Service Centers Finding by Fuzzy Antibases of Fuzzy Graph

Transcription

1 Servce Ceters Fdg by Fuzzy Atbases of Fuzzy Graph Leod Bershte,, Aleader Bozheyu, Igor Rozeberg, Tagarog Isttute of Techology of Souther Federal versty, Nerasovsy 44, 4798, Tagarog, Russa Scetfc ad Techcal Ceter Itech of Souther Federal versty, Otyabrsaya Square 4, 479, Tagarog, Russa Publc Corporato Research ad Developmet Isttute of Ralway Egeers, Nzhegorodsaya Street, 7/, 99, Moscow, Russa Abstract I ths paper the questos of defto optmum allocato of the servce cetres of some terrtory are observed It s supposed that terrtory s descrbed by fuzzy graph I ths case a tas of defto optmum allocato of the servce cetres may be trasformed to the tas of defto of fuzzy atbases of fuzzy graph The method of defto of fuzzy atbases s cosdered ths paper The eample of foudg optmum allocato of the servce cetres as defto of fuzzy atbases s cosdered too Keywords: fuzzy drected way, accessble degree, fuzzy trastve closure, fuzzy recprocal trastve closure, fuzzy set of atbases Itroducto There are may tass of optmum allocato of the servce cetres [] They are a allocato of rado ad TV stato some rego; a allocato of mltary bases, whch cotrol some terrtory; a allocato of shops, whch serve some rego ad so o However, the formato about the allocato of the servce cetres s accurate or ot relable very frequetly [] The calculato of a servce degree (or qualty ca be carred out by several, cludg to cotradctg each other, crtera For eample, the defto of umber ad allocato of shops ca be made by tag to accout qualty of roads, cost of groud the gve area, dstace from other areas, ad other crtera Rag of such crtera s frequetly made subectvely, o the bass of the huma factor We cosder that some terrtory s dvded to areas There are servce cetres, whch may be placed to these areas It s supposed that each cetre may be placed Ths wor has bee supported by the Russa Foudato for Basc Research proect --9а

2 to some statoary place of each area From ths place the cetre serves all area, ad also some eghbor areas wth the gve degree of servce The servce cetres ca fal durg the eplotato (for eample, for plaed or etraordary repar It s ecessary for the gve umber of the servce cetres to defe the places of ther best allocato I other words, t s ecessary to defe the places of servce cetres to areas such that the cotrol of all terrtory s carred out wth the greatest possble degree of servce Ma cocepts ad deftos I ths paper we suppose that the servce degree of rego s defed as the mmal meag of servce degrees of each area Tag to accout, that the servce degree ca ot always have symmetry property (for eample, by specfc character ad relef of the rego the model of such tas s a fuzzy drected graph G=(X, [] Here, set X={ }, I={,,,} s a set of vertces ad ={< <, > / <, >> }, <, > X s a fuzzy set of drected edges wth a membershp fucto :X [,] The membershp fucto, > of graph G = ( X, defes < a servce degree of area the case whe a servce ceter s placed to area We assume, that the servce degree has property of trastvty, e f the servce cetre s the area ad serves area wth a degree, >, ad f the servce cetre < <, s area ad serves area wth a degree > the a degree of servce of area from area ot less tha, > & <, > < For cosderato of questos of optmum allocato of the servce cetres we shall cosder cocepts of a fuzzy drected way ad fuzzy atbase of the fuzzy graph [4] Defto Fuzzy drected way L(,m of fuzzy drected graph G=(X, s called the sequece of fuzzy drected edges from verte to verte m: L (, m =< <, > / <, >>, < <, > / <, >>,, < < l, m > / < l, m >> Couctve durablty of way (L (, s defed as: m (L(, = & m <,β > L(,m <, β > Fuzzy drected way L(,m s called smple way betwee vertces ad m f ts part s ot a way betwee the same vertces Obvously, that ths defto cocdes wth the same defto for ofuzzy graphs Defto Verte y s called fuzzy accessble of verte the graph G=(X, f ests a fuzzy drected way from verte to verte y The accessble degree of verte y from verte, ( y s defed by epresso: γ(,y = ma ( (L (,y, =,,,p,

3 where p - umber of varous smple drected ways from verte to verte y Let's cosder, that each verte X the graph G=(X, s accessble from tself wth a accessble degree γ(,= Eample For the fuzzy graph preseted o Fg, verte 5 s fuzzy accessble verte from wth a accessble degree: γ (, 5 = ma{(,7 &,; (,6 &,8} = ma{,;,6} =,6,7,,6,8 5,4 Fg Fuzzy graph Let a fuzzy graph G=(X, s gve Let's defe fuzzy multple-valued reflectos Γ, Γ, Γ,, Γ as: Γ ( = { < ( /( > } Γ (, here ( X [ ( = <, > ] Γ (, Γ ( = Γ{Γ( }, Γ ( = Γ{Γ ( },, - Γ ( Γ{Γ ( } = { < ( / > }, here = Γ ( ( > Γ ( Γ ( l l l X It s obvous, that Γ ( X [ ( = ( & <, ] 4,5 s a fuzzy subset of vertces, whch t s accessble to reach from, usg fuzzy ways of legth Eample For the fuzzy graph preseted o Fg, we have: Γ ( = { <,7 /( >, <,6 / > }, Γ ( = { <,6 / 5 > } Defto Fuzzy trastve closure ( s fuzzy multple-valued reflecto: Γ Γ( = Γ ( Γ( Γ ( = Γ ( Here, by defto: Γ ( = { < / > } I other words, Γ( s fuzzy subset of vertces, whch t s accessble to reach from by some fuzzy way wth the greatest possble couctve durablty As we cosder fal graphs, t s possble to put, that: =

4 Γ( = Γ ( = Eample For the fuzzy graph preseted o Fg, fuzzy trastve closure of verte s defed as Γ( = { < / >, <,7/ >, <,6/ >, <,5/ 4 >, <,6/ 5 > } Let's defe fuzzy recprocal multple-valued reflectos Γ, Γ, Γ,, Γ as: Γ ( = { < ( /( > } Γ (, here ( X [ ( = <, > ] Γ (, Γ ( = Γ {Γ ( }, Γ ( = Γ {Γ ( },, ( Γ ( Γ {Γ ( } = { < ( / > }, here = Γ ( ( > Γ ( Γ ( l l l X It s obvous, that Γ ( X [ ( = ( & <, ] s a fuzzy subset of vertces, from whch t s accessble to reach verte, usg fuzzy ways of legth Eample 4 For the fuzzy graph preseted o Fg, we have Γ ( = { <,4 / 4 > }, Γ ( = { <,4/ 5 > } Defto 4 Fuzzy recprocal trastve closure Γ ( s fuzzy recprocal multplevalued reflecto: Γ ( = Γ ( Γ ( Γ ( = Γ ( I other words, Γ ( s fuzzy subset of vertces, from whch t s accessble to reach verte by some fuzzy way wth the greatest possble couctve durablty Eample 5 For the fuzzy graph preseted o Fg, fuzzy recprocal trastve closure of verte s Γ ( = { < / >, <,/ >, <,4/ >, <,4/ 4 >, <,4/ 5 > } Defto 5 Graph G=(X, s amed fuzzy strogly coected graph f the codto s satsfed: = ( X(S = X ( Γ( Here S s the carrer of fuzzy trastve closure Γ Γ( ( Dfferetly, graph G=(X, s fuzzy strogly coected graph f betwee ay two vertces there s a fuzzy drected way wth the couctve durablty whch s dstct from It s easy to show, that epresso ( s equvalet to epresso (: ( X(S X ( = - Γ (

5 Here - s the carrer of fuzzy recprocal trastve closure Γ ( S Γ ( Defto 6 Let fuzzy trastve closure for verte loos le Γ ( = {< ( / >,< ( / >,,< ( / }, the the volume > ( =, =, ρ G = & & ( we ame a degree of strog coectvty of fuzzy graph G Let G=(X, G =(X, s fuzzy subgraph wth degree of strog coectvty ρ ( G s fuzzy graph wth degree of strog coectvty ρ (G, ad G Defto 7 Fuzzy subgraph =(X, we ame mamum strog coectvty fuzzy subgraph or fuzzy strog compoet coectvty f there s o other subgraph for whch G ρ ρ G =(X, Defto 8 A subset vertces [,], f some verte G, ad ( G (G B X s called fuzzy atbase wth the degree y B may be accessble from ay verte X / B wth a degree ot less ad whch s mmal the sese that there s o subset B B, havg the same accessble property Let's desgate through R (B a fuzzy set of vertces, from whch the subset B s accessble, e: R(B = Γ (, B Here, Γ ( s a fuzzy recprocal trastve closure of the verte The the set B s fuzzy atbase wth a degree oly case, whe the codtos are carred out: R(B = { < / > X&( =,( }, ( ( B B [R(B ={ < / > X&( =,( < }] (4 The codto ( desgates, that ay verte ether s cluded to set B, or s accessble from some verte of the same set wth a degree ot less The codto (4 desgates that ay subset B B does dot have the property ( The followg property follows from defto of fuzzy atbase: Property I fuzzy atbase B there are o two vertces whch are etered to same strog coectvty fuzzy subgraph wth degree above or equal

6 Let,,, } s a set of all values of membershp fucto whch are { L attrbuted to edges of graph G The the followg propertes are true: Property I ay fuzzy crcut-free graph ests eactly L fuzzy atbases wth degrees {,,, L} Property I ay fuzzy crcut-free graph there s ust oe fuzzy atbase wth degree Property 4 If a fuzzy crcut-free graph a equalty < s eecuted, the cluso B B s carred out Let's ote terrelato betwee fuzzy atbases ad strog coectvty fuzzy subgraphs Followg propertes are true Property 5 If subset B s fuzzy atbase wth degree, the there s such subset X X, that B X, ad fuzzy subgraph G = (X, has degree of strog coectvty ot less Property 6 If subset B s fuzzy atbase wth degree, the there s ot such subset X X, that X B ad fuzzy subgraph G = (X, has degree of strog coectvty The followg cosequece follows from property 6: Cosequece That fuzzy graph G there was fuzzy atbase wth degree, cosstg of oly oe verte, t s ecessary that fuzzy graph G was strog coectvty wth degree Property 7 Let γ(, s a accessble degree of verte from verte The the followg statemet s true: (, B [γ(, <] (5 Dfferetly, the accessble degree of ay verte B from ay other verte B s less tha meag Let a set τ ={X, X,,X l } be gve, where X s a fuzzy -verte atbase wth the degree of We defe as = ma {,,, } I the case τ = we defe = - Volume meas that fuzzy graph G cludes -verte subgraph wth the accessble degree ad does t clude -verte subgraph wth a accessble degree more tha Defto 5 A fuzzy set B = { < />, < / >,, < / } l > s called a fuzzy set of atbases of fuzzy graph G Property 8 The followg proposto s true: =

7 The fuzzy set of atbases defes the greatest degree of servce of all terrtory by the cetres of servce ( {,,,} Thus, t s ecessary to determe a fuzzy set of atbases for a fdg of the greatest degree Method for fdg of fuzzy set of atbases We wll cosder the method of fdg a famly of all fuzzy atbases wth the hghest degree The gve method s a aalogue method for defto of all mmal fuzzy domatg verte sets [5] ad t s a geeralzato of the Maghout s method for crsp graphs [6] Assume that a set B s a fuzzy base of the fuzzy graph G wth the degree The for a arbtrary verte X, oe of the followg codtos must be true a B ; b f B, the there s a verte such that t belogs to the set B wth the degree γ(, I other words, the followg statemet s true: ( X[ B ( B ( B γ(, ] (6 To each verte X we assg Boolea varable p that taes the value, f B ad otherwse We assg a fuzzy varable ξ = for the proposto γ(, Passg from the quatfer form of proposto (4 to the form terms of logcal operatos, we obta a true logcal proposto: Φ = &(p (p ( (p &γ Tag to accout terrelato betwee mplcato operato ad dsucto operato ( β= β, we receve: Φ = &(p p (p &γ Supposg ξ = ad cosderg that the equalty p p &ξ = p ξ s true for ay verte, we fally obta: Φ = &( (p &γ (7 We ope the paretheses the epresso (7 ad reduce the smlar terms the followg rules: a a&b=a; a&b a& b=a; ξ &a ξ &a&b (8 Here, a,b {,}, ξ ξ, ξ,ξ [,] The the epresso (7 may be rewrte as:

8 Φ = p & p & & p & (9 ( =, l Property 9 If epresso (9 further smplfcato o the bass of rules (8 s mpossble, the everyoe dsuctve member defes atbase wth the hghest degree Proof Let's cosder, that further smplfcato s mpossble epresso (9 Let, for defteess, dsuctve member ( p & p & & p &, <, (,] ( s cluded the epresso (9 Let's assume, that the subset {,,, } there s some verte, for eample ' X = s ot atbase wth degree The + X / X ', for whch the statemet ( =, ( γ (, < s true I other words, the accessble degree of ay + verte of subset X from verte + s less tha value We preset the epresso (7 a d: Φ = ( p ( ξ +, p & ( ξ p ξ ξ p ξ p & ( ξ p p ξ p & & ( ξ +, p ξ +, p p p p + ξ +, p & I epresso ( all coeffcets ξ <, Therefore, epresso (9 +, =, all dsuctve members whch do ot cota varables p, p,, p, ecessarly + + cota coeffcets smaller value From here follows, that the dsuctve member ( s ot cluded the epresso (9 The receved cotradcto proves, that subset X ' = {,,, } s atbase wth degree Let's prove ow, that the dsuctve member ( s mmum member We wll assume the retur The followg codtos should be carred out: a subset X ' = {,,, } s atbase wth degree β> ; or b there s a subset X " X ' whch s atbase wth degree Let the codto a s satsfed The the et statemet s true: (, = +, (,, γ (, β Let's preset epresso Φ the form of ( If to mae logc multplcato of each bracet agast each other wthout rules of absorpto (8 we wll receve the dsuctve members cotag eactly of elemets, ad, o oe elemet from each bracet of decomposto ( We wll choose oe of dsuctve members as follows: - From the frst bracet we wll choose elemet p ; - From the secod bracet - elemet p ; ;

9 - From th bracet - elemet p ; - From (к+ th bracet we wll choose elemet ξ p such, that de [, ], ad volume ξ > β ; +, +, - From (к+ th bracet - elemet ξ p, for whch de [, ], ad volume ξ > β, etc; +, +, - From th bracet - elemet ξ, for whch de [, ], ad volume ξ > β,, p sg rules of absorpto (8, the receved dsuctve member ca be led to d p & p & & & ', whch the volume ( p β β ' = m{ ξ +,, ξ +,,, ξ, } β > ad whch wll be ecessarly absorbed dsuctve member ( (Decomposto (9 the dsuctve member caot clude the receved cotradcto ( proves mpossblty of case a Let's assume ow, that the codto s satsfed Let, for defteess, X " = {,,, } Cosderg epresso Φ the form of decomposto (8, we wll choose a dsuctve member as follows: - From the frst bracet we wll choose elemet p, - From the secod bracet - elemet p,, - From (к- th bracet - elemet p, - From th bracet we wll choose elemet ξ p such, that de [, ], ad volume ξ,, - From (к+ th bracet - elemet volume ξ +,, etc, +,, ξ p, for whch de [, ], ad - From th bracet - elemet ξ p, for whch de [, ], ad volume ξ, +, sg rules of absorpto (8 the receved dsuctve member ca be led to d ( p & p & & p &, whch sze β = m{ ξ,, ξ +,, ξ },, + ad whch wll be ecessarly absorbed by a dsuctve member ( (Decomposto (9 the dsuctve member caot clude the receved cotradcto ( proves mpossblty of case b Property 9 s proved The followg method of foudato of fuzzy atbases may be proposed o the base of property 9: - We wrte proposto (7 for gve fuzzy graph G ; - We smplfy proposto (7 by proposto (8 ad preset t as proposto (9; - We defe all fuzzy atbases, whch correspod to the dsuctve members of proposto (9 Eample 6 Fd all fuzzy atbases for the fuzzy graph preseted Fg:

10 ,,8,7,,6 Fg Fuzzy graph The verte matr for ths graph has the followg form:,7 R = 6,8,,6, O the bass of the made above defto of fuzzy accessble verte we ca costruct a accessble matr N, cotag accessble degrees for all pars of vertces: N= = = γ R Here, γ =γ(,,, X, R the verte matr power for graph Matr R s a detty matr: R = We rase the cotguty matr to, powers tg them, we fd a accessble matr: N = R R R R,7 =,6,6,8,6,6,6,6,6,6,6 The correspodg epresso (7 for ths graph has the followg form:

11 Φ = (p p,8 p,6 p4,6 & (p,7 p p,6 p4,6 & & (p,6 p,6 p p & (p,6 p,6 p,6 p,6 4 Multplyg parethess ad, parethess ad 4 ad usg rules (8 we fally obta: Φ = p,6 pp4,7 ppp4 p,6 pp4,8 p,6 p4,6 It follows from the last equalty that the graph G has 7 fuzzy atbases, ad fuzzy set of atbases s defed as: B = { <,6/ >, <,8/ >, < / >, < / 4 > } The fuzzy set of atbases defes the et optmum allocato of the servce cetres: If we have or more servce cetres the we must place these cetres to vertces,, ad 4 The degree of servce equals ths case If we have servce cetres the we must place these cetres to vertces, ad 4 The degree of servce equals,8 ths case If we have oly oe servce cetre the we ca place t ay verte The degree of servce equals,6 last case 4 Cocluso The tas of defto of optmal allocato of the servce cetres as the tas of defto of fuzzy atbases of fuzzy graph was cosdered The defto method of fuzzy atbases s the geeralzato of Maghout s method for ofuzzy graphs It s ecessary to mar that the suggested method s the method of ordered full selecto, because these tass are reduced to the tas of coverg, e these tass are NPcompete tass However, ths method s effectve for the graphs whch have ot homogeeous structure ad ot large dmesoalty Refereces Chrstofdes, N: Graph theory A algorthmc approach Academc press, Lodo (976 Malczews, J: GIS ad multcrtera decso aalyss Joh Wlley ad Sos, New Yor (999 Kaufma, A: Itroducto a la theore des sous-esemles flous Masso, Pars (977 4 Bershte, LS, Bozheu, AV: Fuzzy graphs ad fuzzy hypergraphs I: Dopco, J, de la Calle, J, Serra, A (eds Ecyclopeda of Artfcal Itellgece, Iformato SCI, pp Hershey, New Yor (8 5 Bershte, LS, Bozheu, AV: Maghout method for determato of fuzzy depedet, domatg verte sets ad fuzzy graph erels J Geeral Systems : ( 6 Kaufma, A: Itroducto a la combatorque e vue des applcatos Duod, Pars (968