INTRODUCTION
One of the most important socialeconomic development indices of
every city or country is its transportation system and its functionality
and accessibility. After every disastrous event, such as a hurricane,
an earthquake, or a mass car accident, in the transportation network in
a large and populated city the network capacity will considerably decrease
at local or global level depending on the scale or size of the disaster.
These stochastic events seriously may damage transportation infrastructure,
such as roads and bridges and have longterm effects upon the transportation
system. Undoubtedly, they would require days, weeks and months of hard
work to return the damaged parts of the system to functioning. One type
of such system which is highly vulnerable to damages would be urban transportation
network. Regarding the damages occurred in various part of street network,
for one thing and lack of alternative routes, for another, travelers pose
an increased travel time and cost. This makes upgrading transportation
components in the priority order against the disaster a tremendously vital
problem (Dicleli and Mansour, 2003).
Transportation network components prioritization techniques: The
concept of prioritization has been used by some researchers since early
90`s. Ranf and Eberhard (2007) developed a prioritization scheme after
major bridge damage to inspect the bridges in order to conduct a retrofit
operation. The prioritization strategy was developed incorporating ShakeMaps,
the bridge inventory and newly developed fragility curves. This strategy
might recognize 80% of the moderately damaged bridges for the Nisqually
earthquake by only inspecting 481 (14%) of the 3,407 bridges within the
boundaries of the ShakeMap. However, the conventional method would require
1,447 (42%) bridges to be inspected.
Zonta et al. (2007) developed a bridge prioritization plan for
the Autonomous Province of Trento, Italy based on reliability concepts
as a part of a Bridge Management System (BMS). The reliability of bridges
was conservatively estimated for each bridge. If the condition of bridge
was critical its reliability was investigated in a more formal manner
using multistep procedures. In order to prioritize the retrofit actions
within a certain budget, they assigned the first rank to those actions
that would minimize the risk of occurrence of an unacceptable event in
the whole network.
Accessibility of transportation networks and economical approaches:
Recently the accessibility and costbenefit concepts have been also employed
by researchers who have worked on transportation systems in disasters.
Sohn (2006) evaluated the importance of highway network components in
Maryland under flood damage. The author applied an accessibility index
to incorporate both the distancedecay effect and the traffic volume influence
on the transportation network. A hypothetical disruption of individual
links within the floodplain was assumed and the accessibility level of
each county and the state were checked before and after the disruption.
The investigator reported that the results indicated critical links which
were specified based on the distanceonly. The distancetraffic volume
criteria appeared to be different which resulted in a different prioritization
scheme for retrofit actions depending on what criterion to choose. For
instance, the percentage loss of accessibility due to the disruption of
a link was generally greater in terms of the distancetraffic volume criteria.
It is concluded that some links would be significant in both cases, particularly,
if a specific link did not have an alternative route (e.g., bypass). The
two criteria might obtain a similar outcome if counties connected by the
link were low accessibility counties.
Chang and Nojima (2001) evaluated performance of the urban rail and highway
transportation systems in terms of network coverage and transport accessibility
in the Kobe, Japan, the region devastated by the 1995 HyogokenNanbu earthquake.
The authors reported that the performance degradation was much more critical
for highways and railways than for other lifeline infrastructure systems.
Service restoration proceeded much faster for rail. The restoration of
highway system performance was highly dependent on recovery of highway
traffic volumes. The study finally measured the subarea transport accessibility
and applied this to Kobe`s constituent city wards.
The costbenefit concept was used in earthquake disaster mitigation for
urban transportation systems as an integrated methodology built on the
Midwest states (Sohn et al., 2003). The authors considered two
aspects of cost: final demand loss and transport cost increase. They applied
1812 New Madrid earthquake to develop a scenario for the analysis. The
modeling system included a transportation network loss function, a final
demand loss function and an integrated commodity flow model. The investigators
indicated the most important link on the network in an economic sense
as well as the link with the greatest physical disruption by running the
earthquake scenario. The study concluded that the links with greater physical
disruption would not be always the ones addressing the greater economic
damage. Moreover, a costbenefit analysis was carried. A decisionmaking
process on the optimal retrofit priority of bridges and links on the transportation
network was, also, executed.
Transportation network functionality: For conducting a prioritization
scheme, it is necessary to evaluate the function of the network aftermath of
a disaster, which can be performed based on defining some disaster scenarios
for the probable future events in the corresponding area. Several researchers
have worked on the evaluation of the transportation system functionality (Nicholson
and Du, 1997). Some investigators have looked at the problem from a different
prospective which was a reliability analysis of the system components (Keller,
2002). The stability of the network and the components, particularly bridges,
has been also studied by SanchezSilva and Daniels (2004). In recent years,
the probabilistic vulnerability of the network has been paid more attention.
Several techniques such as graph theoretic, simulation and optimizationbased
techniques have played a significant role in examining potential network vulnerabilities
which result in mitigating facility loss and prioritizing retrofit efforts (Grubesic
et al., 2008).
Risk analysis of transportation network: Some researchers have
taken a particular look at the evaluation and risk analysis of the transportation
system exposed to seismic damage (D`Andrea et al., 2005) and proposed
a methodology for evaluation of seismic risk of road infrastructures according
to study of seismic hazard, investigation of the probability of presence
of road users in various regions, analysis of distribution of the population
and the infrastructures, evaluation of the functional vulnerability with
respect to the potential replaceability of damaged assets and assessment
of structural vulnerability of the assets corresponded to the characteristics
of the different components.
Pitilakis et al. (2007) proposed the RISKUE methodology for the
risk analysis of lifelines with particular emphasis on transportation
networks. The presented methodology might reduce the consequences of lifeline
damage in urban areas and provide an efficient mitigation strategy and
prioritization policies for preearthquake and post earthquake actions.
In order to develop the methodology, the author incorporated a detailed
inventory for every component at risk along with a reliable seismic hazard
assessment, suitable selection of fragility models, estimation of the
"global value" and economical influence of lifeline damage and losses.
Research need: It has been revealed that in spite of several researches
on the investment prioritization, still few works have been conducted
on the efficiency of the proposed prioritization techniques. This study
aims to specify the vulnerable components of the system on the one hand
by using their fragility curves (probability distribution functions) and
to evaluate the functionality of the network on the other hand by defining
various disaster scenarios. The basic idea is to use the risk analysis
approach in combination with investment prioritization and it is believed
that by this approach the reliability of the whole transportation network
can be efficiently increased. The capability of the proposed approach
is demonstrated by solving a numerical example analyzed by a program developed
by the authors. In the following sections of the paper, firstly, the basic
concepts and definitions are presented. Secondly, the network modeling
and related formulation for investment prioritization are expressed. Then,
the efficiency of the proposed method is shown by a numerical example.
Finally, the obtained results are discussed.
THE BASIC CONCEPTS
For evaluation of the transportation network subjected to a disastrous
event the problem is dealing with a network which has some extent of uncertainty
in its various components. Therefore, the prioritized investment process
for upgrading the functionality of the network subjected to a given disaster
should be based on probability theory. On this basis, for prioritization
of the investment various techniques used in the risk analysis can be
employed. These include (Bell, 2000; Beale and Horard, 1995):
• 
The Markov chain 
• 
The fuzzy theory 
• 
The game theory 
• 
The decision tree 
• 
The dynamic programming 
• 
The simulation techniques 
Some of these methods could not be used as a direct method for this problem
regarding its optimization nature (e.g., Decision Tree). Also, Most of
these procedures (except simulation techniques) may represent a constant
value as a network evaluation measure, while according to the probability
nature of the problem; the probability distribution function should be
presented as an evaluation measure. Simulation method would exhibit a
distribution of relevant result which is compatible with problem entity
although it is needed to consider a large number of probable network states
in disaster period and simulation method could not reduce this huge number
of states. Developing a procedure which can decrease the number of states
(e.g., dynamic programming method), firstly and would demonstrate an evaluation
measure as a probability distribution function (simulation techniques),
secondly, might be the solution.
Dynamic programming is an optimization method which can directly give
the best state from some possible states of investment by minimizing the
cost and maximizing the performance of the system, so it can be very useful
for the purpose of this study. On the other hand, simulation techniques,
such as Monte Carlo or LHS (Latin Hypercube Simulation), are very effective
tools for making virtual models of the system without any need to physical
experiments. These techniques have been used in risk analysis problems
by Chen et al. (2002) and can be used in this study as well for
prediction of the network performance in the uncertain conditions like
disastrous situations.
On this basis, the use of dynamic programming in combination with simulation
techniques seems to provide an efficient method for investment prioritization,
which can be conducted in the following stages:
• 
Identification of the network, its components and their
fragility curves 
• 
Defining the disaster scenario based on its intensity and occurrence
time 
• 
Predicting the states of the network and its critical components
based on the defined scenario and the employed fragility curves 
• 
Evaluating the network functionality by appropriate indices (total
travel time) 
• 
Prioritizing the mitigation investment by considering the functionality
of the network critical component 
THE NETWORK MODELING AND EVALUATION FORMULATION
For each component of the network a specific fragility curve (probability
distribution function) can be assumed in the case of any given disaster.
Each component k can have a probabilistic stability function P^{k}
which is affected by the investment amount and the type of the event,
that is:
where, Inv^{k} is the amount of investment for the component
k and Int^{k} is the intensity of the disaster in the location
of component. It is possible that a driver face during his/her travel
a component which seems impassible because of the event or congestion.
Passengers would behave in two different ways. Firstly, by probability
of μ he/she may return back to a location in the route, where it
is possible to choose another path. Obviously, this will cause some delay
in the travel. So, a delay coefficient β can be defined for every
path. After returning to a previous location for choosing another path
the driver will choose one of other existing paths. On this basis the
average travel time for these paths (t_{0ij}) in ij OD pair can
be defined as:
In which is the travel time for the path r (of the paths remained intact)
in ij OD pair, let b the number of intact paths and γ is the travel
time increase coefficient, which shows the effect of the transferred travels
to the path r because of the impassibility of some other paths.
Secondly, he/she would make a detour by probability of (1μ) to
get to proposed destination. For sake of presentation simplicity the road
network is assumed (Fig. 1). Generally, in the road
network, drivers have more information about various routes (e.g., detour
which has been shown on Fig. 1 by dash line). Namely,
there are, usually, specific obvious sign for detour between to nodes
in road network to guide passengers in emergency situations. Therefore,
it is reasonably supposed that just one forth of drivers may not use the
detour to access to their destination. It should be noted that detours
have been exploited, only, when the main roads have been failed.
On the other hand, if all paths are blocked because of the high extent
of the event all drivers have to wait till at least one path is opened.
This delay time is shown by D_{ij}, which is:
where, m is the total number of paths in every ij origin and destination
(OD) pair, the θ is the travel time increase coefficient which indicates
disutility for travelers in case of all paths blocked in one OD pair.
For travel assignment the Logit formula can be used, which is:
where, is the choice probability of the path r from all available paths
between OD pair, considering the minimum travel time as the choosing
measure. If the traffic volume in OD pair is V_{ij} the volume
assigned to the path is given by:

Fig. 1: 
Schematic presentation of the network model 
Now the network evaluation measure, which is the total travel time for
all paths in a OD pair, can be calculated by:
In Eq. 6 m is the number of paths, is volume assigned to the path r,
n is the number of components existing in every OD pair, is
the travel time for the link k, α is the travel time increase coefficient,
μ is the probability of rerouting, β denote the delay increase
coefficient, is
the travel time of links which are used twice because of the end blockage,
(1μ) is the probability of making detour, λ is the increase
coefficient which is embedded to the
travel time of links which are failed, to produce the detour travel time
(it is supposed that each link has one at least one detour), let denote
the probability of failure of path r in ij OD pair and t_{0ij},
and
D_{ij} have been calculated by Eq. 2, 5 and 3, respectively. Then
the total travel time of all paths reaching destination j can be calculated
by adding all corresponding T_{ij}. Finally, after calculating
the travel time for each destination node of the network in order to obtain
the Total Travel Time (TTT) of the entire network, it is necessary to
assign appropriate importance factors (IF_{j}) which are affected
by accessibility to other nodes, center population, connection to emergency
and rescue and relief centre to all nodes and then add up all the factored
travel times. It has been assumed that IF_{4} and IF_{6}
would be 0.4 and 0.6, respectively.
This model has some new advantages on which has been paid little attention,
as follows:
• 
The multi OD pairs are considered to calculate the
network evaluation measure 
• 
The stability function is denoted as probability distribution instead
of uniform value 
• 
Considering the end blockage of damaged component, rerouting and
using detours 
• 
Exclude delays in severe disaster which makes all routes in one
OD pair impassible 
• 
Feasibility of evaluation of importance of information dissemination
about failed links 
USING THE MODEL IN INVESTMENT PRIORITIZATION
An illustrative sample of the network is used for the vulnerability
analysis of the transportation network and then for prediction of the
direct economic losses and travel delays, as well as estimation the financial
resources required for its retrofit is shown in Fig. 1.
In this example the network consists of eight nodes, 11 links and four
OD pairs (O_{1}D_{1}, O_{1}D_{2}, O_{2}D_{1}
and O_{2}D_{2}).
In this study seven levels of performance have been assumed for the network
components, in which level 1 corresponds to the lowest or poorest performance
and level 7 to highest or the best performance. It is assumed that 10
megaunits of currency have been decided to be invested on the example
network upgrading and that each 2 megaunits can upgrade the performance
level of each components of the network one level. The initial performance
level and the average travel time of the network components have been
shown in Table 1, also it has been assumed that 1200
and 500 veh h^{1} is allocated to O_{1}D_{1}
and O_{1}D_{2}, respectively and, simultaneously, 300
and 1000 veh h^{1} is dedicated to O_{2}D_{1}
and O_{2}D_{2}, respectively.
Using dynamic programming method: As it has been mentioned, dynamic
programming is an efficient method for Linear Programming, which can be
used of resource allocation. The authors have developed a program which
gives the optimum amount of investment for each component of the network,
provided that the network performance evaluation measure is specified,
which is in this paper the total travel time in the network. It may be
understood that link 8 is the most critical and effective component in
the network performance (Table 2).
Table 1: 
The specification of the example network 

Table 2: 
Optimum amount of investment on network components by
using dynamic programming 


Fig. 2: 
The Tornado graph for sensitivity analysis 
Using simulation technique: The simulation technique has been
employed by using at the rate of Risk program. This program is capable
of simulation in both Monte Carlo and LHS sampling methods. In both of
these methods various states of network regarding components failure distribution
would be observed in order to develop a probability distribution of network
measure. To find the investment prioritization pattern the step by step
simulation method is employed, so that after each step of simulation a
sensitivity analysis is performed and the most critical component is determined
by a Tornado graph as shown in Fig. 2, which shows that
in the first step link 8 is the most critical one. Therefore, it catches
the first 2 megaunits.
The calculation is repeated till all 10 megaunits of the investment
is utilized. It is noteworthy that the pattern of assigned investment
to the components is the same for both of the sampling methods. Also,
the network measure calculated by LHS technique is more appropriate (roughly
0.5%) than that of Monte Carlo which is not meaningful (Table
3).
Table 3: 
The Results of investment prioritization by Monte Carlo
and LHS techniques 

RESULTS AND DISCUSSION
The effectiveness of dynamic programming and simulation in resource
allocation: After the determination of two mentioned methods for resource
allocation among the other risk analysis methods, in order to improve
the efficiency of investment prioritization procedure of these two techniques,
the various states of founded network would be compared by their network
measure.
It has been shown in Table 4 that the prioritized investment
in both dynamic programming and simulation techniques results in much
more improve in the network performance evaluation measure in comparison
with uniform investment state (almost 7%) and no investment state (roughly
19%) which undoubtedly represent the high efficiency of investment prioritization
in increasing the network reliability by these two techniques. In addition,
Table 4 compares the results of two methods according
to various states of investments.
Comparing the results of dynamic programming and simulation: As
it has been implied, the dynamic programming and simulation methods would
be the most appropriate solution for resource allocation, but now it should
be discussed that which of them is more appropriate. Dynamic programming
could be a powerful resource allocation method to reduce the numerous
amounts of network failure states. Furthermore, it might be exploited
to present a reasonable investment prioritization pattern, however the
constant output (network measure) specified by this method is not match
with probabilistic nature of this study. On the other hand, simulation
technique is more appropriate for analysis of damaged network because
of its probabilistic nature. In addition, the simulation technique can
result in a probability distribution function as the output (Fig.
3) which would be more compatible with the probabilistic network evaluation
measure entity, since the plenty number of network states may not be decreased
by this method.
As a result, it has been indicated that each method has some advantages
and a few disadvantages, due to omit the disadvantages the combined method
shall be specified. The simulation technique should be used because of
its probabilistic nature and the resource allocation pattern has to be
obtained from dynamic programming in case of network states reduction
and strong matching of this method with optimization problem.
Table 4: 
The results of network analysis with various states
of investment by various technique 


Fig. 3: 
Distribution function of the network evaluation measure
by simulation method (LHS) 
In fact, the
combined simulation and dynamic programming method is theoretically more
reasonable than step by step simulation and dynamic programming solely.
Arbitrary in aforementioned illustrative sample, the results of both
methods have been identical. Therefore, the result of combined method
would be the same as simulation method on account of similar resource
allocation patterns. But, generally, the result of combined method would
be different from achieved output of simulation or DP methods. Indeed,
since associated advantages and disadvantages, the combined method would
be completely preferred.
To be more accurate, after developing the new combined simulation and
dynamic programming technique, the optimum sampling method should be demonstrated.
Table 3 obviously shows that the number of iterations
for achieving 1.5% of convergence by LHS technique is around 20% less
than the Monte Carlo technique. Therefore, LHS sampling method would be
the efficient technique.
Table 5: 
The results of network analysis for various states of
investment with and without information dissemination by different
methods 

The effect of information dissemination on the network evaluation
measure: To avoid the travels in the endblocked paths in the network,
which leads to time wasting and therefore, to increase in the total travel
time of the network, it is possible to inform the drivers about the situation
of various paths, specially the endblocked ones. To find out how much
this information dissemination affects the network performance evaluation
measure the same three states of investment have been analyzed again by
both the dynamic programming and the simulation techniques and the results
are shown in Table 5. In so doing, the delay time which
is obtained due to rerouting in each OD pairs would be omitted from
proposed model (Eq. 6).
As it is shown in Table 5, informing drivers about
the damaged or endblocked paths increases in average the network performance
evaluation measure around 20% in the case of uniform investment and almost
23% in the case of prioritized investment. This leads to decrease in travel
time, reduce in travel costs and increase in reliability of the network
as well as increase in ease and satisfaction. Therefore, information dissemination
has a very great role in the improvement of the network performance in
the case of disastrous events. Table 5 compares the
network evaluation measure without and with informing in the case of applying
the Dynamic Programming (DP).
CONCLUSION
Based on the numerical results it can be said that:
• 
For one thing, by using prioritized assigned investment
pattern determined by dynamic programming technique as the initial
values the simulation technique gives a suitable probability distribution
function as an output. On this basis it can be said that this combined
technique based on LHS sampling method is the suitable way for solving
the investment prioritization problems in degradable transportation
network which has not been taken into consideration by other researcher
yet. 
• 
For another, prioritized investment in both dynamic programming
and simulation techniques results in much more improve in the network
performance evaluation measure, where the simulation technique gives
better results to some extent. 
• 
Ultimately, information dissemination has a very great role in the
improvement of the network performance in the case of disastrous events.

• 
To assure that the proposed approach is efficient in the case of
large networks and various disastrous situations and difference fragility
conditions of the network component performance some further studies
are required, which are now being undertaken by the authors. 