Keywords parasitic flow loops, flow networks, dominated parasitic flow loops, optimization 1. PARASITIC FLOW LOOPS IN NETWORKS

Transcription

1 Internatonal Journal of Operatons Research Vol., No., (04) Domnated parastc flow loops n networks M.Todnov * Department of Mechancal Engneerng and Mathematcal Scences, Oxford Brookes Unversty Oxford, Wheatley, OX33 HX, UK Receved November 03; Revsed January 04; Accepted January 04 Abstract The paper ntroduces the concept domnated parastc flow loops and demonstrates that these occur naturally n real networks transportng nterchangeable commodty. The domnated parastc flow loops are augmentable broken loops whch have a domnatng flow n one partcular drecton of traversng. The domnated parastc flow loops are assocated wth transportaton losses, congeston and ncreased polluton of the envronment and are hghly undesrable n real flow networks. The paper derves a necessary and suffcent condton for the non-exstence of domnated parastc flow loops n the case of presence of paths wth zero and non-zero flow. The necessary and suffcent condton s at the bass of a method for determnng the probablty of a domnated parastc flow loop. The results demonstrate that the probablty of a domnated parastc flow loop s very large and ncreases very quckly wth ncreasng the number of flow paths. Domnated parastc flow loops can be draned by augmentng them wth flow, whch results n an overall decrease of the transportaton cost, wthout affectng the quantty of delvered commodty from sources to destnatons. Accordngly, an effcent algorthm for removng domnated parastc flow loops has been presented and a number of mportant applcatons have been dentfed. The presented algorthm has the potental to save a sgnfcant amount of resources to the world economy. Keywords parastc flow loops, flow networks, domnated parastc flow loops, optmzaton Internatonal ournal of Operatons Research. PARASITIC FLOW LOOPS IN NETWORKS Closed parastc flow loops n networks are hghly undesrable because they affect negatvely the qualty of servce of networks. Parastc flow loops crculate commodty unnecessarly, ncrease the cost of transportaton n the networks and cause polluton and congeston. Closed flow loops of matter and energy, whch are benefcal for the system, have been studed n ecosystems (Ulanowcz and Kay, 99). Closed data flow loops n genetc networks have also been recently reported n (Itzhack et al, 03). Undesrable (parastc) closed flow loops, n cases where the transported commodty physcally travels along a closed contour, have been reported n computer networks, where due to faults n the routng, the packets physcally travel along a closed loop (Hengartner et al, 00; Paxson, 997). Sharr (98) consdered strongly connected components of a graph, whch mples the exstence of closed cyclc paths. Todnov (03a) establshed that closed parastc flow loops could appear even f the transported commodty does not physcally travel along a closed contour. Ths work establshed for the frst tme that closed parastc flow loops occur naturally n real networks dspatchng the same type of nterchangeable commodty - fuel, electrcty, gas, chemcals, partcular goods, partcular servces, etc. by followng drected straght-lne paths from sources of flow to destnatons. The example n Fg., taken from (Todnov 03a) llustrates ths phenomenon. Suppose that the network n Fg.a transports nterchangeable commodty (for example, the same type and qualty of petrol, electrcty, the same type goods etc.). The throughput capacty of each edge n the network f Fg.a s 00 unts. Assume that the commodty s transported along straght-lne paths only: along path (,,3,4,5,6), path (7,8,9,4,0,,) and path (3,4,0,3,5,6). Despte that the commodty dspatched from each node travels along a straght lne to ts destnaton and no commodty physcally travels along a closed contour, a drected loop carryng 00 unts of flow essentally appears between the ntersecton nodes 3,4 and 0. Removng 00 unts of flow from the drected flow loop (3,4,0,3) reduces the flow along edges (3,4), (4,0) and (0,3) to zero, wthout dmnshng the amount of total flow sent from the source nodes to the destnaton nodes (Fg.b). Even for very few ntersectng flow paths, the probablty of a drected flow loop s large, whch means that the exstence of drected parastc flow loops n real networks s practcally nevtable. Thus, for 'n' randomly orented ntersectng straght-lne flow paths on a plane, the probablty of a parastc flow loop between some of the ntersecton * Correspondng author s emal: mtodnov@brookes.ac.uk

2 IJOR Vol., No., (04) a) b) 7 s 7 s t3 s3 t3 s t 5 5 s s t t Fgure. a) Closed parastc flow loops may appear even f the transported commodty does not travel along a closed contour; b) Closed parastc flow loops can be removed wthout affectng the throughput flow from sources to destnatons (Todnov 03a). ponts s equal to n / n and approaches very quckly unty wth ncreasng the number of ntersectng paths 'n' (Todnov, 03b). Despte the large number of publshed algorthms on optmsng the flows n networks (Ahua et al.,993; Asano and Asano, 000; Cormen at al. 00), the queston related to dentfyng and removng closed parastc flow loops n flow networks, has evaded the efforts of researchers workng n ths feld. Indeed, despte the years of ntensve research on statc flow networks, a surprsng omsson (dscussed n detal n Todnov (03a)) has been made: the algorthms for maxmsng the throughput flow publshed snce 956 leave hghly undesrable parastc flow loops n the optmsed networks. The treatment of parastc flow loops presented n (Todnov 03a) was lmted to closed parastc flow loops only. In (Todnov 03b), almost-drected flow loops have been ntroduced, where the drecton of the flows n all but one edge s along a partcular traversng drecton. The almost-drected flow loops were also shown to be assocated wth losses. However, no treatment of the probablty of an almost-drected flow loops has been presented n (Todnov 03b) or an algorthm for removng almost-drected flow loops. The present paper goes sgnfcantly beyond the treatment n (Todnov 03a, 03b) by showng that the loops do not have to be drected or almost-drected to be assocated wth losses. The present paper demonstrates that augmentable loops whch have only domnatng flow along a partcular drecton of traversal are also assocated wth transportaton losses and congeston, and are hghly undesrable n real flow networks. Consequently, the purpose of ths paper s to extend the analyses presented n (Todnov 03a and 03b) by: () Introducng and defnng the general concept domnated parastc flow loop ; () Demonstratng that smlar to the drected and almost-drected parastc flow loops, the domnated parastc flow loops are also assocated wth sgnfcant losses and ther dranng/removal has the potental to save a sgnfcant amount of resources to the world economy. () Suggestng a general algorthm for optmsng real networks by removng all drected, almost-drected and domnated parastc flow loops.. TRANSPORTATION LOSSES ASSOCIATED WITH DOMINATED PARASITIC Loops A domnated parastc flow loop s defned along a closed contour wth a specfed drecton of traversng (Fg.a). It s a sequence of n sectons/edges ( n 3 ) n whch the flow along n k sectons/edges (not necessarly sequental) ponts aganst the drecton of traversng of the contour and the rest of the k sectons/edges (the closng edges) are ether empty or partally flled wth forward flow. The n-k edges wth backward flow wll be referred to as backward edges/sectons whle the remanng k edges wll be referred to as closng edges/sectons. For a domnated parastc flow loop to be present, the cyclc path must be augmentable and the sum of the lengths of the n k edges wth backward flow must be greater than the sum of the lengths of the closng k edges: n k k ( b) ( c) l l () t 83-73X Copyrght 04 ORSTW

3 IJOR Vol., No., (04) 3 a) b) n d s d3 s d3 c) 9 0 d) 9 0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 s 0/ d 0/0 0/0 0/ /0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/ /0 0/0 0/0 0/0 s d s3 d s3 Fgure. a) A domnated parastc flow loop; b,c) Removal of a domnated parastc flow loop. where l ( b) s the length of the th backward edge and l ( c) s the length of the th closng edge. Essentally, the sum of the lengths of the edges wth backward flows domnates the sum of the lengths of the closng edges. The domnated flow loop can be thought as an augmentable broken flow loop for whch some of the flow travels n backward drecton and the sum of the lengths of the backward sectons domnates the sum of the lengths of the closng sectons. To be augmentable, the closng edges must ether be empty or partally saturated wth forward flow. Let be the flow magntude by whch a domnated flow loop can be augmented wthout volatng the capacty constrants on the edges. The domnated parastc flow loop can be augmented (essentally draned) by decreasng the flow along the n-k edges wth domnatng (backward) flow by and ncreasng the flow wth the same amount along the closng k edges. The dranng operaton does not volate the flow conservaton at each node and the capacty constrants along the edges and leads to a new feasble flow n the network. Note that the closed parastc flow loops ntroduced n Todnov (03a) and the almost-drected flow loops ntroduced n Todnov (03b) are only specal cases of a domnated parastc flow loop. The concept domnated parastc flow loop s much broader. It ncludes not only drected parastc loops and almost-drected parastc loops (broken at a sngle edge), but also augmentable loops broken at more than a sngle edge. Note that for a domnated parastc flow loop to be present, the sum of the lengths of the edges wth backward flow must be greater than the sum of the lengths of the closng edges (see equaton ()). Ths condton s not requred for the almost-drected parastc flow loops ntroduced n (Todnov, 03b), whch have a sngle ( k ) closng edge (Fg.b). Indeed, the length of a polygonal path between two ponts s always greater than the dstance between the two ponts. Therefore, f the only closng edge s the n-th edge, the nequalty () becomes n ( b ) ( c ) n l l, () whch s automatcally fulflled. As a result, nequalty () s automatcally fulflled for an almost-drected flow loop. However, for a domnated flow loop wth more than a sngle closng edge, nequalty () may not be fulflled. Domnated parastc flow loops are assocated wth sgnfcant losses and ther removal/dranng s hghly benefcal. After the dranng of a domnated flow loop, ether one or more edges wth backward flow wll become empty or one or more of the k closng edges loop wll become fully saturated wth flow. As a result, the domnated parastc loop wll be removed. Fgure c features the domnated parastc flow loop (,7,6,5,4,3,). The backward edges are the edges (,7),(7,6),(6,5) and (5,4). The closng edges are the empty edges (4,3) and (3,). The frst number of the edge labels denotes the edge 83-73X Copyrght 04 ORSTW

4 IJOR Vol., No., (04) 4 a) b) u 3 u O a u x u 4 u 5 Fgure 3. a) Randomly orented ntersectng flow paths; b) All drecton vectors of the source-destnaton paths can be translated to start from a common pont O. capacty whle the second number stands for the actual flow along the edge. The loop can be draned by augmentng t wth 0 unts of flow n the drecton (,7,6,5,4,3,). The result s the network n Fg.d, where no domnated parastc flow loops exst. Domnated parastc flow loops and augmentable broken loops wth domnatng flow along a partcular drecton of traversng are equvalent concepts. If the cost of transportaton per unt dstance does not vary on the dfferent edges, dranng an augmentable broken flow loop wth domnatng flow always results n a reducton of the transportaton cost. In ths case, the followng theorem can be stated. Theorem. Augmentable broken loops, wth domnatng flow along a partcular drecton of traversng, are assocated wth an ncreased transportaton cost whch can be reduced by dranng the loops. Proof. Denote the cost of transportaton per unt length by c. Consder the n-k edgfes wth domnatng backward flow. ( b) ( b) Denote the length of these edges by l, l,..., l 83-73X Copyrght 04 ORSTW ( b) n k ( c) ( c) ( ). Denote the lengths of the k closng edges by l, l,..., l c. Accordng to the defnton of a domnated parastc flow loop, nequalty () holds. Because the cost of transportaton per unt length s the same, augmentng the loop wth the maxmum possble (bottleneck) flow the transportaton cost by k ( c) max n k ( b) max max wll result n a reducton of c l along the backward edges and an ncrease of the transportaton cost by c l along the closng k edges. The augmentaton (dranng) s performed aganst the drecton of the domnatng flow along the backward edges. From nequalty (), the nequalty k n k ( c) ( b) ( ) 0 mn c l l s obtaned, whch means that dranng a broken loop wth domnatng flow along a partcular drecton of traversng results n a net decrease of the cost of transportaton. Consequently, the exstence of a domnatng flow along augmentable broken loops s assocated wth ncreased transportaton costs. 3. ESTIMATING THE PROBABILITY OF A DOMINATED PARASITIC FLOW LOOP The unexpectedly hgh probablty of exstence of domnated parastc flow loops wll be demonstrated by consderng randomly orented ntersectng straght-lne flow paths on a plane (Fgure 3a). All flow paths transport the same type of nterchangeable commodty (e.g. petrol) and along each flow path, there s ether flow wth a partcular drecton (Fg.3a, the lnes wth an arrow) or no flow (Fg.3a, the lnes wthout arrows). It s assumed that there are at least three ntersectng flow paths; there are no parallel paths and no three paths ntersect nto a sngle pont. These condtons are natural and common. Indeed, for randomly orented straght-lne flow paths n a plane, t s very unlkely to fnd two parallel paths or three paths ntersectng nto a sngle pont. The lkelhood that a domnated k

5 IJOR Vol., No., (04) 5 a) b) b B u u 3 b b x C b C' O u A b 3 b 3 b 3 Fgure. 4. a) Drecton vectors of the non-zero flow paths; b) A drected parastc flow loop formed by the three flow paths; parastc flow loop wll be present between the ntersecton ponts of the network, gven that the orentaton of the ntersectng flow paths s random, wll be termed probablty of a domnated parastc flow loop for random flow paths n a plane. The exstence of a domnated parastc flow loop anywhere between the ponts of ntersecton mples the exstence of a trangular domnated parastc flow loop. As a result, the exstence of a trangular domnated parastc flow loop s a necessary condton for the exstence of a domnated parastc flow loop. The exstence of a trangular domnated parastc flow loop s also a suffcent condton for the exstence of a domnated parastc flow loop. Consequently, the probablty of a domnated parastc flow loop for randomly orented flow paths can be estmated by estmatng the probablty of a trangular domnated parastc flow loop. A unt vector can be assgned to each flow path (Fg.3b,4a), pontng n the drecton of the flow along the flow path. The angle whch the unt vector subtends wth the horzontal axs (Fg.3b) gves the orentaton of the flow path and the drecton of the flow along the path, f the path carres non-zero flow. Each flow path s randomly orented, whch means that the angle whch the drecton vector subtends wth the fxed horzontal x-axs s unformly dstrbuted n the nterval (0, π). In other words, all possble flow path orentatons are charactersed by the same probablty. The unt vectors assgned to the flow paths wll be referred to as drecton vectors. They can all be translated at the common orgn O, as shown n Fg.3b and 4a. Two types of drectons wll be dstngushed: non-zero flow drecton vectors, correspondng to flow n paths wth non-zero flow and zero flow drectons - correspondng to the orentaton of flow paths whose flow s zero. It s assumed that the flow through the non-zero flow paths flls the entre avalable throughput capacty of the path. For the sake of brevty, the non-zero flow drecton vectors wll be referred to as nz-vectors. Theorem. If t s possble to select three nz-vectors whch do not le n the same half plane, then a drected parastc flow loop s present. Proof. Suppose that the three nz-vectors u, u and u 3, do not le n a sngle half-plane (Fg.4a). Then, each of the angles,, 3 between the vectors, s smaller than and the sum of the angles s exactly equal to : 0,,,3 ; 3 The ntersectons of the flow paths form a trangle, the external angles of whch correspond to the angles, and between the nz-vectors (Fg.4b). Because of the relatonshp, the flows along the sdes of the trangle 3 3 determned by the ntersectons of the flow paths, always form a closed drected flow loop (Fg.4b). Theorem 3. If all nz-vectors le n the same half plane, a drected parastc flow loop does not exst. Proof. Suppose that the flow paths wth nz-vectors u, u and u 3, defne a trangular drected flow loop (Fg.5a). The sdes of the trangular loop can then be presented by k u, k u and k 3u 3, where k, k and k 3 are some constants. The sum of the vectors k u, k u and k 3u 3 s zero and as result, the relatonshp (Todnov 03a): k u + k u + k 3u 3 = 0 (3) holds. A smlar relatonshp also holds for the proectons of the vectors k u, k u and k 3u 3 along any specfed axs. Thus, for an axs defned by a drecton vector s, expandng (k u + k u + k 3u 3)s = 0 yelds the relatonshp (Todnov 03a): m s + m s + m 3s = 0 (4) b B' 83-73X Copyrght 04 ORSTW

6 IJOR Vol., No., (04) 6 a) b) y ku u u u 3 ku ku 3 O x -y Fgure.5. a) A closed trangular parastc flow loop; b) the drecton vectors of the flow paths wth nonzero flow cannot possbly resde n a sngle half-plane (Todnov, 03b). a) b) C C 0/0 0/0 0/0 0/0 A 0/0 B A 0/0 B c) C 0/0 0/0 A 0/0 B Fgure 6. a) Intersectng two non-zero flow paths wth a zero flow path yelds a domnated parastc flow loop b) The draned domnated parastc flow loop from a); c) Intersecton of two non-zero flow paths wth a zero flow path does not always result n a domnated parastc flow loop. for the proectons m s=k u s, m s=k u s and m 3s=k 3u 3s. Consequently, equaton (4) s a necessary condton for the exstence of a drected trangular loop. Suppose that there s a sngle half plane where all nz-vectors resde (Fg.5b) and a drected parastc flow loop exsts. The exstence of a drected parastc plow loop mples the exstence of a trangular drected parastc flow loop. The nz-vectors u, u and u 3 of the paths formng the drected trangular loop wll also resde n the same half-plane. Wthout loss of generalty, suppose that the axs x and the half-axs (O,y) defne ths half-plane (Fg.5b). The sum of the proectons of the nz-vectors along the y-axs wll now volate the necessary condton (4), because there wll be a non-zero sum of proectons along the axs (O,y) (m y, + m y + m 3y > 0). We arrved at a contradcton. Therefore, f all nz-vectors le n the same half-plane, a drected parastc flow loop does not exst. Here, t needs to be ponted out mmedately, that although three non-zero flow paths may not form a parastc flow loop, two non-zero flow paths ntersected by a zero flow path may form a domnated parastc flow loop. For example, n Fg.6a, the non-zero flow paths BC and CA, each wth capacty 0 unts flow per unt tme and each carryng 0 unts of flow per unt tme, have been ntersected by the zero flow path AB wth capacty 0 unts of flow per unt tme and carryng no flow. (The frst number on the edge labels stands for the capacty of the edge and the second number stands for the actual flow along the edge). The cyclc path ACBA s augmentable. Ths cyclc path s essentally a domnated parastc flow loop from whch 0 unts of flow can be draned. The result s the network n Fg.6b. The non-zero flow paths n Fg.6c have also been ntersected wth a zero flow path. In ths case however, no domnated parastc flow loop s present X Copyrght 04 ORSTW

7 IJOR Vol., No., (04) 7 Lemma. If a sngle path wth zero flow and more than one non-zero flow paths are present, a domnated flow loop s absent f all random nzvectors correspondng to the non-zero flow paths resde n a sngle half-plane defned by the drecton of the zero flow path. Lemma. If a sngle path wth zero flow and more than one non-zero flow paths are present, a domnated parastc flow loop s present f the nonzero flow paths do not resde n a sngle half-plane defned by the drecton of the zero flow path. These lemmas (see Fg.7a) follow from Theorem 3 and wll not be proved here. The probablty that all n random nzvectors wll resde n one of the two half-planes defned by the drecton of the zero flow path s 83-73X Copyrght 04 ORSTW / n. Consequently, for n non-zero flow paths and a sngle zero flow path, the probablty of exstence of a domnated flow loop s p / n. Theorem 4. If more than one zero flow paths and more than one non-zero flow paths are present, domnated parastc flow loops are absent f and only f all nz-vectors resde n one of the sectors defned by the drectons of the zero flow paths. In Fg.7b, the four sectors defned by the drectons of the two zero flow paths are,, and. No domnated 3 4 parastc flow loop s present because all nz-vectors, defnng the drectons of the flows n the nz-paths, resde n sector 3. In Fg.7c, however, the nz-vectors do not resde n a sngle sector defned by the drecton of the sngle zero flow path z. As a result, a domnated parastc flow loop wll be present. Proof of Theorem 4. Gven that a domnated parastc flow loop s absent, suppose the contrary that not all nz-vectors resde n a sngle sector defned by the drectons of the zero flow paths. In ths case, there wll be two nz-vectors, each resdng n a dfferent sector (Fg.7d, the vectors u and u resdng n sectors and ). Because the sectors and do not overlap, there s a zero flow path whose drecton z defnes two half-planes each of whch contans one of the two 0 sectors. As a result, the two nz-vectors resdng n these sectors wll resde n dfferent half planes, wth respect to the drecton z 0 of the zero flow path (Fg.7d). In ths case, accordng to Lemma, a domnated parastc flow loop wll be present, whch contradcts our assumpton that a domnated flow loop s absent. Consequently, the nz-vectors cannot resde n dfferent sectors; they all must resde n a sngle sector. Suppose now that all nz-vectors resde n a sngle sector. We wll show that, n ths case, no domnated parastc flow loop exsts. Indeed, the fact that all nz-vectors resde n a sngle sector mples that they all resde n a sngle half-plane. The exstence of a domnated flow loop mples the exstence of a domnated trangular flow loop. The domnated trangular parastc flow loop cannot contan only nz-paths, because all nz-vectors resde n a sngle half-plane. Accordng to Theorem 3, no drected parastc flow loop exsts n ths case. The only possblty that remans s the trangular domnated flow loop to contan two nz-paths and a sngle zero flow path. However, ths s mpossble because wth respect to any selected zero flow path, both of the nz-vectors wll resde n the same half-plane formed by the drecton of the zero flow path. Accordng to Lemma, the zero flow path and the two nz-paths cannot form a domnated trangular flow loop. Consequently, no domnated parastc flow loop exsts f all nz-vectors resde n a sngle sector. The theorem has been proved. Accordng to Lemma, n the flow path confguratons presented n Fg.8a and Fg.8b, a domnated parastc flow loop exsts. Ths s because the nz-vectors do not resde n one of the two sectors(half-planes) defned the drecton of the zero flow path. In addton, the sum of the lengths of the two edges wth non-zero flow, both pontng aganst the specfed drecton of traversng the loop, are always greater than the length of the edge wth zero flow. In Fg.8c and 8d, both nz-vectors resde solely n one of the two sectors defned by the drecton of the zero flow path and, accordng to Lemma, no domnated parastc flow loop exsts. Domnated parastc flow loops do not exst n the case where two zero flow paths are ntersected wth a sngle non-zero flow path. In ths case, condton () for the exstence of a domnated parastc flow loop s volated because the sum of the lengths of the edges wth zero flow (the closng edges) s always greater than the length of the edge wth non-zero flow (the backward edge). 3. Smulaton algorthm and results The orentaton of the nz-vectors charactersng the non-zero flow paths s determned smply by generatng random numbers, unformly dstrbuted n the nterval (0, ). The orentaton of the zero flow paths s determned by generatng random numbers dstrbuted n the nterval (0, ).

8 IJOR Vol., No., (04) 8 a) b) u 3 u u z z b b b z c) d) u 3 b 3 u u u 3 u u b z z 0 u u b Fgure 7. a) and b) A domnated parastc flow loop does not exsts; c) A domnated trangular parastc flow loop exsts; d) Two nz-vectors n two dfferent sectors. a) b) C C A B A B c) d) C C A B Fgure 8. a,b) Flow confguratons whch: a,b) result n a domnated flow loop; c,d) do not result n a domnated flow loop. Because the exstence of a domnated flow loop between the ponts of ntersecton of the flow paths mples the exstence of a trangular domnated flow loop, determnng the probablty of a domnated parastc flow loop can be reduced to determnng the probablty of a trangular domnated flow loop. If no zero flow paths are present, determnng the A B 83-73X Copyrght 04 ORSTW

9 IJOR Vol., No., (04) Probablty Total number of flow paths Fgure 9. Probablty of a domnated parastc flow loop as a functon of the total number of flow pats. The number of non-zero flow paths s equal to the number of zero flow paths Probablty non-zero flow paths=3 non-zero flow paths= Number of zero flow paths Fgure 0. Probablty of a domnated parastc flow loop as a functon of the number of randomly orented zero flow paths, for n= and n=3 non-zero flow paths. probablty that no parastc flow loop s present reduces to determnng the probablty that all non-zero vectors le n a sngle half-plane. Ths s equvalent to determnng the exstence of an angle greater than between two adacent nz-vectors. Accordng to Todnov (03a) ths probablty s n / n, where n s the number of non-zero flow paths. Subtractng ths probablty from one, gves the probablty ( n / n ) that there wll be a parastc flow loop. If zero flow paths are also present, determnng the probablty that no domnated parastc flow loop s present reduces to determnng the probablty that all nz-vectors wll resde n a sngle sector formed by the drectons of the zero flow paths. Ths check can be performed easly, f the random orentaton angles of the zero flow paths and the nz-vectors are sorted (separately) n ascendng order X Copyrght 04 ORSTW

10 IJOR Vol., No., (04) 0 The algorthm s gven n Appendx A. For randomly orented non-zero flow paths and zero flow paths, such as n Fg.3a, the probablty of a domnated parastc flow loop for a dfferent total number of flow paths has been plotted n Fg.9. The number of zero flow paths s equal to the number of non-zero flow paths. Wth ncreasng the total number of flow paths, the probablty of a domnated flow loop approaches unty very quckly. In Fg.0, the probablty of a domnated flow loop has been plotted as a functon of the number of zero flow paths, for a mxture of zero flow paths and non-zero flow paths, where the number of non-zero flow paths s n= and n=3, respectvely. Agan, wth ncreasng the number of randomly orented zero flow paths, the probablty of a domnated parastc flow loop ncreases very quckly. For three ntersectng non-zero flow paths and two zero flow paths, the probablty of a domnated parastc flow loop s about 87%! These results show that the exstence of domnated flow loops on a set of ntersectng zero and non-zero flow paths s practcally nevtable. 4. COST FACTOR AND CYCLIC PATHS WITH A NEGATIVE COST FACTOR 4. Cost factor of an augmentable path and cancellng domnated parastc flow loops from networks Consder a network wth undrected edges. A path n a flow network s a unque sequence of edges between two nodes. Suppose that the flow through a partcular path can be ncreased by wthout volatng the capacty constrants on the edges and the flow conservaton law at the nodes along the path. Such a path wll be referred to as augmentable path. Consder an edge whose flow drecton s opposte to the drecton of traversng of the augmentable path. Accordng to an earler conventon, such edges are referred to as backward edges. Smlarly, edges whch are empty or whose flow drecton s along the drecton of traversng the augmentable path are referred to as closng edges. Suppose that the cost of transportaton per unt length s c unts. Augmentng the path wth means that the flow along each backward edge s decreased and the flow along each forward edge s ncreased. Because, durng the path augmentaton, the flow along the closng edges s ncreased, the transportaton cost along the closng edges wll ncrease. For the th closng edge, the ncrease of the cost of transportaton (c) l C s equal to C c (5) where l ( c) s the length of the th closng edge. The ncrease of the cost of transportaton for a path composed of c M c ( c) M closng edges only, s gven by c C c l (6) Suppose that a path contanng both closng edges and backward edges s augmented wth flow wth magntude. After the path augmentaton, along any backward edge, the flow s no longer transported but s prevented from beng transported, because the flow along backward edges has been decreased by. Therefore, the ncrease n the transportaton cost For C, along a backward edge wth ndex, has a negatve sgn: C c l (7) () b M backward edges, the total ncrease of the cost of transportaton can be approxmated by b b M b () b C c l (8) where l ( b) s the length of the th backward edge. Consequently, the total ncrease of the cost of transportaton edges can be approxmated by Mc Mb ( c) ( b) f b C along a path ncludng both backward and closng C C C c l l (9) The quantty Mc Mb ( c) ( b) l l (0) 83-73X Copyrght 04 ORSTW

11 IJOR Vol., No., (04) wll be referred to as a cost factor of a path. As t wll be shown later, ths concept s of fundamental mportance to the optmal soluton. A postve sgn of the cost factor means that f the path s augmented wth flow, the transportaton cost wll ncrease. A negatve sgn of the cost factor means that the transportaton cost wll decrease followng the path augmentaton. The larger the magntude of the cost factor, the larger s the amount of the transportaton cost ncrease/decrease followng the path augmentaton. A domnated parastc flow loop s always charactersed by a negatve cost factor. The augmentaton of a domnated parastc flow loop results n a decrease of the transportaton cost. At the same tme, the amount of flow transported from sources to destnatons s not affected. Effectvely, augmentng a parastc flow loop n a drecton opposte to the drecton of ts domnant (backward) flow, s a process of dranng the flow loop. An ndcaton of the presence of a domnated flow loop s the presence of an augmentable cyclc path wth a negatve cost factor, anywhere n the network. Now suppose that there s a set of sources wth specfed generaton and a set of consumers wth specfed consumpton. The generated and consumed commodty s exchangeable, whch means that any gven consumer could be suppled from any gven source. The followng theorem then holds. Theorem 5. A necessary and suffcent condton for a mnmum transportaton cost s the non-exstence of a cyclc path wth a negatve cost factor. Proof. Provng that the non-exstence of augmentable cyclc paths wth a negatve cost factor s a necessary condton for a mnmum transportaton cost s straghtforward. Suppose that for a gven total throughput flow f * from sources to consumers, the transportaton cost s the smallest possble. If there exsts an augmentable cyclc path wth a negatve cost factor, ths path could be augmented, whch wll result n dfferent feasble edge flows, wth the same throughput flow f * from sources to consumers but wth a smaller transportaton cost. However, ths s mpossble because, by assumpton, the transportaton cost s the smallest possble. We arrve at a contradcton. Suppose now, that n a network charactersed by edge flows f (, ) and throughput flow f * from sources to consumers, there s no augmentable cyclc path wth a negatve cost factor. Then, the transportaton cost C, assocated wth edge flows f (, ) s the smallest possble. Indeed, suppose that edge flows f (, ) exst, assocated the same throughput flow f * and a smaller transportaton cost C C. To prove ths part of the theorem, t s necessary to use a theorem, stated by Ahua et al. (993). Here, ths theorem wll be stated as Lemma 3, to reflect the crcumstance that t wll be used to prove Theorem 5. Lemma 3. If there are two dfferent feasble edge flows f (, ) and f (, ), resultng the same throughput flow, the flow f (, ) can be presented as a sum of the flow f (, ) and the augmented flows along a set of augmentable cyclc paths. Accordng to Lemma 3, the set of feasble edge flows f (, ) resultng n throughput flow f *, can be obtaned from the set of feasble edge flows f (, ), resultng n the same throughput flow f *, by a sum of augmentatons along cyclc paths only. Wthout loss of generalty, suppose that the edge flows f (, ) have been obtaned from edge flows f (, ), after addng the augmented flows along k cyclc paths. The expected transportaton cost C s then gven by C C c c... c () k k where s the cost factor of the th cyclc path and s the flow wth whch the th cyclc path s augmented. Because of the assumpton C C (the edge flows f (, ) are assocated wth smaller losses than edge flows f (, ) ), the nequalty c c... c 0 () k k must necessarly hold. Ths nequalty however s mpossble because, accordng to our assumpton, there s no augmentable cyclc path wth a negatve cost factor. Ths completes the proof. Consequently, the transportaton cost C assocated wth edge flows f (, ) s ndeed the smallest possble. Dranng all cyclc paths wth negatve cost factors leads to the smallest transportaton costs n the network. 4. Examples of networks wth domnated flow loops X Copyrght 04 ORSTW

12 IJOR Vol., No., (04) a) 60/0 5 60/0 s 60/60 0/0 40/40 0/0 3 60/60 0/0 40/ /60 0/0 d 3 d s d s 3 b) 60/ /40 s 60/60 0/0 40/0 0/0 3 60/0 40/0 0/0 4 60/60 0/0 d 3 d s d s 3 Fgure. Electrcty supply from three generators s, s, s 3, to three consumers d, d, d. 3 Typcal examples of domnated flow loops exst n cases where a number of consumers are served by a number of sources and the commodtes/servces provded to the consumers are exchangeable. Here are some examples: - A number of customers, lvng at dfferent locatons n a large cty are servced by a number of reparers of a partcular type of equpment. The reparers belong to the same company and are also based at dfferent locatons n the cty (each customer s servced equally well by any reparer). - Generators at dfferent locatons supply electrcty to customers. - Fuel termnals at dfferent locatons supply fuel to dfferent petrol statons. - A contaner supply company, wth branches at dfferent locatons n a country, delvers the same type of contaners to clents. - Warehouses at dfferent locatons, supply supermarkets wth the same type of goods. - Students spendng ther ndustral placement year at dfferent locatons n a country, are vsted by members of staff lvng n dfferent towns; - Volunteers workng for a charty and resdng at dfferent locatons, vst elderly patents; - Agents lvng at dfferent locatons n a cty, travel to advertse the same type of commodtes to prospectve customers. Ths lst can be contnued. As can be seen from the examples, domnated parastc flow loops are present even n socal support networks. Consder the task of supplyng electrcty from three generators wth capactes s 60 MW, s 0 MW and s 0 MW, to three consumers wth consumpton capactes d 0 MW, d 0 MW and d 60 MW. For the 3 3 sake of smplcty, assume that the cost of transmttng MW electrcty for hour, along each secton, s m. One soluton s presented n Fg.a. The frst number on each edge/lne s the lne capacty (n MW) and the second number s the actual power flow transmtted through the lne. The soluton from Fg.a however, can be mproved by notcng that the loop (4,3,,,5,4) s a domnated parastc flow loop. Augmentng ths loop wth 40 MW results n the network flows from Fg.b. The throughput flow from generators to consumers has not been affected but the transportaton cost per hour has been reduced by 40m. For one year of contnuous operaton, ths saved transportaton cost wll accumulate to a very large sum ( m m). As can be verfed, the soluton n Fg.a was not the optmal soluton. Another nterestng applcaton exsts n the case where n volunteers belongng to the same organsaton and lvng n dfferent parts of a cty are allocated to n patents also lvng n dfferent parts of the cty (Fg.). Each volunteer must be assgned to exactly one patent. In ths case, only the total length of the servce routes matters, not the capactes of the routes. Consequently, a common capacty of unt can be assgned to each edge. Because each volunteer must vst exactly 83-73X Copyrght 04 ORSTW

13 IJOR Vol., No., (04) 3 d s d 5 d s s 4 d s d Fgure. A real flow network before the optmsaton. s 5 one patent, the generaton capactes of the sources s, s,..., s 5 and the consumpton capacty of the consumers d, d,..., d must also be set to be unt (Fg.). For the sake of smplcty, t has been assumed that all edges have the same 5 length. The flow n the network s nterchangeable, whch means that the type of commodty dspatched from any gven source can satsfy any gven consumer. One way of satsfyng the demand from consumers s shown n Fg.. Each source has been connected wth the correspondng destnaton through a drect, straght-lne path. As t can be verfed, ths soluton s far from optmal. There are domnated flow loops, whch ncrease unnecessarly the cost of transportaton of the commodty n the network. For example, the network contans the parastc flow loop (,3,4,8,,,0,6,), the domnated parastc flow loop (4,0,9,3,4), etc. If the cost of transportaton of per edge s 00 unts, the cost of transportaton of the commodty to all destnatons amounts to 500 unts. After augmentng the parastc flow loop (,3,4,8,,,0,6,) wth unt flow and the domnated parastc flow loop (4,0,9,3,4) wth unt of flow, the network from Fg.3 s obtaned. No augmentable cyclc paths wth negatve cost factor exst, hence, accordng to Theorem 5, the obtaned schedulng s assocated wth the smallest transportaton cost. The resultant network s charactersed by a transportaton cost of 500 unts only, whch consttutes a 40% reducton of the ntal transportaton costs. Reducng the length of the supply paths n socal support networks reduces not only the transportaton costs and polluton but also the rsk of delays. Suppose that the events causng delay along the supply paths follow a homogeneous Posson process wth ntensty, and the total length of the supply paths s L. The probablty of havng a delay along the supply paths s gven by the negatve exponental dstrbuton p exp( L ) (3) d Decreasng the total length L of the supply paths reduces sgnfcantly the probablty p of a delay n the socal d support networks. 4.3 Algorthm for removng domnated parastc flow loops from networks Suppose that n sources supply s, s,..., s quanttes of nterchangeable commodty to m consumers wth n consumptons c, c,..., c m : 83-73X Copyrght 04 ORSTW n m s c. Step. The frst step of the algorthm for removng domnated parastc flow loops conssts of transformng the ntal flow network. A flow network wth multple sources and consumers (the n sources n Fg.4a wth flow generaton

14 IJOR Vol., No., (04) 4 d s d 5 d s s 4 d s 5 s d Fgure 3. The flow network after the optmsaton. Fgure 4 a) An example of a flow network wth n sources of nterchangeable commodty and m consumers. b) The network (a) has been reduced to a flow network wth a sngle super-source s, and a sngle super-consumer t. quanttes s, s,..., s n and the m consumers wth consumpton quanttes c, c,..., c ) s frst reduced to a flow network m wth a sngle super-source s (Fg.4b) and a sngle super-consumer t. The multple sources n Fg.4a are replaced by a sngle super-source wth nfnte generaton feedng each of the ntal sources through lnes wth throughput flow capactes s, s,..., s, equal to the amounts of commodty ntally suppled by n the sources. In a smlar fashon, the multple consumers n Fg.4a are replaced by a sngle consumer wth nfnte consumpton capacty. The super-consumer s connected through lnes wth throughput flow capactes c, c,..., c equal m to the amount of consumed quanttes (Fg.4b). The sources of flow and the consumers become ordnary throughput edges, wth flow capactes equal to the flows generated by the sources and the flows consumed by the consumers. As a result, the sources of nterchangeable commodty and consumers dsappear, and nstead, ordnary throughput edges appear. Step. Assgn zero transportaton costs to the edges connectng the super-source s and the super-consumer t wth the rest of the network. For the rest of the edges, assgn cost of transportaton proportonal to the length of each edge. Step 3. Ths step conssts of maxmsng the flow n the transformed network (Fg.4b), from the super-source s to the super-consumer t, at a mnmum transportaton cost. A number of algorthms have already been proposed for maxmsng the flow at a mnmum cost (Klen, 967; Bennngton, 973; Fresdorf and Hamacher 98; Tardos 985; Goldberg and 83-73X Copyrght 04 ORSTW

15 IJOR Vol., No., (04) 5 Taran 987,989; Ahua et al. 99; Orln 993), some of whch are charactersed by a strctly polynomal runnng tme (Tardos 985; Orln 993). The successve shortest path method wth modfed weghts could for example be used n Step 3 for maxmsng the throughput flow at a mnmum cost. Step 4. The super-source s, the super-consumer t and ther connectng edges are removed from the transformed network. The nodes correspondng to the ntal sources and consumers become sources and consumers agan. The edge flows n the resultant network defne the optmal soluton Proof of the correctness of the proposed algorthm. After conductng Step 4, t can be shown that there are no domnated parastc flow loops left n the orgnal network. Indeed, suppose that the specfed throughput flow has been guaranteed and the sum of the costs of transportaton along the edges s the smallest possble. Suppose that there exsts a domnated parastc flow loop (, +,..., + c, ) n the network, startng and endng at a partcular node. The bottleneck resdual capacty Δ of the loop s then determned and the loop s augmented wth the bottleneck flow Δ. Because the augmentaton s along a cyclc path, t does not affect the throughput flow n the network. The throughput flow from sources to destnatons wll reman unchanged after the augmentaton. Durng the augmentaton (dranng) of the parastc flow loop, the backward flows along the loop are reduced (the bottleneck flow s prevented from beng crculated), and the flow along the closng edges s ncreased. Because the sum of the lengths of the backward edges s larger than the sum of the lengths of the closng edges, the overall cost of transportaton wll be reduced. However, ths contradcts the assumpton that the transportaton cost assocated wth the ntal edge flows s the smallest possble. Consequently, there can be no domnated parastc flow loops n the transformed network, where the specfed throughput flow from sources to destnatons has been acheved at a mnmum cost. If no domnated parastc flow loops exst n the transformed network, no domnated parastc flow loops wll exst n the network after removng the super-source s, the super-consumer t and ther connectng edges. After conductng Step 3, t can also be shown that the suppled and draned nterchangeable commodty through the edges connectng the super-source s and the super-snk t, correspond one-to-one to the suppled and consumed quanttes of nterchangeable commodty n the ntal network. Because n the ntal flow network, the throughput flow from the sources to the consumers s a feasble flow, the maxmum possble flow n the transformed flow network cannot be smaller than the ntally exstng throughput flow. Smultaneously, the maxmum possble flow n the transformed network cannot exceed the exstng feasble flow n the ntal network, because the sum of the capactes of the edges connectng the super-source are equal to the total generated nterchangeable commodty by the ntal sources. Consequently, the maxmum throughput flow n the transformed network s equal to the sum of the flows from the sources n the ntal network. In addton, n the transformed network, the amount of flow passng through each source (now an ordnary node) s exactly equal to the generated by the source flow n the ntal network. As a result, after conductng Step 3, the suppled quanttes through the edges connectng the super-source n the transformed network correspond one-to-one to the generated quanttes n the ntal network. In a smlar fashon, t can be proved that the draned quanttes through the edges connectng the super-consumer n the transformed network correspond one-to-one to the consumed quanttes n the ntal network. The presented algorthm goes beyond dscoverng domnated parastc flow loops. It removes from the network not only the domnated parastc flow loops but also drected and almost drected parastc flow loops. CONCLUSIONS. Smlar to the drected and almost-drected parastc flow loops, augmentable broken loops wth domnatng flow n a partcular drecton of traversng, are also assocated wth transportaton losses, congeston and ncreased polluton of the envronment.. Domnated parastc flow loops occur naturally n real networks by followng drected straght-lne paths from sources of flow to destnatons. The probablty of a domnated parastc flow loop n real networks s unexpectedly large and ncreases very quckly wth ncreasng the number of ntersectng flow paths. 3. An mportant concept referred to as cost factor of a cyclc path has been ntroduced, wth fundamental mportance to optmsng the flows n real flow networks. The cost factor of a cyclc path s the dfference between the sum of the lengths of the closng edges and the sum of the lengths of the edges wth backward flow. 4. The domnated parastc flow loops are augmentable cyclc paths charactersed by a negatve cost factor. Domnated parastc flow loops can be augmented wth flow, whch s essentally a process of dranng the flow 83-73X Copyrght 04 ORSTW

16 IJOR Vol., No., (04) 6 loops. The result s an overall decrease of the transportaton costs wthout affectng the quantty of the delvered commodty from sources to consumers. 5. The necessary and suffcent condton for nonexstence of domnated flow loops n a network s the nonexstence of cyclc paths wth negatve cost factor. 6. An algorthm has been proposed for elmnatng domnated parastc flow loops n networks. The algorthm nvolves transformng the network nto a sngle source/sngle snk network and a step whch nvolves maxmsng the flow n the transformed network at a mnmal transportaton cost. The proposed algorthm has the potental to save sgnfcant amount of resources to the world economy. 7. If more than one zero flow paths and non-zero flow paths are present, domnated flow loops are absent f and only f all random nz-vectors resde n one of the sectors defned by the zero flow paths. 8. A number of mportant applcatons have been dentfed, related to optmsng flow networks by removng domnated parastc flow loops. REFERENCES. Ahua, R.K., T.L. Magnant, J.B. Orln. (993). Network flows: Theory, Algorthms and Applcatons, Prentce Hall.. Ahua, R.K., A.V. Goldberg, J.B. Orln and R.E. Taran. (99). Fndng mnmum-cost flows by double scalng, Mathematcal Programmng, 53, pp Asano, T., Y. Asano. (000). Recent developments n maxmum Flow algorthms, Journal of the Operatons Research Socety of Japan, 43(), pp Cormen, T.H., T.C.E. Leserson, R.L. Rvest, and C. Sten. (00). Introducton to Algorthms, nd ed., MIT Press and McGraw-Hll. 5. Fresdorf, H. and H. Hamacher. (98). Weghted mn cost flows, European Journal of operatonal research,, pp Goldberg, A.V. and R.E. Taran. (987). Solvng mnmum-cost flow problems by successve approxmaton, Proceedngs of the nneteenth annual ACM symposum on theory of computng, ACM, New York,. 7. Goldberg, A.V. and R.E. Taran. (989). Fndng mnmum-cost crculatons by cancellng negatve cycles, Journal of the ACM, 36(4), pp Goodrch, M.T. and R. Tamassa. (00). Algorthm desgn, John Wley & Sons. 9. Hengartner, U., S. Moon, R. Morter, and C. Dot. (00). Detecton and analyss of routng loops n packet traces. Sprnt ATL Techncal Report TR0-ATL Itzhack, R., L. Tsaban and Y. Louzoun. (03) Long loops of nformaton flow n genetc networks hghlght an nherent drectonalty, Systems Bomedcne, (), pp Klen, M. (967). A prmal method for mnmal cost flows wth applcatons to the assgnment and transportaton problems, Management Scence, 4(3), pp Orln, J.B. (993). A faster strongly polynomal mnmum cost flow algorthm, Operatons Research, 4(), pp Paxson, V. (997). End-to-end routng behavor n the Internet. IEEE/ACM Transactons on Networkng, 5(5), pp Sharr, M. (98). A strong-connectvt;y algorthm and ts applcatons n data flow analyss, Computers and Mathematcs wth Applcatons, 7, pp Sreedhar, V.C., G.R. Gao, Y.F. Lee. (996). Identfyng loops usng DJ graphs. ACM Transactons on Programmng Languages and Systems, 8 (6), pp Tardos, E. (985). A strongly polynomal mnmum cost crculaton algorthm, Combnatorca, 5(3), pp Taran, R.E. (983). Data structures and network algorthms, SIAM, Phladelpha. 8. Todnov, M. (03a). Parastc flow loops n networks, Internatonal Journal of Operatons Research, 0(3), pp Todnov, M. (03b). Drected and almost-drected flow loops n real networks, Internatonal ournal of advanced computer scence and applcatons, 4(8), pp Ulanowcz, R.E., J.J. Kay. (99). A package for the analyss of ecosystem flow networks, Envronmental software, 6(3), pp.3-4. APPENDIX A. Algorthm for determnng the probablty of a domnated parastc flow loop. Ths s the algorthm n pseudo-code for determnng the probablty of a parastc flow loop. The varable count counts the number of tmes a domnated parastc flow loop has not been found and the varable Probablty contans the probablty of a domnated parastc flow loop. The statements wthn braces {...} are treated as a compound statement. // The number n of non-zero flow paths s n>=; 83-73X Copyrght 04 ORSTW