Reliability Based Investment Prioritization in Transportation Networks
Amir Golroo* Department of Civil Engineering University of Waterloo Ontario, Canada, N2L 3G1 Phone: 1-519-888-4567 Ext.33872 E-mail:
[email protected] Afshin Shariat Mohaymany Department of Civil Engineering Iran University of Science and Technology Tehran, Iran Phone: 98-21-73913143 E-mail:
[email protected] Mahmoud Mesbah Institute of Transport Studies Department of Civil Engineering Monash University Victoria, Australia 3800 Phone: 61-3-99054945 E-mail:
[email protected]
* Corresponding author
Submitted for publication and presentation, Transportation Research Record Committee number ADE40 TRB Committee on Critical Transportation Infrastructure Protection July 2009 Words: 4,693 +7 tables/figures =6,443
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ABSTRACT Transport networks are vulnerable to disasters which disrupts their normal function. Collapse of I-35W Bridge in Minneapolis and closure of I-65 in Kentucky due to a freight train derailment in 2007 are two recent examples which underline a need to invest on reliability of network elements. This paper discusses a network perspective for optimizing resource allocation based on reliability of transport infrastructure. For this purpose, an optimization framework is developed to efficiently assign the resources to maximize the network reliability. Two measures are identified for each link. Each link may have several states of degradation. The effect of degradation in a link on the network is also considered. As the decision variables are integer, this optimization problem is tackled by implementing an Integer Programming (IP) technique. The proposed framework is applied to an example network and numerical results are presented. The proposed methodology could successfully allocate resources for network reliability improvement.
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1. INTRODUCTION A disaster such as an earthquake, flood, or a mass car accident is an event that disrupts the everyday life of people and causes several losses of lives, properties, and lifelines (e.g. communication, pipeline, power, and transportation network). A transportation network is substantially vulnerable to damage as a result of a disaster. The disaster could be resulted in failure of bridges or closure of roads which causes interruption in the transportation network flow. After a disaster, the accessibility to the affected areas could be limited. In extreme cases some areas could be isolated in which trapped victims in rubbles may die. Over time after the disaster, the society would undergo extra amount of expenses due to network disruption (e.g. interruption in everyday businesses). In this respect, improving the transportation network reliability is an essence in order to mitigate the adverse effects of the disaster. The concept of Reliability has been widely acknowledged and employed in various research studies such as communication management and computer network design [1, 2]. In particular, for transport network reliability two types of models have been developed. The first type is considering pure networks in which only the topology of the network is characterized [3, 4]. The second type is considering a flow network that is identified by both topological structure and pattern of flow in the network. It might be required that the network is able to maintain a level of flow [5], travel time [6], capacity [7] or any other performance measure under the unexpected disruption ranging from accidents [e.g. 8] to environmental disasters. Sanso and Soumis [9] presented a general framework for assessing the reliability of flow networks such as communication, power, and transportation which more or less is the basis of most of the abovementioned studies. They suggested developing a performance measure Z such that R[Z] is the reliability measure. Since it is not practical to generate all possible states of the network after a disaster, their procedure chooses the m most probable states for which routing and values of Z are determined. These criteria are the inputs for computing the bounds on R[Z] as suggested by Li and Silvester [10]. The average of the upper bound and the lower bound is an estimate of the reliability of network (Equation 1).
R [ Z ]up =
m
∑
Pr{C k }Z (C k ) + (1 −
k =1
m
∑ Pr{C
k
}) Z (C l )
k =1
m
(1)
m
R [ Z ]down = ∑ Pr{C }Z (C ) + (1 − ∑ Pr{C k }) Z (C f ) k
k
k =1
R [Z ] =
k =1
1 ( R[ Z ]up + R[ Z ]down ) 2
l f Let Ck denote kth state of the network, and Z (C ) and Z (C ) be the network performance in the best and worst state. The best state is when all components are in a normal state and the worst state is when all components have failed.
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Identifying the performance measure for presenting the reliability measure is of great importance. Nicholson and Du [5] presented flow as a performance measure for defining reliability. They argued that a transportation network is reliable if the measure of flow of a network remains more than the pre-specified threshold. Asakura [6] suggested the travel time to be taken into account as a performance measure. He indicated that a network state is reliable if the ratio of travel time of an origin destination in an affected network over the corresponding travel time in a normal network is more than the certain threshold. He found this measure consistent with connectivity measures [4] for system reliability. Appropriate reliability models are useful as long as they can support the decision making process to make critical elements more reliable in a transportation network. This is of significance since it may save lives, for instance, if a bridge crossing the Mississippi river in Minneapolis had been adequately invested for making a more reliable bridge, it would not have collapsed, killing at least seven people. A similar investment could have reduced the 20 mile closure of I-65 in the derailment of Freight Train Q231-01 in Glencoe, Kentucky in 2007 [11]. Nojima [12] presented a prioritized resource allocation framework for upgrading the seismic reliability of a transportation network. He proposed a model that optimizes resource allocation in accordance with maximization of accessibility of specific destinations or the whole network. In particular, Basoz and Kiremidjian [13] focused on bridges as vulnerable components of a transportation network. They established seismic risk assessment of bridges which allowed for identification of the most critical components in a transportation network. They developed a method to allocate available resources to retrofit critical affected bridges in prioritized order. In another endeavor, a methodology for risk evaluation of highway systems was proposed by Shariat Mohaymany [14] which was based on the importance analysis of the system components. Accordingly, he applied his methodology to bridges by classifying them and defining various levels of damage. Moreover, Sanchez-Silva et al. [15] suggested a model for optimum assignment of resources in terms of a set of possible actions described in failure and repair rate of entire components. Although the literature in developing reliability measure is extensive, and in fact, some of this research addresses decision making tools, optimized resource allocation on the basis of a reliability measure has received little attention. The objective of this study is to develop an optimization framework to efficiently assign the resources to a transportation network to maximize its reliability. This paper is organized into four sections; after this introduction, the methodology will be presented, then numerical example will be illustrated and finally this study will be closed by conclusion.
2. METHODOLOGY In this section a formulation is presented for optimization of transportation network reliability. It is assumed that no new link will be added to the network and improvement of reliability does not significantly change travel demand. Therefore, the total travel time (T) in the normal state will remain constant before and after the investment. It should be noted that the purpose of investment on a link is to improve its reliability rather than to add a lane. Therefore, T does not change although some links in the network would be invested on. The improvements could be in many different ways. For instance, more resistant road structures, imposing more restrictive standards for buildings that may 4
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collapse in severe events, regular maintenance inspections of road furniture, trees and life lines, protection of road segments against being flooded, and constructing of precautionary facilities to enhance the road safety after a disaster. Generally, the reliability of the transport network is the result of two items: 1. The reliability of each element when isolated from the network 2. The way these elements are connected or topology of the network The first item is regarded as property of the elements, and is discussed in the next section. Element properties are the characteristic of supply only. Furthermore, supply includes the topology as the characteristics of the network. Although, it is possible to estimate the network reliability exclusively based on supply (e.g. the studies introducing connectivity reliability or accessibility reliability), demand is also considered in estimating network reliability in this study. The second item deals with the impacts of both supply (topology) and demand on reliability. In section 2.2 a measure for network reliability is discussed. Then based on this network reliability measure, an importance index is introduced in section 2.3. While reliability of the element reflects the performance of a link in isolation, the importance index represents the role of the link in the network. Section 2.4 formulates an objective function to optimize the network reliability, and to solve this prioritization problem. In section 2.5, the framework is summarized in a four-step algorithm which is suitable for computations. 2.1. Reliability of the Elements Reliability of the network is a function of the reliability of its elements. Basically, the source of risk to the network is the risks subject to network elements. In this study, it is assumed that the links of the network are at risk of capacity reduction or failure. However, the nodes are always operational. After a disaster, a link may switch to different levels of operation, called states, one of which is the failure state.
It is assumed that the capacity of the link may be reduced from c0 in the normal state to c1 to cn as a result of a disaster, where cn is the failure capacity and is equal to zero. Considering the probability of each state, the expected capacity of an element (i,j) is given by Billinton and Allan [2]: M
Eij (G ) = ∑ cijm Pijm
(2)
m =1
Where: Eij : expected capacity cijm : capacity in state m Pijm : the probability of operating at state m
It should be noted that there would be no limitation for the number of states in the present method, and all the calculations could be done for multi-state links.
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In practice, the capacities and their respective probabilities could be obtained from several sources. For example, Hazus-99 [16] has suggested a series of fragility curves for bridges. These curves could be easily converted to the expected values. Moreover, the probabilities and capacities could be estimated, using the recorded data banks of past disasters. 2.2. Evaluation of the Network Performance In the normal state when no disaster is occurred, the network can be analyzed by the traditional User Equilibrium (UE) approach. As a result, flow (x) and travel time (t) are calculated. These variables would reflect how the network would accommodate travel demand.
After a disaster, the network looses a portion of link capacities. Following the general framework suggested by Sanso and Soumis [9], as reviewed in the introduction, many performance measures were used in the literature. In this study, total travel time (T) is chosen as the performance measure. The reliability function of the network R(θT) is the probability that the ratio of travel time in the degraded network over the respective time in the normal (before incident) network remains under a pre-specified threshold level,
θT :
{
R (θ T ) = Pr (T / T (c ° )) ≤ θ T
where T (c o ), T respectively.
}
(3)
are the network total travel time before and after the incident,
One may argue that travel demand may be affected after a disaster; while the above approach implicitly assumes a constant demand. It is important to state that the performance measure derived with this approach (R) is later used to evaluate a measure of network topology (link importance). This measure indicates how many alternative routes the network offers. Nevertheless, the proposed methodology is general and independent of the performance measure. The performance measure in Eq (3) can be replaced by any performance measure reviewed in the introduction. Consequently, up to this stage, the network performance has been evaluated before and after the disaster. The aforementioned flow and travel time are the attributes of the links; however, the travel time reliability is assigned to the network. For priority formulation purposes, the network reliability should be associated to the elements of the network which is discussed in the next section. 2.3. Link Importance For asset allocation purposes, network links should be prioritized. Flow (x) and travel time (t) of the links would demonstrate the relative importance of the link in normal states. However, it is essential to use a variable to take into account the link importance after a disaster. Nojima [12] has introduced the following importance (Equation 4) in order to arrange the links by their role in the evaluation criteria.
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I ij = d ( R) / d ( Pij )
(4)
where I ij is the importance of link (i,j), R is the network reliability, and Pij is the probability by which the link is not effected by the disaster. Wakabayashi [17] argued that this index has defects such as resulting in higher importance value for the less important link in a simple parallel network. In this study, I ij is defined by Equation 5 as used by some researchers [e.g. 18]. I ij = R+ij − R−ij
(5)
Where R+ij is the reliability of the network when link (i, j) operates in its normal state (c0) and R−ij is when link (i, j) has failed (cn). 2.4. Optimization Program The required parameters to formulate the resource allocation problem have been reviewed in previous sections. Having calculated all required parameters, in this section the objective function is developed. Each of the intervening parameters influences the prioritization problem differently. First, the expected capacity (E) or link reliability is the variable to be tuned; i.e. investment would decrease the failure probabilities, and a decrease in the failure probabilities would in turn raise the expected value of a link. The formulation to be presented here aims to find the optimum level of investment for a link (i,j).
Furthermore, for a link in an urban network, the higher the flow or the travel time, the more crucial the role of the link. The product of the parameters of flow (xij) and travel time (tij) indicates the share of the link in the total travel time which is considered as a base for the network reliability. Assume a network which has several alternative paths between all OD pairs. In such a fully connected network, if a new destination with a very minor demand is connected to the network with a single link, that link would be the most important one with respect to connectivity reliability. Therefore, this link would be assigned a considerable investment. If the policy after a disaster is to maintain the network connected, investment on this minor link would be desirable. However, if the policy is to maximize the system benefit, the investment should ensure that the main corridors in the network will remain functional. For instance, if access time for emergency purposes is reduced, more lives would be saved in the network, although a few ones would be lost in a minor isolated location. It is assumed here that the second policy is followed, so the share of link in the total travel time would weight the link importance. Eventually, the importance (Iij) of the link (i,j) is included to take into account the share of link in the network reliability as the performance measure. The parameter Iij also includes the influence of topology. The resource allocation problem with respect to reliability is summarized in Equation 6.
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Max Z =
∑
K
( i , j )∈ A
xij tij I ij (∑ wijk Eijk ) k =1
S .t. K
∑w k =1
k ij
= 1 ; ∀(i, j ) ∈ A K
∑ ∑w e
( i , j )∈ A k =1
k k ij ij
(6)
≤B
wijk = 0 or 1 ; ∀k , ∀(i, j ) ∈ A Where wijk is 1 only if the investment level of k is recommended for link (i,j). The cost of such investment is assumed to be eijk and K is the number of investment levels. The sum of investments has to be less than or equal to the budget (B) as indicated in the first constraint. Moreover, it is evident that just one action should be chosen for investment on link (i,j). The second constraint guarantees this condition. The above problem includes an integer variable that makes the solution computationally costly. This problem can be solved using deterministic and heuristic methods in operational research [19, 20]. Branch and Bound method is employed herein since it is guaranteed to find the optimal answer in an integer program.
2.5. Prioritization Algorithm The following step by step algorithm is suggested on the basis of the proposed method to find the optimum resource allocation in the network. Step 1: Find the expected value (Eij1) for the present conditions of all links (i, j ) ∈ A . Also, calculate the possible upgraded Eijk after investment. Step 2: Find the m most probable states for the whole network (R) and for the network assuming each link is once functional (Rij) and once failed (R-ij). Step 3: Run an assignment for all the set of most probable states. Find total travel time and then find reliability (R, Rij, R-ij) and link importance (I). Step 4: Solve the optimization problem of Equation 6 to find wijk. The above steps are implemented in an example in section 3 and detailed results are presented.
3. NUMERICAL EXAMPLE The optimal resource allocation problem in the transportation network is represented here in an example network, the schematic layout of which is illustrated in Figure 1. The network consists of six nodes and seven links. Four levels of performance are assumed for the links, in which level D corresponds to the lowest or poorest performance and level A to the highest or best performance. Before the investment, all the links are in the lowest level of performance. It is supposed that seven mega-unit of currency have been decided to be invested on the example network upgrading, and that each one mega-unit can upgrade the performance level of each seven links for one level. Travel demand is from
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O1 to D1 (900 veh/hr) and from O2 to D2 (800 veh/hr). The capacity, initial and calculated travel time, and flow of the network links are given in Table 1. 1
O1
3
D1
6
D2
4
2
O2
5
Figure 1: Schematic presentation of the network model Table 1: Initial and calculated characteristics of the network links Link No.
Capacity
(1,5) (1,3) (2,3) (2,6) (3,4) (4,5) (4,6)
700 700 700 700 700 700 700
Travel Time Initial 3 1 1 3 1 1 1
Flow
UE 3.14 1.01 1.00 3.13 1.12 1.01 1.00
527 373 289 511 662 373 289
The cost function of Bureau of Public Roads (BPR) is used as follows: v (7) t = t 0 (1 + α ( ) β ) c where t0 denotes initial travel time, and it is supposed that α is 0.15 and β is 4, respectively. Also, v and c represent flow and capacity as were defined before. The traffic assignment is fulfilled by Frank-Wolfe method [21]. In this example three (discrete) failure states are considered for each link: normal, degraded, and failed states (Table 2). It is assumed that in the normal state, a link could be used in its whole capacity, but a degraded element just is used in half of its capacity, and, finally, the failed component has not any capacity to pass the flow. Table 2: Stability probability levels of the network links Investment Level
Normal (c0)
Degraded (c1)
A B C D
0.95 0.90 0.85 0.80
0.04 0.08 0.12 0.16
Failed
(c2)
0.01 0.02 0.03 0.04
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The solution will be presented separately, in the steps which were indicated in section 2.5. STEP 1. Reliability of elements Using Equation 2, the reliability of links is calculated for each level of performance. Reliabilities are equal to 616, 637, 658, and 679 veh/hr for performance levels of 1 to 4, respectively. STEP 2. Evaluation of the network performance A total of 239 network states were produced using Chiou and Li [22] method. These states would cover 95.47 per cent of all probable states (37 = 2187 states). A sample of these states and their occurrence probability is represented in Table 3. In order to calculate the link importance by Equation 5, the network reliability in two conditions is needed: network with and without the link. It is assumed that, in the first condition, the link is always functional while in the second condition it is failed. So, for each condition, the same table as Table 3 is calculated to derive the m most probable states. STEP 3. Link importance When the most probable states are determined, total travel time for each state is found as illustrated in Table 3. Each state is accepted, provided that it satisfies Equation 3 with θ = 1.5 . If it is accepted, a value of 1 is attributed to that state and 0 otherwise. Then, applying Li & Silvester [10] method, the network travel time reliability is derived (Equation 1). After that, the link importance is calculated using Equation 5. The results of step 3 are summarized in Table 4. Note that reliability of the network (R) is always smaller than the reliability of the network assuming link (i, j) is functional and larger than the reliability of the network when link (i, j) is assumed failed. It is also evident from the table that links (1,5) and (2,6) are the most important links in the network. This can also be verified schematically from Fig 1 as they directly connect the OD pairs.
Table 3: The most probable states of the network
link failure sate Network State (1,5)
(1,3)
(2,3)
(2,6)
(3,4)
(4,5)
(4,6)
Prob.
Total performance Prob. of Reliable Travel θ Time measure =1.5 States
1 2 3
1 2 1
1 1 2
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
20.97 4.19 4.19
5332 5607 5377
1 1 1
20.97 4.19 4.19
239
1
1
1
1
1
3
3
0.05
6433
1
0.05
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Table 4: Link importance and network reliability Link No. (1,5) (1,3) (2,3) (2,6) (3,4) (4,5) (4,6)
Reliability Network R+15 R-15 R13 R-13 R23 R-23 R26 R-26 R34 R-34 R45 R-45 R46 R-46
Number of States 239 153 62 153 62 153 62 153 62 153 62 153 62 153 62
Lower Bound 90.20 94.58 48.23 92.39 73.98 92.39 73.14 94.58 48.23 93.45 62.29 92.39 73.35 92.39 73.77
Coverage 95.47 97.45 90.18 97.45 90.18 97.45 90.18 97.45 90.18 97.45 90.18 97.45 90.18 97.45 90.18
Upper Bound 94.73 97.13 58.06 94.94 83.80 94.94 82.96 97.13 58.06 96.00 72.11 94.94 83.17 94.94 83.59
Reliability
Importance
92.46 95.86 53.15 93.67 78.89 93.67 78.05 95.86 53.15 94.73 67.20 93.67 78.26 93.67 78.68
--42.71 14.78 15.62 42.71 27.53 15.41 14.99
Step 4. The objective function Ultimately, after calculating reliability of elements, flow and travel time, and link importance, the optimization problem can be formulated to find the investment levels on each link (wijk). To solve the optimization problem, Integer Programming (IP) is employed and the results of the allocated money units and final link levels are shown in Table 5. The above steps are coded in a computer program to facilitate running for various examples. Table 5: Optimum investment prioritization on network links Link (i,j)
(1,5)
(1,3)
(2,3)
(2,6)
(3,4)
(4,5)
(4,6)
Investment Cost (eij) Final Link Level (k; wijk=1)
3 A
0 D
0 D
3 A
1 C
0 D
0 D
The solution of IP shows that links (1,5) and (2,6) are the most critical ones to which the maximum re-enforcement concerns are dedicated. So, links (1,5) and (2,6) were upgraded to the highest or the best performance (level A). It should be noted that the largest amount of money which can be spent for each link is 3 units. After those links, link (3,4) is the most critical element in the network. Thus, it could absorb one money unit form the budget. Sensitivity analysis is, also, proposed herein as an adequate tool to express the importance of performance of each link in the overall network reliability. To perform sensitivity analysis, @Risk software (2004) has been utilized to be able to deal with the probabilistic nature of link performance (i.e. levels of failure). This tool provides priority of links in terms of their effect on the network reliability. That is, the first link has the most the significant impact on the network reliability while the last one has the lowest impact (Figure 2).
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Regression Sensitivity for Objective Function (Z) Link (1,5) ==>/C19
0.704
Link (2,6) ==>/F19
0.697
0.202
Link (3,4) ==>/G19
Link (4,5) ==>/H19
0.059
Link (1,3) ==>/D19
0.056
Link (2,3) ==>/E19
0.045
Link (4,6) ==>/I19
0.044
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Figure 2: Sensitivity analysis of network reliability
This graph clearly indicates that the results of both methods are identical. That is, the most critical links, regarding the sensitivity analysis graph, are (1,5) and (2,6) which should be upgraded to the highest link performance. The IP approach has also assigned 3 money units to these links.
4. CONCLUSION Resource allocation is a challenging task in improving the reliability of transport networks that has not been paid enough attention. Many researchers have focused on algorithms to estimate the reliability of the transport networks. Some others have addressed the investment prioritization techniques to find the solution for optimization problems. This paper has proposed the combined framework that efficiently allocates the pre-specified budget to network links in a way that reliability is maximized. The presented step by step algorithm considers probability distributions and travel time to derive reliability of links and network reliability, respectively. The problem formulation includes factors of link importance so that links could be prioritized. The conclusions can be summarized as follows: 1. The optimization problem (i.e., resource allocation) is solved efficiently by applying integer programming: Branch and Bound method. 2. The simulation technique as a sensitivity analysis tool successfully provides priority of links in terms of their effect on the network reliability. 3. The results achieved using both approaches (i.e., integer programming and simulation technique) are consistent.
5. REFERENCES 1.
Sandler, G.H., System reliability engineering. Prentice-Hall international series in space technology. 1963, Englewood Cliffs, N.J.: Prentice-Hall. 12
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2. 3. 4. 5. 6. 7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Billinton, R. and R.N. Allan, Reliability Evaluation of Engineering Systems, Concepts and Techniques. Second ed. 1992: Plenum Press. Bell, M.G.H. and Y. Iida, Transportation network analysis. 1997, Chichester: John Wiley and Sons. ix, 216. Iida, Y. and H. Wakabayashi, An approximation method of terminal reliability of road network using partial minimal path and cut set, in Fifth WCTR 1989: Yokohama. p. 367-380. Nicholson, A. and Z.-P. Du, Degradable transportation systems: an integrated equilibrium model. Transportation Research, Part B (Methodological), 1997. 31B(3): p. 209-23. Asakura, Y. Reliability Measures of an Origin and Destination Pair in Deteriorated Road Network With Variable Flow. in 4th EURO Transportation Meeting. 1996. Newcastle: Pergamon. Chen, A., et al., Capacity reliability of a road network: An assessment methodology and numerical results. Transportation Research Part B: Methodological, 2002. 36(3): p. 225-252. Shariat Mohaymany, A. and M. Mesbah. A New Approach for Reliability Assessment of Urban Transportation Networks. in Applications of Advanced Technology in Transportation. Proceedings of the Ninth International Conference. 2006: American Society of Civil Engineers. Sanso, B. and F. Soumis, Communication and transportation network reliability using routing models. Reliability, IEEE Transactions on, 1991. 40(1): p. 29-38. Li, V. and J. Silvester, Performance Analysis of Networks with Unreliable Components. Communications, IEEE Transactions on [legacy, pre - 1988], 1984. 32(10): p. 1105-1110. Infoplease. "2007 Disasters." 2009 [cited 2009 Jun 26]; Available from: http://www.infoplease.com/ipa/A0934966.html. Nojima, N., Performance-based prioritization for upgrading seismic reliability of a transportation network. Journal of Natural Disaster Science, 1999. 20(2): p. 5766. Basoez, N. and A.S. Kiremidjian. Bridge prioritization for emergency responses. 1995. San Francisco, CA, USA: ASCE. Shariat Mohaymany, A., Evaluation of Transportation Lifelines Subjected to Earthquake, in Civil Engineering. 2001, Iran University of Science and Technology (IUST): Tehran. Sanchez-Silva, M., et al., A transport network reliability model for the efficient assignment of resources. Transportation Research Part B: Methodological, 2005. 39(1): p. 47-63. Hazus-99, Earthquake Loss Estimation Technical Manual. 1999, Washington DC: National Institute of Building Science. Wakabayashi, H. Network Reliability Improvement: Probability Importance and Criticality Importance. in The Second International Symposium on Transportation Network Reliability (INSTR). 2004. Christchurch, New Zealand. Poorzahedy, H. and S.N. Shetab Bushehri, Network performance improvement under stochastic events with long-term effects. Transportation, 2005. 32(1): p. 6585.
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19. 20. 21. 22.
Winston, W.L. and J.B. Goldberg, Operations research : applications and algorithms. 4th ed. 2004, Southbank, Vic.: Thomson Brooks/Cole. xvi, 1418 +. Taplin, J.H.E., Cost-benefit analysis and evolutionary computing : optimal scheduling of interactive road projects. Transport economics, management, and policy. 2005, Cheltenham, UK ; Northampton, MA, USA: E. Elgar Pub. xii, 215. Ortúzar, J.d.D. and L.G. Willumsen, Modelling transport. 3rd ed. 2001, Chichester New York: J. Wiley. xiii, 499. Chiou, S. and V. Li, Reliability Analysis of a Communication Network with Multimode Components. Selected Areas in Communications, IEEE Journal on, 1986. 4(7): p. 1156-1161.
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