Finding of Service Centers in Gis Described by Second Kind Fuzzy Graphs

Transcription

1 World Appled Scences Journal (Specal Issue on Technques and Technologes): 8-86, 03 ISSN IDOSI Publcatons, 03 DOI: 0.589/dos.was.03..tt.44 Fndng of Servce Centers n Gs Descrbed by Second Knd Fuzzy Graphs Aleander Bozhenyuk, Igor ozenberg and Dna Yastrebnskaya Southern Federal Unversty, 44 Nekrasovsky Street, Taganrog, 3479, ussa Publc Corporaton esearch and Development Insttute of alway Engneers, 7/, Nzhegorodskaya Street, Moscow, 0909, ussa Abstract: In ths paper the problem of optmal locaton of servce centers s consdered by mnma crteron. It s supposed that the nformaton receved from GIS s presented lke second knd fuzzy graphs. Method of fndng of optmal locaton k-centers s suggested. Bass of ths method s buldng procedure of reachablty matr of second knd fuzzy graph n terms of reachablty matr of frst knd fuzzy graph. Ths method allows to solve not only problem of fndng of optmal servce centers locaton n transport network but also fndng of optmal locaton k-centers wth the best vtalty degree and selectng of servce center numbers. The eample of fndng optmum allocaton of centers n transport networks wth the largest degree s consdered as well. Key words: Mnma crteron GIS Fuzzy graph Servce centers INTODUCTION placng of hosptals, polce statons, fre brgades, mportant enterprses and servces on some stes of a Locaton of facltes on the regon map and n consdered terrtory s reduced to ths task. In some transport network s one of the optmal problems. Some cases the crteron of optmalty can consst n the varants of these problems are topcal for most practcal mnmzaton of the ourney tme (or n the mnmzaton applcaton. For nstance, optmal locatons of of dstances) from the servce centre to the most constructon unts, repar obects, vantage-ponts have to remote servce staton. In other cases the crteron of be found n concordance wth some knd of crteron or optmalty conssts n the choce of such an allocaton of group of crterons. the centers that the route from them to any other place of Such locaton problems can be solved wth usng of servce s passed on the best way by some crteron. In fuzzy logc and fuzzy set theory. In ths case basc data other words the problem s the optmzaton of "the worst are set by fuzzy numbers or lngustc varables. varant [6]. However, the nformaton represented n the GIS, has an appromate value or nsuffcently authentc Fuzzy Data Usng to Solve the Locaton Problems: The [7]. large-scale ncreasng and versatle ntroducton of the Fuzzness concernng problems of servcng centers geographcal nformaton system (GIS) s substantally locaton can be cause: Informaton naccuracy; Parameter connected wth the necessty of perfecton of the naccuracy; Inaccuracy happens n decson-makng nformaton systems provdng decson-makng. The GIS process; Inaccuracy s called forth by envronmental are appled practcally n all spheres of human actvty. nfluence. Geographcal nformaton technologes have now reached Inaccuracy caused by the frst reason can consst n an unprecedented level, offerng a wde range of powerful mprecse measurement of dstance between servng functons such as nformaton retreval and dsplay, facltes. For eample, n case of dstance measurement analytcal tools and decson support [, ]. Unfortunately, whch equal n several klometer, the measurng accuracy geographcal data are often analyzed and communcated equalng n several centmeters or meters may appear. amd a largely non-neglgble uncertanty. Uncertanty Further f facltes have complcated forms ts parameters ests n the whole process from the geographcal are measured dffcultly. When measurng dstance abstracton, data acquston and geo-processng to the between facltes of comple form, t s necessary to usage [3, 4]. One of the tasks solved the GIS s the task of determne two ponts among a thousand and only the centers allocaton [5]. The search of the optmal thereafter to carry out the measurement. Correspondng Author: Bozhenyuk, Southern Federal Unversty, 44 Nekrasovsky Street, Taganrog, 3479, ussa. 8

2 World Appl. Sc. J., (Specal Issue on Technques and Technologes): 8-86, 03 Impossblty of parameter measurng gets connected Let G = (,U) s a second knd fuzzy graph. Here a set = { < ()/ > }, s the fuzzy set of vertces, defned on set, =n and U = { < U (, )/(, ) > },, s wth the evdence that some characterstcs are qualtatve and haven t numercal values. For eample, qualty of a secton of a route can t be measured but t wll estmate lke good, bad, the worst and etc. Parameter naccuracy brought out n characterstc features of obects happens n the event that obects changes ts parameters tself (for eample, szes of water body change naturally) or ts borders are ndstnct (for eample, boundares of forests or meadows). Inaccuracy of decson-makng process happens n cases when data about targets and some parameters aren t determnate eactly or t allows varatons n the certan range. Then model of problem descrbed by fuzzy theory wll be more approprate nstead of usng tradtonal mathematcs. It wll lead to more adequate descrbng of realty and permts to fnd more sutable decson. Inaccuracy called forth by envronmental nfluence conssts n mpact of unforeseen factors upon parameters of obects or ts nterrelatons that changes ther values. For nstance, passage tme between two obects depends on numbers of traffc lghts and ts condton at the passng moment, weather condtons (rate of moton under a ran, a snow and n a fog s less than n the clear weather) and quanttes of vehcles on the road and etc. Whle consderng locaton problems n mprecse condtons t s sad commonly about naccuracy of parameters measurement of obect because of ther comple structure and about mpact of eternal factors on parameters takng nto account for optmzaton [8]. We consder that a certan way system has n statons. There are k servce centres, whch may be placed nto these statons. Each centre can serve several statons. The degree of a servce staton by a centre depends on a cyclc route whch connects them. It s necessary for the gven number of centers to defne the places of ther best allocaton. In other words, t s necessary to defne the places of k centers nto n statons so that the control of all terrtory (all statons) s carred out wth the greatest possble degree of servce. Locaton Optons of Servce Centers n Second Knd Fuzzy Graph: For choosng of the best locaton of servce centers we can use the noton of conunctve strength and the reachablty degree of verte. In ths case we wll compare all of the movement paths among themselves and fnd routes wth a mamal reachablty degree. It can be carred out by usng of reachablty matr of second knd fuzzy graph. fuzzy set of drected edges. Here µ () [0,] - membershp functon for verte, µ (,) [0,] - membershp functon U for edge (, ). A path of second knd fuzzy graph l (, ) s called a drected sequence of fuzzy edges from verte to verte, n whch the fnal verte of any edge s the frst verte of the followng edge [9, 0]. A conunctve strength of second knd fuzzy graph s defned by epresson: l (, ) = & (, )& ( ) U k t t < k, t> l (, ) t L (, ) (, ) ma{ (, )} Let = l l L V ( Y) = & ( (, )& (, )) \ Y Y s a famly of second knd fuzzy graph paths from verte to verte. Then the value defnes the reachablty degree of verte from verte. For locaton of servce centers n second knd fuzzy graph we should consder fndng problem of optmal stuaton n the sense that all (resdual) vertces are servced wth the greatest vtalty degree []. Let k s servce centers (k<n), placed n the vertces of subset Y, Y =k, Y and (,) s reachablty degree of verte from verte. Value s vtalty degree of second knd fuzzy graph G whch servced by k-centers from verte set Y. Value V ( Y ) determnes mnma strong connectvty value between each verte from set /Y and one center from set Y. Fuzzy set V = { < V ()/ ><, V () / >,..., < V ( n) / n > } defned on verte set, s called fuzzy set of vtalty of graph. Fuzzy set of vtalty determnes the G = ( U, ) V G greatest vtalty degrees of graph n the event that t s servced by,,.., n centers. Here,. G V G ( n ) = 83

3 World Appl. Sc. J., (Specal Issue on Technques and Technologes): 8-86, 03 Values ( k n) sgnfy we can place k-centers V G ( k ) n the graph G so that there s a path from at least one center to any verte of graph G and back. Conunctve strength of the graph wll be not less than V G ( k ). In the paper [] was consdered method of fndng of mnmal sets of servce centers. These sets have the greatest vtalty degree n transport network descrbed frst knd fuzzy graph. Fg. : Transformaton of second knd fuzzy graph nto Let Y s vertces subset of frst knd fuzzy graph frst knd fuzzy graph n the case t=3 = ( U, ) n whch servce centers s located and vtalty degree equals V. Therefore one of the two condtons for any verte can be satsfed: Verte belongs to set Y; There s verte that belongs to set Y and nequaltes (, ) V and (, ) V are met. Usng the quantfer form of notaton we can get the truth of the followng epresson [3-5]: logcal varable p. Ths logcal varable possesses the of fuzzy graph = ( U, ). Here subset Y s mnmal n the Φ = &( p ( p & & )) V =, n =, n Fg. : Second knd fuzzy graph Property: If further smplfcaton n the epresson (3) ( )[ Y ( )( Y & (, ) V & (, ) V)] based on rules () s not possble then totalty of all vertces, conformng to varables, for each dsunctve Each verte can be set n correspondence the term gves vertces subset Y wth vtalty degree V value n case of Y and 0 f not. Epresson (, ) V sense of ts any subset doesn t have ths property. can be set n correspondence the fuzzy varable Consderng method works wth reachablty verte = (, ). matr of ntal fuzzy graph. It doesn t matter what knd of So we can get the truth of logcal epresson: ntal fuzzy graph s taken. So n order to apply the method n case of second knd fuzzy graph we should transform ths graph G nto frst knd fuzzy graph G ' lke that: G Let = and equalty holds for any then: Φ = & ( & & p ) V =, n =, n p p & & = p & & emove brackets n epresson () and collect usng rules of fuzzy capture: a a&b = a, a&b a&b = a, & a & ' ' '' a& b= & a, f Here, a, b {0,} and ', '' [0,]. Consequently epresson () wll be represented as: ' '' () () Let y - verte of ntal second knd fuzzy graph G that s adacent for t vertces and has a vtalty degree µ. epresent verte as drected complete t-subgraph of frst knd wth vtalty degree of nsde edges equal µ. Connect everyone of the ntal outsde t vertces wth one of the vertces of receved subgraph. Eample of transformaton s presented n Fg. : Frst knd fuzzy graph comes out of ths transformaton. Meanwhle, transton from graph G to graph s bunque. Bult reachablty matr () I of C receved frst knd fuzzy graph and pass from t to reachablty matr () I of ntal second knd fuzzy graph C G. Apply the foregong method to the matr ( I). C Φ = ( p & p &...& p & V ) V =, l k (3) Gven procedure s consdered by eample. 84

4 World Appl. Sc. J., (Specal Issue on Technques and Technologes): 8-86, 03 Each marked block of complementary matr conforms to verte of ntal second knd fuzzy graph. So the greatest values are chosen n these blocks. And reachablty matr of second knd fuzzy graph, (II) gven n Fg.., wll be defned as: G Fg. 3: Complementary frst knd fuzzy graph Eample: Let s consder an eample of fndng of servce centers n second knd fuzzy graph G gven n Fg.. Here vertces of graph are obects of transport network but edges are paths that connect the obects. Let s fnd centers of graph supposng that t s located n graph vertces. In other words servce centers should be located n obects of transport network. The adacent matr of fuzzy graph G s presented as: = , 0,5 0,8 0,9 0, 7 0,8 0, , 7 0,6 0 0, ,5 For fndng of reachablty matr ( II ) fuzzy graph G we need bult new complementary frst knd fuzzy graph. Ths graph s presented n Fg.3: The reachablty matr ( II) of fuzzy graph can be (I) = presented:. Here, 0 - dagonal untary = 0, n r () () () = &, r k r k k=, n = () = r,, =, n = 0 () () r r k k of second knd matr, - degree of adacent matr. Adacent matr of frst knd fuzzy graph takes on form: Fnd degree of matr, where matr elements are determned as: matr : Smlarly fnd: Dagonal untary matr where and are elements of, for nde =3,4,5,6,7. for our case equals: (I) = = 0,7 Let s on the matres and fnd matr : ( II ) C = , 6 0,8 0,9 0,7 0,6 0, 0,7 0,7 0, 0,6 0, 0, Epresson () takes on form: Φ V = ( p 0,6 p 0,7 p3 0,6 p4) & (0,6 p p 0,6 p3 0, p4) & (0,7 p 0,6 p p3 0, p4) & (0,6 p 0, p 0, p p ) = 3 4 = ppp3p4 0,7 pp3p4 0,6 pp4 0,6 p3p4 0,6 p 0, p 0,p 0, p. 3 4 Therefore second knd fuzzy graph G, gven n Fg.., has one mamal subset of vertces Y= {,, 3, 4} (.e. all graph vertces). If servce centers are located n these vertces vtalty degree of graph wll be equal. Also graph has one mamal subset of three vertces: Y = {, 3, 4}. In ths case locaton of servce centers n these vertces leads to vtalty degree of graph s equal 0,7. Graph has two mamal subset Y3 = {, 4}, Y4 = { 3, 4} and ts vtalty degree wll be equal 0,6 n case of servce centers locaton eactly n these vertces. And fnally, graph has one mamal subset Y 5= { }, so vtalty degree of graph wll be equal 0,6 by locaton of servce center n verte. Fuzzy set of vtalty of gven graph wll be of the form: V G = { < 0,6/ ><, 0,6/ ><, 0,7/3 ><, /4 > } We can notce that there s no pont n locatng two servce centers n one graph because the greatest vtalty degree n ths case wll concde wth vtalty reached by one center. CONCLUSION After realzaton of transton to characterstcs of nterrelaton between obects and to propertes of obects. 85

5 World Appl. Sc. J., (Specal Issue on Technques and Technologes): 8-86, 03 tself that represented as membershp functons we can 9. Monderson, J.N. and P.S. Nar, 000. Fuzzy Graphs begn to solve the problem of the best locaton of and Fuzzy Hypergraphs. Hedelberg; New-York: servce centers. Then basc defntons of strong Physca-Verl., pp: 383. connectvty and fuzzy graph vtalty should be used. 0. Bershten, L.S. and A.V. Bozhenyuk, 008. Fuzzy The present approach of fndng of vtalty fuzzy set Graphs and Fuzzy Hypergraphs. In: Dopco, J., de la allows us to solve not only problem of k-centers optmal Calle, J., Serra, A. (eds.) Encyclopeda of Artfcal locaton wth mamal vtalty degree but also problem of Intellgence, Informaton SCI, Hershey, New York, selectng of k-servce center numbers. pp: Bozhenyuk, A.V., I.N. ozenberg and T.A. Starostna, ACKNOWLEDGEMENTS 006. Analyss and research flows and vtalty of transport network under fuzzy data. Moscow: Ths work has been supported by the ussan Nauchny Mr, pp: 36. Foundaton for Basc esearch, Proect, Bozhenyuk, A.and I. ozenberg, 0. Allocaton of servce centers n the GIS wth the largest vtalty EFEENCES degree. In the Proceedngs of the IPMU 0, Part II, Communcatons n Computer and Informaton. Clarke, K., 995. Analytcal and Computer Scence, CCIS 98, Sprnger-Verlag, Berln Hedelberg, Cartography. Englewood Clffs, N.J.: Prentce Hall, pp: pp: Bershten, L.S. and A.V. Bozhenuk, 999. Maghout. Longley, P., M. Goodchld, D. Magure and D. hnd, Method for Determnaton of Fuzzy Independent, 00. Geographc Informaton Systems and Scence. Domnatng Verte Sets and Fuzzy Graph Kernels. John Wley & Sons, Inc., New York, pp: 56. Journal General Systems, 30: Zhang, J. and M. Goodchld, 00. Uncertanty n 4. Bershten, L.S. and A.V. Bozhenuk, 999. Geographcal Informaton. Taylor & Francs, Inc., Determnaton of Fuzzy Internal Stable, Eternal New York, pp: 66. Stable Sets and Kernels of Fuzzy Orented Graphs. 4. Goodchld, M., 989. Modellng Error n Obects and Journal of Computer and Systems Scences Felds. In.: Accuracy of Spatal Databases, M. Internatonal. 38(): Goodchld and S. Gopal (eds.) Taylor & Francs, Inc., 5. Bershten, L.S., A.V. Bozhenyuk and I.N. ozenberg, Basngstoke, pp: Fuzzy Colorng of Fuzzy Hypergraph. In the 5. Kaufmann, A., 977. Introducton a la theore des Proceedngs of the Computaton Intellgence, Theory sous-ensemles flous. Masson, Pars, pp: 346. and Applcatons. Internatonal Conference 8th Fuzzy 6. Chrstofdes, N., 976. Graph theory. An algorthmc Days n Dortmund, Germany, Sept. 9-Oct. 0, 004. approach. Academc press, London, pp: 43. Bernd eusch (ed.): Sprnger-Verlag Berln 7. Malczewsk, J., 999. GIS and multcrtera decson Hedelberg, pp: analyss. John Wlley and Sons, New York, pp: ozenberg, I.N. and T.A. Starostna, 006. Solvng of locaton problems under fuzzy data wth usng GIS. Moscow: Nauchny Mr, pp: