AN APPROXIMATE RELIABILITY EVALUATION METHOD FOR IMPROVING TRANSPORTATION NETWORK PERFORMANCE

Considering the importance of maintaining network performance at desired levels under uncertainty, network reliability, as a new approach to assessing the performance of degradable urban transportation networks, has become increasingly developed in two recent decades. In this paper, a method for optimizing resource allocation to meet the required levels of transportation network reliability is proposed. !e worked out method consists of two stages: at stage one, a method for computing the reliability of network connectivity based on the reliability of computing arc performance with an assumption that capacities are random variables for each arc is presented. !ese random variables are assumed to be conformed to especial probability density functions which can be modi#ed through investing to improve the performance reliability of the arcs. At stage two, a mixed integer nonlinear programming model is developed to optimize resource allocation in the network. Numerical results are also provided in a simple network to demonstrate the capability of the employed method.


Introduction
Economic issues of a city or region necessitate a transportation system to be e cient, safe and reliable to provide an accessible network for trip-makers. In order to achieve such expected goals in a network, engineers need powerful scienti c tools whereby they can improve network components with appropriate performance Bogdevičius 2007, 2009;Daunoras et al. 2008;Basu and Maitra 2007;Beasley and Christo des 1997, etc.).
Under the present conditions, the level of service (LOS) is the most common measure for assessing the performance of a road segment de ned based on the density of vehicles per mile per lane (Highway Capacity Manual 2000). On a large spatial scale, vehicle-miles travelled (VMT), vehicle-hours travelled (VHT) and total delay can be used to evaluate the performance of a segment or an entire freeway system (Chen et al. 2003). When designing a network is considered under uncertain conditions, we can hardly use aforementioned performance measures; consequently, network performance reliability has to be evaluated. For example, an earthquake pertaining to its destructive power is a probabilistic event and we are not able to predict if it occurs and how much intensity it will have. erefore, in degradable networks dealing with uncertain conditions, it is necessary to employ such e ective methods that can consider uncertainty associated with their evaluation parameters.
In this paper, for analyzing network performance reliability, instead of using the presumed arc performance reliability applied in some previous researches, arc capacity functions have been used; therefore, arc performance reliability is computed during network analysis based on a more real assumption. In other words, by means of this method, it would be possible to use the actual quantity of arc performance reliability in any special condition for solving the problem (e. g. considering the expected LOS and arc ow volumes) rather than using the presumed values for arc performance reliability.
is paper addresses the problem of investing in a degradable transportation network in order that when dealing with uncertain conditions, the network operates appropriately and can maintain performance index at a desired level. e proposed method uses the performance functions of arcs that are a probabilistic distribution of capacities. e method also has two main aspects: rst, a method for assessing connectivity reliability is presented and second, a mixed-integer nonlinear model is developed for optimizing the allocation of resources, so as network performance reliability can be maximized for di erent levels of available funds. Some existing studies on road network reliability analysis along with the de nitions of di erent aspects of transportation network reliability will be presented in the Section 2 of this paper. Section 3 presents a new framework for reliability analysis consisting of arc capacity probability distributions, arc reliability functions, capacity reliability and connectivity reliability. Section 4 discusses investment functions applied to modify arc reliabilities. e optimization programming model for resource allocation is extended in Section 5. Section 6 applies the proposed method to a numerical example.
e Section 6 provides a conclusion and identi es directions for future research.

Literature Review
e majority of researches have been done on transportation network reliability focusing on reliability assessment. On the contrary, no great attention has been paid to reliability optimization problems encountered in this eld.
However, research in the area of transportation hazards generally aids governments in allocating limited resources to the four phases of risk management: mitigation, preparedness, response and recovery (Kutz 2003). e present investigation can be related to those research works in which preparedness or recovery is under consideration.
Reliability is de ned as the ability of an item to perform a required function under given environmental and operational conditions and for a stated period of time (Rausand and Høyland 2003). Unfortunately, the reliability analysis of road networks, compared with some other systems (like electric power systems, water distribution systems and communication networks), have received little attention in spite of its importance. However, the existing reliability studies on road networks mainly contain four aspects: travel time reliability, connectivity reliability, capacity reliability and performance reliability.

Connectivity Reliability
Possibly, the major impulse for establishing serious researches on road network reliability was natural disasters (e. g. earthquake) that can severely disrupt the network by disconnecting paths. Hence, connectivity reliability was the rst measure of performance reliability taken into account to evaluate the performance of degradable transportation networks. Connectivity reliability considers the probability that a pair of nodes in a network remains connected. is measure, in other words, is similar to the reliability measure of communication networks commonly called 'Two-terminal Reliability' (e. g. Grosh 1989). A special case of this index is terminal reliability that is concerned with the existence of at least one path between each origin-destination (OD) pair (Iida and Wakabayashi 1989). Some methods for analyzing the connectivity reliability of transport networks could be found in the previous works like Asakura (1999); Wakabayashi and Iida (1992); Bell and Iida (1997); Asakura et al. (2001); Sumalee and Watling (2003); Du and Nicholson (1997). All of these studies assume links to have a probabilistic and binary state. e state of a link is shown by an integer variable equal to 1 if the link operates normally and 0 when it fails. Note that the states of a link may be considered as different de nitions determined by the planner.
As it will be seen, this paper presents a new approach for computing the states of arcs, whereby the performance reliability of an arc is de ned considering the probability density function of its capacity and especial v C ratio as the level of service (LOS). Having the performance reliability of the arcs, the reliability of network connectivity can be computed through a closed formula. Furthermore, since travel time is a function of v C ratio, there may be an interrelation between connectivity reliability obtained applying this method and travel time reliability.

Travel Time Reliability
Travel time reliability considers the probability that a trip between a given origin-destination (OD) pair can be completed successfully within a speci ed time interval (Asakura and Kashiwadani 1991;Bell et al. 1999).
is measure is useful to evaluate network performance under normal daily ow variations (Chen et al. 2002a). One of the important approaches was the de nition of travel time reliability regarding the degradation state of a network in which travel time reliability is de ned as a function of the ratio of travel times under degraded and non-degraded state (Asakura 1999). Some travel time related aspects of transportation network reliability has recently been introduced taking into account 'travel time budget' like those by Lo et al. (2006); Siu and Lo (2008), and 'schedule reliability' by Li and Huang (2005).

Capacity Reliability
Recently, a capacity-related measure for assessing the performance of degradable road networks has been introduced which is concerned with the probability that a network can accommodate certain tra c demand at a required service level (Chen et al. 1999;Yang et al. 2000). is measure, capacity reliability, may also be dened as the probability that reserve the capacity (largest multiplier applied to a given basic OD demand matrix that can be allocated to a network without violating arc capacity) of the network is greater than or equal to the required demand for given capacity loss due to degradation. Capacity degradation is subject to various events and can practically be obtained as a random variable. e only method presented in previous researches to compute capacity reliability was a basis for Monte Carlo simulation (Chen et al. 2002b).

Behavioural-Related Reliability
In addition to the three aforementioned types of performance reliability, some other measures were addressed to assess the performance of a transport network. ese measures in many cases are related to the behavioural responses of users and are based on different aspects of utility or disutility. Bell (1999Bell ( , 2000 de ned a performance reliability index as the utility of users when they are extremely pessimistic about the state of the network in which the utility was obtained in mixed-strategy Nash equilibrium. Yin and Ieda (2001) and Yin et al. (2004) considered performance reliability as the total disutility of commuters on the basis of minimal expected disutility between each OD pair for di erent classes of commuters.

Resource Allocation
Investing in a transportation network may be considered with di erent points of view, from enhancing the service level of a freeway to adding supplementary links into the existing network. Sánchez-Silva et al. (2005) proposed a resource allocation model de ning the network reliability index as changes in the accessibility of the network. e model maximized the network reliability index based on a set of possible actions described in terms of failure and repair rates of each link. However, considering the complexity of resource allocation in transportation networks under uncertainty, especially in the case of large networks, an approximation method may provide the most appropriate solution taking account of elapsed time for the analyzing procedure. Unfortunately, very little attempt has been made with respect to the methods of approximation resource allocation in transportation networks. Recently, Mansour-Khaki et al. (2006) have presented an approximation method to optimize connectivity reliability on the basis of dynamic programming.
In this paper, the optimization model re ects how the performance of a network can be maximized taking into account a special level of budget. In other words, by means of this model, it can be determined what links and level of investment must be modi ed to provide maximum connectivity reliability of a degradable transportation network.

Reliability Analysis
As mentioned in the previous section, di erent works have been done on the reliability of transportation network connectivity in most of which (e. g. Iida and Wakabayashi 1989;Wakabayashi and Iida 1992;Sumalee and Watling 2003) the performance reliability of each arc has been assumed as prede ned values. In this paper, the basis of analysis is a random distribution function for the capacity of each arc, which indicates the actual performance of network arcs. Here, the performance reliability of each arc is estimated separately during network analysis with respect to the real condition happening in the network, and then, the reliability of network connectivity is calculated based on the reliability of arc performance.
However, a transportation network is susceptible to a broad range of internal and external incidents that can a ect its normal functioning. External incidents mainly include weather conditions and natural phenomena like snowfall, ice formation, hurricane, earthquake and av-alanche. Internal degradations are due to tra c related factors such as a failure of control systems, accidents as well as maintenance and construction operations. All aforementioned circumstances can severely decrease arc capacities. As a result, the network cannot perform as it would in a normal situation. In such circumstances, it would be important to know about the performance reliability of the network.
In this paper, the e ects of all incidents are assumed to be included in the capacity function of the network arcs. Further, it is accepted that these capacity functions are independent functions. Arc capacity functions are discussed in the following part of this section.

Arc Capacity Functions
An arc capacity function re ects the relation between di erent levels of arc capacity and their probabilities. In this paper, the capacity function presumed for each arc is a normal probability density function (PDF) for its capacity in which the probability of occurring di erent capacities is shown as probability density. e purpose of selecting normal distribution is that in some previous works, this type of performance function is used. Such similarity allows us to compare the results obtained from these similar works. A normal PDF for the capacity of arc i, ( ) i f c is shown as follows: 2 where: variable c is the value taken by random arc capacity C and i P and i V are respectively the mean and standard deviations of the capacity of arc i . Note that ( ) i f c does not indicate occurrence probability for C c but shows a special manner of spreading the probability of variable C . In general, a normal random variable can take every quantity in interval ( , ) f f while the capacity of arc i is limited between upper bound max i C and lower bound min i C . An important issue about PDFs is that the integration of the PDF from f to f must be equal to 1. If normal distribution is assumed to have a determined lower and upper bound, then, the sum of areas under the curve of the PDF cannot be equal to 1. In order to overcome this problem, a pseudo-normal PDF is utilized and denoted as p i f c . For this purpose, a constant parameter is de ned as follows: Now, we can de ne the pseudo-normal PDF as: (3) erefore, the aforementioned characteristic is kept because: Hence, p i f c is considered as a probability density function which can hold all characteristics of normal distribution with the exception of in nite bounds. Furthermore, adding constant quantity to a random variable does not a ect the spread of distribution (Ross 2009 It implies that the shape of the second PDF p i f c is kept as well as the rst one i f c .

Arc Performance Reliability
In this paper, the performance reliability of arc i is de ned as the probability that the capacity of arc i is greater than or equal to its ow volume. In other words, the performance reliability of arc i can be formed as follows: min max , ; 0, ; where: i v and i C are ow volume and the random capacity of arc i . In fact, when arc ow volume is lower than the upper bound of capacity, the integration of i f c dc from i v to max i C expresses arc reliability; otherwise, arc reliability is 0; namely, there is no chance for arc capacity to be greater than or equal to ow volume assigned to the arc.

A Challenge on Equilibrium Flow Consideration in Networks with Stochastic Supply
In congested urban transportation road networks, arc ow volumes are computed with respect to the relation between travel cost (i.e. travel time) and volume on arcs, like: where: i TC is travel cost on arc i . Assuming that all driver (user) travelling in the network from origin O to destination D ( O and D are speci c nodes of the networks) are familiar with the whole network and know well the expected travel cost for traversing each arc based on Wordrop's user equilibrium principle (She 1985), the following mathematical programming can be applied to assign Origin-Destination (OD) travel demands for the network and to compute volumes on arcs i v : Subject to: , where: R -the set of network paths, W -the set of OD pairs, w R -the set of paths, w q -ow between OD pair w W , r f -user equilibrium ow on path r R and r i X is 1 if arc i belongs to path r ; otherwise 0. e above problem has been known as user equilibrium (UE) assignment. In many previous related literature (e. g. Chen et al. 1999Chen et al. , 2002aChen et al. , 2002bYang et al. 2000), with simpli cation assumptions, professionals have tend to consider the UE with respect to stochastic arc capacities, i. e. all users are assumed to be aware of variations in arc capacities or exact arc travel times. Hence, the UE is converted to stochastic user equilibrium (SUE) with the following objective: Note that i C is a random variable used in equation 12 rather than i c used in equation 8 that is non-random capacity of arc i . Although we can rely on such assumption where variations are due to special incidents (like accident, control device failure, etc.), talking about dayto-day capacity variations, this assumption would not be realistic. erefore, in this paper, we assume that users are ordinarily aware of average travel times (or arcs capacities) at a speci c hour of workdays, and therefore, for the sake of assigning the existing tra c to the network arcs, the expected values of random arc capacities in this particular case accepted as the average capacity values i P , will be considered. us, the objective function of the UE problem is de ned as: When a real network is under consideration, we try to design arcs with such properties that they can operate at a desired LOS in the network. Each level of service (from A to E) takes a special amount of i v C (or i D ), which is considered as the LOS index of arc i ; therefore, the reliability of arc performance can be considered as the probability that the capacity of arc i is greater than or equal to i i v D . is de nition may help us to eval-uate the reliability of network performance at a special LOS. Hence, relation 6 can be rewritten as following: max min max , ; 0, .
It is to say, i 1 D indicates that all arcs of the network can operate with their maximum capacities.

Connectivity Reliability
is section discusses how the connectivity reliability of a network (or OD pairs) can be calculated based on the reliability of arc performance and considering a required service level in the form of v C ratio. On the other hand, since the travel time of an arc is subject to its ow volume to capacity ratio, there may be an interrelation between travel time reliability and connectivity reliability presented in this paper.
ere exist various ways to compute the exact connectivity reliability for a special OD pair in a network. Generally, due to a huge computation volume arisen from the exact solution procedures in large networks, these methods cannot be used in a realistic scale network. A relevant approximate method is to compute an upper and lower bound of reliability using minimal cut and path sets (precise information about minimal cut and path sets can be found in reliability related texts, e. g. Bedford and Cooke 2001;Grosh 1989). Using this approximation method, connectivity reliability between OD pair w can be computed as follows: where: connectivity w R , max w R and min w R are respectively average, lower bound, and upper bound of connectivity reliability between OD pair w . To evaluate the connectivity reliability of all OD pairs as a unique index, the weighted average of OD pair connectivity reliability may be utilized with respect to the ratio of the required demand between OD pairs to the whole required demand: where: w q is the required demand between OD pair w . Hence, the reliability of network connectivity is a function of arc reliability: 1 2 .. connectivity net.
is method provides a closed formula to evaluate the reliability of network connectivity in the calculation procedure of which not only service level of each arc but also random arc capacities are taken into account. e most important advantages of employing the closed formula to evaluate network performance reliability are simple to use in optimization techniques and have very little elapsed time for the computation procedure com-pared with simulation based techniques such as Monte Carlo and Latin Hypercube Sampling (LHS).

Investment Functions
An investment function shows the amount of investment (or investment) needed to promote the reliability of a link from a base level to a higher level and/or from the higher level to an even higher one and so forth. Pertaining to the structure of arcs in a transportation network, it is assumed that investment functions can take only especial levels of investments, and therefore are discrete functions. We further have assumed that the investments have assigned to transform arc capacity PDFs to modify their performances in a manner that each investment level provides a special modi ed PDF for the capacity of each arc. ij E represents the quantity of the investment function of arc i for investment level j and shows how much investment it needs to meet level j of reliability, ij R when the performance reliability of arc i is at its base level, i.e. ij E will result in ij R . Recall that we have assumed that investment ij E alters the form of base arc capacity PDF, where: ij C is the random capacity of arc i at investment level j and max ij C and min ij C are respectively upper and lower bounds for capacity of arc i at investment level j .

Resource Allocation Model
is section discusses the objective and constraints of the resource allocation problem. e objective is to maximize the reliability of network connectivity . connectivity net R : . 1 1 2 2 .. , ,... , where: . connectivity net R is a mixed integer nonlinear function of network performance reliability obtained from rela-obtained from relation 17 replacing i R with ij ij j J R Z ¦ , n is the number of arcs in the network and ij Z is a binary state variable in case if investment level j J is assigned to arc i I , then its value equals 1, otherwise is 0.
Constraints of the model are: where: ij E is maximum budget or the allowable total investment. Constraint 20 indicates that the sum of investments must not be greater than maximum budget. Constraint 21 shows that each arc can achieve only one investment level and constraint 22 guarantees that decision variable ij Z is binary. It is to be noted that the reliability of arc performance alter along with arc capacities because the reliability of arc performance depends on arc volumes (e. g. equation 18) and arc volumes are due to the conservation of user equilibrium condition provided by equations 13, 9, 10 and 11. us, a bi-level approach must be considered to keep optimality. We utilized a standard Genetic Algorithm (GA) to solve the proposed bi-level resource allocation model the detail algorithm of which will be presented in another article.

Considerations on Network Size
When a large-scale network is under consideration, not only the exact calculation of connectivity reliability but also its lower an upper approximation will be computationally intractable. To concern this problem, some approximation methods may be applied, like those presented by Shier and Liu (1992); Li and Silvester (1984); Rosenthal (1977). For even more approximations, these methods can be used along with di erent types of kshortest path algorithms. Interested readers may nd kshortest path algorithms in Takaoka (1998).

Numerical Results
As explained in the previous sections, this paper concerns with two aspects of network reliability evaluation -performance reliability analysis and resource allocating with respect to reliability optimization. e subjects of those parts dealt with reliability analysis are similar in some existing works, and therefore for the sake of comparing the numerical results obtained from the proposed and other methods, we have examined a simple test network employed repeatedly in network reliability analysis, particularly in the case of capacity reliability studies (e. g. Chen et al. 2002b). As shown in Fig. 1, the network consists of ve nodes, seven arcs and two OD pairs. e base demand for OD pairs (1,4) and (1,5) are 20 and 25, respectively. e Bureau of Public Road (BPR) arc travel time function is used: where: i v , f i t and i C are ow, free-ow travel time and random capacity on arc i , respectively. Table 1 gives the results of user equilibrium tra c assignment and general information about the network arcs.

Reliability Analysis
Because Monte Carlo simulation is a powerful technique to analyze probabilistic networks, especially when enough samples are drawn, it may result in an acceptable solution. For comparisons, therefore, we will consider Monte Carlo simulation method. Reliability assessment procedure presented in this paper has been applied to the test network with the assumption of / 1 i v C . Fig. 2 compares the connectivity reliability of OD pair (1,4) obtained from the proposed method and Monte Carlo simulation (with 5000 samples) for di erent demand levels.
As mentioned in sub-section 3.3, the LOS (in the form of / v C ratio) can be considered to calculate network reliability in the proposed method.  Fig. 3 shows how the reliability of network connectivity reacts to the di erent amounts of / v C . When / 0.75 v C (the network is expected to be connected with a higher LOS), the network is nearly unreliable. On the other hand, for / 0.9 v C ! , lower service quality is expected, and accordingly, the network is almost reliable. Di erently from the above amounts of / v C , when / v C is between 0.75 and 0.9, the rate of change in the reliability of network connectivity is considerable, thus it would be important to know the LOS or its related / v C the network is expected to operate at and, therefore, how much the decision maker must spend to ascertain the desired performance of the network.

Resource Allocation
Before computing the reliability of arc performance for di erent investment levels, we need to determine investment functions. It is assumed that all arcs can be modi ed by investment quantities 1, 2.5 and 5 each of which can alter only the mean of capacity PDFs and all standard deviations would be constants. For di erent investment levels, the mean capacities of the arcs and related calculated reliability for di erent amounts of v C are displayed in Table 2. Fig. 5 re ects the result summary of implementing the proposed reliability optimization model to the test network for various amounts of budget from 0 to 35. e gure also exhibits the sensitiveness of the resource allocation model for di erent v C s. As shown in Fig. 5, when v C decreases from 1 to 0.7, the reliability of network connectivity declines rapidly and for amounts lower than 0.7, the reaction of connectivity reliability is not so considerable.
In Table 3, the arcs that ought to be invested for some possible budget levels and their related investment levels are presented. Table 3 shows that various budget levels would result in di erent arc investment combinations. For example, with the budget level of 1 unit ( max 1 E ), the proposed optimization model necessitates arc 1 to be invested at investment level 1, whereas, for budget level 2, we have to invest 1 unit in both arcs 2 and 6. Fig. 4 re ects a summary of results implementing the proposed reliability optimization model to the test network for various amounts of budget from 0 to 35. e gure also exhibits the sensitiveness of the resource allocation model for di erent v C s. As shown in the gure, when v C decreases from 1 to 0.7, the reliability of network connectivity declines rapidly and for amounts lower than 0.7, the reaction of connectivity reliability is not so considerable.
In order to display how much capability the proposed reliability optimization method has, we have applied uniform investment allocation in all arcs for three possible levels of budget: 7 (1 investment unit for each arc), 17.5 (2.5 investment units for each arc), and 35 (5 investment units for each arc). Fig. 5 compares the proposed method and uniform allocation which clearly indicates that the capability of the proposed method will extremely appear when we are confronted with investment limitation. Namely, the di erence between the results obtained from di erent methods for budget level 7 is too much higher than those obtained from 17.5 and 35.

Concluding Remarks
We have presented a reliability optimization method concerned with resource allocation modelling so as it would be possible to identify which arcs with how much e ort must be modi ed to meet the required levels of connectivity reliability. is study has two main properties. e rst one is dealt with a new approach to performance reliability computed based on the reliability of arc performance and taking into account the required LOS as a function of v C ratio and arc capacity PDFs. e second property is the ability to adopt the above discussed ap-proach to the evaluation of performance reliability in a way it can be embedded in an optimization model. However, the most important result of this research is that it establishes a tool enabling approximate improvement in transport networks under uncertainty. In the proposed reliability optimization model, the arcs can be modi ed with di erent levels of investment which consequently would make di erent combinations of arc capacities in a manner that selecting one of these combinations would result in approximate improvement in the network. Although this paper presents a method comprising  Table 3. e arcs that should be invested in various budgets to achieve optimum resource allocation in the network, v/C=1 Budget Level, E max E ij 1 2 3.5 4.5 5 5.5 6.5 8.5 9.5 16 35 1 1 2,6 5 2,7 2,5,7 2,5,6,7 6 5,6 5 2.5 1 1 1,7 1 1 1,2,7 1,2,7 2,6 5`1,7 All Arcs possible in uencing factors, reliability optimization in transportation networks is still at its preliminary stages. However, future research may take advantage of extending the proposed method considering the interdependency of arc capacity distributions, the role of multiple objective decision making methods in the optimization model and time scheduling to minimize tra c disturbance during the execution of improvement actions on the arcs.