Journal of Natural Disaster Science, Volume 40，Number 2，2020，pp26-43

Characteristic Analysis of Various Importance Indicesfor Effective Highway Network Reliability Improvement

Takahiro NAGAE*
Hiroshi WAKABAYASHI***Doctor course student, Graduate School of Faculty of Urban Science, Meijo University**Professor, Faculty of Urban Science, Meijo University(Received: Jul. 29, 2019 Accepted: Feb. 11, 2020)

ABSTRACT
Threats of potential natural disasters have necessitated the urgency to construct and sustain a highly reliablenetwork for promotion of national resilience. However, improving all the links simultaneously is difficult due tobudget constraints. Therefore, network reliability can be improved effectively by improving the most important
key link. Our previous research revealed that improved criticality importance (CIW) is better than reliability
importance (RI). However, as link reliability increases, the difference between the values of both indicators
shrinks in terms of fairness of link improvement. In this study, we compared four importance indices. The newly
added indices are Fussell-Vesely (FV) and the risk achievement worth (RAW), which are used in highly reliable
systems (e.g., nuclear power plants). CIW, RI, FV, and RAW are compared by terminal reliability, difference
between maximum and minimum values of link reliability (DBMM), and number of improved links. First, RIand RAW improve terminal reliability better compared with CIW and FV; however, the difference is small. RIand RAW give a larger DBMM than CIW, which indicates that using RI implies more gaps between parallel
routes than using CIW and FV. Moreover, CIW is the best in terms of fairness of link improvement, as it has thelargest number of improved links. Therefore, CIW is a recommended importance index.
Keywords: Highway network reliability, Importance index, Cost-benefit analysis, Fussell-Vesely, Risk
achievement worth.

1. Introduction
Japan is frequently struck by natural disasters, i.e., earthquake or unusual weather, potentially leading to someunusable segments of the highway and requiring a highly reliable highway network that offers alternative routes
for a sure and stable traffic service. Such a scheme is called “connectivity reliability,” whose improvement
essentially fosters the network’s anti-disaster capabilities. On the other hand, constructing highly reliable andefficient routes is subject to budget constraints and may be extremely difficult to achieve. Moreover, such an
objective necessitates the establishment of an adequate quantitative index to identify the links improved in what
is known as “importance analysis.” Previous studies have found some issues on this matter, which are highly
stressed herein as follows:(1) When using importance indices, calculation of a network containing nine nodes and twelve links
26

T.NAGAE, H.WAKABAYASHI

resembles calculation of a parallel network. Therefore, improved criticality importance (CIW) is better thanreliability importance (RI) when analyzing the importance indices of a link in a highway network.
(2) Higher link reliability is associated with the problem of reduced difference between the indicators ofindices in terms of the cost-benefit functions.
(3) In this study, in order to solve the above problem, new importance indices are added for comparison.
These indices are Fussell-Vesely (FV) and risk achievement worth (RAW). These indices are used inhigh-reliability systems (e.g., aviation and nuclear power).
Moreover, necessary requirements for the importance indices of a road network are explored in this paperby newly estimating reliability improvement together with these indicators.
In this study, we simulate highway network improvement using four importance indices. As aprerequisite, it is necessary to explain existing research on highway network connectivity reliability andimportance indices. Therefore, citations are made as needed within Sections 2 to 4 and Section 5 describes new
research. The remainder of this paper is structured as follows: Section 2 reviews a reliability analysis of ahighway network and the study of importance analysis as well as highlighting the characteristics of andproblems with previously proposed importance indices. Section 3 provides a definition of connectivity reliability
and explains the problems of conventional importance analyses. Section 4 presents three cost reliability
functions. Section 5 compares terminal reliability improvement using four importance indices with cost-benefit
functions for a network with nine nodes and twelve links. Section 6 presents concluding remarks on the problem
discussed in Section 2.

2. Review of reliability and importance indices of a highway network
Connectivity, travel time, and capacity reliabilities (Nicholson et al., 2003) have been proposed as methods thatmeasure the reliability of a highway network. Our study focuses on connectivity reliability and importanceindices. Reliability means that “systems are in a condition to be able to accomplish a predetermined function
during a prescribed period of service” (Barlow and Proschan, 1965); thus, it is defined as an expression ofprobability. The CPU time and the memory size required for reliability analysis increase exponentially as thesize of the highway network increases. This is called a non-deterministic polynomial-time hard (NP-hard)
problem. Wakabayashi and Iida (1992) proposed an efficient and practical reliability analysis to solve thisproblem; nonetheless, efficient and practical importance analysis remains unsolved. The purpose of animportance index is to effectively improve the reliability of the system, e.g., the probability importance(Birnbaum, 1969) and criticality importance (CI) (Henley and Kumamoto, 1981) that have previously been
proposed and that are discussed later. Importance indices can be interpreted as the degree to which improvementin the reliability of a component contributes to the reliability of the overall system. Moreover, in this study, theimportance of the connectivity reliability of a highway network is defined as the degree to which a component
contributes to the reliability between nodes when link reliability is improved. The following statements explain
the problem of importance indices of a highway network as they are currently used:(1) It is more difficult to improve a more reliable link than a less reliable link. This is termed “diminishingmarginal utility.” Birnbaum’s importance index, RI, does not reflect this fact.
(2) In a parallel network, use of RI and CI results in only the more reliable links being improved and in lessreliable links remaining unimproved.
(3) Whether proper improvement in terminal reliability is obtained by only an importance index is unknown.
Therefore, considering the problem from a cost-benefit viewpoint is also necessary.
An improved criticality importance (CIW) index has recently been proposed to solve problems (1) and27

Journal of Natural Disaster Science, Volume 40，Number 2，2020，pp26-43(2) (Wakabayashi, 2004); nevertheless, until today, the results of applying this index to a network reveal
insufficient calculation and evaluation. Furthermore, this CIW index has not yet been compared with the RI
index, and this paper attempts to analyze this CIW index and compare it with the RI index. For problem (3), only
a few comparisons have been made to determine what difference in terminal reliability exists when the network
is improved using a conventional importance index. We can form the following hypothesis: if the cost reliability
function is set such that it expresses the expense and relations of link reliability (higher link reliability), then thecosts for anti-disaster measures are higher, while the expense to strengthen a link of low reliability is low.
Furthermore, we developed some link reliability cost functions and compared a rational solution with anequitable solution (using a cost-benefit analysis), after which we clarified how to change using a relation withlink reliability functions. Next, we established a small network and calculated the “improved path,” determinedby importance indices and a cost-benefit analysis. The index for comparison uses terminal reliability and theratio of terminal reliability to cost.
On this basis, we reviewed conventional studies. A sample article regarding importance evaluation in the
highway network is highlighted by a method that uses a cost–link-reliability function (Nicholson, 2007) as is
described below. Nicholson (2007) proposed that link reliability be provided by the function of traffic volume;
thus, the cost required for reliability improvement cannot be considered. In addition, we question whether this isbeneficial because the improvement cost did not increase significantly after we compared the growth rates of the
improvement cost and the link reliability.

3. Reliability and importance indices of a highway network (Wakabayashi and Fang, 2013)
3.1. An overview of connectivity reliability
This study is basically the application of reliability graph analysis (RGA) (Barlow and Proschan, 1965) to road
networks. RGA analyzes the connectivity between two points of input and output by using a directed graphcalled a reliability block diagram for the relationship between system reliability and unit (system component)
reliability. The theory underlying RGA is what is called “coherent system theory”. An example of a reliability
graph is shown in Figure 1.

Fig. 1 An example of a reliability graphIn terms of the reliability graph, the reliability of the entire system is defined as follows:
The reliability of the entire system is the probability that all units (path set or tie set) included in at least one path
between input and output will work as specified when the arrow in the figure is followed. In Fig. 1, when any
one of the unit combinations {1, 2}, {1, 5, 4}, or {3, 4} functions simultaneously, the entire system functions. In
28

T.NAGAE, H.WAKABAYASHI

this way, RGA expresses reliability using graph theory. Since the road network can also be described using graphtheory, reliability graph analysis and road network reliability analysis have many common points in theoreticalhandling. In reliability analysis, devices composed of a relatively large number of elements (components) arecollectively referred to as a system, and the components are referred to as a unit. Assume that the system and theunit have only two states: function and failure.
For unit 𝑎,binary variable 𝑥𝑎 is defined as follows:

𝑥𝑎 = {

1, unit 𝑎 is functioning,
0, unit 𝑎 is failing.

(3.1.1)

With the number of units as l, the system state can be expressed by the following equation:

𝒙 = (𝑥1 , 𝑥2 , ⋯ , 𝑥𝑙 ).

(3.1.2)This vector is called a system state vector. System state ∅ can be defined using 𝒙 as follows:

1, system is functioning,∅(𝒙) = {0, system is failing.

(3.1.3)The structure function can know the function / failure state of the system using the state vector that
represents the function / failure state of each unit. A system whose structure function has the following two
properties is called a coherent system.
(i) Each unit a(a=1,⋯,l) is associated with ∅( 𝒙).
(ii) ∅( 𝒙) is non-decreasing for each variable 𝑥𝑎 (a=1,⋯,l).
In the following, in this study, the system is a coherent system.
The system reliability is obtained as follows. The system reliability with structure function ∅( 𝒙) is R ,and the reliability of each unit is 𝑟𝑎 . Define random variable 𝑋𝑎 for each unit as follows:

𝑋𝑎 = {

1, unit 𝑎 is functioning,
0, unit 𝑎 is failing.

(3.1.4)and𝑃𝑟 {𝑋𝑎 = 1} = 𝑟𝑎 ,(3.1.5)

𝑃𝑟 {𝑋𝑎 = 0} = 1 − 𝑟𝑎.

(3.1.6)

𝐸[𝑋𝑎 ] = 1 × 𝑃𝑟 {𝑋𝑎 = 1} + 0 × 𝑃𝑟 {𝑋𝑎 = 0} = 𝑟𝑎.

(3.1.7)furthermore,

A vector composed of random variables 𝑋𝑎 is defined by the following equation:

𝑿 = (𝑋1 , ⋯ , 𝑋𝑙 ).
29(3.1.8)

Journal of Natural Disaster Science, Volume 40，Number 2，2020，pp26-43

Function / failure of the system can be expressed by random variable ∅(X) using structure function ∅( 𝒙), andthe system reliability is as follows:

𝑅 = 𝑃𝑟 {∅(𝑿) = 1} = 𝐸[∅(𝑿)].

(3.1.9)These are definitions in reliability graph analysis. In order to divert RGA to a road network, it isnecessary to make the unit correspond to the link and the system correspond to the probability of travelingbetween node pairs. Thus, reliability analysis of a road network can be handled in the same way as RGA.

3.2. Connectivity reliability of a highway network
Before we discuss our proposal of a comparative analysis for improving network reliability, we introduce thework of Wakabayashi and Fang (2013). In their work, the connectivity reliability of a highway network is
defined as the probability that two given nodes within the network are connected with a certain service level oftraffic for a given time period. Similarly, link reliability in the network is defined as the probability that the traffic
reaches a certain service level for a given time period. However, since this study assumes an alternative route in
the event of a disaster, it is simply defined as the “probability of can traveling links between the departure node
to the arrival node” (travel time required is not considered).
Terminal reliability, R, is given by the minimal path sets expression as follows (Iida and Wakabayashi,1988):
𝑝

𝑅(𝐫) = 𝐸 [1 − ∏(1 − ∏ 𝑋𝑎 )],𝑠=1(3.2.1)𝑎∈𝑃𝑠

where P_s is the S th minimal path set, and p is the number of minimal path sets. This calculation method iscalled the Boolean absorption method (Wakabayashi, and Iida, 1992).
Here, ∅(X), X and r are a structural function and are vector representations for 𝑋𝑎 and 𝑟𝑎 ･ 𝑋𝑎 is a binary
indicator variable for link 𝑎 as:

𝑋𝑎 = {

1, if link 𝑎 provides the certain service level,0, otherwise.

(3.2.2)

Link reliability, 𝑟𝑎 , is:𝑟𝑎 = 𝐸[𝑋𝑎 ].

(3.2.3)In order to calculate the exact value of terminal reliability, it is necessary to use all minimal path sets.
This means that the reliability value of the same link is included in the calculation process. In the following, adescription will be made using a specific network. A minimal path set is a subset that involves a system failure if
any one of the units (links) that make up the route from the departure node to the arrival node of the system fails.
For example, in the nine-node and twelve-link network shown in Figure 2, the minimal path between node pairs(1, 9) are {1, 2, 5, 10}, {3, 8, 11, 12}, {1, 4, 7, 10}, {3, 6, 7, 10}, {1, 4, 9, 12}, {3, 6, 9, 12}, {1, 2, 5, 7, 9, 12}, {3,8, 11, 9, 7, 10}, {1, 4, 6, 8, 11, 12}, {3, 6, 4, 2, 5, 10}, {1, 2, 5, 7, 6, 8, 11, 12}, and {3, 8, 11, 9, 4, 2, 5, 10}. Thus,
the identical link is included in several minimal paths. To avoid overlap of probability, a Boolean absorption forthe logical conjunction is required.
30

T.NAGAE, H.WAKABAYASHIThe terminal reliability of a traffic network depends on the network structure and the link reliabilities.
There are, therefore, two basic approaches to improving network reliability, namely, to improve the network
structure or to improve the reliability of the links. The focus here is identifying which links should be improved,
so as to maximize the improvement in network reliability. In this study, calculation is performed assuming thateach link reliability is independent.

Fig.2. Nine-node and twelve-link network

3.3. Definition of current importance indicesTo find the key links for the most efficient improvement of terminal reliability, the RI index (Birnbaum, 1969)
was proposed as
𝜕𝑅(𝐫)あ𝑅𝐼𝑎 = 𝜕𝑟 .

(3.3.1)𝑎

RI indicates the impact of an improvement in link reliability, i.e., an increase or decrease in the reliability of linka affects a corresponding increase or decrease in terminal reliability. RI is also known as Birnbaum’s structural
importance. In Eq. (3.3.1), R(r) is the terminal reliability between the origin node and the destination node, a isthe link number, and 𝑟𝑎 is the value of the link reliability. When it is not a specific calculation example, we
abbreviate importance index 𝑅𝐼𝑎 as RI. In the following, the same notation is used for other importance indices(CI, CIW, FV, and RAW). The result from RI indicates that, in the case of a parallel network, improving the morereliable link will be more effective for improving terminal reliability. Usually, however, it is difficult to improvea more reliable link, whereas it is rather easy to improve a less reliable one. This result of improving, managing,
and reconstructing a network contradicts what would be expected.
The CI index was proposed to counter the shortcoming of RI. By definition, it is the ratio of proportional
improvement in network reliability to proportional improvement in link reliability (Henley and Kumamoto,1992) as given by∆𝑅(𝐫)⁄𝑅(𝐫)}∆𝑟𝑎 →0∆𝑟𝑎 ⁄𝑟𝑎𝑟𝑎= 𝑅𝐼𝑎 ×.
𝑅(𝐫)𝐶𝐼𝑎 = lim {(3.3.2)31

Journal of Natural Disaster Science, Volume 40，Number 2，2020，pp26-43

The result from the CI index is the same for both links in a series network, but it justifies strengthening
the less reliable link. This is a shortcoming of the CI index. Moreover, the index provides no information to
distinguish between the two links for improving network reliability. Additionally, it indicates that, in the case of aparallel network, improving the more reliable link gives a greater increase in the terminal reliability of the
network. The results for a parallel network provided by both RI and CI suggest that a less reliable link should be
ignored.

Fig. 3. Series network

Fig. 4. Parallel network

3.4. Definition of improved criticality importance
Changing the definition of the equation in reliability engineering, improved criticality importance (CIW),
proposed by Wakabayashi (2004), is expressed by Eq. (3.4.1);

𝐶𝐼𝑊𝑎 = lim {−∆𝑞𝑎 →0=−∆𝑅(𝐫)⁄𝑅 (𝐫)}∆𝑞𝑎 ⁄𝑞𝑎

𝜕𝑅(𝐫)
𝑞𝑎
×
𝜕𝑞𝑎
𝑅(𝐫)
(1−𝑟 )= 𝑅𝐼𝑎 × 𝑅(𝐫)𝑎 ，(3.4.1)and𝑞𝑎 = 1 − 𝑟𝑎 ,(3.4.2)where 𝑞𝑎 (=1 - 𝑟𝑎 )is the unreliability of link 𝑎.Hereafter, R uses the same meaning as R (r).

𝐶𝐼𝑊𝑎 =(1−𝑟𝑎 )
𝑅

𝑅𝐼𝑎 ,

32

(3.4.3)

T.NAGAE, H.WAKABAYASHI

where two links, 1 and 2, are connected in series; it follows from Eqs. (3.3.1), (3.3.2), and (3.4.1) that

𝐶𝐼𝑊1 =

1−𝑟1
𝑟1,(3.4.4)and

𝐶𝐼𝑊2 =

1−𝑟2.

(3.4.5)

𝐶𝐼𝑊1 > 𝐶𝐼𝑊2, if 𝑟1 < 𝑟2 .

(3.4.6)𝑟2

furthermore,Thus, in a series network, the CIW has the same property as Birnbaum’s structural importance, which isexactly that from Eq. (3.4.4), as would be expected. Similarly, where links 1 and 2 are connected in parallel, it
follows from Eqs. (3.3.1), (3.3.2), and (3.4.1) that(1−𝑟 )(1−𝑟 )𝐶𝐼𝑊1 = 𝑟 +𝑟1 −𝑟 ∙𝑟2 = 𝐶𝐼𝑊2 .
1

2

1 2

(3.4.7)Here, although CIW made more progress than CI proposed by Henley and Kumamoto (1981), this index
is the same for both links in a parallel network, whereby it does not provide any distinguishable information
between them in terms of improving network reliability. Thus, when using CIW, a different index is necessary
for the selection of an improved link. In this paper, we consider an “improvement path” of terminal reliability
using the “cost-link reliability function,” which is similar to the cost-benefit ratio (Nicholson, 2007). Highly
designated links should be chosen as important links using indices RI, CI, and CIW for improving network
reliability. See the lines of “Desirable result” in Table 1.
Table 1. Summary of the merits and demerits of three importance indices.

33

Journal of Natural Disaster Science, Volume 40，Number 2，2020，pp26-43

3.5. Cost reliability function and cost-benefit analysis
As stated, a different index is required with CIW for choosing an improved link. In this regard, we establishedthe reliability of the improved link and three cost reliability functions to emphasize the relationship of the cost it
requires.
(1)
Case A: Type of constant—Cost
𝐶𝑜𝑠𝑡 𝐴 = 1000.

(3.5.1)In this case, the cost to improve the reliability is the same regardless of the value of the link reliability.
(2)
Case B: Type of increase—Linear
𝐶𝑜𝑠𝑡 𝐵 = 5000 × (𝑟𝑎 + 0.1).

(3.5.2)

Here, the cost for improvement increases linearly with the link reliability.
(3)
Case C: Type of increase—Quadratic
𝐶𝑜𝑠𝑡 𝐶 = 500 × (50 × 𝑟𝑎 2 + 15 × 𝑟𝑎 + 1).

(3.5.3)In this case, the cost increases as a quadratic function of the link reliability.
The effect of an improvement in network reliability, which requires a cost increase, might not beapparent in the short term. For a long-term improvement, Eq. (3.5.4) shows a simple cost–benefit function thatshows how the network reliability will improve against the cost increase:
𝑅 −𝑅

𝐸𝑓𝑓(𝑌, 𝐹) = 𝐶𝑜𝑠𝑡 0 × 𝑌 × 𝐹,

(3.5.4)

where
Y : Number of years to invest;
Fn : The conversion cost-benefit of the increased traffic volume obtained by the reliability improvement in then-th unit of time (the unit of time is one year);
R0 : Original network reliability;
Cost : Cost increase to improve the network reliability from 𝑅0 to R;
Eff(Y): The efficiency of the cost-benefit obtained by the reliability improvement of traffic systems in Y years.

4. Comparison of RI and CIW in a network consisting of nine nodes and twelve links(Nagae and Wakabayashi, 2016)
Before the comparative analysis in Section 5, where four indices are considered, we state the characteristics of
RI and CIW in this section. In both RI and CIW, we calculated the dispersion of link reliability, terminal
reliability, and cost-benefit analysis through a method showcasing a parallel network of nine nodes and twelve
links (Fig. 2). First, we determined in the network that RI had better terminal reliability improvement than CIW,
while CIW provided smaller dispersion among link reliability than RI (Wakabayashi, 2004). Second, we
compared both indices. We assumed an initial value of the link reliability for six cases and the set of linkreliability values by generating random variables. For this value, both cases were prepared for 100 trials, after
34

T.NAGAE, H.WAKABAYASHI

which we selected the improved link yielding the maximal value of importance. We improved the value of
reliability by 0.01 each time and repeated that six times. The results are shown in Table 2 and Fig.5. The results
indicate that terminal reliability improvements by RI are better than those determined by CIW. Moreover, thevariance of the link reliability for CIW was smaller than that for RI, whereas cost-benefit (Eff) results by CIW
were better than those determined by RI. From these, we could assume that CIW is generally better than RI.
However, as the link reliability value increases, the difference between RI and CIW decreases. Figure 5 shows
how the calculation result of the total cost and Eff varies depending on the link reliability in Case C. As the initialvalue of link reliability increases, the cost for improvement increases, so the total cost increases for both RI and
CIW. In terms of total cost, CIW’s superiority has not changed. On the other hand, with regard to Eff, thedifference between RI and CIW narrows as the link reliability increases, and RI is better than CIW at (0.8 ≦ r<0.9). In a high-reliability road network, it can be considered that the improvement in terminal reliability has agreater effect on Eff than the improvement of links with low reliability.

Fig. 5. Comparison of RI and CIW (total cost and cost-benefit function)

35

Journal of Natural Disaster Science, Volume 40，Number 2，2020，pp26-43

Table 2. Calculation result of reliability improvement determined by RI and CIW
RI

CIW

0.3≦r＜0.4
Initial value
Terminal reliability ( R )0.081075
Link
max-min0.084694
reliability
var0.000744
Link reliability
CaseB-cost function CaseC
Cost-benefit
CaseB
function ( Eff ) CaseC

0.086887
0.097449
0.000903
12428.21
37934.29
0.234416
0.077248

0.4≦r＜0.5
Initial value
Terminal reliability ( R )0.197628
Link
max-min0.084694
reliability
var0.000744
Link reliability
CaseB-cost function CaseC
Cost-benefit
CaseB
function ( Eff ) CaseC

0.207803
0.099669
0.000938
15533.63
55519.18
0.328121
0.0922100.5≦r＜0.6
Initial value
Terminal reliability ( R )0.374920
Link
max-min0.084694
reliability
var0.000744
Link reliability
CaseB-cost function CaseC
Cost-benefit
CaseB
function ( Eff ) CaseC

0.389087
0.103755
0.000995
18702.90
76770.99
0.379289
0.092709

0.6≦r＜0.7
Initial value
Terminal reliability ( R )0.588454
Link
max-min0.084694
reliability
var0.000744
Link reliability
CaseB-cost function CaseC
Cost-benefit
CaseB
function ( Eff ) CaseC

0.604520
0.108641
0.001047
21860.41
101277.18
0.367998
0.079663

0.7≦r＜0.8
Initial value
Terminal reliability ( R )0.790794
Link
max-min0.084694
reliability
var0.000744
Link reliability
CaseB-cost function CaseC
Cost-benefit
CaseB
function ( Eff ) CaseC

RI

CIW

0.805442
0.111902
0.001090
24988.23
128823.05
0.293502
0.057073

0.804356
0.083736
0.000703
23829.55
118179.58
0.284725
0.057479

0.8≦r＜0.9
Initial value
Terminal reliability ( R )0.933145
Link
max-min0.084694
reliability
var0.000744
Link reliability
CaseB-cost function CaseC
Cost-benefit
CaseB
function ( Eff ) CaseC

RI

CIW

0.943236
0.113137
0.001102
28024.91
158689.58
0.180299
0.031917

0.941806
0.083269
0.000697
26810.32
146314.41
0.161657
0.029658

36

RI

RI

RI

0.086834
0.087645
0.000769
12027.39
35868.34
0.239767
0.080730

CIW
0.207632
0.085930
0.000741
14941.05
51870.52
0.335083
0.096762

CIW
0.388720
0.084733
0.000719
17875.81
70847.65
0.386219
0.097611

CIW
0.603825
0.084231
0.000710
20848.34
93001.81
0.368869
0.082801

T.NAGAE, H.WAKABAYASHI

5. Application of PSA importance analysis to road networks
5.1 Definition of the FV and RAW indices
The previous sections have so far dealt with importance indices suitable for a road network, where we clearly
understood the origin of the issue due to which differences between CIW and RI tend to decrease from theperspective of the cost-benefit functions of the link reliability. To reiterate, CIW is better than RI; nevertheless, inan item for comparison, it is necessary to measure the degree of superiority or inferiority of the item as there are
multiple elements describing each.
In this study, we will investigate whether other new importance indices used in high-reliability systems inother fields should be applied to road networks. In practice, importance indices are widely used in systems ofaerospace, space, nuclear power, etc. These fields are characterized by complicated systems often involving ahigh risk of failure, where very high reliability is necessary. In particular, in the nuclear power industry,
probabilistic safety assessment (PSA) is used to determine which equipment in the system is important for
prevention of serious events, i.e., core damage. Herein, we estimated whether the importance index used in PSA
can be used in a highly reliable road network. Table 3 summarizes the importance indices used in PSA. 𝑄(base) is the original failure probability of the whole system, 𝑄 (qi=1) is the failure probability of the wholesystem assuming that the failure probability of the target unit is 1, and 𝑄 (qi=0) is the failure probability of thewhole system assuming that the failure probability of the target unit is 0. FV and RAW are the most common
importance indices in PSA. These use the notation of the citation source. “i” does not include the meaning of“-th”. However, “i” and “a” are the same in terms of one particular unit that calculates the importance. Hence,
this paper treats “i” as equivalent to “a” of 𝑅𝑎 . The relationship between the failure probability and thereliability can be expressed by the following equation:𝑞 = 1 − 𝑟,

(5.1.1)

𝑄 = 1 − 𝑅.

(5.1.2)The Fussell Vesely (FV) index shows how low the failure probability of the entire system is; i.e., the degree towhich the reliability of the system improves when the failure probability of the target unit is assumed to be 0. In
other words, a unit with a large FV reflects a great effect of the improvement measures. The risk achievement
worth (RAW) index shows the extent to which the failure probability of the whole system increases when the
failure probability of the target unit is assumed to be 1, i.e., the degree to which the reliability of the systemdecreases. In other words, a unit having a large value of RAW indicates that contribution to reliability is high.

37

Journal of Natural Disaster Science, Volume 40，Number 2，2020，pp26-43

Table 3. PSA importance indicesName

Abbr. Formula

Fussell-Vesely

FV

Q(base)  Q(q i  0)
Q(base)Risk reduction
worth

RRW

Q (base)Q ( q i  0)

Risk achievement
worth

RAW

Q(qi  1)
Q(base)Birnbaum
importance

BI＝RI

Q(qi  1)  Q(qi  0)

5.2. Significance of the FV and RAW indices in a road network
FV and RAW have features that make them applicable to calculating the value of importance for a highly reliableroad network.
First, FV has similar characteristics to CIW in series and parallel networks. In a series network, thedegree of importance of a link with low reliability is high whereas, in a parallel network, the importance value isthe same regardless of link reliability. The calculation results of FV and RAW in series and parallel networks is
discussed below (refer to Table 3 for the calculation formula of FV and RAW).
For two links in a series network (Fig. 3), terminal reliability R is given by𝑄(base) = 1 − (1 − 𝑞1 )(1 − 𝑞2 ),(5.2.1)

where 𝑄 (base) is system failure probability, and 𝑞1 and 𝑞2 are the values of failure probability for links 1 and 2,
respectively. Moreover, 𝑄(𝑞1 = 1) and 𝑄(𝑞1 = 0)become

𝑄(𝑞1 = 1) = 1 − (1 − 1)(1 − 𝑞2 ) = 𝑄(𝑞2 = 1),(5.2.2)

𝑄(𝑞1 = 0) = 1 − (1 − 0)(1 − 𝑞2 ) = 𝑞2・(5.2.3)

𝐹𝑉1 > 𝐹𝑉2 if 𝑞1 > 𝑞2・(5.2.4)

It follows that

Therefore, in a series network, FV has the same property as CIW. Furthermore,

𝑅𝐴𝑊1 = 𝑅𝐴𝑊2′(5.2.5)which suggests an equal value for RAW in both links, in a series network, For two links in a parallel network(Fig. 4), terminal reliability R is given by38

T.NAGAE, H.WAKABAYASHI

𝑄(base) = 𝑞1 ∙ 𝑞2 .

(5.2.6)𝑄(𝑞1 = 1) = 𝑞2 ,(5.2.7)𝑄(𝑞1 = 0) = 0 = 𝑄(𝑞2 = 0).

(5.2.8)𝐹𝑉1 = 𝐹𝑉2 .

(5.2.9)Moreover, 𝑄(𝑞1 = 1)and 𝑄(𝑞1 = 0)are

It follows that

Therefore, in a parallel network, FV has the same property as CIW. Furthermore,

𝑅𝐴𝑊1 < 𝑅𝐴𝑊2 if 𝑞1 > 𝑞2 .

(5.2.10)

Next, RAW is applicable as an importance index for a road network because the calculation results forseries and parallel networks indicate that it has similar characteristics to CI; therefore, there is a problem withusing RAW alone. Nevertheless, as RAW is defined by the expression in Table 3, there is the characteristic thatthe calculation result would change greatly due to a slight difference in failure probability. In this regard, RAW
can be a useful index when finding important units in a highly reliable system.
In the succeeding section, we will introduce a calculation example where link reliability is improved
when FV and RAW are utilized.

5.3. Comparison of the FV and RAW indices for a network consisting of nine nodes and twelve
links
We used the FV and RAW indices to calculate dispersion of link reliability and terminal reliability and to conduct
cost-benefit analysis, through a method for a parallel network of nine nodes and twelve links (Fig.2). Here, we
assumed an initial value of the link reliability for Case 1 (0.3≤r<0.7), Case 2 (0.4≤r<0.6), Case 3(0.8≤r<0.85) and Case 4 (0.85≤r<0.9) and the set of link reliability by generating random variables. For thisinitial value, we prepared all cases for 100 trials, after which we chose the improved link yielding a maximal
value of importance. We adjusted the improvement value per time to 0.01 and repeated the improvement six
times. Table 4 shows the results.

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Journal of Natural Disaster Science, Volume 40，Number 2，2020，pp26-43

Table 4. Calculation result of the reliability improvement determined by RI, CIW, FV, andRAW (average of 100 trials)
0.3≦r＜0.7
Terminal reliability（R )Linkreliability
Link reliability-cost function

max-min
var
CaseB
CaseC

Cost-benefit

CaseB

function( Eff )CaseC

0.4≦r＜0.6
Terminal reliability（R )Linkreliability
Link reliability-cost function

max-min
var
CaseB
CaseC

Cost-benefit

CaseB

function( Eff )CaseC

0.8≦r＜0.85
Terminal reliability（R )Linkmax-minreliability
varLink reliability CaseB

Initial value

0.285939
0.338775
0.011910

Initial value

0.281817
0.169387
0.002978Initial value

0.904052
0.042347
0.000186-cost function CaseC
Cost-benefit

CaseB

function( Eff )CaseC

0.85≦r＜0.9
Terminal reliability（R )Linkreliability

max-minvar

Link reliability CaseB-cost function CaseC
Cost-benefit CaseB
function( Eff )CaseC

Initial value

0.955265
0.042347
0.000186

RI
0.360909
0.428488
0.016829
19185.38
82421.92
1.985003
0.480095

CIW
0.353997
0.332619
0.011235
15829.26
58630.60
2.162511
0.596737

FV
0.360909
0.428488
0.016829
15829.26
58630.61
2.572620
0.709951

RAW
0.369156
0.538928
0.022142
22373.40
107713.96
1.897583
0.409050

RI
0.348665
0.305951
0.007370
18815.00
78362.04
1.782364
0.431773

CIW
0.345081
0.211266
0.004249
16942.73
64924.73
1.870238
0.490750

FV
0.357623
0.211266
0.004249
15829.26
58630.61
2.419196
0.669506

RAW
0.357771
0.420757
0.011610
21359.44
98399.49
1.787287
0.391583

RI
0.914819
0.083644
0.000491
27102.40
149218.06
0.198716
0.036115

CIW
0.914005
0.046325
0.000215
26275.75
140995.91
0.189435
0.035314

FV
0.914604
0.067306
0.000345
26212.20
140375.18
0.201323
0.037606

RAW
0.916952
0.087509
0.000522
27197.77
150175.34
0.237234
0.042990

RI
0.963202
0.085654
0.000504
28643.88
165082.00
0.138609
0.024066

CIW
0.962293
0.046213
0.00021427773.61
155986.35
0.126567
0.022543

FV
0.962293
0.044458
0.000199
27758.77
155833.02
0.126630
0.022564

RAW
0.964787
0.086443
0.000512
28666.04
165315.54
0.166163
0.028831

First, as a general trend, the numerical value of the calculation result is roughly RI and RAW, and CIW
and FV are similar regardless of the value of link reliability. The items compared for the calculations included, as
in Section 4.2., terminal reliability, link reliability (difference between maximum and minimum, variance),values of the cost-reliability function, and cost–benefit results. With regard to terminal reliability, RAW always
tended towards the value of RI by the best result. Similarly, no significant difference was observed for FV and
CIW. In link reliability, CIW showed a good value generally, which implies that it was advantageous from theperspective of equitableness of link improvement, where this value is considered superior. However, there was aslight difference in the domain of high reliability, and FV reversed CIW to give the best value. Specifically, thedifference between the max-min value in Case 4(0.85≤r<0.9) shows the same value up to two decimal places
for both FV and CIW. In addition, the value of var for RI and RAW is more than doubled; however, the value of
CIW and FV is less than 1.2 times. Furthermore, in terms of cost, RI and RAW were not good overall. Cases B
and C yielded good CIW at low reliability, although at high reliability its difference from FV shrank and reversed.
However, in Case C (0.85≤r<0.9), both RI and RAW cost 1.05 times or more based on FV, while CIW is less40

T.NAGAE, H.WAKABAYASHI

than 1.004; hence, CIW is not much different from FV. With respect to cost–benefit, FV and RAW gave a goodvalue at low reliability and high reliability, respectively. At this point, the value in Case C with high reliability,
which is considered to be the most realistic numerical value, did not reflect much difference with its numericalvalue in any importance indices.
Figure 6 shows the number of the links (as an improvement target) improved for 100 trials using RI, CIW,
FV, and RAW, respectively. For example, when this number is one, the same link was improved six times. RAW
always selected the same link for improvement regardless of the link reliability value. Therefore, RAW is inferiorin view of fairness of link improvement. RI often chose the same link for improvement even with low linkreliability. This tendency became prominent at higher reliability, and was similar to RAW in (0.85≤r<0.9). FV isbetter in terms of fairness because the link to be improved is dispersed with low link reliability. However, thehigher the link reliability, the stronger the tendency to improve the same link. This point is similar to RI. Amongthe four indices, CIW had the largest number of improvement target links. As a striking difference against otherindices, the number of improvement links was secured even at high reliability. From these results, CIW is thebest index in terms of fairness of link reliability improvement.

Fig. 6. Comparison of the number of improved links (RI, CIW, FV, and RAW).

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Journal of Natural Disaster Science, Volume 40，Number 2，2020，pp26-43

Table 5. Merits and demerits of importance indices (RI, CIW, FV, and RAW)
RI

CIW

FV

RAW

Range of link reliability

Low

High

Low

High

Low

High

Low

High

Terminal reliability（R ）Linkmax-min

○
△

○
△

△
◎

○
○

○
○

○
◎

◎
△

◎
△

reliability

var

△

△

◎

○

○

◎

△

△

Link reliability-cost
Cost-benefit( Eff )

○
△

△
○

◎
○

○
△

○
◎

◎
○

×
△

△
◎

Number of improved links

△

×

○

◎

○

△

×

×

6. Conclusion
In this paper, importance indices to improve the connectivity reliability of the highway network efficiently were
examined.
First, issues related to connectivity reliability and importance indices in the highway network wereorganized. Next, the procedure for applying the calculation method to reliability analysis of the road network
was introduced. Third, we compared three conventional importance indices with a simple network and analyzed
their merits and demerits. Our previous research revealed that CIW is better than RI for analyzing the importanceindices of a link in a highway network. However, when the reliability of the link becomes high, there arises a
problem that the difference between the indicators of both is reduced in terms of cost-benefit. Therefore, in thisstudy, other importance indices compared are FV and RAW. Both are used in fields such as nuclear power.
As a result, RAW is the best index for improving terminal reliability. For an actual high-reliability road
network, FV yields the best cost, while RAW gives a cost-effective value. However, FV and RAW are inferior to
CIW in terms of fairness of link improvement. Overall, CIW is the best for fairness of link improvement. In thisregard, note that the CIW index may somewhat be inferior to other indicators by virtue of total cost andcost-benefit. These results are calculation examples in a model network, and do not guarantee the same result inan actual road network. However, because it is difficult to calculate the exact value of reliability in a large-scale
network, it is necessary to use an approximate calculation method. In addition, the calculation may be performed
with the network shape simplified. If the network figure is similar to the nine-node and twelve-link network, theCIW index is still advantageous.
In future studies, larger and more complex networks should be simulated, and the efficiency of traffic
network reliability improvement analyzed. Moreover, there is a method of giving a probability value of linkreliability and conducting efficient calculation of reliability in a large-scale network. Regarding the method of
estimating the link reliability value, we believe it is better to change the method according to the type of disaster.
It will be easy to calculate for a certain amount of past data. One example is an assumption of flood damage due
to seasonal heavy rain. For efficient calculation of network reliability, there is a method to calculate theapproximate value using a partial minimal path set. The practicality of the method needs to be verified.

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