The Methods of Maximum Flow and Minimum Cost Flow Finding in Fuzzy Network

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1 The Methods of Mamum Flow and Mnmum Cost Flow Fndng n Fuzzy Network Aleandr Bozhenyuk, Evgenya Gerasmenko, and Igor Rozenberg 2 Southern Federal Unversty, Taganrog, Russa AVB002@yande.ru, e.rogushna@gmal.com 2 Publc Corporaton Research and Development Insttute of Ralway Engneers, Moscow, Russa I.rozenberg@gsmps.ru Abstract. Ths artcle consders the problems of mamum flow and mnmum cost flow determnng n fuzzy network. Parameters of fuzzy network are fuzzy arc capactes and transmsson costs of one flow unt represented as fuzzy trangular numbers. Conventonal rules of operatng wth fuzzy trangular numbers lead to a strong blurrng of ther borders, resultng n loss of selfdescrptveness of calculatons wth them. The followng technque of addton and subtracton of fuzzy trangular numbers s proposed n the presented paper: the centers are added (subtracted) by the conventonal methods, and the borders of the devatons are calculated usng lnear combnatons of the borders of adacent values. The fact that the lmts of uncertanty of fuzzy trangular numbers should ncrease wth the ncreasng of central values s taken nto account. To llustrate the proposed method numercal eamples are presented. Keywords: Fuzzy arc capacty, lnear combnaton of borders, fuzzy trangular number, fuzzy flow. Introducton Ths artcle deals wth flow problems arsng n networks. The network corresponds to a drected graph G ( X, A), where X the set of nodes, A the set of arcs wth dstngushed ntal (source) and fnal (snk) nodes. Each arc (, ) A has capacty determnng the mamum number of flow unts, whch can pass along the arc. The relevance of the tasks of mamum and mnmum cost flow determnng les n the fact that the researcher can effectvely manage the traffc, takng nto account the loaded parts of roads, redrect the traffc, and choose the cheapest route. Suppose a network, whch arcs have capactes ( q ). Formulaton of the problem of mamum flow fndng n the network s reduced to mamum flow determnng that can be passed along arcs of the network n vew of ther capactes []:

2 2 A. Bozhenyuk et al. s kt Г( s) - k Г ( t), s, k, t, ( ) Г k Г ( ) 0, s, t, 0 q, (, ) A. ma, () In () the amount of flow, passng along the arc (, ) ; the mamum flow value n the network; s ntal node (source); t fnal node (snk); ( q arc capacty, Г ) the set of nodes, arcs from the node X go to, Г ( ) the set of nodes, arcs to the node cars, gong from the node X go from. represents, for eample, the amount of X to the node X. The frst equaton of () defnes that we should mamze the total number of flow unts emanatng from the source ( ), whch s equal to the total number of flow unts enterng the snk ( ). The second equaton of () s a flow conservaton constrant, whch means that the total number of flow unts emanatng from the source ( ) must be equal to the total number of flow unts enterng the snk ( ) and the total number of flow unts emanatng from any node s, t must be equal to the total number of flow unts enterng the node s,t. The thrd nequalty of () s a bound constrant, whch ndcates that the flow of value, passng along any arc (, ) must not eceed ts arc capacty. The task of mnmum cost flow determnng n a network can be formulated as follows: suppose we have a network, whch arcs have two assocated numbers: the arc capacty ( q ) and transmsson cost ( c ) of one flow unt passng from the node X to the node X. The essence of ths problem s to fnd a flow of the gven value from the source to the snk, whch doesn t eceed the mamum flow n the graph and has mnmal transmsson cost. In mathematcal terms the problem of mnmum cost flow determnng [2] n the network can be represented as follows: c (, ) A Г ( ) 0 q mn,, s, k, t, k Г ( ) 0, s, t,, (, ) A. (2)

3 Mamum Flow and Mnmum Cost Flow Fndng 3 In (2) c transmsson cost of one flow unt along the arc (, ), gven flow value, that doesn t eceed the mamum flow n the network. In practce, the arc capactes, transmsson costs, the values of the flow enterng the node and emanatng from the node cannot be accurately measured accordng to ther nature. Weather condtons, emergences on the roads, traffc congestons, and repars nfluence arc capactes. Varatons n petrol prces, tolls can ether nfluence transmsson costs. Therefore, these parameters should be presented n a fuzzy form, such as fuzzy trangular numbers [3]. Thus, we obtan a problem statement of mamum and mnmum cost flow problems n fuzzy condtons. 2 Lterature Revew of the Mamum and Mnmum Cost Flow Determnng Tasks The problem of the mamum flow fndng n a general form was formulated by T. Harrs and F. Ross [4]. L. Ford and D. Fulkerson developed famous algorthm for solvng ths problem, called augmented path algorthm [5]. Mamum flow problem was consdered n [, 6]. There are dfferent versons of the Ford-Fulkerson s algorthm. Among them there s the shortest path algorthm, proposed by J. Edmonds and R. Karp n 972 [7], n whch one can choose the shortest supplementary path from the source to the snk at each step n the resdual network (assumng that each arc has unt length). The shortest path s found accordng to the breadth-frst search. Determnng the mamum flow n the transportaton network n terms of uncertanty has been studed less. In [8] a soluton takng nto account the nterval capactes of arcs was proposed. S. Chanas [9] proposed to solve ths problem by usng socalled fuzzy graphs. There are contemporary artcles whch solve the problem by the smple method of lnear programmng [0]. Many researchers have eamned the task of mnmum cost flow fndng n crsp condtons n the lterature. Methods of ts soluton can be dvded nto graph technques and the methods of lnear programmng. In partcular, solutons by the graph methods are consdered n [, 6]. The advantages of ths approach are great vsualzaton and less cumbersome. The mnmum cost flow s proposed to fnd by Busacker- Gowen and M. Klen s algorthms n [2]. In [2, 6] a task of mnmum flow determnng s consdered as a task of lnear programmng. Ths approach s cumbersome. The methods of mnmum cost flow fndng n networks n fuzzy condtons can be dvded nto two classes. The frst class nvolves the use of conventonal flow algorthms for determnng the mnmum cost flow, whch operate wth fuzzy data nstead of crsp values and requre cumbersome routnes wth fuzzy numbers. The second class of problems mples the use of fuzzy lnear programmng, whch was wdely reported n the lterature [, 2]. Authors [3] consder the tasks of fully fuzzy lnear programmng. These tasks are cumbersome and can not lead to optmal solutons n the mnmum cost flow determnaton. The soluton of fuzzy lnear programmng tasks by the comparson of fuzzy numbers based on rankng functons s eamned n [4].

4 4 A. Bozhenyuk et al. 3 Presented Method of Operatng wth Fuzzy Trangular Numbers Researcher s faced wth the problem of fuzzness n the network, when consderng the problems of mamum and mnmum cost flow fndng. Arc capactes, flow values, passng along the arcs, transmsson costs per unt of goods cannot be accurately measured, so we wll represent these parameters as fuzzy trangular numbers. We wll represent the trangular fuzzy numbers as follows: ( a,, ), where a the center of the trangular number, devaton to the left of the center, devaton to the rght of center, as shown n Fg.. ( ) α 0 a a ( ) a a ( ) a 2 Fg.. Fuzzy trangular number. Conventonal operatons of addton and subtracton of fuzzy trangular numbers are as follows: let A and A 2 be fuzzy trangular numbers, such as A a,, and A2 a2, 2, 2. Therefore, the sum of trangular numbers can be wrtten as: A A2 a a2, 2, 2 and the dfference represented as: A A2 a a2, 2, 2 [3]. The dsadvantage of the conventonal methods of addton and subtracton of fuzzy trangular numbers s a strong blurrng of the resultng number and, consequently, the loss of ts self-descrptveness. For eample, when addng the same trangular number wth tself, the borders of ts uncertanty ncrease: (2,,) + (2,,) = (4, 2, 2) and (2,,) + (2,,) + (2,,) = (6, 3, 3). Generally, t s not true, because the center of the trangular number should ncrease, whle ts borders must reman constant. The fact that the degree of borders blurrng of fuzzy number depends on the sze of ts center s not usually consdered, when specfyng the trangular fuzzy numbers. Therefore, the more the center, the more blurred the borders should be (whle measurng kg of materal, we are talkng about kg, mplyng the number from 900 to 00 g, but whle measurng t. of materal, mply that about t. s the number from 990 kg to 0 kg ). Comparson of fuzzy trangular numbers accordng to varous crtera s also very dffcult and tme-consumng. Consequently, followng method s proposed to use

5 Mamum Flow and Mnmum Cost Flow Fndng 5 when operatng wth trangular fuzzy numbers. Suppose there are the values of arc capactes, flows or transmsson costs n a form of fuzzy trangular numbers on the number as. Then when addng (subtractng) the two orgnal trangular fuzzy numbers ther centers wll be added (subtracted), and to calculate the devatons t s necessary to defne requred value by adacent values. Let the fuzzy arc capacty (flow or ' transmsson cost) near s between two adacent values near and near 2, ' ( 2) whch membershp functons ) and ) have a trangular form, as shown n Fg. 2. ( 2 ( 2 ' 2 L l R l L l ' R L 2 l l 2 R l 2 Fg. 2. Gven values of arc capactes (flows or transmsson costs). Thus, one can set the borders of membershp functon of fuzzy arc capacty (flow ' or transmsson cost) near by the lnear combnaton of the left and rght borders of adacent values: L ( 2 ) L ( ) 2 L l l l2, ( 2 ) ( 2 ) R ( 2 ) R ( ) 2 R l l l2. ( 2 ) ( 2 ) (3) L R In (3) l s the left devaton border of requred fuzzy number, l s the rght devaton border. In the case when the central value of trangular number resultng by addng (subtractng) repeats the already marked value on the number as, ts devaton borders concde wth the devaton borders of the number marked on the number as. If requred central value s not between two numbers, but precedes the frst marked value on the number as, ts devaton borders concde wth those of the frst marked on the as. The same apples to the case when the requred central value follows the last marked value on the as.

6 6 A. Bozhenyuk et al. 4 Solvng the Task of Mamum Flow Fndng n Fuzzy Network The task of mamum flow fndng n fuzzy network can be formulated as follows: 0 Г ( ) k kг ( ) q, (, s kt - Г( s) kг ( t), s,, t, 0, s, t, ) A. ma, In (4) s requred mamum fuzzy flow value n the network; fuzzy amount of flow, passng along the arc (, ); q fuzzy capacty of the arc (, ); 0 s fuzzy number of the form (0, 0, 0), as t reflects the absence of the flow. Let s consder an eample, llustratng the soluton of ths problem, represented n Fg. 3. Let network, representng the part of the ralway map, s gven n a form of fuzzy drected graph, obtaned from GIS Obect Land [5, 6]. Let the node s a source, node 2 s a snk. The values of arc capactes n the form of fuzzy trangular numbers are defned above the arcs. It s necessary to calculate the mamum flow value between statons Kemerovo ( ) and Novosbrsk-Gl. ( 2 ) accordng to Edmonds-Karp s algorthm [7] and the method, descrbed for operatons wth fuzzy trangular numbers. Determnng of mamum flow s based on sendng flows along the arcs of the network untl one cannot send any addtonal unt of flow from the source to the snk. Edmonds-Karp s algorthm represents the choce of the shortest supplementary path from the source to the snk at each step n the resdual network (assumng that each arc has unt length). Fuzzy resdual network contans the arcs of the form (, ) wth the fuzzy resdual arc capacty q, f the arcs (, ) have the flow value q n the ntal network; and the arcs of the form (, ) wth the resdual arc capacty, f the arcs, ) have the flow value 0. ( The arc, ) s called saturated when the flow, passng along t, equals to arc capacty ( q. Other words, resdual arc capacty defnes how many flow unts can be ( sent along the arc, ) to reach arc capacty. Resdual arc capacty of arc saturated arc, ) s zero. ( (4)

7 Mamum Flow and Mnmum Cost Flow Fndng 7 2 (32, 6, 7) (45, 8, 8) (20, 2, 3) (20, 2, 3) Novosbrsk-Gl. Sokur Inskaya B-p. 49 km. 0 (32, 6, 7) 9 Proektnaya (45, 8, 8) (32, 6, 7) (6, 2, 2) (22, 4, 4) (6, 2, 2) Urga II (6, 2, 2) Urga I 5 Topk (25, 5, 4) (30, 5, 6) 4 Ishanovo (40, 7, 7) Kemerovo-Sort. 3 2 (20, 2, 3) q Predkombnat (45, 8, 8) Kemerovo Fg. 3. Intal network. Therefore, the algorthm proceeds as follows: the frst teraton of the algorthm performs an augmentng chan Push the flow, equals to unts along t. The arc 9, becomes saturated, consequently, fuzzy resdual capacty of the arc 9, equals to (0, 0, 0). Let s defne the fuzzy resdual capactes of the remanng arcs of augmentng chan. The arc, 3 has fuzzy resdual capacty equals to (45, 8, 8). Thus, the central value of the resultng number s 7. It s located between adacent arc capactes: (6, 2, 2) and (20, 2, 3) of the orgnal graph as shown n Fg. 4. ( ) Fg. 4. Fuzzy trangular number wth a center equals to 7 and ts adacent numbers. Compute the left and the rght devaton borders of the fuzzy trangular number wth a center of 7 accordng to (3). Thus, we obtan a fuzzy trangular number of the form (7, 2, 2.25) unts.

8 8 A. Bozhenyuk et al. Fuzzy resdual capacty of the arc, ) s (30, 5, 6). Consequently, ( 3 8 we obtan a fuzzy trangular number wth a center of 2, located to the left of fuzzy trangular number of the form (6, 2, 2). Devaton borders of the requred number concde wth devaton borders of the number (6, 2, 2). Thus, we obtan a fuzzy trangular number of a type (2, 2, 2) unts. Fuzzy resdual capacty of the arc ( 8, 9) equals to (7, 2, 2.25) unts, smlarly wth the arc, 3. Fnally, fuzzy resdual capacty of the arc, 2 s equal to (32, 6, 7) (28, 5, 5),.e. we obtan fuzzy number (4, 2, 2) unts and fuzzy resdual capacty of the arc 2, equals to unts. Fuzzy resdual capactes of the arcs, ),(, ),(, ),(, ),(, ) are unts. ( The second teraton of the algorthm gves the augmentng chan Push the flow equals to (20, 2, 3) unts along t. The arc, 2 becomes saturated, consequently, fuzzy resdual capacty of the arc, 2 equals to (0, 0, 0). Fuzzy resdual capacty of the arc 2, 4 s (40, 7, 7) (20, 2, 3),.e. we obtan a fuzzy trangular number (20, 2, 3) unts. Fuzzy resdual capacty of the arc 4, 5 s the dfference between the numbers (25, 5, 4) and (20, 2, 3). Thus, we get a number wth a center of 5, located to the left of the number (6, 2, 2),.e. (5, 2, 2) unts. Fuzzy resdual capacty of the arc 5, 6 s equal to (22, 4, 4) (20, 2, 3),.e., (2, 2, 2) unts. Fuzzy resdual capacty of the arc 6, 0 s equal to (32, 6, 7) (20, 2, 3),.e. (2, 2, 2) unts. Fuzzy resdual capacty of the arc 0, 2 s (45, 8, 8) (20, 2, 3),.e., (25, 5, 4) unts. Fuzzy resdual capactes of the arcs are (20, 2, 3) unts. ( 2, ),( 4, 2),( 5, 4),( 6, 5),( 0, 6),( 0, 2) The thrd teraton of the algorthm performs the augmentng chan Push the flow equals to (2, 2, 2) unts along. The arc 5, 6 becomes saturated. Let s defne fuzzy resdual capactes of the remanng arcs of the augmentng chan. Fuzzy resdual capacty of the arc, 3 s (7, 2, 2.25) (2, 2, 2),.e. (5, 2, 2) unts. Fuzzy resdual capacty of the arc 3, s (30, 5, 6) unts. Fuzzy resdual capacty of the arc 3, 4 s equal to (2, 2, 2). We get the number wth a center of 26, located between adacent values (25, 5, 4) and. Compute the left and the rght devaton borders of the fuzzy trangular number wth a center of 26 accordng to (3). Thus, we obtan a fuzzy trangular number of the form (26, 5, 4.33) unts. Fuzzy resdual capacty of the arc 4, 5 s equal to (5, 2, 2) (2, 2, 2),.e. (3, 2, 2) unts. Fuzzy resdual capacty of the arc 6, 0 s equal to (2, 2, 2) (2, 2, 2),.e. (0, 2, 2) unts. Fuzzy resdual capacty of the arc 0, 2 s equal to (25, 5, 4) (2, 2, 2),.e. we obtan a fuzzy number wth a center of 23, located between adacent values (22, 4, 4) and (25, 5, 4), therefore, the left devaton border of the number wth a center of 23 equals to 4.33, the rght devaton border s 4. We obtan fzzy trangu-

9 Mamum Flow and Mnmum Cost Flow Fndng 9 lar number (23, 4.33, 4) unts. Fuzzy resdual capactes of the arcs ( 5, 4),( 6, 5),( 0, 6),( 0, 2) are (22, 4, 4) unts. After eecuton of three teratons of the algorthm t s mpossble to pass any sngle addtonal flow unt. The total flow s + (20, 2, 3) + (2, 2, 2) unts. Therefore, we obtan a fuzzy trangular number wth a center of 50, located to the rght of the number (45, 8, 8) wth the borders, repeated devatons of the number 45: (50, 8, 8) unts. Thus, the mamum flow value between the statons Kemerovo and Novosbrsk-Gl. s (50, 8, 8) unts. Let us carry out an nterpretaton of the results: the mamum flow between the gven statons can not be less than 42 and more than 58 unts, wth the hghest degree of confdence t wll be equal to 50 unts. But wth changes n the envronment, repars on the roads, traffc congestons the flow s guaranteed to le n the range from 42 to 58 unts. Fuzzy optmal flow dstrbuton along the arcs and labels of the nodes s shown n Fg. 5. Saturated arcs are bold. Novosbrsk-Gl. (+0, (5, 2, 2)) 2 (22, 4, 4) Inskaya Sokur (+0, (2, 2, 2)) 0 (+6, (2, 2, 2)) (22, 4, 4) B-p. 49 km. 9 (+7, (5, 2, 2)) (+5, (2, 2, 2)) Proektnaya Urga I (+5, (5, 2, 2)) 6 (22, 4, 4) (+7, (5, 2, 2)) 8 7 Urga II Topk Ishanovo Kemerovo-Sort. Predkombnat Kemerovo 3 (+, ) (22, 4, 4) 5 4 (+4, (5, 2, 2)) (+3, (7, 2, 2.25)) (2, 2, 2) (20, 2, 3) 2 (+, (7, 2, 2.25)) (30, 5, 6) (20, 2, 3) (+, (20, 2, 3)) Fg. 5. Network wth mamum flow of (50, 8, 8) unts. 5 Solvng the Task of Mnmum Cost Flow Determnng n Fuzzy Network Consder the problem of mnmum cost flow fndng n a network accordng to fuzzy values of arc capactes, flows and transmsson costs of one flow unt.

10 0 A. Bozhenyuk et al. (, ) A c mn,, s, k, t, ( ) ( ) Г kг 0, s, t, 0 q, (, ) A. (5) In (5) c fuzzy transmsson cost of one flow unt along the arc (, ), gven fuzzy flow value, that doesn t eceed the mamum flow n the network. Let us turn to the graph, shown n Fg. 3. Fuzzy values of transmsson costs n addton to fuzzy arc capactes are gven n ths task: (2,3,3); (6,,2); (0,2,3); (8,4,5); (4,,); (2,3,3); (20,5,6); (5,4,4); (5,4,4); (2,6,7); (0,2,3); (30,8,9); (8,2,2); (9,5,5); (32,7,2); (32,7,2); (25,7,8); (20,5,6) It s necessary to fnd a flow value of (45, 8, 8) unts from the source to the snk, whch has a mnmal cost. Consder the Busacker-Gowen s [2] algorthm, takng nto account the fuzzy capactes and costs to solve ths problem: Step. Assgn all arc flows and the flow rate equal to zero. * Step 2. Determne the modfed arc costs that depend on the value of the already found flow as follows: c, f 0 q, * c, f q, c, f 0. Step 3. Fnd the shortest chan (n our case the chan of mnmal cost) [2] from * the source to the snk takng nto account arc costs, found n the step. Push the flow along ths chan untl t ceases to be the shortest. Receve the new flow value by addng the new flow value, passng along the consdered chan, to the prevous one. If the new flow value equals to, then the end. Otherwse, go to the step 2. Solve ths problem, takng nto account fuzzy arc capactes costs. Step. Assgn all 0. * Step 2. Determne c c. Step 3. Fnd the shortest path by the Ford s algorthm []: of the total cost of (75, 8, 8) standard unts. Push the flow, equals to unts along ths chan.

11 Mamum Flow and Mnmum Cost Flow Fndng Step 2. Defne the new modfed fuzzy arc costs: * * * * * (6,,2); (6,,2); (4,,); (4,,); (2,6,7); * (2,6,7); * ; * (9,5,5); * (25,7,8); * (25,7,8) Step 3. Fnd the shortest path usng the obtaned modfed costs: of the total cost of (86, 8, 8) standard unts. Push the flow, equals to (7, 2, 2.25) unts along ths chan. As a result, we obtan the total flow equals to (45, 8, 8) unts, havng a total transmsson cost along the network, equals to ((75, 8, 8) + (86, 8, 8)) = (3562, 8, 8) standard unts. There are fuzzy flow values under the arcs and fuzzy transmsson costs c of optmal fuzzy flow values above the arcs of the graph, saturated arcs are bold as shown n Fg. 6. Novosbrsk-Gl. 2 (7, 2, 2.5) (340, 8, 8) (700, 8, 8) Inskaya 0 Sokur B-p. 49 km. (532, 8, 8) (7, 2, 2.5) (36, 8, 8) 9 (588, 8, 8) 6 Proektnaya 8 7 (7, 2, 2.5) (50, 8, 8) Urga I 5 Urga II (28, 5,5) (2, 8, 8) Topk (7, 2, 2.5) (204, 8, 8) Ishanovo 4 (7, 2, 2.5) Kemerovo-Sort. 3 (70, 8, 8) 2 Predkombnat (45, 8, 8) (270, 8, 8) ) c ( Kemerovo Fg. 6. Network wth the flow of (45, 8, 8) unts and transmsson costs of each arc of the total cost (3562, 8, 8) standard unts. 6 Concluson Ths paper eamnes the problems of mamum and mnmum cost flow determnng n networks n terms of uncertanty, n partcular, the arc capactes, as well as the transmsson costs of one flow unt are represented as fuzzy trangular numbers. The technque of addton and subtracton of trangular numbers s consdered. Presented technque suggests calculatng the devaton borders of fuzzy trangular numbers based on the lnear combnatons of the devaton borders of the adacent values. The fact that the lmts of uncertanty of fuzzy trangular numbers should ncrease wth the ncreasng of central values s taken nto account. Advantage of the proposed method les n the fact that operatons wth fuzzy trangular numbers don t lead to a strong

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