Research Article Capacity Expansion and Reliability Evaluation on the Networks Flows with Continuous Stochastic Functional Capacity

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1 Applied Mahemaics, Aricle ID 87660, 9 pages hp://dx.doi.org/0.55/04/87660 Research Aricle Capaciy Expansion and Reliabiliy Evaluaion on he Neworks Flows wih Coninuous ochasic Funcional Capaciy F. Hamzezadeh and H. alehi Fahabadi Deparmen of Mahemaics, College of Basic cience, Karaj Branch, Islamic Azad Universiy, Alborz, Karaj 485-, Iran Correspondence should be addressed o H. alehi fahabadi; hsalehi@u.ac.ir Received May 04; Acceped Augus 04; Published 9 Augus 04 Academic Edior: Long Zhang Copyrigh 04 F. Hamzezadeh and H. alehi fahabadi. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. In many sysems such as compuer nework, fuel disribuion, and ransporaion sysem, i is necessary o change he capaciy of some arcs in order o increase maximum flow value from source s o sink, while he capaciy change incurs minimum cos. In real-ime neworks, some facors cause loss of arc s flow. For example, in some flow disribuion sysems, evaporaion, erosion or sedimen in pipes wase he flow. Here we define a real capaciy, or he so-called funcional capaciy, which is he operaional capaciy of an arc. In oher words, he funcional capaciy of an arc equals he possible maximum flow ha may pass hrough he arc. Increasing he funcional arcs capaciies incurs some cos. There is a cerain resource available o cover he coss. Firs, we consruc a mahemaical model o minimize he oal cos of expanding he funcional capaciies o he required levels. Then, we consider he loss of flow on each arc as a sochasic variable and compue he sysem reliabiliy.. Inroducion Given a direced nework flow wih unique source s and unique sink, one of he common problems is he maximum flow problem where he value of flow from s o should be maximized. Various algorihms have been presened o solve his problem. The firs algorihm developed by Ford and Fulkerson [] was called Augmening Pah Algorihm. Karzanov [] inroduced he firs preflowpush algorihm on layered neworks. He obained an O(n ) algorihm. Capaciy scaling is anoher polynomial algorihm developed by Ahuja and Orlin []. I seems ha he bes available ime bounds algorihms o solve his problem are based on Alon [4] and Ahuja and Orlin [5]. The budgeed capaciy expansion problem has been invesigaed for differen sysems. One of hem is flow expansion on ransporaion neworks wih budge consrains. This problem was sudied by Elalouf e al. [6]. The general budge-resriced max flow problem was firs discussed by Eisel and Frajer [7]. Also, anoher algorihm for opimal expansion problems was provided by Ahuja e al. [8]. Many physical flow sysems are regarded as sochasic nework flows. In such a sysem, he sysem reliabiliy is an imporan crierion for sysem performance. everal sudies have evaluaed nework reliabiliy of maximum flow. Yeh evaluaed he reliabiliy facor of a nework, where arc capaciies are sochasic. He defined a simple approach o evaluae reliabiliy of a mulisae nework [9]. Xiao e al. [0] presenedamodelforreliabiliycompuaionin power disribuion sysem. Laer, he presened approach in [9] was considerably improved by alehi Fahabadi and Forghani-elahabadi []. In anoher paper, Yeh [] designed a survivable nework under cos consrains and compued is reliabiliy. For his case, alehi Fahabadi and Khodaei [] presened a differen mehod o find he opimal sysem parameers and compued is reliabiliy. Lin and Yeh [4] proposed a geneic algorihm o find maximal reliabiliy of neworks. Also Lin sudied sochasic flow nework wih wo commodiies [5] and nework reliabiliy wih budge consrain [6]. In all of he above sudies he probabiliy funcion of he arcs capaciy was assumed o be discree. However, in many real-life sysems, where he flow is liquid, his assumpion is no applicable.

2 Applied Mahemaics In his paper, we consider nework flow sysems in which he funcional capaciies of he arcs are coninuous random variables. The invesigaed problem is o expand he nominal capaciies of arcs so ha he sysem would be capable of securing he demands, while he expansion cos is minimal. Then, we evaluae he reliabiliy of he sysems as he probabiliy ha he nework can ransfer a desired amoun of flow from he source, s, ohesink,. Theformergoalis achieved by running he algorihm CEA, which is explained in ecion and he laer goal is accomplished in ecion 4. Regarding he previous sudies [7, 8], our conribuion has been compiled based on he following. (i) We define and use he arcs funcional capaciies which are quie differen from arcs nominal capaciies in real-life sysems. (ii) The probabiliy funcions of he sochasic funcional capaciies are coninuous. (iii) The capaciy expansion cos is minimum. The capaciy expansion akes place in a discree manner unil he maximum flow reaches he required level. This paper has been arranged as follows. ecion defines noaions, primary assumpions, and he relaed heorems. Also, he mahemaical model of he problem is consruced in ecion. ecion explains he capaciy expansion algorihm. ysem reliabiliy is compued in ecion 4 and evenually a numerical example will be considered in ecion 5.. Noaions and Assumpions We use he following definiions and assumpions hroughou his paper. G(V, A): A direced nework flow wih a unique source s and a unique sink, wherev is he se of nodes and A V Vis he se of arcs. u ij : Nominal capaciy of (i, j) A (he iniial buil-in capaciy of arc(i, j)). u f ij :Funcionalcapaciyofarc(i, j) A, (hecurren operaional capaciy of arc(i, j) A). w ij : Amoun of capaciy expansion on arc(i, j) A. M ij : Maximum capaciy expansion for arc(i, j) A. c ij : Expansion cos per uni of capaciy on arc(i, j) A. α ij : Loss facor on (i, j) A. The fracion of arc s capaciyislosdueosomesochasicevens.then α ij = α ij is fracion of arc s capaciy ha is able o ransmi he flow. We call hem ransmission facor. P i :Theih simple pah from s o. A simple pah does no include repeaed arcs and/or nodes. Ω r :Therh caegory r =,,...,R. Each caegory includes several subses of ransmission facors. In addiion, each caegory is relaed o se of simple pahs ha are capable of sending maximum flow from s o. The nework flow saisfies he following assumpion. (i) Each node is perfecly reliable. This means ha he amoun of flow passing hrough a node does no change. (ii) The loss facors and ransmission facors of differen arcs are saisically independen random variable.. Problem aemen Assume G = (V, A) is a nework flow wih node se, V; arc se, A;sourcenodes; and sink node. We assume G includes n nodes and m arcs. Each arc, (i, j), hasanominalcapaciy of u ij.therearesomefacorsinhisneworkhacauseloss of flow on each arc. For example, in a waer disribuion sysem, evaporaion, erosion, or sedimen in pipes wase he flow in some pars. For each arc we define a real capaciy or he so-called funcional capaciy and denoe i by u f ij. uf ij equals he real amoun of maximum flow ha can pass along he arc(i, j). According o he curren funcional capaciies, he real maximum flow value from s o is V. Thisvalue is equal o he funcional capaciy of he curren minimum cu, (, ) [9]. We are going o expand he nominal capaciy, u ij, such ha he real maximum flow value is increased by V 0 and he oal expansion cos is minimized. We know ha he expansion cos on arc(i, j), for each uni of capaciy, is c ij and he maximum capaciy expansion is M ij unis. For each arc, say, (i, j),hereisalossfacor,α ij,whichisaconinuous random variable in inerval [0, ] and shows he fracion of nominal capaciy ha is wased. Thus ransmission facor (α ij = α ij,forall(i, j) A) is compuable for any arc. Noe ha his facor is a coninuous random number in inerval [0, ] as well. Therefore, he funcional capaciy of arc(i, j) is compued as u f ij =α ij u ij. u f ij s saisfy 0 uf ij =α ij u ij u ij, (i, j) A. Now using he above noaions and assuming (, ) is he curren minimum s cu, he mahemaical model of he problem is consruced as min z= c ij w ij, (i,j) A s.. j V V + V, if i=s, x ij x ji = 0, if i =s,, j V (V + V 0 ), if i=, (i,j) (,) α ij u ij = V, () () () 0 x ij α ij (u ij +w ij ), (i,j) A, (4) α ij [0, ], (i, j) A. (5) Consrains () show he flows are balanced a each node oher han he source and sink; for he source and sink nodes he amoun of ou-flow from s equals he amoun of inflow o and equals V + V 0. Consrain () shows ha he maximum flow, before changing he arcs capaciies, is equal o V. The consrains (4) guarany he arc flows do no exceed hefuncionalarcs capaciies.finally,coninuiyofα ij s,and herefore coninuiy of x ij are he impac of consrains (5).

3 Applied Mahemaics.. The Capaciy Expansion Algorihm (CEA). I should be noed ha, according o he curren funcional capaciies, he value of he maximum flow is V andisequalohecapaciyof he minimum s cu, (, ). Usinghefuncionalcapaciy, we assume ha he curren maximum flow is denoed by x= x ij } (i,j) A and generaes he residual nework G(x). uppose P k is a pah from s o in G. The curren amoun of flow passing hrough his pah is equal o δ k = minu f ij (i, j) P k }. For some arcs, (i, j) P k say, i may be δ k <u f ij.inhis case, (i, j) has some residual (free) capaciy ha may be used, wihou any cos, o increase he flow on (i, j). Ifδ k = u f ij, for an arc(i, j) P k, here is no free capaciy lef on (i, j) and he capaciy of his arc mus be increased incurring some cos. According o his poin, we define capaciy expanding cos for each arc and find he cheapes pah from he source node o he sink node. For his purpose he residual nework and an auxiliary nework called expansion-cos-nework are consruced. We denoe he laer one by G(cos ) and i is consruced as follows. G(cos ) include all nodes of G(x). Foreveryarc(i, j) G(x) wo direced arcs in opposie direcions beween i and j are drawn in G(cos ). Noe ha, if here already are wo arcs beween i and j wih opposie direcions, no more arcs are drawn beween hese nodes. The arc(i, j) shows he possibiliy of expanding capaciy wih cos c ij on (i, j) G(cos ). The arificial arc(j, i) represens he possibiliy of reducing he expanded capaciy on (i, j) G(cos). Iisimporano noe ha, when here is a posiive amoun of flow going from node i o node j hrough he arc(i, j), wemaydecreaseiby assuming an arificial arc from j o i wih some posiive flow x ji and cos c ji.eacharcing(cos ) is labeled by c ij,which is he expansion or narrowing cos per uni of capaciy. If he flow on arc(i, j) is smaller han he funcional capaciy, namely, δ k <u f ij, hen his arc seel has room o increase he flow. Hence here is no need o increase he curren capaciy. The labels for his case are se as c ij = c ji = 0.Oherwise, he capaciy of (i, j) should be expanded. A his sage of he algorihm, if he capaciy of his arc has no been expanded so far, namely, w ij =0, hen increasing he capaciy is possible. Hence we se c ij =c ij and c ji =0.If0<w ij <M ij,hen, for a beer soluion, we may increase he capaciy or reduce he increased capaciy. Therefore he labels are se as c ij =c ij, c ji = c ij. Evenually, if he expanded capaciy value on arc(i, j) is equal o M ij he capaciy expansion is no possible and we can only reduce i. o, he coss are se as c ij =+ and c ji = c ij [6]. Afer seing he labels in G(cos), hecheapessimple pah from he source o he sink is found; he capaciies are exended on his pah as much as possible and he curren ieraion is ended wih compuing he maximum flow. This process is repeaed unil he value of he maximum flow becomes equal o V + V 0... Pseudocode of CEA. Formoredeailssee Pseudocode... Algorihm s Complexiy. We now consider he complexiy of CEA. We migh compue he compuaional ime of his algorihm as allocaed o he following four basic operaions. () Consrucionof G(x) and G(cos): concerning residual nework, he algorihm allocaes coss o he arcs of G(cos ). If he nework conains m arcs, coss of m arcs would be calculaed and assigned o he arcs. Thus he algorihm performs his operaion in O(m) ime. () Finding he shores pah: he algorihm finds he cheapes (shores) pah from he source o he sink according o he cos vecor c.fifoimplemenaion of he modified label correcing algorihm is he bes known algorihm for his purpose. The runime of his algorihm is O(nm),[9]. () Capaciy expansions: for each arc belonging o he cheapes pah, i would be examined wheher capaciy expansions are required or no. This operaion would be implemened in O(m) ime. (4) Compuing he maximum flow: a his sage, he maximum flow algorihm is run. The order of he bes known algorihm (highes label preflow push) is O(n m) [9]. Theabovesepsarerepeaedunilhemaximumflowfor he firs ime becomes equal o or greaer han V+V 0.uppose (V 0 ) is he maximum number overall repeiion needed o reach his hreshold. Finally we have he following resul. Theorem. The capaciy expansion algorihm (CEA) runs in O((V 0 )(m + nm + n m)) ime. 4. ysem Reliabiliy In order o increase he maximum flow value o he required value, he above algorihm compues he opimal capaciy expansion while assuming deerminisic ransmission facors. Bu, in case of randomness he algorihm may be used only for observaion values of hese facors. ysem reliabiliy is defined as he probabiliy ha he maximum flow value saisfies a cerain condiion, such as appraising some demands [, ]. ince he loss facors, ransmission facors, and, herefore, he funcional capaciies are sochasic, he final value of maximum flow is also sochasic. In order o esablish an approach for evaluaing he sysem reliabiliy of such sysems, we assume he loss facors are coninuous uniform random variables and generae N uniformly disribued pseudo-random numbers in he inerval [0, ]. The mehod o generae hese numbers is muliplicaion congruen [0]. Then we calculae he ransmission facors for any arc using loss facors. Noe hese facors are coninuous uniform random numbers in he inerval [0, ] as well. We denoe hem by α,α,...,α N. We consider several subses of hese numbers and denoe hem by A j, j =,,...,L. Each subse includes m members. A he beginning of each run of

4 4 Applied Mahemaics ar Read he nominal capaciies, u ij s, and subse of he ransmission facors, α ij s, of he original nework, G. Calculae funcional capaciy, u f ij s, of each arc. Find he maximum flow, MAX. While MAX < V + V 0 Do Define he residual graph G(x) and G(cos) as follow V G(cos ) =V G(x) and A G(cos ) = (i, j), (j, i) (i, j) or (j, i) A G(x) } andlealabelforeacharcwhichshowhecosofarcexpansionasfollowing: For each (i, j) A G(cos ) Do if u ij =uf ij x ij >0and u ji =x ij >0hen se c ij =0and c ji =0 else if u ij =0and u ji =0 if w ij =0hen se c ij =c ij and c ji =0 else if 0<w ij <M ij hen se c ij =c ij and c ji = c ij else if w ij =M ij hen se c ij = and c ji = c ij end if end if end if Find he shores pah P in G(cos) regarding c value if (sum of he coss on he shores pah < + ) (i,j) Preplace he new capaciy in G as follow: if c ij =0and c ji =0hen se w new ij =0and u new ij =u ij else if c ij >0and c ji 0or c ij 0and c ji >0hen min M pq w pq (p,q) P,c pq >0,c qp 0}, se w new ij = min min u pq x pq (p,q) P,c pq =0,c qp =0}, min u pq +w pq (p,q) P,c pq 0,c qp >0} end if end if end For Calculae funcional capaciy using new capaciy Find maximum flow, MAX. end while end. } } } and u new ij =u ij +w new ij Pseudocode CEA, we ake one of hese subses as he arcs ransmission facors and run he algorihm. A he end of CEA a se of pahs ha ransfer a leas V+V 0 uni of flow from s o is generaed. I is clear ha he generaed pahs and heir number depend on he given subse of ransmission facors. Furhermore, here may be subses of ransmission facors ha generae similar se of pahs. We consider he subses of ransmission facors ha generae he same se of pahs as a caegory and assume ha he number of caegories is R (R L). We denoe he caegories by Ω r, r =,,...,R.Alsoweassumehahe numberofpahsassociaedoherh caegory is K r. Now we consider V r as he final value of maximum flow in each caegory and compue he densiy funcion of V r.ince 0 α ij, uf ijs are coninuous uniform random variables on inerval [0, u ij ] wih densiy funcion of f(u f ij )=(/u ij)i [o,uij ]. The number of pahs generaed a each run of CEA depends on he applied subse of ransmission facors. For each pah P i associaed wih he rh caegory, he flow value on P i is compued as δ r i = min u f pq (p,q) P i}, i=,,...,k r, r=,,...,r. The oal flow value V r is equal o he sum of all flow values on he generaed pahs. Tha is, V r = Kr i= δr i.thedisribuion funcions of δ r i may be compued according o he following heorem. Then we deermine densiy funcion of V r. Theorem (see []). If,,..., n are independen random variables wih disribuion funcion F i () and T () = min,,..., n },henf T() () = n i= [ F i ()],[]. (6)

5 Applied Mahemaics 5 Now,weapplyhefollowingprocedureocompuehe nework reliabiliy. 4.. Compuing he Nework Reliabiliy. Afer running CEA L imes wih inpu of A j, j =,,...,L, and deermining he caegories Ω r, r =,,...,R, he following procedure is performed o compue he nework reliabiliy. ep. From each caegory, Ω r, r =,,...,R,selecone subse of ransmission facors, A r say. ep. For each r=,,...,rand i=,,...,k r,compue δ r i = min u f pq (p,q) P i}, i=,,...,k r, r=,,...,r. ep. Find he probabiliy funcion of δ r i, where i =,,,..., K r and r=,,...,r. ep 4. Calculae V r = Kr i= δr i and is probabiliy funcions []. ep 5. Compue he condiional probabiliy; ha is, p r (V r > V α i, i=,,...,m), r=,,...,r. Now he sysem s reliabiliy is compued using he following equaion: (7) p (V > V) = R r= p r (V r > V α i, i=,,...,m). (8) R 5. Numerical Example In order o illusrae how he algorihm works, we apply i o he nework flow shown in Figure. In his example, we assume V =.89 and V 0 =.8. We generae he following sequence of pseudo-random numbers, ransmission facor, and is subsequences: (α,α,α,α 4,α 5,α 6 ) = (0., 0., 0.4, 0.0, 0., 0.8), (α,α,α,α 4,α 5,α 6 ) = (0.89, 0.9, 0.66, 0.99, 0.8, 0.), A = (0.89, 0.9, 0.66, 0.99, 0.8), A = (0.89, 0., 0.9, 0.8, 0.99 ), A = (0.89, 0.9, 0., 0.99, 0.8), A 4 = (0.89, 0.66, 0.9, 0.8, 0.99 ), A 5 = (0., 0.8, 0.99, 0.66, 0.89). Noe ha A j, j =,,...,5, has been arranged as (α s,α,α,α s,α ). (9) We use A j, j =,,...,5, as ransmission facors and run CEA o find he relaed pahs from node s o node. For example, we consider A and run algorihm CEA. For more explanaion, we describe one sage of implemening his algorihm using A in Figure. Noe ha we show he shores (cheapes) pah in Figure (b) by solid line and oher arcs by doed line. The resul of running CEA using A is shown in Figure. Also noe ha he doed lines in neworks Figures (a) and (b) from noe o node in he las sage indicae ha no increase has aken place on his arc and ha no flow passes hrough his arc. We see ha A generaes he same se of pahs wih A ;hais,(s,, ) and (s,, ). Therefore we pu A and A in caegory, Ω.Also, A and A 4 generae he same pahs: (s,,), (s,,,),and (s,, ).HencewepuA and A 4 in caegory, Ω.Theresul of running CAE using A is shown in Figure 4. We discard he facor A 5, because his subse of he ransmission facors reduces he funcional capaciy of he arcs originaing from source o he exen ha hese arcs are unable o carry a leas V + V 0 uni of flow. For more explanaion, he resul of running CEA using A 5 is shown in Figure 5. Now, we apply he following seps o compue he nework reliabiliy. ep. We choose he subse A from Ω. ep. Le δ = minuf s,uf } and δ = minuf s,uf } where u f ij are random variables ha have uniform disribuion on [0, u ij ];hais,u f s U[0,], uf U[0,5], uf s U[0,6],and u f U[0,6]. ep. Densiy funcions of δ and δ are compued as f δ (δ ) = (8/5) (/5)δ and f δ (δ ) = (/) (/8)δ.Alsof δ,δ(δ,δ )=f δ (δ ). f δ (δ ) = ((8/5) (/5)δ )I (0,)(δ )((/) (/8)δ )I (0,6)(δ ). ep 4. Le V =δ +δ and 8 45 V 7 (V ) + 80 (V ), 0 V <, 54 f V (V 5 8 V, V <6, )= V (V ) 5 6 (V ), 6 V <9. (0) ep 5. Now we can calculae he following condiional probabiliy: P (V >.89 α = 0.89, α = 0.9, α = 0.66, () α 4 = 0.99, α 5 = 0.8) = 0.9.

6 6 Applied Mahemaics (,, $) (5,, 5$) (,, ) (,, $) (5,, 4$) Figure : An example of nework flow; arc labels presen as (u ij,m ij,c ij ) $ $ 4 (a) Funcional capaciy maximum flow:.89 uni (b) hores pah in G(cos ): $ += 5+0= (c) Exended capaciies (d) Funcional capaciy maximum flow: 5.57 uni Figure : The nework afer one sage of running CEA using A. Also we implemen he above seps for A from Ω and ge he following probabiliy: P (V >.89 α = 0.89, α = 0., α = 0.9, () α 4 = 0.8, α 5 = 0.99 ) = 0.7. Finally, we can calculae he sysem reliabiliy: p (V >.89) = 0.. () Alhough he reliabiliy nework in his example was equal o 0., ye his is an effecive algorihm which presens a new mehod for calculaing he reliabiliy in neworks wih coninuous componens. Alhough previous algorihms [ 4] also have presened mehods for calculaing he reliabiliy in he nework, bu in all of hese mehods, componens, including arcs capaciy, have been assumed o be discree, while his assumpion is no rue for all neworks. By presening his algorihm, we have creaed an innovaion in regard wih reliabiliy of neworks wih coninuous capaciies. Finally, we would give anoher example and calculae he reliabiliy of a nework in order o elucidae efficiency of hese algorihms. 5.. Anoher Example. In order o elucidae efficiency of CEA, we consider he nework flow shown in Figure 6. Alsowe assume V = 7.5 and V 0 =5.05. We generae he following pseudo-random numbers and ransmission facors: (α,α,α,α 4,α 5,α 6,α 7,α 8,α 9,α 0 ) = (0.5, 0., 0., 0., 0., 0.4, 0.8, 0.7, 0.6, 0.0), (α,α,α,α 4,α 5,α 6,α 7,α 8,α 9,α 0 ) = (0.85, 0.68, 0.9, 0.7, 0.89, 0.66, 0., 0., 0.4, 0.99). (4)

7 Applied Mahemaics 7 += 5+0= =6 5+0+= (a) Exended capaciies (b) Funcional capaciies Figure : The resul of running CEA using A. += 5+= =.8 +=6 5+= (a) Exended capaciies (b) Funcional capaciies Figure 4: The resul of running CEA using A. += 5+0= =6 5+0= (a) Exended capaciies (b) Funcional capaciies Figure 5: The resul of running CEA using A 5. (5, 4, $) (7,, $) (6,, 4$) (, 7, 7$) (4, 5, 7$) (6,, $) (5, 4, $) 4 (8,, $) Figure 6: An example of he nework flow; arc labels presen as (u ij,m ij,c ij ).

8 8 Applied Mahemaics Table : Final resul of example. r Caegory The subse of A j relaed o Ω r The corresponding se of pahs wih Ω r Condiional probabiliy Ω (0.85, 0.9, 0.89, 0.99, 0., 0.7, 0., 0.68) (s,,,) 0.4 (0.66, 0.68, 0.7, 0.85, 0., 0.89, 0., 0.9) (s,, 4, ) (0.99, 0.85, 0.89, 0.4, 0.66, 0.7, 0., 0.68) Ω (0.85, 0.89, 0.9, 0.4, 0., 0.7, 0., 0.68) (0.89, 0.9, 0.4, 0.85, 0., 0.68, 0.66, 0.99) Ω (0.89, 0.9, 0.4, 0.7, 0., 0.68, 0.66, 0.99) 4 Ω 4 (0.99, 0.9, 0.66, 0.4, 0., 0.7, 0.68, 0.89) (s,,,) (s,,4,) (s,,,4,) (s,,,) (s,,4,) (s,,,4,) (s,,,) (s,,4,) (s,,,4,) (s,,,4,) We consider several subses A j of hese numbers ha each subse includes m members. Noe ha A j has been arranged as A j = (α s,α,α,α s,α,α 4,α 4,α 4 ).Also,weimplemen CEA using A j. In order o classify he resul, we place he resul of implemening CEA and reliabiliy algorihm in Table. We discard he subse (0., 0.9, 0.7, 0., 0.66, 0.7, 0.68, 0.4) or (0.85, 0.9, 0., 0.7, 0.8, 0.9, 0.4, 0.) because hese subses of ransmission facors reduce he funcional capaciy of he arcs whichoriginaefromhesourceorreachhesink,ohe exen ha hese arcs are unable o carry a leas V + V 0 uni of flow. Finally we can calculae he sysem reliabiliy as follows: 6. Conclusion p (V > 7.5) = (5) In his sudy, firs, we proposed an algorihm o change he arcs capaciies so ha he maximum flow value exceeds a predefinedvalueandheoalcosofchangeisminimum. In he algorihm (CEA), arc capaciies were expanded on he cheapes pahs from he source node o he sink node. Then, sysem reliabiliy was evaluaed when he arcs funcional capaciies were sochasic wih coninuous probabiliy funcion. There are several differen mehods in he lieraure for evaluaing of nework reliabiliy, when he random componens are mosly discree. In conras, we sudied ha sysem which has coninuous random componens. For each arc, a funcional capaciy was defined ha had coninuous uniform disribuion on [0, u ij ] andhen,heneworkreliabiliyis calculaed. Conflic of Ineress The auhors declare ha here is no conflic of ineress regarding he publicaion of his paper. Acknowledgmen The auhors are pleased o appreciae he respecful referee(s) forheirvaluablecommensandguidancehamadehis paper more accurae and credible. References [] L. Ford and D. Fulkerson, Maximal flow hrough a nework, Canadian Mahemaics, vol.8,no.,pp , 956. [] A. V. Karzanov, Deermining he maximal flow in a nework by he mehod of pre-flows, ovie Mahemaics Doklady, vol.5, pp , 974. [] R. K. Ahuja and J. Orlin, A capaciy scaling algorihm for he consrained maximum flow problem, Neworks, vol. 5, no., pp , 995. [4] N. Alon, Generaing pseudo-random permuaions and maximum flow algorihms, Informaion Processing Leers, vol.5, no. 4, pp. 0 04, 990. [5] R. K. Ahuja and J. B. Orlin, A fas and simple algorihm for he maximum flow problem, Operaions Research, vol. 7, no. 5, pp , 989. [6] A. Elalouf, R. Adany, and A. Ceder, Flow expansion on ransporaion neworks wih budge consrains, ocial and Behavioral ciences, vol. 54, pp , 0. [7] H. Eisel and H. von Frajer, On he budge-resriced max flow problem, OR pekrum, vol., no. 4,pp. 5, 98. [8] R.K.Ahuja,J.L.Bara,.K.Gupa,andA.P.Punnen, Opimal expansion of capaciaed ransshipmen neworks, European Operaional Research, vol.89,no.-,pp.76 84, 996. [9]W.C.Yeh, Asimpleapproachosearchforalld-MCs of a limied-flow nework, Reliabiliy Engineering & ysem afey, vol.7,no.,pp.5 9,00. [0] J. Xiao, G.-Q. Zu, X.-X. Gong, and C.-. Wang, Model and opological characerisics of power disribuion sysem securiy region, Applied Mahemaics, vol.04,aricle ID7078,pages,04. [] H. alehi Fahabadi and M. Forghani-elahabadi, A noe on A simple approach o search for all d-mcs of a limied-flow

9 Applied Mahemaics 9 nework, Reliabiliy Engineering and ysem afey, vol. 94, no., pp , 009. [] W. C. Yeh, A new approach o evaluae reliabiliy of mulisae neworks under he cos consrain, Omega, vol., no., pp. 0 09, 005. [] H. alehi Fahabadi and M. Khodaei, Reliabiliy evaluaion of nework flows wih sochasic capaciy and cos consrain, Inernaional Mahemaics in Operaional Research, vol. 4, no. 4, pp , 0. [4] Y. K. Lin and C. T. Yeh, Maximal nework reliabiliy wih opimal ransmission line assignmen for sochasic elecric power neworks via geneic algorihms, Applied of Compuing Journal, vol., no., pp , 0. [5] Y. K. Lin, Using minimal cus o sudy he sysem capaciy for a sochasic-flow nework in wo-commodiy case, Compuers & Operaions Research, vol. 0, no., pp , 00. [6] J.. Lin, Reliabiliy evaluaion of capaciaed-flow neworks wih budge consrains, IIE Transacions, vol.0,no.,pp , 998. [7] T. Aven, ome consideraions on reliabiliy heory and is applicaions, Reliabiliy Engineering and ysem afey, vol., no., pp. 5, 988. [8] M. O. Ball, Compuaional complexiy of nework reliabiliy analysis: an overview, IEEE Transacion on Reliabiliy, vol.5, no., pp. 0 9, 986. [9] R. K. Ahuja and T. L. Magnani, Nework Flows, Theory, Algorihms, and Applicaions, Prenice Hall, Englewood Cliffs, NJ, UA, 99. [0] J. Banks and J. Carson, Discree-Even ysem imulaion, Prenice Hall, New York, NY, UA, 984. [] A. M. Mood, F. M. Graybill, and D. C. Boes, Inroducion o Theory of aisics, McGraw Hill, 974.

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