A Multi Objective Graph Based Model for Analyzing Survivability of Vulnerable Networks. Saeedeh Javanmardi 1, Ahmad Makui 2.

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1 Journl of Inustrl n Systms Engnrng Vol., No., pp Summr 00 Mult Otv Grp Bs Mol for nlyzng Survvlty of Vulnrl Ntworks S Jvnmr, m Mku Dprtmnt of Inustrl Engnrng, Irn Unvrsty of Sn n Tnology, Trn, Irn mku@ust..r BSTRT In t vrous fls of sstr mngmnt, oosng t st loton for t Emrgny Support & Supply Srv ntrs (ESSSs) n t survvlty of t ntwork tt provs t lnks twn ESSSs n tr nvronmnt s grt rol to p noug ttnton. Ts ppr ntrous grp s mol to msur t survvlty of t lnkng's ntwork. By vlus omput for tm n ost of rovry of lnk flurs, t propos lotons for ESSSs n rnk. By onsrng t onflts tt n rs twn mxmzng t survvlty of t ntwork n mnmzng t tm n ost of rovry of lnk flurs, n lgortm s propos tt us Smpl tv Wgtng (SW) Mto. numrl xmpl s prov n solv to llustrt ow t lgortm works. Hvng solv t prolm wt ffrnt wgtng vtors, susson s m on t snstvty nlyss of t soluton. Kywors: Survvlty, Dsstr mngmnt, Smpl tv wgtng mto. INTRODUTION Dtrmnng t st loton for n ESSS s n mportnt prolm, splly n t rgons wt grt lvl of sstrs n unprtl vnts. In t s of lnk flurs ftrmt sstrs, t most mportnt prolm s to fn on or mor routs to r t ESSS. Howvr t mount of tm n ost spnt n rtl stutons v somtms no mportn t ll, of ours notng s mu mor sgnfnt tn umn ngs. On t otr n, somtms t trg s of t vnts s tt support & supply or mrgny ml srv (EMS) ntrs xst, ut us of lk of survvlty of t ntwork, nooy n r tm. Trfor, n fftv wy to rs t strutons ftr vnts tt us grt rm or mgs, splly unprtl ons su s rtquk, floos, frs n so on, s to v mor survvl ntworks tt oul prov ffrnt routs to r t n pls. Tr r lot of fntons of survvlty n Grgory (00), Korzk n Grgory (007) n Zsung n Lng Sun (998). In ts ppr t fnton of Grötsl t l. (99) for survvl ntwork s onsr. Ty lm tt t survvl ntworks r t ons tt r stll funtonl ftr t flur of rtn ntwork omponnts. In t prolm onsr n ts ppr, survvl ntwork s t on tt s stll funtonl ftr som lnk flurs. ntwork w s orrsponng utor ISSN: 787, opyrgt 00 JISE. ll rgts rsrv.

2 Mult Otv Grp Bs Mol for noug survvlty, n lnk ny two or mor of ts ffrnt prts n t stutons tt t mn routs r not ppll nymor. frmwork rv from trtonl ntrprs molng tools s propos to ptlz umntrns' knowlg, to nlyz ot gps n st prts n lrn from on oprton to notr, rls t l. (009). Yun n Dngw (009) propos two mtmtl mols for pt slton n mrgny logsts mngmnt. Tr frst mol s snglotv mol n t son on s multotv pt slton mol. u t l. (008) formult n ntgr progrmmng mol tt optmzs t smultnous lloton of multpl typs of mrgny vls mong st of nt sttons to mxmz t srv ovrg to rtl trnsportton nfrstruturs. flty loton mol sut for lrgsl mrgns s propos. Grp tory provs ttr wy to nlyz su ntwork, Hrry (969). So t grp onpt n trnsform t prolm to ffrnt sp. ftr solvng t prolm s grp, t n rougt to ts orgnl sp n orr to rv t rl soluton, Krngs n zmns (00). Ts ppr prsnts mult otv mol to formlz survvlty prolms, s on grp n son mkng mtos. Ston ovrvws t mol n xplns ts pss n tl. Ston outlns s stuy n fnlly n ston onlusons r prsnt.. MODEL OVERVIEW T propos mol s sown n Fgur. Prmtrzton omputng orrsponng survvlty of vry loton Ntwork Trnsformton to smpl nrt grp Ntwork mprovmnt oosng t st loton for ESSS Fgur Gnrl Mol Suppos tt ntwork of n mgnry rgon s prov. Du to t prvous rtr, xprts stt som propos lotons pproprt for ESSS. T stgs r s follows. Ntwork Trnsformton: t frst t ntwork s trnsform to smpl nrt orrsponng grp G = (V, E), wr V s fnt st of vrts n E s st of gs, rprsntng rltons twn prs of vrts. T propos lotons for ESSS r spf on t grp.

3 6 Jvnmr n Mku Prmtrzton: s t rgon s mpp to vrts n gs of G, nformton su s survvlty, tm n ost of rovry of lnks must ssgn to gs. omputng survvlty: On wgt grp G s fn, t lvl of survvlty of ny prt of t ntwork n omput. Frst of ll, ll gsont pts twn propos lotons n ll otr nos must fnng. Hvng ll gsont pts, t survvlty of vry g, vry gsont pt n fnlly t totl orrsponng ntwork of vry propos loton r omput. Tm n ost of rovry of lnks n otn n t sm wy. oosng t st loton: By vng survvlty, tm n ost of rovry for vry propos loton, SW (Smpl tv Wgtng) mto n us to fn t st loton for t ESSS n rnk tm. T followng lgortm sows ow t st loton for ESSSs n foun... lgortm. Trnsform t ntwork to t orrsponng smpl n unrt grp.. Sow t propos lotons for ESSSs y ptl lttrs n put tm n st nm R.. Sow t otr nos y smll lttrs n put tm n st nm N.. Slt on of nonslt mmr of R, nm K. f ll of tm r slt sn now, go to stp 0.. Fn ll gsont pts twn K n otr nos, n put tm n Egsont pts(k) = {Egsont pts (K, ) N} 6. For g, (, ), omput survvlty of (, ) For gsont pt ( ), omput survvlty ( ) Sur ( ) = Mn Vlu { Sur (, )} (, ) ( ) 7. For propos loton omput survvlty of tt loton Sur (s) = Mn Vlu {Sur (s )} Vs 8. Do t sm for omputng Tm n ost of rovry of gs, gsont pts n ntwork of vry propos loton. T(r( )) = Mn Vlu { T(r (,)) } (,) ( ) (r( )) = Mn Vlu { (r (,)) } (,) ( ) T (r (s)) = Mn Vlu {T (r (s ))} Vs (r (s)) = Mn Vlu{ (r (s ))} Vs 9. Go to stp 0. Dvlop t orrsponng ntl SW (Smpl Wgtng Mto) mtrx, usng vlus omput for vry propos loton us of t rtr.. St t wgtng vtor, orng to t mportn of rtr.. pply SW to t prolm.. T output s t st soluton for t prolm. Hr s t orrsponng Psuo o of t propos lgortm:

4 Mult Otv Grp Bs Mol for 7 Psuo o Inputs: N, orgnl vrts st. R, propos lotons st. E, gs. Sur, tl of gs survvly Tr, tl of gs rovry tm r, tl of gs rovry ost, rtr vtor W, rtr wgts vtor Output: Bst loton Mto: For K R o If R = Ø tn ll SW (N, R, E, W); Els ll omputnt (K); Prour omputnt (K) () for v N o () Egsontpts(K) ll of Egsont pts twn K n v; () For ( ) Egsontpts(K) o () Sur ( ) Mn Vlu { Sur } (,) ( ) () Sur (K) Mn Vlu{Sur (K )} (6) T(r ( )) Mn Vlu { Tr } (,) ( ) (7) T(r (K)) Mn Vlu{T(r (K ))} (8) (r ( )) Mn Vlu { r } (,) ( ) (9) (r (K)) Mn Vlu{(r (K ))} (0) R RK () Rturn; Prour SW () for = to R o () SW Sur(K ) () SW T(r (K )) () SW (r (K )) () for = o (6) SW = SW /mx(r ) (7) for = or = o (8) SW = mn(r )/SW (9) for = to R o (0) for = to o () SW SW *W () for = to R o () Vlu(K ) = SW () Fn Mx{Vlu(K )} () Bst Loton K (6) Rturn;

5 8 Jvnmr n Mku.. Numrl xmpl Suppos t grp sown n Fgur s otn usng t nformton of n orgnl ntwork. Propos lotons r sown wt ptl lttrs n otr rgons wt smll lttrs. Fgur Smpl Grp So t nputs r s follows: N :{,,,,,,, } R: {, } Sur, tl of gs survvlty Tl Survvlty of gs Sur(,) Tr, tl of gs rovry tm Tl Tm of rovry of fl gs T(r(,)) r, tl of gs rovry ost

6 Mult Otv Grp Bs Mol for 9 : (Sur, Tr, r) W: (0., 0., 0.) Mto: (r(,)) Tl ost of rovry of fl gs K = omputnt (K) Sur () = Mn Vlu{8, 6, 7., 0.6, 0, 7, 7., 9.67} = 8.6 T (r ()) = Mn Vlu {T (r ( ))} Vs = Mn Vlu {7.,.67, 8,,.67, 7.67, 7, } = 8.9 (r ()) = Mn Vlu{ (r ( ))} Vs = Mn Vlu {6.,, 7., 0, 9, 6.67, 6.67, 8.67} = 7. R R R K = omputnt () Sur () = Mn Vlu {Sur ( )} Vs = Mn Vlu {6.67, 6., 9., 0, 9., 6.,, 6.67} = 7.6 T (r ()) = Mn Vlu {T (r ( ))} Vs = Mn Vlu {7., 7,.,.67,., 7.,, 7.67} = 8.7 (r ()) = Mn Vlu{ (r ( ))} Vs = Mn Vlu{6., 6.67,,67, 0.67, 9.67, 6.67,, 6.} = 707 R Ø SW

7 0 Jvnmr n Mku Sum =. Sum =.8 Mx {Vlu (K )}. Bst Loton T soluton s prsnt n mor tl n ppnx... Ntwork Improvmnt In t s wr t totl survvlty of t ntwork s low, t ntwork potntlly n mprov. So t gs wt low lvls of survvlty must otn n ftr fnng tm, ty r mprov to o ttr n lp t totl ntwork to v ttr srvs n sstrs n unprtl vnts.. ONLUSIONS In ts ppr mult otv grp s mol s prsnt n orr to fn t st on mong svrl propos lotons for ESSS. smpl tv wgtng mto lps t mol n ts wy. Ts mol n usfully ppl n fnng t st loton of otr mrgny n vtl ntrs of ommunts w ntwork survvlty s n mportnt ftor for lotng tm. T propos mol n vlop y usng rt orrsponng grps wn s nssry. It n lso us ffrnt wgts to tst n rv soluton tt s nrr to rl stutons. REFERENES [] Lvtn, Grgory (00), Mxmzng survvlty of yl trnsmsson ntworks wt multstt rtrnsmttrs n vulnrl nos; Rllty Engnrng n Systm Sfty 77; [] Korzk, Ewr, Lvtn, Grgory (007), Survvlty of systms unr multpl ftor mpt; Rllty Engnrng n Systm Sfty 9; [] Zsung, Zu, Lng, Sun (998), strtgl mol for nlyzng survvlty of nvronmntl rsour mngmnt systm; IPRS ; 68. [] M. grotsl,.l. Monm, M. Stor (99), Hnooks n OR & MS;. 0, Vol. 7, Elsvr Sn B.V.. [] rls., Lurs M., Tomsn R. (009), Lrnng from prvous umntrn oprtons, usnss pross rngnrng ppro; Prong of 6 t Intrntonl ISRM onfrn; Swn. [6] Yun Y., Dngw W. (009), Pt slton mol n lgortm for mrgny logsts mngmnt; omputrs n nustrl ngnrng 6; [7] u R., Hung Y., Hung B. (008), llotng mrgny srv vls to srv rtl trnsportton nfrstruturs; Journl of Intllgnt Trnsportton Systms; 89. [8] J H., Ornz F., Dssouk M. (00), molng frmwork for flty loton of ml srvs for lrgsl mrgns; rt rport, unvrsty of soutrn lforn. [9] Hrry, F. (969), Grp Tory. Rng, M: sonwsly.

8 Mult Otv Grp Bs Mol for [0].W. Krngs,. zmns (00), Grp Bs Mol for Survvlty ppltons; Europn Journl of Oprtonl Rsr 6; ppnx W prov tl soluton for t numrl xmpl.: T nputs r s follows: N :{,,,,,,, } R: {, } Sur, tl of gs survvly Tr, tl of gs rovry tm r, tl of gs rovry ost : (Sur, Tr, r) W: (0., 0., 0.) Mto: K = omputnt(k) Egsont pts(k) = {Egsont pts (,) N} Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,, Sur( ) = Mn Vlu { Sur } (, ) ( ) = Mn Vlu {Sur(,.) + + Sur(., )} Sur( ) = Mn Vlu {Sur(), Sur(), Sur()} = Mn vlu {, 7, } = 8 Sur( ) = Mn Vlu {Sur(), Sur(), Sur()} = Mn vlu {, 8, 7} = 6 Sur( ) = Mn Vlu {Sur(), Sur(), Sur()} = Mn vlu {,, } = 7. Sur( ) = Mn Vlu {Sur(), Sur(), Sur()} = Mn vlu {8, 6, 8} = 0.6 Sur( ) = Mn Vlu {Sur(), Sur(), Sur()} = Mn vlu {, 0, } = 0 Sur( ) = Mn Vlu {Sur(), Sur(), Sur()} = Mn vlu{9,, 8} = 7 Sur( ) = Mn Vlu {Sur(), Sur(), Sur()} = Mn vlu {7,, 0} = 7. Sur( ) = Mn Vlu {Sur(), Sur(), Sur()} = Mn vlu {0, 7, } = Sur(s) = Mn Vlu {Sur(s )} Vs Sur() = Mn Vlu {Sur( )} Vs Sur() = Mn Vlu{8, 6, 7., 0.6, 0, 7, 7., 9.67} = 8.6 T(r( ) = Mn vlu { T(r(,)} (,) ( )

9 Jvnmr n Mku = Mn vlu {T(r(,.) + + T(r(., )} T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {,, } = 7. T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {, 6, 9} =.67 T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {, 7 } = 8 T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {9, 8, 9} = T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {,, 7} =.67 T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {9, 6, 8} = 7.67 T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {6,, } = 7 T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {, 8, 7} = T(r(s)) = Mn Vlu {T(r(s ))} Vs T(r()) = Mn Vlu {T(r( ))} Vs = Mn Vlu {7.,.67, 8,,.67, 7.67, 7, } = 8.9 (r( ) = Mn vlu { (r(, )} (, ) ( ) = Mn vlu {(r(,.) + + (r(., )} (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {,, } = 6. (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {, 6, 7} = (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {, 6, } = 7. (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {6, 9, } = 0 (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu{, 8, 6} = 9 (r( )) = Mn vlu{(r()), (r()), (r())} = Mn vlu {9,, 6} = 6.67 (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu{6,, 9} = 6.67 (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {8, 8, 0} = 8.67 (r(s)) = Mn Vlu{(r(s ))} Vs (r()) = Mn Vlu{(r( ))} Vs = Mn Vlu {6.,, 7., 0, 9, 6.67, 6.67, 8.67} = 7. R R R K = omputnt () Egsont pts (K) = {Egsont pts (, ) N} Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,,

10 Mult Otv Grp Bs Mol for Egsont pts (, ) =,, Egsont pts (, ) =,, Egsont pts (, ) =,, Sur( ) = Mn vlu { Sur(, )} (, ) ( ) = Mn vlu {Sur(,.) + + Sur(., )} Sur ( ) = Mn vlu {Sur(), Sur(), Sur()} = Mn vlu {, 7, 0} = 6.76 Sur ( ) = Mn vlu {Sur(), Sur(), Sur()} = Mn vlu {7, 6, 6} = 6. Sur ( ) = Mn vlu {Sur(), Sur(), Sur()} = Mn vlu {9, 9, 0} = 9. Sur ( ) = Mn vlu {Sur(), Sur(), Sur()} = Mn vlu {, 8, 0} = 0 Sur ( ) = Mn vlu {Sur(), Sur(), Sur()} = Mn vlu {, 7, 7} = 9. Sur ( ) = Mn vlu {Sur(), Sur(), Sur()} = Mn vlu{8, 6, } = 6. Sur ( ) = Mn vlu {Sur(), Sur(), Sur()} = Mn vlu{, 6, } = Sur ( ) = Mn vlu {Sur(), Sur(), Sur()} = Mn vlu{, 7, } = 6.67 Sur (s) = Mn Vlu {Sur (s )} Vs Sur () = Mn Vlu {Sur ( )} Vs = Mn Vlu {6.67, 6., 9., 0, 9., 6.,, 6.67} = 7.6 T(r( ) = Mn vlu { T(r (,)} (,) ( ) = Mn vlu {T (r (,.) + + T (r (., )} T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {,, } = 7. T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {6,, } = 7 T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {, 9, 7} =. T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {6, 7, } =.67 T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {0, 6, 8} =. T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {0,, 7} = 7. T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {, 6, 7} = T(r( )) = Mn vlu {T(r()), T(r()), T(r())} = Mn vlu {, 7, } = 7.67 T(r(s)) = Mn Vlu {T(r(s ))} Vs T(r()) = Mn Vlu {T(r( ))} Vs = Mn Vlu {7., 7,.,.67,., 7.,, 7.67} = 8.7 (r( ) = Mn vlu { (r(, )} (, ) ( ) = Mn vlu {(r(,.) + + (r(., )} (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {, 6, } = 6. (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {, 7, 9} = 6.67 (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {, 0, } =.67 (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {,, 0} = 0.67

11 Jvnmr n Mku (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {, 8, 8} = 9.67 (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {7, 7, 6} = 6.67 (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {,, } = (r( )) = Mn vlu {(r()), (r()), (r())} = Mn vlu {, 6, 0} = 6. (r(s)) = Mn Vlu{(r(s ))} Vs (r()) = Mn Vlu{(r( ))} Vs = Mn Vlu{6., 6.67,,67, 0.67, 9.67, 6.67,, 6.} = 707 R Ø SW Sum =. Sum =.8 Mx {Vlu (K )}. Bst Loton

Lecture 20: Minimum Spanning Trees (CLRS 23)

Lecture 20: Minimum Spanning Trees (CLRS 23) Ltur 0: Mnmum Spnnn Trs (CLRS 3) Jun, 00 Grps Lst tm w n (wt) rps (unrt/rt) n ntrou s rp voulry (vrtx,, r, pt, onnt omponnts,... ) W lso suss jny lst n jny mtrx rprsntton W wll us jny lst rprsntton unlss