S. FANG, H. WAKABAYASHI primrily depends on locl government support fter disster in Chin. In prctice, the decisions of locl government in selecting th

Journl of Nturl Disster Science, Volume 34, Number 2, 2013, pp115-125 Improvement of Relibility fter Disster Bsed on Locl Government Support Shuming FANG* Hiroshi WAKABAYASHI** *Doctorl course student, Grdute School of the Fculty of Urbn Science, Meijo University **Professor, Fculty of Urbn Science, Meijo University (Received October 2, 2012 Accepted November 5, 2013) ABSTRACT Highly relible trffic networks on highwys nd rodwys re very importnt during bnorml periods, such s during dissters. relibility cn be improved effectively by focusing on the most importnt key link in the network. Although mny indices hve been proposed for identifying the most importnt key link, numerous shortcomings ssocited with these indices mke it difficult to obtin good solution for evluting network relibility improvement. In ddition, n investment strtegy bsed on locl government support is lso importnt for improving network relibility. However, the cpitl vilble from locl government for trffic restortion is usully limited fter disster. This pper therefore proposes n optiml method of network relibility improvement bsed on locl government support combined with limited cpitl. To determine the efficiency of the model for network relibility improvement, simultions for simple prllel network nd simple bridge network re conducted. A generl lgorithm for identifying the most importnt key link in ny network is then developed bsed on these simultions. Lstly, rtionl conclusion for network relibility improvement is presented. Keyword: relibility,, Locl government support, Importnce index, Disster. 1. Introduction It is importnt to mintin highly relible trffic network of highwys nd rodwys for bnorml periods, especilly during disster. However, the trffic systems of disster res my be seriously dmged, nd deciding which dmged rod should be improved first to mintin or improve trffic network relibility is very difficult. In generl, network relibility cn be gretly improved by improving the key link in the network; the most importnt key link is the link to improve network relibility to the gretest degree. Thus, indices such s relibility importnce (RI) (Brlow et l., 1975) nd criticlity importnce (CI) (Henley et l., 1981) hve been proposed. However, these indices hve their own shortcomings for finding the most importnt key link in some types of networks. In ddition, t the first stge of disster, the cost of repiring the dmged trffic system is importnt for improving trffic network relibility. In Chin, locl government is responsible for reconstruction fter disster, even though the government most likely hs limited cpitl. Therefore, mximizing trffic network relibility s soon s possible 115

S. FANG, H. WAKABAYASHI primrily depends on locl government support fter disster in Chin. In prctice, the decisions of locl government in selecting the most importnt key link re ffected by mny fctors, such s the popultion of the disster re ner dmged rod, the dmge level of other rods, the economic sttus of industril res ner dmged min rod, nd so on. For exmple, the Chinese government first lloctes the most resources to repir dmged rods in disster regions tht re considered the most essentil for strtegic or economic resons. However, locl government support ws first overlooked in current indices such s probbility importnce nd criticlity importnce. Nevertheless, to improve network relibility fter disster, the effect of locl government support on finding the most importnt key link should be tken seriously nd discussed crefully. 2. Current Importnce Indices The concept of importnce indices hs long been proposed in the field of systems engineering, but hs ppered in only few ppers in the trnsporttion field (Brlow, 1969). Importnce is defined s the degree of mgnitude to which n improvement in link relibility contributes to system relibility. The indices of importnce proposed in this pper re bsed on connectivity relibility (lso referred to s terminl relibility). 2.1. Terminl Relibility The terminl relibility of highwy network is defined s the probbility tht two given nodes over the network re connected with certin service level of trffic for given time period (Iid nd Wkbyshi, 1989; Wkbyshi nd Iid, 1992). Similrly, link relibility in the network is defined s the probbility tht the trffic reches certin service level for given time period. Terminl relibility, R, is given by n expression using miniml-pth sets, s follows (Iid nd Wkbyshi, 1990): p R ( r) = E[1 (1 X )], S= 1 Ps (1) where P S is the S th miniml-pth set nd p is the totl number of miniml-pth sets. This clcultion method is bsed on the Boolen bsorption method (Wkbyshi nd Iid, 1992). Here, tor vrible for link : X X is the binry indic- 1, if link provides certin service level, = 0, otherwise. (2) Link relibility, r, is defined s r = E[ X ]. (3) The terminl relibility of trffic network depends on the network structure nd link relibilities. Therefore, two bsic pproches hve been tken to improve network relibility: to improve the network structure or to improve the relibility of the links. The focus here is on identifying which links should be improved to mximize the improvement in network relibility. 2.2. Birnbum Structurl Importnce (1) Definition of Birnbum structurl importnce To find the key link for improving terminl relibility efficiently, the Birnbum structurl importnce index ws proposed (Birnbum, 1969). Birnbum structurl importnce is defined s PI R( r) = r A, 0 PI 1. r 1 Link 1 A () S eries n etw ork r 1 Link 1 r 2 Link 2 Link 2 (b ) P rllel n etw ork Fig. 1 Simple series nd prllel networks (4) Birnbum structurl importnce indictes the impct of link, such tht n increse or decrese in the relibility of the link ffects increse or decrese in terminl relibility. Birnbum structurl importnce is lso known s relibility importnce. (2) Advntges nd shortcomings of Birnbum structurl importnce Although Birnbum structurl importnce hs the potentil to improve network relibility, it hs disdvntge, which is discussed in this section. For the cse of two links in series network s shown in Fig. 1., terminl relibility RAB is shown in Eq. (5): r 2 B B 116

IMPROVEMENT OF TRAFFIC NETWORK RELIABILITY AFTER A DISASTER BASED ON LOCAL GOVERNMENT SUPPORT R = rr. AB 1 2 (5) Here, r 1 nd r 2 re the relibilities for nd link 2, respectively. For the cse of two links in prllel network s shown in Fig. 1.b, terminl relibility RAB is shown in Eq. (6): R AB = 1 (1 r 1 )( 1 r ). 2 (6) The two vlues of Birnbum structurl importnce for series network, RI1 nd RI2, re obtined from Eq. (4) nd Eq. (5) s RI1 = r2 nd RI2 = r1. (7) It follows tht RI1 > RI2, if r1 < r2. (8) Eq. (8) indictes tht improving the lest relible link in series-type network is most effective for improving terminl relibility. This fct is esily expnded for lrge series-type networks. This result for improving, mnging, nd reconstructing network is the expected result. RI1 nd RI2 for these two links in prllel network re obtined from Eq. (4) nd (6) s RI1 =1- r2 nd RI2 =1- r1. (9) It follows tht RI1 < RI2, if r1 < r2. (10) The result from Eq. (10) indictes tht improving the most relible link in prllel-type network is more effective for improving terminl relibility. Usully, however, it is difficult to improve more relible link, wheres it is rther esy to improve less relible link (Brlow nd Proschn, 1975). This result is counter to wht one would expect for improving, mnging, nd reconstructing network. 2.3. Criticlity Importnce (1) Definition of criticlity importnce Becuse of the shortcoming of Birnbum structurl importnce, the criticlity importnce index (CI) ws proposed. CI is the rtio of the proportionl improvement in network relibility to the proportionl improvement in link relibility (Henley nd Kummoto, 1992): CI R / R = r / r = PI r. R (11) (2) Advntges nd shortcomings of criticlity importnce Criticlity importnce lso hs shortcoming, which is discussed in this section. For the cse of two links in series network, it follows from Eq. (4), (5), (7), nd (11) tht r1 r2 CI 1 = = CI. 2 (12) R This result suggests tht the criticlity importnce index is the sme for both links in series network. However, in series network, it is resonble to strengthen less relible link, nd this is thus shortcoming of the criticlity importnce index. In ddition, it does not provide informtion to distinguish between two links in terms of improving network relibility. For the cse of two links in prllel network, it follows from Eq. (4), (6), (9), nd (11) tht r1 r1 r2 CI1 =, (13) R nd r2 r1 r2 CI 2 =. (14) R It follows tht CI1 < CI2, if r1 < r2. (15) Therefore, the criticlity importnce index lso indictes tht in the cse of prllel-type network, improving more relible link further increses the terminl relibility of the network. The results for prllel network provided by both RI nd CI suggest tht less relible link should be ignored in prllel system. In other words, people who live long less relible link would be neglected fter disster. This is not resonble plnning for disster prevention nd reduction. Thus, this result is not expected. 2.4. Advnced Criticlity Importnce (1) Definition of dvnced criticlity importnce proposed by Wkbyshi Bsed on the shortcomings of Eq. (10), (12), nd (15), Birnbum structurl importnce nd criticlity importnce do not reflect the fct tht it is more difficult to improve more relible link thn to improve less relible link. Thus, it is convenient to define importnce s the proportion of the mrginl chnge in terminl relibility ginst the mrginl chnge in link relibility. Chnging the definition of the eqution in 117

S. FANG, H. WAKABAYASHI relibility engineering, the criticlity importnce index proposed by Wkbyshi (CIW) is defined s Eq. (16) (Wkbyshi, 2004): ΔR( r)/ R( r) 1 r CIW (16) = lim = PI, Δq 0 Δq / q R where q=1-r is the unrelibility of link. (2) Advntges nd shortcomings of dvnced criticlity importnce For the cse of two links in series network, it follows from Eq. (4), (5), (7), nd (16) tht nd CIW CIW 1 r 1 1 =, r1 1 r 2 2 =. r2 (17) (18) It lso follows tht CIW1 > CIW2, if r1 < r2. (19) Thus, in series-type network, dvnced criticlity importnce hs the sme property s Birnbum structurl importnce, nd this property from Eq. (19) is exctly s one would expect. For the cse of two links in prllel network, it follows from Eq. (4), (6), (9), nd (16) tht (1 r )(1 r ) CIW = 1 2 1 = CIW2. r1 + r2 r1 r2 (20) From Eq. (20), lthough the dvnced criticlity importnce index is better thn tht proposed by Henley nd Kummoto (1981), this index is the sme for both links in prllel network, so it does not provide informtion to distinguish between them in terms of improving network relibility. The indices RI, CI, nd CIW discussed bove, becuse of their own shortcomings, cnnot be used to select the most importnt key link of trffic network. Therefore, good solution cnnot be obtined by these indices for evluting the improvement of network relibility. In ddition, lthough the cost-benefit rtio is lso importnt (Nicholson, 2007), these indices cnnot predict the increse in cost for improving link relibility when link relibility increses. Thus, trffic network relibility increse in ccordnce with different investment strtegy by locl government should be discussed. 3. Model of Improvement of Relibility Bsed on Locl Government Support As described in Chpter 2, different investment strtegy by locl government is useful for selecting the most importnt key link. In fct, only limited cpitl my be vilble for improving link relibility t the first stge of disster. In generl, different locl governments use different investment strtegies nd provide different support to improve network relibility in Chin. Thus, the effect of locl government support on repiring dmged trffic systems should be discussed bsed on limited cpitl. 3.1. Locl Government Support In Chin, locl government plys n importnt role in reconstructing dmged trffic network nd mintins trffic order fter disster. Not only is the limited cpitl pproprited from the Ministry of Finnce nd tht from the locl finnce deprtment distributed to disster res by locl government, but ll kinds of nonmteril support such s volunteers, psychologicl counseling for victims, nd so on re lso rrnged by locl government. In other words, locl government cn provide mteril support nd nonmteril support. In generl, mteril support is usully considered to be money or goods, nd nonmteril support includes mny spects such s civil technology support, volunteers for reconstructing the trffic system or mintining trffic order, nd so forth. In ddition, piring support policy is being implemented in Chin. Piring support is system to support disster res by ctivting ntionwide help. Other locl governments of non-disster res provide specil prtnership ssistnce support, especilly nonmteril support, to the locl government of disster re when disster hppens. 3.2. Assumptions before Modeling Some ssumptions re given so s to discuss the efficiency of locl government support in mximizing improvement of network relibility in the cse of limited cpitl: 1) The cost of mking the link relibility of link rech 1.0 tends to infinity when locl gov ernment does not provide ny support for link. 2) The mximum cpitl for improving network relibility is limited. 118

IMPROVEMENT OF TRAFFIC NETWORK RELIABILITY AFTER A DISASTER BASED ON LOCAL GOVERNMENT SUPPORT 3) Although relibility equls zero, the originl cost does not equl zero becuse of the bsic work for incresing the link relibility. 4) The support provided by locl government for certin link is limited. 5) The support provided by locl government for certin link is only nonmteril support becuse nonmteril support is more vried thn other fctors t the first stge of disster. 3.3. Modeling Bsed on the bovementioned ssumptions, the reltionship between the increse in cost nd the increse in link relibility cn be formulted. If the vilble cpitl for link with relibility r is C, the cost of improving link relibility increses s link relibility increses by ssuming the proportion of mrginl cpitl dc ginst mrginl link relibility increse dr, nd the reltionship between dc nd dr is shown s follows: r dc β = αe. (21) dr α : Positive constnt, where dc / dr equls α when r equls zero. β : Non-negtive constnt tht is locl government support for link, where β equls zero when locl government cnnot provide ny support, nd dc / dr tends to infinity. Bsed on Eq. (21), the cost-relibility function is shown s Eq. (22): C r β = e + C 0 αβ. (22) Becuse of ssumption 3), C 0 is the vlue of ( C αβ ) when r equls zero. When the link relibility of link increses from r to r + Δr, the required cpitl is clculted s δc = αβ r +Δr β e αβ e (23) If N links re in the trffic network of disster re, the totl cpitl for improving network relibility is shown in Eq. (24): N C increse = δ C. = 1 r β (24). At the first stge of disster, the cpitl for improving network relibility is invribly limited; thus, mximizing network relibility increse bsed on limited cpitl becomes nonliner optimiztion problem, shown s the following: Mximize: δr = R R 0 ; Subject to: ri 0 ri 1, i (1, n; ) C = C; B: An exogenous vrible tht reflects entire locl government support in the dmged trffic network; C: A constnt tht mens the limited cpitl for trffic restortion; D: A constnt; n: Number of links in the whole network; r i0 increse β B; r i i0 = D (D1). (25) : Originl relibility of link i ; R : Current trffic network relibility; R 0 : Originl trffic network relibility; δr: relibility increse. 4. Simultion of the Model In this chpter, to find the most importnt key link, the most optimized improvement of the whole network relibility s stndrd nonliner optimiztion problem is simulted by using the mthemtic softwre LINGO for simple prllel network nd simple bridge network. 4.1. Effect of Government Support on Simple Prllel To find the most importnt key link, the following three cses show the difference in the originl link relibility between two links in prllel network bsed on different levels of government support between two links: very smll difference ( = 0.45 nd r = 0.55), r10 20 r10 20 gret difference ( = 0.3 nd r = 0.7), nd very gret difference ( = 0.1 nd r = 0.9). r10 20 (1) A very smll difference in the originl link relibility between two links Figure 2 includes four cses tht show improvement of link relibility nd network relibility bsed on locl government support for two links when = r 20 0.45 nd = 0.55. The primry horizontl xis of every cse in Fig. 2 is locl government support for link 1, the secondry horizontl xis is locl government r 10 119

S. FANG, H. WAKABAYASHI Link 2 r 10 = 0.45 r 20 = 0.55 Link 1 Fig. 2. Locl government support for two links is the sme Link 2 r 10 = 0.45 r 20 = 0.55 Link 1 Fig. 2.c Locl government support for is greter Link 2 r 10 = 0.45 r 20 = 0.55 Link 1 Fig. 2.b Locl government support for is little greter Link 2 r 10 = 0.45 r 20 = 0.55 Link 1 Fig. 2.d Locl government support for is smller Fig. 2 Results for very smll difference in originl link relibility between two links support for link 2, nd the primry verticl xis is the relibility increse of the links nd trffic network. As shown in Fig. 2., if locl government support for two links is the sme nd reltively smll, less relible link should be selected s the most importnt key link. On the other hnd, more relible link should be selected if locl government support for two links is the sme but reltively gret s shown in Fig. 2.. From Fig. 2.b nd Fig. 2.c, when locl government support for two links is different, the link providing more locl government support should be improved first nd the other link should be ignored. (2) A gret difference in originl link relibility between two links Figure 3 shows the relibility increse bsed on locl government support for two links when = 0.3 nd r 20 = 0.7. Link 1 should be selected s the most importnt key link if it stisfies one of the following conditions: r 10 1) Locl government support for is much greter thn tht for link 2 s shown in Fig. 3.d; 2) Locl government support for two links is reltively smll s shown in Fig. 3., Fig. 3.b, Fig. 3.c nd Fig. 3.e when locl government support for link 2 is not greter thn tht for. Link 2 should be selected s the most importnt key link when the bove two conditions cnnot be stisfied. Any link cn be selected to improve t the point t which the curve of crosses tht of link 2 in Fig. 3.~c. (3) A very gret difference in originl link relibility between two links Figure 4 shows the relibility increse bsed on locl government support for two links when r 10 = 0.1 nd r 20 = 0.9. Link 1 should be selected s the most importnt key link only when locl government support for the two links is reltively smll nd locl Link 2 Link 2 Link 2 r 10 = 0.3 r 20 = 0.7 r 10 = 0.3 r 20 = 0.7 r 10 = 0.3 r 20 = 0.7 Link 1 Link 1 Link 1 Fig. 3. Locl government support for two links is the sme Fig. 3.b Locl government support for is little greter Fig. 3.c Locl government support for is greter Link 2 r 10 = 0.3 r 20 = 0.7 Link 1 Fig. 3.d Locl government support for is much greter Link 2 r 10 = 0.3 r 20 = 0.7 Link 1 Fig. 3.e Locl government support for is little smller Link 2 r 10 = 0.3 r 20 = 0.7 Link 1 Fig. 3.f Locl government support for is smller Fig. 3 Results for gret difference in originl link relibility between two links 120

IMPROVEMENT OF TRAFFIC NETWORK RELIABILITY AFTER A DISASTER BASED ON LOCAL GOVERNMENT SUPPORT Link 2 Fig. 4. Locl government support for two links is the sme r 10 = 0.1 r 20 = 0.9 Link 1 Link 2 r 10 = 0.1 r 20 = 0.9 Fig. 4.b Locl government support for is little greter Link 1 Link 2 Fig. 4.c Locl government support for is greter r 10 = 0.1 r 20 = 0.9 Link 1 Link 2 Fig. 4.d Locl government support for is much greter r 10 = 0.1 r 20 = 0.9 Link 1 Link 2 r 10 = 0.1 r 20 = 0.9 Fig. 4.e Locl government support for is little smller Link 1 Link 2 Fig. 4.f Locl government support for is smller r 10 = 0.1 r 20 = 0.9 Link 1 Fig. 4 Results for very gret difference in originl link relibility between two links government support for link 2 is not greter thn locl government support for s shown in Fig. 4.~e. On the other hnd, link 2 should be selected nd should be ignored when locl government support for the two links is reltively gret s shown in Fig. 4.~e. or when government support for link 2 is greter thn tht for s shown in Fig. 4.f. A specil point for ttention is tht the relibility of begins to increse when the relibility of link 2 becomes 1.0 s shown in Fig. 4.e nd Fig. 4.f ( r 20 = 0.9; the relibility increse of link 2 reches 0.1). Obviously, fter the relibility of higher-relibility link ( r 20 = 0.9) becomes 1.0, the lower-relibility link will be improved no mtter to wht extent it receives locl government support s shown in Fig. 4. In other words, there must be positive relibility increse of if tht of link 2 reches 0.1. In generl, locl government support for different disster res will not be considerbly different. Therefore, s less relible link should be selected s the most importnt key link when government support for two links is reltively smll s in the bovementioned three cses. On the other hnd, more relible link should be selected when locl government support is reltively gret s shown in Fig. 3.~c, Fig. 3.e, Fig. 4.~c, nd Fig. 4.e. 4.2. Effect of Government Support on Simple Bridge As shown in Fig. 5, simple bridge network is used here. The network hs four nodes nd five links. The miniml-pth sets of this bridge re shown s: P1= {1, 2}, P2= {3, 4}, P3= {1, 5, 4}, nd P4= {3, 5, 2} where P1 nd P2 re the primry miniml-pth sets (Iid, Y. et l., 1988, 1990) (Wkbyshi, H. et l., 1991, 1992). Since the independent pth set is set of links in series system, the relibility of one pth set is combintion of link relibility (Wkbyshi et l., 1992). When the most importnt key pth set is found, the most importnt key link belonging to the most importnt key pth set cn be found ccording to RI nd CIW. The relibility of every pth set is shown in the following: R ( P) = rr, R ( P ) = r r, R ( P) = rr r, R ( P ) = r r r. 1 1 2 2 3 4 3 1 5 4 4 3 5 2 (26) The exct vlue of network relibility for the bridge network is shown in Eq. (27) by using Boolen bsorption (Wkbyshi, H. et l., 1992). Fig. 5 A simple bridge network R( r) = rr + r r + rr r + r r r rr r r rr r r 1 2 3 4 1 5 4 3 5 2 1 2 3 4 1 2 4 5 r r r r rr r r r r r r + 2rr r r r. 1 3 4 5 1 2 3 5 2 3 4 5 1 2 3 4 5 (27) To simplify the simultion, only the originl relibility of nd link 3 vries, nd the originl relibility of the is fixed s 0.5. Three cses of difference in originl relibility 121

S. FANG, H. WAKABAYASHI between nd link 3 re presented bsed on locl government support for ll links, nd the three cses re s follows: smll difference (r10= 0.4 nd r30= 0.6), gret difference (r10= 0.3 nd r30= 0.7), nd very gret difference (r10= 0.1 nd r30= 0.9). The originl relibility of every miniml-pth set is different s R(P1) > R(P2) > R(P3) > R(P4) in ll cses. (1) Cse 1: A smll difference in originl link relibility between nd link 3 Figure 6 shows the relibility increse bsed on the level of locl government support for ll links when r10= 0.4 nd r30= 0.6. The primry horizontl xis of every cse in Fig. 6 is locl government support for or link 3, the secondry horizontl xis is locl government support for, nd the primry verticl xis is the relibility increse of every miniml-pth set nd the trffic network. From Fig. 6, if locl government support for ll links is the sme nd reltively smll, less relible primry miniml-pth set, P1, should be selected s the most importnt key pth set, nd of P1 is the most importnt key link. On the other hnd, more relible primry miniml-pth set, P2, should be selected if locl government support for ll links is the sme nd reltively gret, nd link 4 of P2 should be selected s the most importnt key link. When locl government support for nd link 3 is different, miniml-pth set P1 should be selected s the most importnt key pth set by stisfying one of the following conditions: 1) Locl government support for the links of P1 is much greter thn tht for the links of P2; 2) Locl government support for ll links is reltively smll nd locl government support for the links of P2 is smller thn locl government support for the links of P1. P2 should be selected s the most importnt key pth set when the bove two conditions cnnot be stisfied. (2) Cse 2: A gret difference in originl link relibility between nd link 3 Figure 7 shows the relibility increse bsed on locl government support for ll links when r10= 0.3 nd r30= 0.7 nd the relibility of is 0.5. In this cse, if locl government support for ll links is the sme, the results for finding the most importnt key link re sme s those for Cse 1. When locl government support for nd link 3 is different, P2 should be selected by stisfying one of the following conditions: 1) Locl government support for the links of P2 is greter thn tht for the links of P1; 2) Locl government support for the links is reltively gret nd locl government support for the links of P1 is not much greter thn locl government support for the links of P2. P1 should be selected s the most importnt key pth set when the bove two conditions cnnot be stisfied. (3) Cse 3: A gret difference in originl link relibility between nd link 3 Figure 8 shows the relibility increse bsed on the vriety of locl government support for ll links when r10 = 0.1 nd r30 = 0.9 nd the relibility of other 0.18 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.20 0.30 0.40 0.50 0.60 0.70 ohter links 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.30 0.40 0.50 0.60 0.70 0.80 0.90 Fig. 6. Locl government support for ll links is the sme Fig. 6.b Locl government support for is little greter Fig. 6.c Locl government support for is greter 0.20 0.30 0.40 0.50 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.20 0.30 0.40 0.50 0.50 0.60 0.70 0.80 0.90 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 link 3 0.50 0.60 0.70 0.80 0.90 link 3 Fig. 6.d Locl government support for is much greter Fig. 6.e Locl government support for link 3 is little greter Fig. 6.f Locl government support for link 3 is much greter Fig. 6 Results for smll difference in originl link relibility between nd link 3 122

IMPROVEMENT OF TRAFFIC NETWORK RELIABILITY AFTER A DISASTER BASED ON LOCAL GOVERNMENT SUPPORT 0.20 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.18 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.20 0.30 0.40 0.50 0.60 0.70 ohter links 0.18 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.30 0.40 0.50 0.60 0.70 0.80 0.90 Fig. 7. Locl government support for ll links is the sme Fig. 7.b Locl government support for is little greter Fig. 7.c Locl government support for is greter 0.20 0.30 0.40 0.50 0.18 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.20 0.30 0.40 0.50 0.50 0.60 0.70 0.80 0.90 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 link 3 0.50 0.60 0.70 0.80 0.90 link 3 Fig. 7.d Locl government support for is much greter Fig. 7.e Locl government support for link 3 is little greter Fig. 7.f Locl government support for link 3 is much greter Fig. 7 Results for gret difference in originl link relibility between nd link 3 links is 0.5. Miniml-pth set P2 should be selected s the most importnt key pth set by stisfying one of the following conditions: 1) Locl government support for the links of P2 is much greter thn tht for the links of P1; 2) Locl government support for links is reltively gret. The miniml-pth set P1 should be selected s the most importnt key pth set when the bove two conditions cnnot be stisfied. According to the bovementioned three cses of smple bridge network, when miniml-pth set P1 is selected s the most importnt key pth set, should be selected s the most importnt key link to improve trffic network relibility becuse r10< r20. When miniml-pth set P1 is selected s the most importnt key pth set, link 4 should be selected s the most importnt key link to improve trffic network relibility becuse r40< r30. 4.3. An Algorithm for Finding the Most Importnt Key Link in the In this section, n lgorithm for finding the most importnt key link in ny trffic network is proposed. The bsic procedure for finding the min miniml-pth set nd clculting the exct vlue of the terminl relibility of the miniml-pth set ws proposed by Wkbyshi. Thus, we strt with the lgorithm for the miniml-pth set. The lgorithm is shown s follows. Step 1: Let N be the number of miniml-pth sets. The originl relibility of ll links in every miniml-pth set is first stored in memory. For exmple, the originl relibility of the links in the ith miniml-pth set is stored in memory s two-dimensionl rry. Origin [i][]. Then, the originl relibility of ll the miniml-pth sets is clculted nd stored in memory s rry Rops[]. Then, locl government support for ll links is stored in memory s rry GSFL[]. 0.25 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.25 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.20 0.20 0.30 0.40 0.50 0.60 0.70 ohter links 0.18 0.20 0.15 0.05 0.20 0.15 0.05 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.30 0.40 0.50 0.60 0.70 0.80 0.90 Fig. 8. Locl government support for ll links is the sme Fig. 8.b Locl government support for is little greter Fig. 8.c Locl government support for is greter 0.18 0.20 0.30 0.40 0.50 0.25 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.20 0.30 0.40 0.50 0.20 0.15 0.05 0.50 0.60 0.70 0.80 0.90 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 link 3 0.50 0.60 0.70 0.80 0.90 link 3 Fig. 8.d Locl government support for is much greter Fig. 8.e Locl government support for link 3 is little greter Fig. 8.f Locl government support for link 3 is much greter Fig. 8 Result for very gret difference in originl link relibility between nd link 3 123

S. FANG, H. WAKABAYASHI Step 2: The primry miniml-pth sets re scertined first. The lest relible primry miniml-pth set nd the most relible primry miniml-pth set re selected by reding rry Rops[]. The lest relible primry miniml-pth set is mrked LP nd the most relible primry miniml-pth set is mrked HP. Step 3: The problem of finding the most importnt key miniml-pth set is chnged into the problem of finding the most importnt key link in prllel network. LP cn be considered the lest relible link in prllel network nd HP cn be considered the most relible link. The criticl point for locl government support bsed on the vlue of LP nd HP is clculted by using Eq. (23) nd Eq. (24). Then, locl government support for ll links is judged to be reltively gret or reltively smll. Step 4: Locl government support for the links of LP nd HP is compred. If the difference in locl government support between the group of links belonging to LP nd the group of links belonging to HP is not gret (in generl, locl government support for every link is not considerbly different), then one proceeds to step 5, or else proceeds to step 6. Step 5: According to step 3, if locl government support for ll links is reltively smll, LP is the most importnt key miniml-pth set nd the lest relible link of LP is the most importnt key link. If locl government support for ll links is reltively gret, HP is the most importnt key miniml-pth set nd the lest relible link of HP is the most importnt key link. After this judgment, one typiclly proceeds to step 7. Step 6: This step is the specil sitution fter disster. In generl, this sitution will not often hppen. When locl government support for the links of HP is greter thn tht for the links of LP, HP is the most importnt key miniml-pth set nd the lest relible link of HP is the most importnt key link. When locl government support for the links of LP is greter thn tht for the links of HP, two selections re vilble: If the originl relibility of LP is not much smller thn HP, LP is the most importnt key miniml-pth set; or else one proceeds to step 5. Step 7: End. This lgorithm is not suitble for ll kinds of network, nd so this is potentil unknown shortcoming. 5. Conclusion In generl, becuse of economic regionl differences nd limited resources, the centrl government cnnot provide sufficient support for every disster re fter mss disster; locl government support sometimes becomes more importnt for disster prevention nd reduction. In ddition, becuse of the lck of mteril support fter disster, post-disster reconstruction my need to be sustined for somewht longer; locl government support is prticulrly importnt in this cse. In this pper, method for improving trffic network relibility of rodwys nd highwys ws proposed bsed on locl government support. First, the current indices of trffic network relibility including RI, CI, nd CIW were introduced nd the shortcomings of these indices were identified. Second, n improvement model of trffic network relibility bsed on locl government support ws proposed becuse of the circumstnce in which only limited funds re vilble to reconstruct dmged trffic system t the first stge of disster. Third, to determine the model bsed on locl government support for improving trffic network relibility, simultions of prllel network nd simple bridge network were crried out bsed on limited cpitl for trffic restortion s defined in Eq. (24). From these simultions, the following conclusion is obtined. When locl government support for ll links is reltively smll, the lest relible link locted in the lest relible primry miniml-pth set should be selected s the most importnt key link to improve the trffic network. Conversely, when locl government support for ll links is reltively gret, the lest relible link locted in the most relible primry miniml-pth set should be selected s the most importnt key link to improve the trffic network. Bsed on this model simultion, n lgorithm for finding the most importnt key link in generl network ws developed. However, only two simple networks were simulted for determining the proposed model in this reserch. It is uncler whether the bovementioned conclusions cn be obtined from lrger nd more 124

IMPROVEMENT OF TRAFFIC NETWORK RELIABILITY AFTER A DISASTER BASED ON LOCAL GOVERNMENT SUPPORT complex networks. Furthermore, it is lso uncler whether the bovementioned conclusions cn be obtined when there is chnge of entire limited cpitl for trffic restortion. In future studies, lrger nd more complex networks should be simulted, the efficiency of trffic network relibility improvement when there is chnge of entire limited cpitl for trffic restortion should be further discussed, nd the lgorithm for finding the most importnt key link in generl networks should be further developed. REFERENCES Askur, Y., Kshiwdni, M. nd Fujiwr, K., 1998. Functionl Hierrchy of Rod nd Its Reltions to Time Relibility, Journl of JSCE, No. 583/IV- 38, 51-60. Brlow, R.E. nd Proschn, F., 1975. Sttisticl Theory of Relibility nd Life Testing: Probbility Models. Holt, Rinehrt nd Winston, New York, USA. Birnbum, Z.W., 1969. On the Importnce of Different Components in Multi-Component System. Multivrite Anlysis II, Acdemic Press, New York, USA. Fng, S.M. nd Wkbyshi, H., 2010. Proceedings of the Fourth Interntionl Symposium on Trnsport Relibility. Minnesot University, USA. Henley, E.J. nd Kummoto, H., 1981. Relibility Engineering nd Risk Assessment. Prentice-Hll, Inc., Englewood Cliffs, USA. Henley, E.J. nd Kummoto, H., 1992. Probbilistic Risk Assessment: Relibility Engineering, Design nd Anlysis. Institute of Electricl nd Electronics Engineers, New York, USA. Iid,Y. nd Wkbyshi, H., 1988. An Efficient Clcultion Method to Obtin Upper nd Lower Bounds of Terminl Relibility of Rod s using Boolen Algebr. Proceedings of JSCE, No. 395/IV-9, 75-84 (in Jpnese). Iid, Y. nd Wkbyshi, H., 1989. An Approximtion Method of Terminl Relibility of Rod s using Prtil Miniml Pth nd Cut Sets, Trnsport Policy, Mngement nd Technology Towrds 2001. Western Periodicls, Vol. 4, 2185-2198. Lei, Q.X., Chen, W.F., Hung, D.F., Guo, H.M. nd Wei, C.S., 2011. Decision-mking Support System for Deployment of the Erthquke On-site Serching nd Rescue Force. Journl of Seismologicl Reserch, Vol. 3, Yunnn, Chin. Nicholson, A., 2007. Optimizing Terminl Relibility, Proceedings of the Third Interntionl Symposium on Trnsport Relibility. Delft University, Netherlnds. Wkbyshi, H. nd Iid, Y., 1991. An Efficient Evlution Method for Rod Relibility in Disster, Interntionl Symposium on Nturl Disster Reduction nd Civil Engineering. JSCE, 397-405 Wkbyshi, H. nd Iid, Y., 1992. Upper nd Lower Bounds of Terminl Relibility of Rod s: An Efficient Method with Boolen Algebr, Journl of Nturl Disster Science, 14(1), 29-44. Wkbyshi, H. nd Iid, Y., (1994). Improvement of Rod Relibility with Mngement (Edited by Liu, B. nd Blosseville, J.M.). Trnsporttion Systems: Theory nd Applictions of Advnced Technology. Pergmum Elsevier Science, United Kingdom, pp. 603-608. Wkbyshi, H., 2004. Relibility Improvement: Probbility Importnce, Proceedings of Second Interntionl Symposium on Trnsport Relibility. University of Cnterbury, Christchurch, New Zelnd. 125