzato; Tmewdows.. Itroducto I recet years there has bee a very actve research dfferece of covex fuctos programmg (ocovex optmzato), because most of real lfe optmzato problems are ocovex. --------------------------------------------- Correspodg author. 67
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 The feld of etwor optmzato flows has a rch ad log hstory, a dfferece of covex fuctos programmg ad a dfferece of covex fuctos algorthms troduced by Pham Doh Tao 985. Such early wor establshed the foudato of the ey deas of etwor optmzato flow theory. The ey tas of ths fled s to aswer such questos as, whch way to use a etwor s most cost effectve? Maxmum flow problem ad mmum cost flow problem are two typcal problems of them. However, from the pot of vew of practcal cases, we have aother d of problems whch are heretly dfferet form the typcal oes. For stace, Fgure ad portrays a etwor wth edge flow capacty oe ut o all edges, each edge has a trast tme t R +,,, =,,...,. For each vertex V, a tme-wdows a, b ] wth whch the vertex may [ be served ad a t b, t T s a oegatve servce ad leavg tme of the vertex. A source vertex s, wth tme wdows, ] [ b s s source vertex see [,,3,4,5,6,7,5]. a, a s vertex τ wth tme-wdows, b ] [ τ τ a ad t s s a departure tme of the Fgure Fgure Mmum Maxmal Networ Flow Problem wth Tme-Wdows (MMNFPTW) The fgure llustrates the maxmum flow of the etwor, that s, the flow o all edges s oe except the edge x 3, whose flow s zero. O the other had, f the flow o x 3 s fxed at oe ad we caot reduce t by some reasos such as emergecy, the the etwor caot be exploted at the most ecoomcal stuato. I ths case, we ca sed two ut of flow from a source vertex s to a s vertex τ whch satsfy a tmewdows costrat. I the fgure, the flow o x 3 s fxed at oe, the possble flow value we ca sed betwee s ad τ s oe ut. The flow value, we ca sed betwee s ad τ reduces from two ( fgure ) to oe ( fgure ) due to fact that the flow value of x 3 s udrected. It meas that the maxmum flow value s ot attaable f the users o the etwor are dsobedet. Form the pot of vew of modelg, the above two fgures cases are essetally dfferet though they bear some resemblace. Assumg that the flow s drectly, the fgure ams at a optmal value flow. The fgure also 68
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 searches for a optmal value of flow, wthout the drectly of a etwor flow. The stadard etwor flow wth the drectly has bee well studed for several decades. Wthout the drectly, may problems etwor flow, the maxmum flow problem, become more dffculty. Compared to the stadard etwor flow theory s a ew fled, hece s stll ts facy. The atural questo ths ew feld s: gve a etwor N, how to calculate the attaable maxmum value flow of N whe the flow s udrected. To aswer the questo, ths paper we cosder mmum maxmal etwor flow problem wth tme-wdows whch fds out the mmum value ad mmum total tme satsfy a tme-wdows costrat amog the maxmal flows the etwor N from a source vertex s to a s vertex τ. Ir [] gave the defto of udrected flow (u-flow) ad preseted fudametal problems related u-flow. Although the cocept of u-flow s qute dfferet from maxmal flow ad ther relatoshp s ot ow yet so much, the optmal value of mmum maxmal u-flow of a etwor N s equal to the optmal value of mmum maxmal flow uder some assumpto. I Ir [8] profoud essay, several fudametal theorems ad ew research topcs are descrbed, but o algorthms for the correspodg problems are proposed. To the author's owledge, o algorthms for the mmum maxmal flow were ow utl Sh-Yamamoto []. As poted out [4], Sh-Yamamoto's algorthm s ot effcet eough. After that, some algorthms for solvg the problem were proposed such as Shgeo-Yamamoto [3] ad others. Sce the theory dealg wth the etwor flow problems wthout assumg the ameablty of flows s stll ts facy, ths paper we focus o the developmet of algorthm for Mmum Maxmal Networ Flow Problem wth Tme-Wdows (MMNFPTW) vrtue of dfferece covex fuctos optmzato. The remder of ths wor s orgazed as follows. I Secto, we gve the mathematcal models of the problem ad ts equvalet formulatos. I Secto 3, we the outle the propertes of the dfferece covex programmg ad a dfferece covex algorthm. We descrbe the framewor of the dfferece covex algorthm wth tme-wdows. I Secto 4, we gve a ew algorthm of a dfferece covex Mmum Maxmal Networ Flow Problem wth Tme-Wdows (MMNFPTW). Fally, the cocluso s gve Secto 5.. Mathematcal Models ad Equvalet Formulatos. Basc Cocepts ad Deftos Cosder a drected etwor ( V, E), o-egatve trast tme t,,, V. the vertex may be served ad a sgle source vertex, N = where V s a set of m + vertces, E s a set of edges wth a For each vertex V, a tme wdows, b ] [ a wth whch t b, t T s a o-egatve servce ad leavg tme of the vertex. A s a sgle s vertex τ wth tme wdows a, b ],[ a τ, b ] vector of the edge capacty. Let deote the set of feasble flow, [ s τ s respectvely, ad c s the 69
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 = { x x R, Ax = 0,0 x c () where the matrx A stads for a vertex edge cdet relatoshp the etwor. Obvously, s a compact covex set. Defto.. A vector x z. z s sad to be maxmal flow f there does ot exst x such that x z ad Let f be the flow value fucto, f s assumed to be lear o. For stat, t usually fed by f ( = + δ ( s) x x δ ( s) () where + (s) δ ad δ (s) are the sets of edges whch leaves ad eters the source vertex, s respectvely. The f s a lear fucto. Let otherwse. d R where, =, + d f δ (s), d =, f δ (s) ad d = 0, f I ths wor, let R deotes the set of -dmesoal real colum vectors, R = { x x R ; x 0 R = { x x R ; x 0. Let R deotes the set of -dmesoal real row ad vectors, ad R = { x x R ; x 0, R = { x x R ; x 0. We deote e to both a row vector ad a colum vector of oes, ad e to deote the set of extreme pots of S. Let th wth row vector of a approprate dmeso. For a set, M deote the set of all maxmal flows, S V (S) s the M = { z there does ot exst x such that x z, ad x z (3) We cosder the problem s gve by m{ f ( x (4) M Let problem E deote the effcet set of the vector optmzato problem, the the problem (4) s equvalet to the m{ f ( x (5) E. Prmal Formulato Model We defe the fucto r as; 70
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 r( max{ e( y y x, y = (6) It s easy to see that r s a cocave fucto o. I fact, (0,) Clearly, r( 0, x. x x, we have, α ad αr( x ) + ( α) r( x = α max{ e( y x ) y x, y + ( α)max{ e( y x = αe( y = e( αy max{ e( y ( αx = r( αx r( x ) x ) + ( α) e( y + ( α) y + ( α) x ) = ( αx + ( α) x ), r( x ) + ( α) x x ) )) )) y αx + ( α) x, y ) y x, y (7) where arg max{ e( y x, y. Moreover, ( y r ( x ) slac z such that, r s pecewse lear o. I fact, addg a A 0 0 I I 0 y 0 z = c, I 0 I x z R Ay = 0, x y c (8) The for a gve x R, r( ) s a soluto of the followg lear programmg: x max ey ex A 0 0 subect to E E 0 y 0 z = c, y 0, z 0 (9) E 0 E x where E s a matrx. As r ( s a soluto of a lear maxmzato, we assume that 0 r( = cbb c ex x (0) where cb a correspodg coeffcet s vector of obectve fucto, ad B s a basc matrx of problem (9). Lemma.. If the capacty c s tegral ad satsfy the tme-wdows costrat, the so s r ( for ay teger x. 7
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 Proof: Obvously, because of the cocave fucto, we assume that r( = m{ l ( I V, a t b, t T () where t + t t,, V, t, t T ad l ( = l, x + γ are lear fucto o R. It s easy to see that, r( = 0 x f ad oly f x. Hece, problem (4) ca be rewrtte equvaletly as E m{ f ( x, r( 0 () Cosder the followg pealzed problem for a fxed umber u. m{ f ( + ur( x (3) The, we proof the followg lemma Lemma.. Let u = max{ f ( x m{ f ( x, the there exsts a fte umber u 0, such that for every u, the problem () s equvalet to the problem (3). u Proof: For ay u r ad u ad r ( 0, we must have ( m{ f ( + ur( x m{ f ( + u r( x m{ f ( x + max{ f ( x m{ f ( x m{ ( x, r( = 0 f (4) ad whe r ( x 0 ) = 0 the we see, f ( x 0 ) m{ f ( x, r( = 0. Hece, m{ ( + ur( x m{ f ( x, r( = 0 f (5) O the other had, a feasble soluto of m{ f ( x, r( = 0 s also a feasble soluto of (3). We have m{ ( + ur( x m{ f ( x, r( = 0 f (6) It mples that m{ ( + ur( x = m{ f ( x, r( = 0 f (7) 7
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 We ote that, r (.) 0, the ay u u 0 we have m{ f ( + ur( x = m{ f ( x, r( = 0 = m{ f ( x, r( 0 (8) The, () s equvalet to (3). New, we deote by δ the dcator of ad g( f ( + δ (, = also, u r( ; x h( = ; x (9) The g ( ad ( h are covex ad problem (4) s rewrtte as m{ f ( + δ ( = m{ g( (0) Ths s a dfferece of covex fuctos programmg. Hereafter, we use the formato for local search a dfferece of covex fuctos algorthms..3 Dual Formulato Model I Phlp [], t follows that there exsts a smplex f there exsts λ Λ such that Λ R such that a vector x s maxmal flow f ad oly x λy, y λ () Thus, the mmum maxmal flow problem wth tme-wdows to be cosdered ca also be formulated as: m f ( subect to λ ( y 0, y, Λ, x λ () t + t t, a t b, t, t T,,, V Ths s a specal case of mathematcal programmg wth varato equalty ad tme-wdows costrat. We deote that, g x λ = x + λ + vx + vλ v (3) (, ) max{ v 73
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 ad h ( x, ) = x + λ + x λ (4) The, we proof the followg lemma Lemma.3. The costrats () ca be cast to the form g( x, ) x, λ) = 0, λ Λ, x λ (5) Proof: We ote that, g( x, λ ) x, λ) = max{ λ( v v x (6) v Sce s a covex set. Suppose that () holds for some 0 max{ v λx v x v max{ λv λx v = 0 x ad some Λ, λ (7) λ we have that; whch yelds (, ) (, ) = 0 g x λ h x λ g x λ h x λ for some x ad Λ. Suppose that (, ) (, ) = 0 λ The we have that, max{ v ( v v x = 0 λ (8) whch mples that ( v 0 λ for all v. I fact, f we have some v 0 such that λ ( v 0 0, the we ca tae a pot v o le segmet [ v 0, x] satsfyg v x λ cosθ, where θ s the acute agel betwee λ ad v0 x. Sce s covex, the v but ( v v x 0. cotradcts (8). λ It Note that: The fuctos g ad h are covex ad dfferetable. From lemma.3., t follows that the problem ca be formulated by the followg a dfferece of covex fuctos of dfferetable programmg wth tme-wdows costrat: m f ( g, Λ, x subect to ( x, λ) x, λ) = 0 λ (9) t + t t, a t b, t, t T,,, V 74
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 By Shgeo-Taahash-Yamamoto [], we see that the Λ (9) could be replaced by { λ λ R, λ e, λe. The we tae the above set as Λ to desg the algorthms. = 3. A Dfferece Covex Programmg ad a Dfferece Covex Algorthm A dfferece covex programmg ad a dfferece covex algorthms troduced by Pham Dh Tao 985 ad extesvely developed other wors. A dfferece covex algorthms was successfully appled to a lot of dfferet ad varous ocovex optmzato problems to whch t qute ofte gave a global solutos ad proved to be more robust ad more effcet tha related stadard methods, especally the large scale settg. I [0] a dfferece covex algorthm s a prmal-dual approach for fdg local optmum a dfferece covex programmg. More detaled results o a dfferece covex algorthms ca be foud such as [9]. Some umercal expermets are reported that t fds a global mmzer ofte f oe chose a good start pot. Cosder the followg geeral problem: v p = f{ g( x R (30) where (.), (.) : R g h R {, are low sem-cotuous covex fuctos o R. It s easy to see that problem (0) s a specal case of (30) as show (0) uder the coservato +. We also suppose that g( s bouded below o R. The ε -subgradet of g at pot 0 x are defed by: ε g( x0) = { y R g( g( x0) + x x0, ε, x (3) ad g x ) = g( ). The cougate fucto of g s gve by: ( 0 0 x0 g ( y) = sup{ x, g( x R (3) From low sem-cotuous of g ad h, we see that (30): g = g ad h = h hold. Cosder a dual problem of v d = f{ h ( y) g ( y) y R (33) v = f{ g( h x We have that ( ) p x R = f{ g( sup{ x, h ( y) y R x R = f{ g( + f{ h ( y) x, y R x R 75
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 = (34) f{ h ( y) + f{ g( x, x y R = f{ h ( y) + sup{ x, g( x y R = f{ h ( y) g ( y) y R = v d For a par (, y), g. If ) x Fechel's equalty g ( + g ( y) x, y g(x the g ( + g ( y) = x,. holds for ay proper covex fucto g ad Defto 3. A pot x s sad to be local mmal of g h f there exsts a eghborhood N of x such that ( g h)( ( g h)( x ), x N Lemma 3. A pot x s local mmal for h, g the h( x ) g( x ). Proof: Let ( g h)( ( g h)( x ), x N. The g( g( x ) h( x ). Tag z h( x ), we have h ( h( x ) + x x, z x R Therefor, we see that g ( g( x ) + x x, z for all. for x N. We ote that g s covex, the g ( g( x ) + x x, z x R holds for. Lemma 3.3 If h s a pecewse lear covex fucto o dom (h) ad h( x ) g( x ), mmal for g h. the x s local Proof: It s eough to cosder x dom(g). Let h s pecewse lear covex. The there exst a eghborhood N ( x ) such that for ay x N( x ) we ca choose z h( x ) such that h ( x ) = x x, z. For h( x ) g( x ) we have g ( g( x ) + x x, z holds for ( x N x ). It mples that g( g( x ) x ) for x N( x ). The g h. x s local mmal for A Dfferece Covex Algorthm wth Tme-Wdows We descrbe a framewor of the Dfferece Covex Algorthm wth Tme-Wdows s the frst algorthm. 0 0: pc up a pot x dom( h), calculate y 0 h( x 0 ); = ; : each pot has satsfed a tme-wdows costrat,.e., 76
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 t + t t, a t b, t, t T,,, V ; : calculate x argm{ g( ( h( x ) + x x, y ) x R ; calculate y argm{ h ( y) ( g ( y ) + x, y y ) y R ; 3: If h ( x ) g( x ) φ, stop; otherwse, = + go to step. Lemma 3.4 Suppose that the pots x ad above frst algorthm, the ( x h y ) ad y g( x ). y are satsfed a tme wdows-costrat ad geerated the Proof: Assume that x ad y are satsfed a tme wdows costrat ad had. We have m{ g( ( h( x ) + x x, y ) x R = m{ g ( x, y x R x ) + x, y ad m{ h ( y) ( g ( y ) + x, y y ) y R = m{ h ( y) x, ) y R g ( y ) + x, y. (35) Thus, from step the above frst algorthm, g ( x, y g( x ) x, y for all x, ad h ( y) x, h ( y ) x, y for all y. It yelds y g( x ) ad x h ( y ). 4. Methods ad Algorthm Now we go bac to problem (0). I ths secto, we gve a algorthms to solve the problem. A Geeral Framewor of brach-ad-boud algorthm wth tme-wdows ca be stated follows. Algorthm Geeral Framewor: 0: tal settg ad calculatg; : brachg operato wth tme-wdows costrat,.e., t + t t, a t b, t, t T,,, V; : local search for a smaller upper boud; 77
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 3: fd a larger lower boud; 4: remove some regos, do to step. We descrbe the step, ad 3 as the followg explaed: - Descrbe step : Brachg operato wth tme-wdows costrat A smplex based dvso s usually exploted brach-ad-boud method. At some step, a cotemporary smplex S s dvded to two smaller oes S ad. S Tag to accout the covergece of the algorthms ad a tme-wdows costrat, we eed the dvso to be exhaustve,.e., a ested sequece of smplexes { S, =,,... has the followg propertes:. t( S ) t( S ) = φ f ; a t b respectvely ad t t t, V has a tme wdows, b ],[ a, b ] +, S + S for all, [ a where a t b,. lm = S = x 0 0 for some x. At each step, we chose dvde a smplex S to two smaller oes of S. The sequece { S, =,,... such process s exhaustve. S ad + S by bsectg the logest edge - Descrbe step : Local search for a smaller upper boud There are may methods to do local search. Here we explot the frst algorthm ths step. Eve the frst algorthm s ot gog to fd a global optmum theoretcally, but may umercal expermets, t fds a global optmum practcally. As show problem (0) ca be rewrtte as a dfferece covex programmg m{ h, g the we ca use the frst algorthm to obta a locally optmal soluto. The we assume that o = S of the frst algorthm s a local optmal soluto satsfy the tme-wdows costrat o algorthm. S by usg the frst - Descrbe step 3: Fd a larger lower boud Assume that l ( s a affe fucto such that l v ) = h( v ) for all vertces v V S ) wth tmewdows a, b ], a t b, T [ v v v v l ( h( for all x. The, S v t ( ( v be a o-egatve tme. From the covex of (, h we have 78
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 L( S ) = m{ f ( + δ ( l ( x R S m{ f ( + δ ( x R (36) S Moreover, f V S ) = { v,..., s had ad satsfy the tme-wdows costrat, the t s easy to ( v p calculate L S ) because ( m{ f ( δ ( l ( x R = m{ d p + S = v + u p = λ r( v ) λ =, λ 0, A λ v = b,0 λ (37) p = = p λ v c Based o the above dscusso, we gve the followg algorthm of the dfferece covex algorthm of the Mmal Maxmum Networ Flow Problem wth Tme-Wdows (MMNFPTW). The Dfferece Covex Algorthm of the Mmal Maxmum Networ Flow Problem wth Tme- Wdows 0: let ε ad S 0 such that S 0. let x 0 = 0, y 0 = (,..., ), b U = basdca(, b L = m{ f ( Ax = b,0 x b, M = S0; : select S 0 M such that b L = L( S0) ad dved S 0 to S ad S ; : o = S from the frst algorthm for all =, f U o b the o = b U ; U L 3: f L( S ) bl the bl = L( S), f b ε b the Stop; U 4: M = { S M L( S) b, f M = φ the Stop, otherwse, go to step. The covergece of the above ew algorthm of the Mmum Maxmal Networ Flow Problem wth Tme Wdows (MMNFPTW) s from the exhaustve partto. 5. Cocluso A brach-ad-boud algorthm va a dfferece covex algorthm subroute for solvg problem (4) s proposed ths wor. A part from the algorthm, we also dscussed a dual formulato for problem (4) ad vestgated some propertes of a geeral dfferece covex programmg. Though we have ot proposed a algorthm for problem (30), t ca be solved by dfferetal programmg. Due to that problem (3) s a cocave mmzato over a covex set, we ca solve t by may exstg methods drectly or drectly. Amog these methods, t 79
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 mght be terestg to compare the behavor of the dfferet algorthms. Also, we troduce a ew algorthm of the Mmum Maxmal Networ Flow Problem wth Tme-Wdows (MMNFPTW). Acowledgmet The author would le to tha the aoymous referees for ther commets ad careful readgs. Refereces [] N. El-Sherbey. (05). "Mmum Covex ad Dfferetable Cost Flow Problem wth Tme- Wdows". Iteratoal Joural of Sceces: Basc ad Appled Research, Vol. (0), No. (), pp. 39-50. Avalable: http://gssrr.org/dex.php?oural=jouralofbascadappled [] N. El-Sherbey. (05). "The Dyamc Shortest Path Problems of Mmum Cost Legth Tme Wdows ad Tme-Varyg Costs". Iteratoal Joural of Scetfc ad Iovatve Mathematcal Research, Vol. (3), No. (3), pp. 47-55. Avalable: www.arcoural.org [3] N. El-Sherbey. (04). "The Algorthm of the Tme-Depedet Shortest Path Problem wth Tme- Wdows". Appled Mathematcs, Vol. (5), No. (7), pp. 764-770. Avalable: http://dx.do.org/0.436/am.04.5764 [4] N. El-Sherbey. (0). "Imprecso ad flexble costrats fuzzy vehcle routg problem". Amerca Joural of Mathematcal ad Maagemet Sceces, Vol. 3, pp. 55-7. Avalable: http://dx.do.org/0.080/096634.0.0737800 [5] N. El-Sherbey. Ad D. Tuyttes. ''Optmzato multcrtera of routg problem''. Troseme Jouree de Traval sur la Programmg Mathematque Mult-Obectve, Faculte Polytechque de Mos, Mos, Belgque, (00). [6] N. El-Sherbey. ''Resoluto of a vehcle routg problem wth multobectve smulated aealg method''. Ph.D., Mathematcs Departmet, Faculty of Scece, Mos Uversty, Mos, Belgum, 00. [7] R. Horst ad H. Tuy. (993). Global optmzato, Determstc Approaches. ( d edto). Sprger-Verlag. [8] M. Ir. "A essay the theory of ucotrollable flows ad cogesto". Techcal Report. Departmet of Iformato ad System Egeerg, Faculty of Scece ad Egeerg, Chuo Uversty, TRISE 94-03, 994. [9] D. Pham ad T. Le Th. "Covex aalyss approach to d. c. programmg theory, algorthms ad applcatos". Acta Mathematca Vetamca, Vol., pp. 89-355, 997. [0] D. Pham. "Dualty d. c. (dfferece of covex fuctos) optmzato. Sub-gradet methods". Treds Mathematcal Optmzato, Iteratoal Seres of Numercal Mathematcs 84, Brhauser pp. 77-93, 988. 80
Iteratoal Joural of Sceces: Basc ad Appled Research (IJSBAR)(06) Volume 5, No, pp 67-8 [] J. Phllps. "Algorthms for the vector maxmzato problem". Mathematcal Programmg, Vol., pp. 07-9, 97. [] J. Sh ad Y. Yamamoto. "A global optmzato method for mmum maxmal flow problem". ACTA Mathematca Vol.,, pp. 7-87, 997. [3] M. Shgeo, I. Taahash ad Y. Yamamoto. "Mmum maxmal flow problem: a optmzato over the effcet set". Joural of Global Optmzato, Vol. 5, pp. 45-443, 003. [4] Y. Yamamoto. "Optmzato over the effcet set, Overvew". Joural of Global Optmzato, Vol. -4, pp. 85-37, 00. [5] D. Tuyttes, J. Teghem, N. El-Sherbey. (004). "A Partcular Multobectve Vehcle Routg Problem Solved by Smulated Aealg". Lecture Notes Ecoomcs ad Mathematcal Systems, Vol. 535, pp. 33-5, Sprger-Verlag, Berl, Germay. Avalable: http://dx.do.org/0.007/978-3-64-744-4 8