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Differences in Network Reliability Improvement by Several Importance Indices
Available online at www.sciencedirect.com
ScienceDirect Transportation Research Procedia 10 (2015) 155 – 165
18th Euro Working Group on Transportation, EWGT 2015, 14-16 July 2015, Delft, The Netherlands
Differences in network reliability improvement by several importance indices Takahiro Nagaea, Hiroshi Wakabayashib* a
Graduate School ofUrban Science, Meijo University, Nijigaoka 4-3-3, Kani, Gifu 509-0261, Japan b Faculty of Urban Science, Meijo University, Nijigaoka 4-3-3, Kani, Gifu 509-0261, Japan
Abstract For constructing a highly reliable highway network under a disaster environment, this paper presents a comparative study of the importance indices and proposes an approximate method for improving highway network reliability. Three indices are compared. Two of them are conventional indices of reliability importance (RI) and criticality importance (CI), and the other is the improved criticality importance (CIW) index recently proposed by Wakabayashi (2004). First, the significance of reliability importance is discussed. Second, their definitions, merits, and demerits are explained. Third, several cost functions for improving link reliability are introduced. Fourth, RI and CIW are compared for a small-sized network and differences in improved link reliability are discussed. Fifth, an approximate method is proposed using partial minimal path sets. Lastly, after calculation experiments, the advantages of the CIW index and the effectiveness of the proposed approximate method are demonstrated. © 2015 2015The TheAuthors. Authors. Published by Elsevier B. V.is an open access article under the CC BY-NC-ND license © Published by Elsevier B.V. This (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of Delft University of Technology. Peer-review under responsibility of Delft University of Technology Keywords: Highway network reliability, Importance index, Probability importance, Partial minimal path set, Cost-benefit analysis
1. Introduction Under the threat of a potential disaster such as the 2011 Japan earthquake, constructing and sustaining a highly reliable highway network for national resilience is an urgent necessity. However, improving all the links at once is
* Corresponding author. . Tel.: +81 052-832-1151 E-mail address: [email protected]
2352-1465 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Delft University of Technology doi:10.1016/j.trpro.2015.09.065
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difficult because of budget constraints. Therefore, identifying key links to improve network reliability is important; this process is called the importance analysis. Importance analysis of a highway network mainly entails two concerns: (1) the development of an effective importance index and (2) the reduction of the amount of calculations. Therefore, this study verifies the relevance of an index proposed by the authors and develops an efficient method for approximate calculations. To resolve these two concerns, this study presents both a comparative analysis of importance indices for actual use and an efficient method for approximate calculations. The remainder of this paper is structured as follows: Section 2 reviews the reliability analysis of a highway network and the study of importance analysis. Section 2 also highlights the characteristics of and problems with importance indices that have been previously proposed. Section 3 provides a definition of connectivity reliability and introduces the conventional and newly proposed importance indices. Section 4 presents three cost-reliability functions. Section 5 compares terminal-reliability improvement using two importance indices, i.e. CIW and RI with cost-benefit functions for a network with nine nodes and twelve links. Section 6 uses examples to show an approximate calculation method that chooses partial minimal path sets to reduce calculations. Section 7 presents concluding remarks for the problem discussed in Section 2. 2. Reliability and Importance Indices of a Highway Network Connectivity, travel-time, and capacity reliabilities (Nicholson et al., 2003) have been proposed as methods to measure the reliability of a highway network. Our study focuses on the improvement of connectivity reliability and importance indices. 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); it is defined as an expression of probability. The CPU-time and memory size required for reliability analysis increases exponentially as the size of the highway network increases. This is called a non-deterministic polynomial-time hard (NP-hard) problem (US NRC, 1983). For solving this problem, efficient and practical reliability analysis has been proposed by Wakabayashi and Iida (1992). However, efficient and practical importance analysis remains unsolved. The purpose of the importance index is to effectively improve the reliability of the system. Probability importance (Birnbaum, 1969) and criticality importance (CI) (Henley and Kumamoto, 1981) are previously proposed and discussed later. Importance indices can be interpreted as the degree to which a component contributes to the reliability of the system when the reliability of the component is improved. Moreover, in this study, the importance of the connectivity reliability of the highway network is defined as the degree to which a component contributes to the reliability between nodes when link reliability is improved. Next, we 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 to improve a less reliable link. This is called diminishing marginal utility. Birnbaum’s importance index, i.e., Reliability Importance (RI), does not reflect this fact. (2) In a parallel network, use of the RI and CI results in only the more reliable links being improved, and the less reliable links will remain unimproved. (3) The importance index does not always result in an economical highway network improvement. A cost-benefit perspective should also be considered. (4) Importance analysis also is an NP-hard problem. An efficient and approximate method for practical use is required for actual highway networks. An improved criticality importance (CIW) index has recently been proposed to solve the problems (1) and (2) (Wakabayashi, 2004). However, till date, the results of applying this index to a network are not calculated and evaluated enough, and the index is not yet compared with the RI index; this paper attempts this analysis and comparison. For problem (3), only a few comparisons have been made to determine what difference in the terminal reliability exists when the network is improved using a conventional importance index. We can form the following hypotheses: if we set the cost-reliability function that expresses an expense and relations of the link reliability (the higher link
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reliability), the costs for anti-disaster are more expensive; however, the expense to strengthen a link of low reliability is low. For problem (4), when a network expands, the CPU-time and memory size increase exponentially for the importance analysis in addition to the reliability analysis stated above. Therefore, efficient and approximate calculation method is necessary. In this study, we propose an approximate method to calculate them with partial path sets in a highway network. The effectiveness of the approximate calculation method is evaluated by comparing the exact and approximate values irrespective of the whether the order of link importance has changed. Furthermore, we developed some link reliability cost functions and compared a rational solution with an equitable solution (using a cost-benefit analysis), and we clarified how to change using a relation with link reliability functions. Next, we established a small network and calculated the “improved path,” determined by importance indices and a cost-benefit analysis. The index for comparison uses terminal reliability and the ratio of terminal reliability to cost. On the basis of these, we reviewed conventional studies. As an example of the articles about the importance evaluation in the highway network, we discuss the method that uses a cost-link reliability function (Nicholson, 2007). Nicholson (2007) proposes that the link reliability is provided by the function of traffic volume; thus, the cost required for reliability improvement cannot be considered explicitly. 3. Reliability and Importance Indices of a Highway Network (Wakabayashi and Fang, 2014) 3.1. The Connectivity Reliability of a Highway Network In this study, we compare CIW with RI in a small network and propose a calculation reduction method using partial minimal path sets. Before we discuss our proposal for the comparative analysis for improving network reliability, we introduce the work 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 of traffic 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. In a highway network, link reliabilityrais defined as ra
E[ X a ] (1)
Terminal reliability, i.e. node-to-node reliability, R is given by an expression using minimal-path sets as follows:
R(r )
E[I ( X)]
p
E[1 (1 X a )] s 1
aPs
(2)
, where Ps is the s-th minimal-path set, p is the total number of minimal-path sets, Xa is the binary indicator variable for link a (Wakabayashi and Iida, 1992) and I (X) , X , and r are a structural function and vector representations for
Xa
and
Xa
ra : 1, if link a provides a certain service level, ® ¯0, otherwise.
(3)
For example, calculating the connectivity reliability (approximate value: when using only three path sets {1,2,5,10},{1,4,9,12},{3,8,11,12}) between nodes 1 to 9 (Fig. 1), in the calculating formula, is possible as follows:
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Fig.1.㻌A network with nine nodes and twelve links. p
R(r )
E[1 (1 X a )] E[1 (1 X1 X 2 X 5 X10 )(1 X1 X 4 X 9 X12 )(1 X 3 X 8 X11X12 )] s 1
(4)
a p s
In this equation, the variables X1 and X12 appear more than once. To avoid the product of the same indicator, variables used in the calculation of the expected value of reliability, a Boolean absorption such as X1 × X1 = X1 is required. The connectivity reliability of a traffic network depends on the network structure and the link reliabilities. Therefore, two basic approaches have been taken to improve network reliability: improvement of the network structure or improvement of the reliability of the links. The focus here is on identifying which links should be improved to maximize the improvement in network reliability. 3.2. Definition of Reliability Importance To find the key links for most efficiently improving the terminal reliability, the Reliability Importance (RI) index (Birnbaum, 1969) was proposed as
RI a
wR(r ) , 0 d RI a d 1 .(5) wra
RI indicates the impact of an improvement in link reliability, i.e., the increase or decrease in the reliability of link a affects the increase or decrease in the terminal reliability. RI is also known as Birnbaum’s structural importance. R(r) is terminal reliability between nodes A to B, a is link number, and ra is the value of link reliability for link a. For a series network with two links, the terminal reliability RAB is given by Eq. (6): (6) RAB r1r2 , where r1 and r2 are the values of reliability for links 1 and 2, respectively. When determining the reliability importance for a series network, RI1 and RI2, are obtained from Eq. (5) and (6) as RI1 r2 (7) (8) . RI 2 r1 It follows that (9) RI ! RI if r r . 1
2
1
2
Equation (9) indicates that in the case of a series network, improving the less reliable link is most effective for improving terminal reliability. This fact is easily expanded for a large series networks. This result for improving, managing, and reconstructing a network is the expected result. However, for the case of two links in a parallel network, the terminal reliability, RAB, is given by Eq. (10); (10) RAB 1 (1 r1 )(1 r2 ) RI1 and RI2, for these two links in a parallel network, are obtained from Eq. (5) and (10) as
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Fig. 2. Series Network
Fig. 3. Parallel Network
1 r2 and RI 2
RI1
1 r1
(11)
It follows that (12) RI1 RI 2 if r1 r2 The result from Eq. (12) indicates that in the case of a parallel network, improving the more reliable link will be more effective for improving terminal reliability. Usually, however, it is difficult to improve a more reliable link, whereas it is rather easy to improve a less reliable link. This result for improving, managing, and reconstructing a network is counter to what one would expect. 3.3. Definition of Criticality Importance Because of the shortcoming of reliability importance, the criticality importance (CI) index was proposed as the ratio of the proportional improvement in the network reliability to the proportional improvement in the link reliability (Henley and Kumamoto, 1992):
CI a
'R (r ) / RAB (r ) ½ lim ® AB ¾ 'ra / ra ¯ ¿
'ra o0
ra wRAB (r ) u RAB (r ) wra
RI a
ra RAB (r )
(13)
CI also has shortcomings, which we discuss in this section. For the case of two links in a series network, CI is given from Eq. (5), (6), (7), (8), and (13) such that:
CI1
r1r2 R
CI 2
(14)
This result suggests that the criticality importance (CI) index is the same for both links in a series network. However, in a series network, it would be reasonable to strengthen a less reliable link. Therefore, this is a shortcoming of the CI index. In addition, the index provides no information to distinguish the link between the two links for improving network reliability. For the case of two links in a parallel network, CI is given from Eq. (5), (10), (11), and (13) such that:
CI1
r1 r1r2 R
(15)
CI 2
r2 r1r2 R
(16)
and
It follows that
CI1 CI 2 if r1 r2
(17)
Therefore, the CI index also indicates that in the case of a parallel network, improving a more reliable link gives a greater increase in the terminal reliability of the network. The results for a parallel network provided by both the RI and the CI suggest that a less reliable link should be ignored in a parallel system.
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3.4. Definition of Improved Criticality Importance Changing the definition of the equation in the reliability engineering, the CIW, proposed by Wakabayashi (2004), is introduced as Eq. (18);
CIWa
'R(r ) / R(r ) ½ lim ® ¾ 'qa o0 'qa / qa ¿ ¯
wR(r ) qa u R(r ) wqa
RI a u
(1 ra ) R(r )
(18)
qa 1 ra
(19)
where qa ( 1 ra ) is the unreliability of link a . For the case of two links in a series network, CIW is given from Eq. (5), (6), (7), (8), and (18) such that:
CIW1
1 r1 r1
(20)
CIW2
1 r2 r2
(21)
and
it follows that
CIW1 ! CIW2 if r1 r2
(22)
Thus, in a series network, the CIW proposed by Wakabayashi (2004) has the same property as RI, and this property from Eq. (22) is exactly as one would expect. For the case of two links in parallel, CIW is given from Eq. (5), (10), (11), and (18) such that:
CIW1
1 (1 r1 )(1 r2 ) R
(23)
and
CIW2
1 (1 r2 )(1 r1 ) R
it follows that CIW1=CIW2
(24) (25)
, although the criticality importance proposed by Wakabayashi made more progress than the index proposed by Henley and Kumamoto (1981), this index is the same for both links in a parallel network. When using CIW, a different index is necessary for the selection of an improved link.In this paper,we consider “an improvement path” of the terminal reliability using “the cost-link reliability function,” which is like the costbenefit ratio (Nicholson, 2007). 3.5. Merits and Demerits of Importance Indices We summarize the merits and demerits of these three kinds of importance indices: a) Merits and Demerits of Reliability Importance (RI) Judging from a definition and formality, the RI is rational as a general rule. However, it does not reflect that it is more difficult to improve a higher reliable link than to improve a less reliable link. b) Merits and Demerits of Criticality Importance (CI) CI has a shortcoming that gives the same importance for a series network. In addition, for a parallel network, use of both CI and RI result in only the more reliable links being improved, and the less reliable
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links will not be improved. c) Merits and Demerits of Improved Criticality Importance CIW overcomes a deficiency of the RI because it can reflect that the improvement of a link having high reliability is difficult. In addition, CIW overcomes a fault of the CI in the series network, because it can fairly determine which links in a parallel network should be improved. 4. Cost-Reliability Function and Cost-Benefit Analysis (Wakabayashi and Fang, 2014) In the case of CIW, a different index is needed, therefore, to choose an improved link. Therefore, the reliability of the improved link and three kinds of cost-reliability functions were established to show the relationship of the cost it requires. (1) Case A: Type of Constant—Cost Cost A = 1000
(26)
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 + 1)
(27)
In this case, the cost for improvement increases as a linear function as the link reliability increases. (3) Case C: Type of Increase—Quadratic Cost C = 500 × (50 × ra2 + 15 × ra + 1)
(28)
In this case, the costs for improvement increase as a quadratic function as link reliability increases. The effect of an improvement in network reliability, which requires a cost increase, might not be obvious in the short term. Therefore, a simple cost-benefit function that shows the improvement of the network reliability against the cost increase for a long time is defined as follows:
Eff (Y , F )
RAB RAB 0 uY u F Cost AB
(29)
where, Y: Number of years to invest; Fn: The conversion cost benefit of the increased traffic volume obtained by the reliability improvement in the n-th unit of time (In this paper, the unit of time is one year); R AB0 : Original network reliability; Cost AB : Cost increase to improve the network reliability from R AB0 to R AB . Eff (Y ) : The efficiency of the cost benefit obtained by the reliability improvement of traffic systems in Y years. 5. Comparison of RI and CIW in a Network Consisting of Nine Nodes and Twelve Links We calculate and compare the terminal reliability improvement by CIW and RI. The comparison is evaluated using terminal reliability, difference between maximum and minimum values of link reliability (DBMM), variance of value of link reliability, and cost benefit after improvement. From the discussion in 3.2, 3.4, and 3.5, DBMM is expected to be large by RI and vice versa by CIW.
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First, in the parallel network, RI provides better terminal reliability improvement than CIW, and CIW provides smaller dispersion among link reliability than RI (Wakabayashi, 2004). Second, we compare CIW with RI in the network consisting of nine nodes and twelve links (Fig. 3). We assume that an initial value of the link reliability for Case 1 (0.3 d r 0.7) and Case 2 (0.4 d r 0.6) and the set of link reliability values are given by generating random variables. As for the initial value, both Cases 1 and 2 are prepared for 100 trials. The link with the maximum value of importance is chosen as the improved link. Link reliability is increased by 0.05 for every improving phase till the sixth phase. The results are shown in Tables 1 and 2 and Fig. 4. Table 1 lists the average results of 100 trials using the calculation results of the reliability improvement determine by RI and CIW of Cases 1 and 2. It presents terminal reliability, link reliability (difference between maximum and minimum variance), and the value of cost-reliability function and a cost-benefit results. Table 1. Calculation result of the reliability improvement determined by and of Case 1 and Case 2. 㻯㼍㼟㼑㻝 㼀㼑㼞㼙㼕㼚㼍㼘 㻜㻚㻟䍺㼞䠘㻜㻚㻣 㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥 㻵㼚㼕㼠㼕㼍㼘㻌㼂㼍㼘㼡㼑 㻜㻚㻞㻤㻡㻥㻟㻥 㻵㼙㼜㼞㼛㼢㼑㼐㻌㼎㼥㻌㻾㻵 㻜㻚㻟㻢㻜㻥㻜㻥 㻵㼙㼜㼞㼛㼢㼑㼐㻌㼎㼥㻯㻵㼃 㻜㻚㻟㻡㻟㻥㻥㻣
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㻸㼕㼚㼗㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥㻙㻯㼛㼟㼠㻌 㻯㼛㼟㼠㻌㻮㼑㼚㼑㼒㼕㼠㻌 㻯㼍㼟㼑㻞 㼀㼑㼞㼙㼕㼚㼍㼘 㻲㼡㼚㼏㼠㼕㼛㼚 㻲㼡㼚㼏㼠㼕㼛㼚㻔㻱㼒㼒㻕 㻜㻚㻠䍺㼞䠘㻜㻚㻢 㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥 㻯㼍㼟㼑㻮 㻯㼍㼟㼑㻯 㻯㼍㼟㼑㻮 㻯㼍㼟㼑㻯 㻵㼚㼕㼠㼕㼍㼘㻌㼂㼍㼘㼡㼑 㻜㻚㻞㻤㻝㻤㻝㻣 㻝㻥㻝㻤㻡㻚㻟㻤 㻤㻞㻠㻞㻝㻚㻥㻞 㻝㻚㻥㻥 㻜㻚㻠㻤 㻵㼙㼜㼞㼛㼢㼑㼐㻌㼎㼥㻌㻾㻵 㻜㻚㻟㻠㻤㻢㻢㻡 㻝㻡㻤㻞㻥㻚㻞㻢 㻡㻤㻢㻟㻜㻚㻢㻜 㻞㻚㻝㻢 㻜㻚㻢㻜 㻵㼙㼜㼞㼛㼢㼑㼐㻌㼎㼥㻯㻵㼃 㻜㻚㻟㻠㻡㻜㻤㻝
㻸㼕㼚㼗㻌㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥 㼙㼍㼤㻙㼙㼕㼚 㼢㼍㼞 㻜㻚㻝㻢㻥㻟㻤㻣 㻜㻚㻜㻜㻞㻥㻣㻤 㻜㻚㻟㻜㻡㻥㻡㻝 㻜㻚㻜㻜㻣㻟㻣㻜 㻜㻚㻞㻝㻝㻞㻢㻢 㻜㻚㻜㻜㻠㻞㻠㻥
㻸㼕㼚㼗㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥㻙㻯㼛㼟㼠㻌 㻯㼛㼟㼠㻌㻮㼑㼚㼑㼒㼕㼠㻌 㻲㼡㼚㼏㼠㼕㼛㼚 㻲㼡㼚㼏㼠㼕㼛㼚㻔㻱㼒㼒㻕 㻯㼍㼟㼑㻮 㻯㼍㼟㼑㻯 㻯㼍㼟㼑㻮 㻯㼍㼟㼑㻯 㻝㻤㻤㻝㻡㻚㻜㻜 㻣㻤㻟㻢㻞㻚㻜㻠 㻝㻚㻣㻤 㻜㻚㻠㻟 㻝㻢㻥㻠㻞㻚㻣㻟 㻢㻠㻥㻞㻠㻚㻣㻟 㻝㻚㻤㻣 㻜㻚㻠㻥 㻌
Table 2 compares the link number, which becomes an improvement target, and the number of improvements to five trials from 1 to 5 using RI and CIW. Figure 4 shows the number of links improved for 100 trials using RI and CIW for Cases 1 and 2. For example, when this number is 1, it shows that the same link is improved six times. Table 2. Link number and the number of improvements, which become an improved target for Trials 1–5. 㻜㻚㻟䍺㼞䠘㻜㻚㻣 㻵㼙㼜㼞㼛㼢㼑㼐 㻺㼡㼙㼎㼑㼞 㼛㼒㻌㼀㼕㼙㼑㼟
㻜㻚㻠䍺㼞䠘㻜㻚㻢 㻵㼙㼜㼞㼛㼢㼑㼐 㻺㼡㼙㼎㼑㼞 㼛㼒㻌㼀㼕㼙㼑㼟
㻾㻵 㻸㼕㼚㼗㻌㻺㼛㻚 㼀㼞㼕㼍㼘㻌㻝㻚 㼀㼞㼕㼍㼘㻌㻞㻚 㼀㼞㼕㼍㼘㻌㻟㻚 㼀㼞㼕㼍㼘㻌㻠㻚 㼀㼞㼕㼍㼘㻌㻡㻚
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㻯㻵㼃 㻸㼕㼚㼗㻌㻺㼛㻚 㼀㼞㼕㼍㼘㻌㻝㻚 㼀㼞㼕㼍㼘㻌㻞㻚 㼀㼞㼕㼍㼘㻌㻟㻚 㼀㼞㼕㼍㼘㻌㻠㻚 㼀㼞㼕㼍㼘㻌㻡㻚
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Fig. 4. Comparison of the number of improved links ( , ).
As a result, terminal reliability improvements determined by RI are better than those determined by CIW; however, for the numerical value, there is not a large difference between RI and CIW. Next, the variance of the link reliability determined by CIW is smaller than that determined by RI, and the cost-benefit results determined by CIW are better than those determined by RI. Thus, generally, CIW is better than RI.
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6. Comparative Analysis of Calculation Methods of Approximate Values It is necessary to reduce computational work for both connectivity reliability and importance analyses, because the computational work increases exponentially when the size of network expands. In other words, the CPU time and memory size increase exponentially with the number of path sets. Inversely, if the number of path sets is reduced, they decrease exponentially. Therefore, we propose an efficient and approximate method to calculate them with partial path sets in a highway network. Note that the order of importance indices remain unchanged. In this study, we compare two calculation results of the choice method using the distance of the path sets and the occurrence probability of the path sets in the network. It is important for the importance analysis of the connectivity reliability in the highway network not to change rank value of link reliability between the exact value and the approximate value. If it reduces the number of paths from 12 (all path sets) to approximately seven as a calculation result of CIW, the change in the order is not large. This result suggests that there are no significant changes in the order of importance if a path set is reduced, provided all links are included in the path sets. In this study, the path sets are chosen in two ways using path-set occurrence probability and path-route distance, and the approximate value of terminal reliability was clarified in a network consisting of nine nodes and twelve links. We added distance order as a path criteria for selection, because we thought that it was used in short distances in the real highway network. We assumed that an initial value of the link reliability was prepared (0.3 d r 0.7) and made a random number occur in each. As for the initial value, we prepared five trials. We reduced the number of the paths by occurrence-probability order and distance order and compared the value of the terminal reliability and the importance of links. We show those results in Fig. 5.
Fig. 5. Importance of the links compared to occurrence-probability order and distance order (Trial 1).
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Figure 6 compares the behavior of a path of the occurrence-probability order and the distance order. The right end of each curve is the exact value for which all paths are used. The lines indicate the value of importance of each link.
Fig. 6. Terminal reliability compared to occurrence-probability order and the distance order (Trials 1–5).
As a result, we recognized that two choice methods made little difference even if we reduced the number of path sets to approximately seven among all 12 path sets. There is a possibility that we can use not only the occurrence probability but also the method using the distance to choose paths.
7. Conclusion and Future Subjects The concluding remarks are as follows: (1) For network reliability improvement, the use of importance index is effective. (2) Definitions and characteristics of reliability importance (RI), criticality importance (CI), and improved criticality importance (CIW) are introduced. (3) CIW overcomes the demerits of RI and CI, discussed in 3.5. CI has a serious shortcoming, discussed in 3.3. (4) CIW and RI are compared by generating random numbers for sets of link reliability with respect to terminal reliability, difference between maximum and minimum values of link reliability (DBMM), variance, and cost benefit after improving terminal reliability. (5) Consequently, RI gave better improvement in terminal reliability than CIW, although the difference is small. RI gave a larger DBMM than CIW. This result indicates that RI has more gaps between paralleled routes than CIW. (6) Therefore, CIW is a recommended importance index. (7) Reliability analysis represents an NP-hard problem. When the network expands, the computational work increases exponentially. Thus, an efficient and approximate calculation method using partial minimal path sets is proposed. As the number of path sets decreases, the computational work is reduced exponentially. (8) No problem is encountered provided the order of link importance indices is unchanged when the number of path sets is reduced. A test calculation is performed.
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The future work is as follows: Cumulative calculation is required to verify the advantages of CIW. It is also needed to verify the effectiveness of the approximate method, including how to choose path sets. Moreover, application to a large-scale network is required. This study greatly contributes to the national resilience in constructing a highly reliable highway network under disaster environment. References Barlow, R. E. and Proschan, F. (1965). Mathematical Theory of Reliability, p.6. John Wiley & Sons, Inc., New York. Birnbaum, Z. W. (1969) On the Importance of Different Components in Multi-Component System. Multivariate Analysis II (P. R. Krishnaiah Ed.), Academic Press, New York. Henley, E.J. and Kumamoto, H. (1981):Reliability Engineering and Risk Assessment, Prentice-Hall, Inc., 418-436. Nicholson, A. (2007). Optimising Network Terminal Reliability. Proceedings of the 3rd International Symposium on Transportation Network Reliability, CD-ROM, 2007. Nicholson, A. Schmoecker, J. Bell, M.G.H. and Iida, Y (2003). Assessing Transport Reliability: Malevolence and User Knowledge In: Michael G. H. Bell and Yasunori Iida(Ed.) The Network Reliability of Transport, Proceedings of the 1st International Symposium on Transportation Network Reliability (INSTR), pp.1-22, Pergamon, 2003. US NRC: PRA Procedures Guide: A Guide to the Performance of Probabilistic Risk Assessments for Nuclear Power Plants, NUREG/CR-2300 (1983) Wakabayashi, H. (2004). Network Reliability Improvement: Reliability Importance. Proceedings of Second International Symposium on Transport Network Reliability. University of Canterbury, Christchurch, New Zealand. Wakabayashi, H. and Fang, S. (2014). Comparison of Importance Indices for Highway Network Reliability Improvement Combined with CostBenefit Analysis. In: Jorge Freire de Sausa and Riccardo Rossi (Ed.). Computer-based Modelling and Optimization in Transportation, Advances in Intelligent Systems and Computing 262, 265-280. Springer, Switzerland. (ISBN 978-3-319-04629-7, ISSN 2194-5357, DOI: 10.1007/978-3-319-04630-3㸧(ISBN 978-3-319-04630-3 (eBook), ISSN 2194-5365 (electronic)). Wakabayashi, H. and Iida, Y. (1992). Upper and Lower Bounds of Terminal Reliability of Road Networks: An Efficient Method with Boolean algebra. Journal of Natural Disaster Science, Volume 14, No.1.
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18th Euro Working Group on Transportation, EWGT 2015, 14-16 July 2015, Delft, The Netherlands
Differences in network reliability improvement by several importance indices Takahiro Nagaea, Hiroshi Wakabayashib* a
Graduate School ofUrban Science, Meijo University, Nijigaoka 4-3-3, Kani, Gifu 509-0261, Japan b Faculty of Urban Science, Meijo University, Nijigaoka 4-3-3, Kani, Gifu 509-0261, Japan
Abstract For constructing a highly reliable highway network under a disaster environment, this paper presents a comparative study of the importance indices and proposes an approximate method for improving highway network reliability. Three indices are compared. Two of them are conventional indices of reliability importance (RI) and criticality importance (CI), and the other is the improved criticality importance (CIW) index recently proposed by Wakabayashi (2004). First, the significance of reliability importance is discussed. Second, their definitions, merits, and demerits are explained. Third, several cost functions for improving link reliability are introduced. Fourth, RI and CIW are compared for a small-sized network and differences in improved link reliability are discussed. Fifth, an approximate method is proposed using partial minimal path sets. Lastly, after calculation experiments, the advantages of the CIW index and the effectiveness of the proposed approximate method are demonstrated. © 2015 2015The TheAuthors. Authors. Published by Elsevier B. V.is an open access article under the CC BY-NC-ND license © Published by Elsevier B.V. This (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of Delft University of Technology. Peer-review under responsibility of Delft University of Technology Keywords: Highway network reliability, Importance index, Probability importance, Partial minimal path set, Cost-benefit analysis
1. Introduction Under the threat of a potential disaster such as the 2011 Japan earthquake, constructing and sustaining a highly reliable highway network for national resilience is an urgent necessity. However, improving all the links at once is
* Corresponding author. . Tel.: +81 052-832-1151 E-mail address: [email protected]
2352-1465 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Delft University of Technology doi:10.1016/j.trpro.2015.09.065
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difficult because of budget constraints. Therefore, identifying key links to improve network reliability is important; this process is called the importance analysis. Importance analysis of a highway network mainly entails two concerns: (1) the development of an effective importance index and (2) the reduction of the amount of calculations. Therefore, this study verifies the relevance of an index proposed by the authors and develops an efficient method for approximate calculations. To resolve these two concerns, this study presents both a comparative analysis of importance indices for actual use and an efficient method for approximate calculations. The remainder of this paper is structured as follows: Section 2 reviews the reliability analysis of a highway network and the study of importance analysis. Section 2 also highlights the characteristics of and problems with importance indices that have been previously proposed. Section 3 provides a definition of connectivity reliability and introduces the conventional and newly proposed importance indices. Section 4 presents three cost-reliability functions. Section 5 compares terminal-reliability improvement using two importance indices, i.e. CIW and RI with cost-benefit functions for a network with nine nodes and twelve links. Section 6 uses examples to show an approximate calculation method that chooses partial minimal path sets to reduce calculations. Section 7 presents concluding remarks for the problem discussed in Section 2. 2. Reliability and Importance Indices of a Highway Network Connectivity, travel-time, and capacity reliabilities (Nicholson et al., 2003) have been proposed as methods to measure the reliability of a highway network. Our study focuses on the improvement of connectivity reliability and importance indices. 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); it is defined as an expression of probability. The CPU-time and memory size required for reliability analysis increases exponentially as the size of the highway network increases. This is called a non-deterministic polynomial-time hard (NP-hard) problem (US NRC, 1983). For solving this problem, efficient and practical reliability analysis has been proposed by Wakabayashi and Iida (1992). However, efficient and practical importance analysis remains unsolved. The purpose of the importance index is to effectively improve the reliability of the system. Probability importance (Birnbaum, 1969) and criticality importance (CI) (Henley and Kumamoto, 1981) are previously proposed and discussed later. Importance indices can be interpreted as the degree to which a component contributes to the reliability of the system when the reliability of the component is improved. Moreover, in this study, the importance of the connectivity reliability of the highway network is defined as the degree to which a component contributes to the reliability between nodes when link reliability is improved. Next, we 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 to improve a less reliable link. This is called diminishing marginal utility. Birnbaum’s importance index, i.e., Reliability Importance (RI), does not reflect this fact. (2) In a parallel network, use of the RI and CI results in only the more reliable links being improved, and the less reliable links will remain unimproved. (3) The importance index does not always result in an economical highway network improvement. A cost-benefit perspective should also be considered. (4) Importance analysis also is an NP-hard problem. An efficient and approximate method for practical use is required for actual highway networks. An improved criticality importance (CIW) index has recently been proposed to solve the problems (1) and (2) (Wakabayashi, 2004). However, till date, the results of applying this index to a network are not calculated and evaluated enough, and the index is not yet compared with the RI index; this paper attempts this analysis and comparison. For problem (3), only a few comparisons have been made to determine what difference in the terminal reliability exists when the network is improved using a conventional importance index. We can form the following hypotheses: if we set the cost-reliability function that expresses an expense and relations of the link reliability (the higher link
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reliability), the costs for anti-disaster are more expensive; however, the expense to strengthen a link of low reliability is low. For problem (4), when a network expands, the CPU-time and memory size increase exponentially for the importance analysis in addition to the reliability analysis stated above. Therefore, efficient and approximate calculation method is necessary. In this study, we propose an approximate method to calculate them with partial path sets in a highway network. The effectiveness of the approximate calculation method is evaluated by comparing the exact and approximate values irrespective of the whether the order of link importance has changed. Furthermore, we developed some link reliability cost functions and compared a rational solution with an equitable solution (using a cost-benefit analysis), and we clarified how to change using a relation with link reliability functions. Next, we established a small network and calculated the “improved path,” determined by importance indices and a cost-benefit analysis. The index for comparison uses terminal reliability and the ratio of terminal reliability to cost. On the basis of these, we reviewed conventional studies. As an example of the articles about the importance evaluation in the highway network, we discuss the method that uses a cost-link reliability function (Nicholson, 2007). Nicholson (2007) proposes that the link reliability is provided by the function of traffic volume; thus, the cost required for reliability improvement cannot be considered explicitly. 3. Reliability and Importance Indices of a Highway Network (Wakabayashi and Fang, 2014) 3.1. The Connectivity Reliability of a Highway Network In this study, we compare CIW with RI in a small network and propose a calculation reduction method using partial minimal path sets. Before we discuss our proposal for the comparative analysis for improving network reliability, we introduce the work 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 of traffic 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. In a highway network, link reliabilityrais defined as ra
E[ X a ] (1)
Terminal reliability, i.e. node-to-node reliability, R is given by an expression using minimal-path sets as follows:
R(r )
E[I ( X)]
p
E[1 (1 X a )] s 1
aPs
(2)
, where Ps is the s-th minimal-path set, p is the total number of minimal-path sets, Xa is the binary indicator variable for link a (Wakabayashi and Iida, 1992) and I (X) , X , and r are a structural function and vector representations for
Xa
and
Xa
ra : 1, if link a provides a certain service level, ® ¯0, otherwise.
(3)
For example, calculating the connectivity reliability (approximate value: when using only three path sets {1,2,5,10},{1,4,9,12},{3,8,11,12}) between nodes 1 to 9 (Fig. 1), in the calculating formula, is possible as follows:
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Fig.1.㻌A network with nine nodes and twelve links. p
R(r )
E[1 (1 X a )] E[1 (1 X1 X 2 X 5 X10 )(1 X1 X 4 X 9 X12 )(1 X 3 X 8 X11X12 )] s 1
(4)
a p s
In this equation, the variables X1 and X12 appear more than once. To avoid the product of the same indicator, variables used in the calculation of the expected value of reliability, a Boolean absorption such as X1 × X1 = X1 is required. The connectivity reliability of a traffic network depends on the network structure and the link reliabilities. Therefore, two basic approaches have been taken to improve network reliability: improvement of the network structure or improvement of the reliability of the links. The focus here is on identifying which links should be improved to maximize the improvement in network reliability. 3.2. Definition of Reliability Importance To find the key links for most efficiently improving the terminal reliability, the Reliability Importance (RI) index (Birnbaum, 1969) was proposed as
RI a
wR(r ) , 0 d RI a d 1 .(5) wra
RI indicates the impact of an improvement in link reliability, i.e., the increase or decrease in the reliability of link a affects the increase or decrease in the terminal reliability. RI is also known as Birnbaum’s structural importance. R(r) is terminal reliability between nodes A to B, a is link number, and ra is the value of link reliability for link a. For a series network with two links, the terminal reliability RAB is given by Eq. (6): (6) RAB r1r2 , where r1 and r2 are the values of reliability for links 1 and 2, respectively. When determining the reliability importance for a series network, RI1 and RI2, are obtained from Eq. (5) and (6) as RI1 r2 (7) (8) . RI 2 r1 It follows that (9) RI ! RI if r r . 1
2
1
2
Equation (9) indicates that in the case of a series network, improving the less reliable link is most effective for improving terminal reliability. This fact is easily expanded for a large series networks. This result for improving, managing, and reconstructing a network is the expected result. However, for the case of two links in a parallel network, the terminal reliability, RAB, is given by Eq. (10); (10) RAB 1 (1 r1 )(1 r2 ) RI1 and RI2, for these two links in a parallel network, are obtained from Eq. (5) and (10) as
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Fig. 2. Series Network
Fig. 3. Parallel Network
1 r2 and RI 2
RI1
1 r1
(11)
It follows that (12) RI1 RI 2 if r1 r2 The result from Eq. (12) indicates that in the case of a parallel network, improving the more reliable link will be more effective for improving terminal reliability. Usually, however, it is difficult to improve a more reliable link, whereas it is rather easy to improve a less reliable link. This result for improving, managing, and reconstructing a network is counter to what one would expect. 3.3. Definition of Criticality Importance Because of the shortcoming of reliability importance, the criticality importance (CI) index was proposed as the ratio of the proportional improvement in the network reliability to the proportional improvement in the link reliability (Henley and Kumamoto, 1992):
CI a
'R (r ) / RAB (r ) ½ lim ® AB ¾ 'ra / ra ¯ ¿
'ra o0
ra wRAB (r ) u RAB (r ) wra
RI a
ra RAB (r )
(13)
CI also has shortcomings, which we discuss in this section. For the case of two links in a series network, CI is given from Eq. (5), (6), (7), (8), and (13) such that:
CI1
r1r2 R
CI 2
(14)
This result suggests that the criticality importance (CI) index is the same for both links in a series network. However, in a series network, it would be reasonable to strengthen a less reliable link. Therefore, this is a shortcoming of the CI index. In addition, the index provides no information to distinguish the link between the two links for improving network reliability. For the case of two links in a parallel network, CI is given from Eq. (5), (10), (11), and (13) such that:
CI1
r1 r1r2 R
(15)
CI 2
r2 r1r2 R
(16)
and
It follows that
CI1 CI 2 if r1 r2
(17)
Therefore, the CI index also indicates that in the case of a parallel network, improving a more reliable link gives a greater increase in the terminal reliability of the network. The results for a parallel network provided by both the RI and the CI suggest that a less reliable link should be ignored in a parallel system.
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3.4. Definition of Improved Criticality Importance Changing the definition of the equation in the reliability engineering, the CIW, proposed by Wakabayashi (2004), is introduced as Eq. (18);
CIWa
'R(r ) / R(r ) ½ lim ® ¾ 'qa o0 'qa / qa ¿ ¯
wR(r ) qa u R(r ) wqa
RI a u
(1 ra ) R(r )
(18)
qa 1 ra
(19)
where qa ( 1 ra ) is the unreliability of link a . For the case of two links in a series network, CIW is given from Eq. (5), (6), (7), (8), and (18) such that:
CIW1
1 r1 r1
(20)
CIW2
1 r2 r2
(21)
and
it follows that
CIW1 ! CIW2 if r1 r2
(22)
Thus, in a series network, the CIW proposed by Wakabayashi (2004) has the same property as RI, and this property from Eq. (22) is exactly as one would expect. For the case of two links in parallel, CIW is given from Eq. (5), (10), (11), and (18) such that:
CIW1
1 (1 r1 )(1 r2 ) R
(23)
and
CIW2
1 (1 r2 )(1 r1 ) R
it follows that CIW1=CIW2
(24) (25)
, although the criticality importance proposed by Wakabayashi made more progress than the index proposed by Henley and Kumamoto (1981), this index is the same for both links in a parallel network. When using CIW, a different index is necessary for the selection of an improved link.In this paper,we consider “an improvement path” of the terminal reliability using “the cost-link reliability function,” which is like the costbenefit ratio (Nicholson, 2007). 3.5. Merits and Demerits of Importance Indices We summarize the merits and demerits of these three kinds of importance indices: a) Merits and Demerits of Reliability Importance (RI) Judging from a definition and formality, the RI is rational as a general rule. However, it does not reflect that it is more difficult to improve a higher reliable link than to improve a less reliable link. b) Merits and Demerits of Criticality Importance (CI) CI has a shortcoming that gives the same importance for a series network. In addition, for a parallel network, use of both CI and RI result in only the more reliable links being improved, and the less reliable
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links will not be improved. c) Merits and Demerits of Improved Criticality Importance CIW overcomes a deficiency of the RI because it can reflect that the improvement of a link having high reliability is difficult. In addition, CIW overcomes a fault of the CI in the series network, because it can fairly determine which links in a parallel network should be improved. 4. Cost-Reliability Function and Cost-Benefit Analysis (Wakabayashi and Fang, 2014) In the case of CIW, a different index is needed, therefore, to choose an improved link. Therefore, the reliability of the improved link and three kinds of cost-reliability functions were established to show the relationship of the cost it requires. (1) Case A: Type of Constant—Cost Cost A = 1000
(26)
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 + 1)
(27)
In this case, the cost for improvement increases as a linear function as the link reliability increases. (3) Case C: Type of Increase—Quadratic Cost C = 500 × (50 × ra2 + 15 × ra + 1)
(28)
In this case, the costs for improvement increase as a quadratic function as link reliability increases. The effect of an improvement in network reliability, which requires a cost increase, might not be obvious in the short term. Therefore, a simple cost-benefit function that shows the improvement of the network reliability against the cost increase for a long time is defined as follows:
Eff (Y , F )
RAB RAB 0 uY u F Cost AB
(29)
where, Y: Number of years to invest; Fn: The conversion cost benefit of the increased traffic volume obtained by the reliability improvement in the n-th unit of time (In this paper, the unit of time is one year); R AB0 : Original network reliability; Cost AB : Cost increase to improve the network reliability from R AB0 to R AB . Eff (Y ) : The efficiency of the cost benefit obtained by the reliability improvement of traffic systems in Y years. 5. Comparison of RI and CIW in a Network Consisting of Nine Nodes and Twelve Links We calculate and compare the terminal reliability improvement by CIW and RI. The comparison is evaluated using terminal reliability, difference between maximum and minimum values of link reliability (DBMM), variance of value of link reliability, and cost benefit after improvement. From the discussion in 3.2, 3.4, and 3.5, DBMM is expected to be large by RI and vice versa by CIW.
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Takahiro Nagae and Hiroshi Wakabayashi / Transportation Research Procedia 10 (2015) 155 – 165
First, in the parallel network, RI provides better terminal reliability improvement than CIW, and CIW provides smaller dispersion among link reliability than RI (Wakabayashi, 2004). Second, we compare CIW with RI in the network consisting of nine nodes and twelve links (Fig. 3). We assume that an initial value of the link reliability for Case 1 (0.3 d r 0.7) and Case 2 (0.4 d r 0.6) and the set of link reliability values are given by generating random variables. As for the initial value, both Cases 1 and 2 are prepared for 100 trials. The link with the maximum value of importance is chosen as the improved link. Link reliability is increased by 0.05 for every improving phase till the sixth phase. The results are shown in Tables 1 and 2 and Fig. 4. Table 1 lists the average results of 100 trials using the calculation results of the reliability improvement determine by RI and CIW of Cases 1 and 2. It presents terminal reliability, link reliability (difference between maximum and minimum variance), and the value of cost-reliability function and a cost-benefit results. Table 1. Calculation result of the reliability improvement determined by and of Case 1 and Case 2. 㻯㼍㼟㼑㻝 㼀㼑㼞㼙㼕㼚㼍㼘 㻜㻚㻟䍺㼞䠘㻜㻚㻣 㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥 㻵㼚㼕㼠㼕㼍㼘㻌㼂㼍㼘㼡㼑 㻜㻚㻞㻤㻡㻥㻟㻥 㻵㼙㼜㼞㼛㼢㼑㼐㻌㼎㼥㻌㻾㻵 㻜㻚㻟㻢㻜㻥㻜㻥 㻵㼙㼜㼞㼛㼢㼑㼐㻌㼎㼥㻯㻵㼃 㻜㻚㻟㻡㻟㻥㻥㻣
㻸㼕㼚㼗㻌㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥 㼙㼍㼤㻙㼙㼕㼚 㼢㼍㼞 㻜㻚㻟㻟㻤㻣㻣㻡 㻜㻚㻜㻝㻝㻥㻝㻜 㻜㻚㻠㻞㻤㻠㻤㻤 㻜㻚㻜㻝㻢㻤㻞㻥 㻜㻚㻟㻟㻞㻢㻝㻥 㻜㻚㻜㻝㻝㻞㻟㻡
㻸㼕㼚㼗㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥㻙㻯㼛㼟㼠㻌 㻯㼛㼟㼠㻌㻮㼑㼚㼑㼒㼕㼠㻌 㻯㼍㼟㼑㻞 㼀㼑㼞㼙㼕㼚㼍㼘 㻲㼡㼚㼏㼠㼕㼛㼚 㻲㼡㼚㼏㼠㼕㼛㼚㻔㻱㼒㼒㻕 㻜㻚㻠䍺㼞䠘㻜㻚㻢 㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥 㻯㼍㼟㼑㻮 㻯㼍㼟㼑㻯 㻯㼍㼟㼑㻮 㻯㼍㼟㼑㻯 㻵㼚㼕㼠㼕㼍㼘㻌㼂㼍㼘㼡㼑 㻜㻚㻞㻤㻝㻤㻝㻣 㻝㻥㻝㻤㻡㻚㻟㻤 㻤㻞㻠㻞㻝㻚㻥㻞 㻝㻚㻥㻥 㻜㻚㻠㻤 㻵㼙㼜㼞㼛㼢㼑㼐㻌㼎㼥㻌㻾㻵 㻜㻚㻟㻠㻤㻢㻢㻡 㻝㻡㻤㻞㻥㻚㻞㻢 㻡㻤㻢㻟㻜㻚㻢㻜 㻞㻚㻝㻢 㻜㻚㻢㻜 㻵㼙㼜㼞㼛㼢㼑㼐㻌㼎㼥㻯㻵㼃 㻜㻚㻟㻠㻡㻜㻤㻝
㻸㼕㼚㼗㻌㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥 㼙㼍㼤㻙㼙㼕㼚 㼢㼍㼞 㻜㻚㻝㻢㻥㻟㻤㻣 㻜㻚㻜㻜㻞㻥㻣㻤 㻜㻚㻟㻜㻡㻥㻡㻝 㻜㻚㻜㻜㻣㻟㻣㻜 㻜㻚㻞㻝㻝㻞㻢㻢 㻜㻚㻜㻜㻠㻞㻠㻥
㻸㼕㼚㼗㻾㼑㼘㼕㼍㼎㼕㼘㼕㼠㼥㻙㻯㼛㼟㼠㻌 㻯㼛㼟㼠㻌㻮㼑㼚㼑㼒㼕㼠㻌 㻲㼡㼚㼏㼠㼕㼛㼚 㻲㼡㼚㼏㼠㼕㼛㼚㻔㻱㼒㼒㻕 㻯㼍㼟㼑㻮 㻯㼍㼟㼑㻯 㻯㼍㼟㼑㻮 㻯㼍㼟㼑㻯 㻝㻤㻤㻝㻡㻚㻜㻜 㻣㻤㻟㻢㻞㻚㻜㻠 㻝㻚㻣㻤 㻜㻚㻠㻟 㻝㻢㻥㻠㻞㻚㻣㻟 㻢㻠㻥㻞㻠㻚㻣㻟 㻝㻚㻤㻣 㻜㻚㻠㻥 㻌
Table 2 compares the link number, which becomes an improvement target, and the number of improvements to five trials from 1 to 5 using RI and CIW. Figure 4 shows the number of links improved for 100 trials using RI and CIW for Cases 1 and 2. For example, when this number is 1, it shows that the same link is improved six times. Table 2. Link number and the number of improvements, which become an improved target for Trials 1–5. 㻜㻚㻟䍺㼞䠘㻜㻚㻣 㻵㼙㼜㼞㼛㼢㼑㼐 㻺㼡㼙㼎㼑㼞 㼛㼒㻌㼀㼕㼙㼑㼟
㻜㻚㻠䍺㼞䠘㻜㻚㻢 㻵㼙㼜㼞㼛㼢㼑㼐 㻺㼡㼙㼎㼑㼞 㼛㼒㻌㼀㼕㼙㼑㼟
㻾㻵 㻸㼕㼚㼗㻌㻺㼛㻚 㼀㼞㼕㼍㼘㻌㻝㻚 㼀㼞㼕㼍㼘㻌㻞㻚 㼀㼞㼕㼍㼘㻌㻟㻚 㼀㼞㼕㼍㼘㻌㻠㻚 㼀㼞㼕㼍㼘㻌㻡㻚
㻝 㻠 㻢 㻢 㻜 㻜
㻞 㻜 㻜 㻜 㻜 㻜
㻟 㻜 㻜 㻜 㻜 㻢
㻠 㻞 㻜 㻜 㻜 㻜
㻡 㻜 㻜 㻜 㻜 㻜
㻢 㻜 㻜 㻜 㻜 㻜
㻣 㻜 㻜 㻜 㻜 㻜
㻤 㻜 㻜 㻜 㻜 㻜
㻥 㻜 㻜 㻜 㻝 㻜
㻝㻜 㻜 㻜 㻜 㻜 㻜
㻝㻝 㻜 㻜 㻜 㻜 㻜
㻝㻞 㻜 㻜 㻜 㻡 㻜
㻜㻚㻟䍺㼞䠘㻜㻚㻣
㻾㻵 㻸㼕㼚㼗㻌㻺㼛㻚 㼀㼞㼕㼍㼘㻌㻝㻚 㼀㼞㼕㼍㼘㻌㻞㻚 㼀㼞㼕㼍㼘㻌㻟㻚 㼀㼞㼕㼍㼘㻌㻠㻚 㼀㼞㼕㼍㼘㻌㻡㻚
㻝 㻡 㻠 㻡 㻜 㻜
㻞 㻜 㻜 㻜 㻜 㻜
㻟 㻜 㻜 㻜 㻝 㻡
㻠 㻝 㻜 㻜 㻜 㻜
㻡 㻜 㻜 㻜 㻜 㻜
㻢 㻜 㻜 㻜 㻜 㻜
㻣 㻜 㻜 㻜 㻜 㻜
㻤 㻜 㻜 㻜 㻜 㻜
㻥 㻜 㻜 㻜 㻜 㻜
㻝㻜 㻜 㻞 㻝 㻜 㻝
㻝㻝 㻜 㻜 㻜 㻜 㻜
㻝㻞 㻜 㻜 㻜 㻡 㻜
㻵㼙㼜㼞㼛㼢㼑㼐 㻺㼡㼙㼎㼑㼞 㼛㼒㻌㼀㼕㼙㼑㼟
㻜㻚㻠䍺㼞䠘㻜㻚㻢 㻵㼙㼜㼞㼛㼢㼑㼐 㻺㼡㼙㼎㼑㼞 㼛㼒㻌㼀㼕㼙㼑㼟
㻯㻵㼃 㻸㼕㼚㼗㻌㻺㼛㻚 㼀㼞㼕㼍㼘㻌㻝㻚 㼀㼞㼕㼍㼘㻌㻞㻚 㼀㼞㼕㼍㼘㻌㻟㻚 㼀㼞㼕㼍㼘㻌㻠㻚 㼀㼞㼕㼍㼘㻌㻡㻚
㻝 㻝 㻠 㻞 㻞 㻟
㻞 㻜 㻜 㻜 㻜 㻜
㻟 㻠 㻝 㻝 㻜 㻞
㻠 㻜 㻜 㻜 㻜 㻜
㻡 㻜 㻜 㻜 㻜 㻜
㻢 㻝 㻜 㻜 㻜 㻜
㻣 㻜 㻜 㻜 㻜 㻜
㻤 㻜 㻜 㻜 㻜 㻜
㻥 㻜 㻜 㻝 㻜 㻜
㻝㻜 㻜 㻝 㻜 㻠 㻜
㻝㻝 㻜 㻜 㻜 㻜 㻜
㻝㻞 㻜 㻜 㻞 㻜 㻝
㻯㻵㼃 㻸㼕㼚㼗㻌㻺㼛㻚 㼀㼞㼕㼍㼘㻌㻝㻚 㼀㼞㼕㼍㼘㻌㻞㻚 㼀㼞㼕㼍㼘㻌㻟㻚 㼀㼞㼕㼍㼘㻌㻠㻚 㼀㼞㼕㼍㼘㻌㻡㻚
㻝 㻝 㻠 㻞 㻞 㻟
㻞 㻜 㻜 㻜 㻜 㻜
㻟 㻠 㻝 㻝 㻜 㻞
㻠 㻜 㻜 㻜 㻜 㻜
㻡 㻜 㻜 㻜 㻜 㻜
㻢 㻝 㻜 㻜 㻜 㻜
㻣 㻜 㻜 㻜 㻜 㻜
㻤 㻜 㻜 㻜 㻜 㻜
㻥 㻜 㻜 㻝 㻜 㻜
㻝㻜 㻜 㻝 㻜 㻠 㻜
㻝㻝 㻜 㻜 㻜 㻜 㻜
㻝㻞 㻜 㻜 㻞 㻜 㻝
Fig. 4. Comparison of the number of improved links ( , ).
As a result, terminal reliability improvements determined by RI are better than those determined by CIW; however, for the numerical value, there is not a large difference between RI and CIW. Next, the variance of the link reliability determined by CIW is smaller than that determined by RI, and the cost-benefit results determined by CIW are better than those determined by RI. Thus, generally, CIW is better than RI.
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6. Comparative Analysis of Calculation Methods of Approximate Values It is necessary to reduce computational work for both connectivity reliability and importance analyses, because the computational work increases exponentially when the size of network expands. In other words, the CPU time and memory size increase exponentially with the number of path sets. Inversely, if the number of path sets is reduced, they decrease exponentially. Therefore, we propose an efficient and approximate method to calculate them with partial path sets in a highway network. Note that the order of importance indices remain unchanged. In this study, we compare two calculation results of the choice method using the distance of the path sets and the occurrence probability of the path sets in the network. It is important for the importance analysis of the connectivity reliability in the highway network not to change rank value of link reliability between the exact value and the approximate value. If it reduces the number of paths from 12 (all path sets) to approximately seven as a calculation result of CIW, the change in the order is not large. This result suggests that there are no significant changes in the order of importance if a path set is reduced, provided all links are included in the path sets. In this study, the path sets are chosen in two ways using path-set occurrence probability and path-route distance, and the approximate value of terminal reliability was clarified in a network consisting of nine nodes and twelve links. We added distance order as a path criteria for selection, because we thought that it was used in short distances in the real highway network. We assumed that an initial value of the link reliability was prepared (0.3 d r 0.7) and made a random number occur in each. As for the initial value, we prepared five trials. We reduced the number of the paths by occurrence-probability order and distance order and compared the value of the terminal reliability and the importance of links. We show those results in Fig. 5.
Fig. 5. Importance of the links compared to occurrence-probability order and distance order (Trial 1).
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Takahiro Nagae and Hiroshi Wakabayashi / Transportation Research Procedia 10 (2015) 155 – 165
Figure 6 compares the behavior of a path of the occurrence-probability order and the distance order. The right end of each curve is the exact value for which all paths are used. The lines indicate the value of importance of each link.
Fig. 6. Terminal reliability compared to occurrence-probability order and the distance order (Trials 1–5).
As a result, we recognized that two choice methods made little difference even if we reduced the number of path sets to approximately seven among all 12 path sets. There is a possibility that we can use not only the occurrence probability but also the method using the distance to choose paths.
7. Conclusion and Future Subjects The concluding remarks are as follows: (1) For network reliability improvement, the use of importance index is effective. (2) Definitions and characteristics of reliability importance (RI), criticality importance (CI), and improved criticality importance (CIW) are introduced. (3) CIW overcomes the demerits of RI and CI, discussed in 3.5. CI has a serious shortcoming, discussed in 3.3. (4) CIW and RI are compared by generating random numbers for sets of link reliability with respect to terminal reliability, difference between maximum and minimum values of link reliability (DBMM), variance, and cost benefit after improving terminal reliability. (5) Consequently, RI gave better improvement in terminal reliability than CIW, although the difference is small. RI gave a larger DBMM than CIW. This result indicates that RI has more gaps between paralleled routes than CIW. (6) Therefore, CIW is a recommended importance index. (7) Reliability analysis represents an NP-hard problem. When the network expands, the computational work increases exponentially. Thus, an efficient and approximate calculation method using partial minimal path sets is proposed. As the number of path sets decreases, the computational work is reduced exponentially. (8) No problem is encountered provided the order of link importance indices is unchanged when the number of path sets is reduced. A test calculation is performed.
Takahiro Nagae and Hiroshi Wakabayashi / Transportation Research Procedia 10 (2015) 155 – 165
The future work is as follows: Cumulative calculation is required to verify the advantages of CIW. It is also needed to verify the effectiveness of the approximate method, including how to choose path sets. Moreover, application to a large-scale network is required. This study greatly contributes to the national resilience in constructing a highly reliable highway network under disaster environment. References Barlow, R. E. and Proschan, F. (1965). Mathematical Theory of Reliability, p.6. John Wiley & Sons, Inc., New York. Birnbaum, Z. W. (1969) On the Importance of Different Components in Multi-Component System. Multivariate Analysis II (P. R. Krishnaiah Ed.), Academic Press, New York. Henley, E.J. and Kumamoto, H. (1981):Reliability Engineering and Risk Assessment, Prentice-Hall, Inc., 418-436. Nicholson, A. (2007). Optimising Network Terminal Reliability. Proceedings of the 3rd International Symposium on Transportation Network Reliability, CD-ROM, 2007. Nicholson, A. Schmoecker, J. Bell, M.G.H. and Iida, Y (2003). Assessing Transport Reliability: Malevolence and User Knowledge In: Michael G. H. Bell and Yasunori Iida(Ed.) The Network Reliability of Transport, Proceedings of the 1st International Symposium on Transportation Network Reliability (INSTR), pp.1-22, Pergamon, 2003. US NRC: PRA Procedures Guide: A Guide to the Performance of Probabilistic Risk Assessments for Nuclear Power Plants, NUREG/CR-2300 (1983) Wakabayashi, H. (2004). Network Reliability Improvement: Reliability Importance. Proceedings of Second International Symposium on Transport Network Reliability. University of Canterbury, Christchurch, New Zealand. Wakabayashi, H. and Fang, S. (2014). Comparison of Importance Indices for Highway Network Reliability Improvement Combined with CostBenefit Analysis. In: Jorge Freire de Sausa and Riccardo Rossi (Ed.). Computer-based Modelling and Optimization in Transportation, Advances in Intelligent Systems and Computing 262, 265-280. Springer, Switzerland. (ISBN 978-3-319-04629-7, ISSN 2194-5357, DOI: 10.1007/978-3-319-04630-3㸧(ISBN 978-3-319-04630-3 (eBook), ISSN 2194-5365 (electronic)). Wakabayashi, H. and Iida, Y. (1992). Upper and Lower Bounds of Terminal Reliability of Road Networks: An Efficient Method with Boolean algebra. Journal of Natural Disaster Science, Volume 14, No.1.
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