MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES


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1 MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES Romn V. Tyshchuk Informtion Systems Deprtment, AMI corportion, Donetsk, Ukrine Emil: 1 INTRODUCTION During the considertion of the min problems in the fields of trnsporttion nd communiction, the following tsk is very often should to be solved [FF78], [PG81]. The commodity should to be trnsported from the source of supply to the consuming node. The huling units re situted in different plces. It is supposed tht the trnsporttion network hs limited cpcity. In this tsk it is necessry to find the optimum route of the huling units from their initil plces to the source of supply nd then to the consuming node. There is nother tsk with the similr structure: for the existing communiction network it is necessry to build centrl communiction node through which ll the informtion should be trnsferred nd then it is necessry to get mximum informtion flow. In ll this exmples to solve the tsk the specil node is need to be chosen, through which ll the flow should be trnsferred from the source to the sink. This node is clled funnelshped node. In prctice during the nlysis of such systems it is often the mny forms of uncertinty, described in [B96] should to be tking into considertion. The most of the existing methods, which re solving the considered problem, re not ble to work with such forms of uncertinty like dispersion, vgue dt defining, mesurement errors nd so on. In this connexion it is necessry to develop new methods, which cn to operte with the bovementioned forms of uncertinty. There re some pproches, which re proposed to solve the network flow tsks with fuzzy input dt [D01], but this problem is not investigted completely nd still ctul. 2 METHODOLOGY 2.1 Preliminry remrks of pplying fuzzy sets theory to the network flow problems In mny cses some spects of the network flow theoretic problems re uncertin. In these cses it cn be useful to del with this uncertinty using the methods of fuzzy logic. There re severl wys in which network cn be fuzzy [B96], [T01b], [MN99]. In this pper one of much interest type of network fuzziness is considered, which is occurred when the
2 network hs known nodes nd rcs, but fuzzy weights (or cpcities) on the rcs. Thus only the weights re fuzzy. According to [T01b], [D01] fuzzy flow f in the network N with the source node s nd the sink node t with rcs cpcities { c i } is function, defining on the set of rcs E s following: 0 f (e) c (e) (1) where 0 f R i (e) c R i (e) 0 (2) f L i (e) c L i (e) where i is the number of the corresponding αlevel, L is the index, representing the lower bound of the αlevel intervl, R is the index, representing the upper bound of the αlevel intervl; f (x, y) = f (y, x), x s,t (3) (x, y) E (y,x) E One of the min problems in solving fuzzy flow network flow tsks is defining the rnking procedure for the fuzzy numbers. One of the fuzzy numbers rnking methods, proposed by Dubois nd Prde, is bsed on fuzzy mx nd fuzzy min opertions, which re the mximum nd minimum opertions of rel numbers extended to fuzzy numbers. Note tht when nd b re fuzzy numbers, m x(, b ) nd m i n(, b ) re not lwys one of the or b nd they my be built with prts of both nd b. This pproch is proposed to use in ugmenting flow building procedure nd in the mximum flow finding opertion. Another method is ssuming tht the decisionmker priory chooses degree of conformity for which the inequlity my be considered s true by himself. As vrint of this method the four coefficients PSE, PS, NSE, NS [DP88] re proposed to use to describe the reltive position of two fuzzy numbers from the point of view of possibility theory. This pproch is proposed to use in the mximum network flow lgorithm with fuzzy input dt in choosing the mximum of ll ugmenting pths to decrese in generl the totl number of opertions. As well known the network flow tsks usully re very complex nd one of the wys to solve such problems is simplifying or modifiction the existing network to use the known methods nd pproches. For the considering tsk the lgorithm is proposed in which the mximum network flow lgorithms with fuzzy input dt for the networks with one source, one sink nd one commodity re solved sequentilly [T01]. 2.2 Generl formultion of the mximum flow finding tsk in fuzzy networks with funnelshped nodes Let N be fuzzy undirected network with rc cpcities c ij representing by fuzzy numbers. Let s, nd t be the source, the funnelshped node nd the sink respectively. Let f 1 ij be flow from node i to node j, which hs direction to funnelshped node nd let f 2 ij be flow from node i to node j, which hs direction from funnelshped node. Let v 1 be fuzzy
3 2 flow vlue from source to funnelshped node nd let ~ v be fuzzy flow vlue from funnelshped node to sink. It is necessry to find mximum vlue of the flow ~ v : v mx (4) with the following constrints: f 1 ij = f 1 (5) ji (i, j ) E ( j,i) E f 2 ji (j,i ) E f 2 = (6) ij (i, j ) E f 1 ij + f 2 ij c ij, i, j (7) v = v 1 = v 2, (8) f k ij > 0. (9) It is esy to see tht this problem is similr to the tsk of the twocommodity flow finding in nonoriented network. But in this tsk we hve two different types of the sme commodity, not two flows of two commodities. It is necessry lso tht these two types of the flow should be equl. And if in the twocommodity tsk it is necessry to mximize the totl flow of two commodities, in this tsk it is necessry to mximize the 1/2( v 1 + v 2 ) vlue. Also it is esy to show tht the mximum flow through the funnelshped node cn be find s the min[ v 1, v 2,1/ 2mx( v 1 + v 2 )], where v 1 nd v 2 re the fuzzy mximum flows from s to nd from to t respectively. Note tht in this cse it is necessry to use fuzzy min opertion. 2.3 Algorithm One of the min fuzzy flow network fetures is concerning to the fct tht the result of the solving the mximum flow tsk in fuzzy network is directly depending on the ugmenting flow vlue on ech itertion. This feture is directly determining the functioning of this lgorithm nd lso determining the thesis tht tht the mximum flow through the funnelshped node cn be find s the min[ v 1, v 2,1/ 2mx( v 1 + v 2 )]. Therefore during the ugmenting flow vlue finding in given tsk it is necessry to follow the rule mentioned below: if it is possible to mke the current rc Fsturted for the lower bound of the αlevel intervl then the upper bound of the αlevel intervl of the ugmenting flow vlue should be equl to the lower bound vlue of the αlevel intervl of the ugmenting flow vlue. For exmple, if the rc cpcity is represented by the intervl fuzzy number nd equl to (2,5) then it is not llowed to choose the ugmenting flow vlue equl to (1,5), it is necessry to choose sequentilly (1,1), (1,1), (0,3) or (2,2), (0,3) etc. Step 1. Find vlues v 1 nd v 2 using the mximum network flow lgorithm with fuzzy input dt (Fig. 1).
4 s v~1 t v~2 Figure 1. Finding the mximum flows from s to nd from to t. Step 2. Build new network N with dditionl node s nd dditionl rcs (s, s) nd (s, t) with unlimited cpcities. Find fuzzy mximum flow from s to (Fig. 2). s s v~ t Figure 2. Finding the mximum flow from s to Step 3. Compute the vlue v * = min[ v 1, v 2,1/ 2 v ]. If v * = 0 then stop lgorithm. The result is equl to 0. Step 4. Build new network N with dditionl node s nd dditionl rcs (s, s) nd (s, t) with cpcities v *. Find fuzzy mximum flow from s to (Fig. 3). s s s t Figure 3. Finding the mximum flow from s to Decompose this flow on the s  flow through the node s nd  s flow through the node t. Remove ll dditionl nodes nd rcs from the network. 2.4 Illustrting exmple As n exmple to illustrte lgorithm we consider smll network shown in Fig.4. The node is funnelshped node. Intervl fuzzy numbers presents ll the rc cpcities, shown in the network ner the rcs. (3,4) 1 (2,3) S (3,4) T (4,5) (2,3) (5,6) (3,4) 2 3 (5,6) Figure 4. Exmple network S1 S T 3 T v (2,3) (5,6) (2,3) (4,5) (1,1) (1,1) v (3,3) (3,4) (5,6) (2,3) (3,4)
5 Tble 1. Results of the first step of the lgorithm S S S T S1 S T3 T v (7,9) (4,6) (2,3) (5,6) (2,3) (4,5) (1,1) (2,3) (1,2) (3,4) Tble 2. Results of the second step of the lgorithm After the first two steps the lgorithm gve the following results: ~ 1 v = (7,9), ~ 2 v = (8,10), ~ v = (11,15). Thus ~ * v = (5.5,7.5) S S S T S1 S2 T3 T v (5.5,7.5) (5.5,7.5) (2,3) (3.5,4.5) (2.5,3.5) (3,4) (2,3) (4,5) (0.5,0.5) (2,3) Tble 3. Results of the fourth step of the lgorithm Note, tht on the 2d step the vlue 1/2mx( v 1 + v 2 ) hs been defined using the mximum flow finding lgorithm with fuzzy input dt for the networks with one source, one sink nd one commodity. 3 CONCLUSIONS Our ides nd conceptions of the rel world re often vgue nd ill defined. In prticulr, imprecise observtions or possible perturbtions men tht fuzzy numbers my well better represent cpcities nd flows in network thn crisp quntities. In this pper n pproch bsed on fuzzy set theory hs been presented to solve the mximum flow finding tsk in the fuzzy network with funnelshped node. The min two rnking methods of the fuzzy numbers hve been described, which re used in fuzzy network flow tsks. The min fuzzy flow network fetures re described nd the lgorithm hs been proposed using mximum network flow lgorithms with fuzzy input dt for the networks with one source, one sink nd one commodity. REFERENCES [B96] M.Blue, B.Bush, J.Puckett (1996). Applictions of fuzzy logic to grph theory, Technicl report in Los Almos ntionl lbortory, LAUR [D01] P.Dimond (2001). A fuzzy mxflow mincut theorem, Fuzzy sets nd systems, 119. [DP88] D.Dubois, H. Prde (1988). Applictions l representtion des connissnces en informtique, Msson
6 [FF78] H.Frnk, I.T.Frish (1978), Communiction, trnsmission, nd trnsporttion networks, AddisonWesley [MN99] Y.E.Mlshenko, N.M.Novikov (1999), Models of uncertinty in multiuser networks, Russin Acdemy of Sciences, Moscow [PG81] D.T.Phillips, A.GrciDiz (1981), Fundmentls of Network Anlysis, PrenticeHll Inc. [T01] R.Tyshchuk (2001), Mximum network flow lgorithm with fuzzy input dt, 5th Interntionl Conference on KnowledgeBsed Intelligent Informtion Engineering Systems nd Allied Technologies, Osk, Jpn, 2001 [T01b] R.Tyshchuk (2001), Network flow tsks with fuzzy input dt, Donetsk Ntionl Technicl University scientific works. Series: Computer engineering nd utomtion, 38
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