Capacity uncertainty on urban road networks: A critical state and its applicability in resilience quantification

Computers, Environment and Urban Systems 54 (2015) 108–118

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Capacity uncertainty on urban road networks: A critical state and its applicability in resilience quantification B.K. Bhavathrathan, Gopal R. Patil ⁎ Transportation Systems Engineering, Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

a r t i c l e

i n f o

Article history: Received 14 June 2014 Received in revised form 22 July 2015 Accepted 23 July 2015 Available online xxxx Keywords: Urban road network Capacity uncertainty Network resilience Minimax optimization Genetic algorithm

a b s t r a c t There are many aspects of urban transportation that represent sources of uncertainty in the design of roadways, such as the level of capacity needed to ensure efficient traffic flow. As a result of uncertainty in roadway capacity, an urban road network can be deemed to operate at different capacity levels. Some of these levels will have unused capacity, whereas some others will not be enough to cater traffic from all origins to all destinations. Past models assume knowledge over the pattern of these uncertainties. However, it is difficult to gather such knowledge from field observations, and it is absent for majority of the world's urban areas. We present an alternative methodology in which the capacities are considered as variables that can take any value from zero to a practically realizable maximum. Using a minimax optimization formulation, we determine bounds on urban roadway capacity levels, below which the traffic demand will go unmet. We call this the critical state, and define it as a state of link capacities which effects in the maximum irreducible operational cost on the network with the demand getting fulfilled. We prove that at a critical state, the total travel time (or cost) of the system will be a unique value; i.e. for a given urban road network and a given traffic demand, there is an associated unique critical travel time. We illustrate that this unique travel time—which is an aggregate value of the travel times from all roads on the network—can be used as a benchmark to create various metrics for the urban road network. As an illustrative example on the applicability of critical state, we compare the unique travel time with the best possible travel time on the network, and develop a metric for network resilience. Network resilience is calculated as a normalized difference of the critical and best operation costs. Two-space genetic algorithm is used to solve the problem formulation. The formulation and the solution methodology are illustrated on test networks and results are presented. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction On urban road networks, capacity uncertainty exists at two levels: uncertainty in estimation and uncertainty in realization. Though the Highway Capacity Manual—inspired from the vehicular behavior on American cities—advises realizable capacities on urban facilities, the recommendations are yet to be validated for majority of the world's cities where traffic behaves contrarily. Many countries do not readily possess such manuals or the data required to accurately predict capacity of their roadways. On one hand, this causes the uncertainty in estimation of roadway capacity. On the other hand, urban road networks are inherently subject to frequent capacity disruptions due to traffic incidents, snowing, flooding, road space reallocations, road space infiltrations, etc., and their combinations. This causes uncertainty in realization of capacity. A conceptualization of various eventualities arising from capacity disruptions can be seen in Jenelius and Mattsson (2015). ⁎ Corresponding author. E-mail addresses: [email protected] (B.K. Bhavathrathan), [email protected] (G.R. Patil).

http://dx.doi.org/10.1016/j.compenvurbsys.2015.07.005 0198-9715/© 2015 Elsevier Ltd. All rights reserved.

Many existing models do not account for capacity uncertainty and result in suboptimal network design decisions (Ng & Waller, 2009). As observed by Bell (2000), most studies that do consider uncertainty in link performance, assume knowledge of a disruption pattern (for example Chen et al., 2002; Du & Nicholson, 1997; Lo et al., 2006; Lo & Tung, 2003; Ng et al., 2011; Nicholson & Du 1997). Nicholson and Du (1997) incorporated the inherent degradable nature of roadway capacities to analysis of transportation networks and developed an integrated equilibrium model. Based on the model, the performance sensitivity of transportation networks with degradable roadway capacities were analyzed by Du and Nicholson (1997). Lo and Tung (2003) analyzed networks with uncertain capacity-realizations and introduced the notion of probabilistic user equilibrium (PUE). They assumed uniform distribution on capacity-realizations, and provided a relaxation methodology on this assumption using known distributions of incident severity. Lo et al. (2006) modeled risk aversion behavior of travelers on networks susceptible to uncertainty in capacity-realizations, assuming standard distributions with known values of parameters on capacity degradation. Chen et al. (2002)—while assessing capacity reliability of a road network—assumed roadway capacities as random variables that follow probability distribution with specified correlation among them. Recently,

B.K. Bhavathrathan, G.R. Patil / Computers, Environment and Urban Systems 54 (2015) 108–118

Ng et al. (2011) introduced a distribution-free methodology in which the model required N-moments of travel time data. All these instances involved explicit or implicit assumption of familiarity on capacity realization pattern which in most cases is absent (for instance, Kwan and Ransberger (2010) describes some practical issues and institutional obstacles involved in collecting road network disruption data), thus creating a gap that is addressed by this paper. We first determine a critical state of capacities, at which the network's operational cost will be unique. This unique cost is compared with the best possible operational cost on the network to establish its resilience. In the next section we provide a description on the critical state and the effecting unique operational cost. The problem is explicitly defined followed by a game theoretical interpretation of the critical state. The minimax optimization formulation is described in the subsequent section. Also, a proposition is made and proven, which exhibit the properties of existence, stability and more importantly, the uniqueness of the network's operational cost at critical state. The formulation is solved using two-space genetic algorithm which is also briefed in Section 3. The minimax formulation and the solution methodology are exemplified on four networks in Section 4. In Section 5, network resilience is quantified on the four networks, using the unique operation cost at critical capacity state. Summary and conclusions are presented in Section 6. 2. Problem definition and game theoretic interpretation 2.1. Critical state

Definition. Given a road network and a traffic demand, critical state is a state of link capacities which effects in the maximum irreducible operational cost on the network with the demand getting fulfilled. The formulation presented in Eqs. (1)–(5) solves for the critical state. It can be seen in the following section that the formulation in Eqs. (1)–(5) determines a set of realized capacities. The network is in a critical state at these realized capacities. If any one of these realized capacities goes lower than that at the critical state, demand will go unmet. The operational cost on the network is accounted as the system travel time (STT), which is an aggregate of all vehicles travel time on all links of the network. STT depends on the vehicle routing on the road network. There is more than one option for feasible routing. A capacity state has to be identified that would result in maximum STT that cannot be reduced by changing the routing. The problem is formally presented in the next subsection. 2.2. Problem definition For the remainder of the paper, the urban road network and its components will be addressed using the standard terminology in graph theory. The urban road system will be addressed as a network; the roads as links and the zones of traffic generation/attraction as nodes. Consider the network as a digraph G (N, A), where N is the set of nodes and A is the set of links. Let R denote the set of origins and S denote the set of destinations. Let the vector D denote the capacities, and F denote the flows on each link of the set A. The problem is to find a degraded capacity vector D⁎ which would cause a critical operation cost STT⁎ on the network G such that no modified flow vector F⁎ can reduce it. The issue that makes the problem complex is that D and F vary with mutually contradicting effect on the STT. Thus, D has to be determined to maximize cost of network operation, and simultaneously F has to be determined to minimize it. The problem can be interpreted in the perspective of game theory, and also that of robust optimization. We describe the game theoretic perspective in the next section, because it aids in proving the existence, stability and uniqueness of the critical state network operation cost.

109

2.3. Game theoretic interpretation Consider an urban road network. The capacities at which the roads of the network are functioning at an instance define the state of the network at that instance. Corresponding to this state, the traffic demand can be routed in different ways on the network. Among these routing options, the one which involves the least STT is the best possible routing scheme for this state. For this routing scheme, a simple maximization problem with capacities as decision variables is not enough to solve for the critical state of capacities; as it may still be possible that another routing scheme exists which is a best response to the current level of capacities. A minimizing routing scheme is possible and yet another capacity state would exist as a best response to the new routing scheme; so on and so forth. It is thus a parallel problem of maximizing and minimizing the STT, and can be envisioned as a virtual non-cooperative game involving two imaginary players. The problem can thus be interpreted in a game theoretic perspective. In a game theoretic perspective, the problem is to find strategy sets for the players of a non-cooperative game, as one player is trying to minimize and another to maximize, the cost of network operation. For this situation, a game is envisaged with two players: a friend and a foe of the network. Note that the entity ‘foe’ is an imaginary player. One may attribute the role of an imaginary player to that of a network tester, who tests the network under the most pessimistic cases. Colony (1970) was the first to include such an imaginary attacker in transport analysis, envisaging a game with nature as an imaginary player. Hollander and Prashker (2006) points out that in non-cooperative game theory, the inclusion of an imaginary player to analyze pessimistic consequences of uncertainty is not uncommon; many examples may be found in the literature (Bell, 2000, 2003; Bell & Cassir, 2002; Brown et al., 2006; Karaa & Lau, 2006; Lou & Lihui, 2011; Lownes et al., 2011; Satayapiwat et al., 2008; Szeto, 2011). To satiate imagination, the friend can be thought of as a centralized network operator who responds to capacity state set by the foe, in the best possible way by directing the users to choose paths to reduce the network operation cost. In a game involving i and −i as the two players—say i is the friend and −i is the foe—friend's (player i) strategy si will be the link flow vector and foe's (player −i) strategy s − i will be the degradation vector. Player −i's minimax strategy against player i is that strategy, which − i plays to maximize its own payoff and minimize the payoff to i, while i tries to maximize it. Conversely, the minimax strategy for player i against player −i is arg min max u−i ðsi ; s−i Þ, and player i's minimax si

s−i

value is min max u−i ðsi ; s−i Þ, where u−i(si, s−i) denotes the utility/paysi

s−i

off that −i receives by playing s − i while i plays si. A player−i would want to play a minimax strategy to punish the other player as much as possible. In the context presented here, we model STT as u−i(si, s−i), the link flows as s − i, and the link capacity levels as si. Hence, the STT at critical capacity state equals min max u−i ðsi ; s−i Þ . Note that the si

s−i

game strategy space in the game is finite, as the strategies of either player is bounded at upper and lower limits; this is further discussed in the following section. Due to the properties of two-player finite games, at the minimax value, both players' strategy sets would reach an equilibrium such that they will not be able to gain any further in STT by unilaterally changing strategies. 3. Minimax optimization formulation and solution methodology 3.1. Model formulation It is well known in the literature of game theory that minimax optimization formulations can be employed to solve two-person zero sum games (Parthasarathy & Raghavan, 1971), and many such instances can be seen in different domains (Lai & Tanaka, 1982; Lung & Dumitrescu,

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Fig. 1. Flowchart of the coevolutionary two-space genetic algorithm.

2011; Vamvoudakis & Lewis, 2012). The setup of one party maximizing and another party minimizing a same objective straightforwardly explains how minimax optimization and two-player zero sum games are related (Hollander & Prashker, 2006). In this study, the strategies played by both players are optimized using a minimax formulation, which finds the minimum possible value of an objective function with respect to one set of decision variables that coexists with the maximum possible value of the same function with respect to a different set of decision variables. Following are the other notations associated with the formulation (note that link specific variables exist only if the corresponding link exists): qrs ij xij fijs Kij yij cij(xij, yij) − A+ i , Ai

demand between origin r and destination s (r ∈R, s ∈S) directed link between node i and j, ij∈A flow on link ij flow specific to destination s on link ij maximum physically possible capacity on link ij such that 1/yij is the realized proportion of Kij link cost function — travel time experienced on a link as a function of flow and realized capacity set of outgoing links and the set of incoming links, respectively, at node i

The minimax optimization formulation is presented below. X

z ¼ min max xi j

yi j

  xi j :ci j xi j ; yi j

ð1Þ

ij

Subject to: X

s

i j∈Aþ i

X

f i j−

s

X ji∈A− i

f i j ¼ xi j

s

f ji ¼ qis

∀i j

∀i ; i≠s

ð2Þ

ð3Þ

s

  ∀ij 0≤xi j ≤ K i j =yi j

ð4Þ

1≤yi j ≤mi j

∀ij:

ð5Þ

Note that xij and yij are the decision variables, and in Eq. (1), minimization is carried out with respect to xij and maximization with respect to yij. The degree of the objective function depends on the degree of the link cost function. mij is an upper limit on the variables yij. yij are decision variables such that 1/yij will be the portion of the capacity that remains after a degradation. mij can be set by the planner such that 1/mij is a guaranteed minimum proportion of capacity realization. In other words, mij is a finitely large value such that (100(mij − 1)/mij)% is the maximum allowable capacity degradation on any link ij. It can be seen that if mij is set as 100, it imparts a limit of 99% link capacity degradation. To allow complete degradation of a link, the planner/analyst may use a very high value of mij. This constraint in Eq. (5) helps in attaining compactness on all variables in the set and thus improves solvability. The objective function (Eq. (1)) represents the STT of the network as a function of flow and realized capacity on each link. Eq. (2) is the flow conservation constraint at node i, which maintains that the difference of outgoing and incoming flow is the destination s bound demand generated at node i. Note that the flow variables are destination specific in Eq. (2). This makes the formulation applicable on networks with multiple origin–destination pairs. The flow variables sum up to total link flows, as maintained by the definitional constraint in Eq. (3). Eq. (4) constrains the flow variables between an upper and lower bound for two reasons. Firstly, most of the cost functions are highly sensitive when the ratio ‘volume to maximum physically possible capacity’ (v/c) goes above unity, and the conditions soon approach hard operability or inoperability. Note that an upper bound of 1.0 is thus applied to the volume to capacity ratio through Eq. (4). Secondly, it aids solvability as discussed in the next section. The idea of using minimax formulations to minimize the maximum possible value of an objective on a network is very popular in the network interdiction literature (Smith et al., 2013). For instance, in studies by Dai and Poh (2002), Wood (1993), and Cormican et al. (1998), they focus on minimizing the maximum flow of prohibited goods through a capacitated network. However, the chief difference that the interdiction models have with the model proposed here is that the cost of

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degradations are constrained within a budget. This is so because the very objective of the interdiction literature is to suggest strategies that optimize the cost of interdiction. On the other hand, in this work, the intention is to find an upper bound of all possible degradation states. A budgetary constraint on degradation would then underestimate the upper bound.

which makes it bounded. Similar to elements of X, the elements of Y can also bear the values at the bounds, thus making it closed. Thus, Y is also a compact space. The function x ij . cij (x ij , yij ) maps from ℜ2 → ℜ and is jointly continuous in the intervals as specified by the constraints. As continuity retains over summation, ∑ xi j ci j ðxi j ; yi j Þ is also continuous. Further, x ij . c ij (xij , y ij ) is convex

3.2. Existence, stability and uniqueness of the minimax value

with respect to X and Y separately. Since both X and the value of cost function are always positive, the summation ∑ xi j ci j ðxi j ; yi j Þ on

ij

In this section, we discuss the existence of stable and unique minimax value for the problem formulated in the previous section. An extended form of the classical minimax theorem due to Neumann (1945) as stated by Simons (1995) is: “Let X and Y be nonempty convex subsets of Euclidean space and f be jointly continuous. Suppose that f is quasiconcave on X and quasiconvex on Y, then min max f ðx; yÞ ¼ max min f ðx; yÞ”. If y∈Y

x∈X

x∈X

y∈Y

the function f (x, y) holds the conditions mentioned in the previous statement, it is then certain that there exists at least one saddle point on the function. For the formulation in Eqs. (1) to (5) to hold minimax theorem, it is thus required that: (1) xij ∈ X ∈ ℜn,{xij ∈ ℜ|xa ≤ xij ≤ xb},∀ ij and yij ∈ Y ∈ ℜn,{yij ∈ ℜ|ya ≤ yij ≤ yb}, ∀ ij, (2) ∑ xi j ci j ðxi j ; yi j Þ is a ij

jointly-continuous function which maps ℜn → ℜ, and (3) ∑ xi j ci j ðxi j ; ij

yi j Þ is quasiconvex with respect to xij on X and quasiconcave with respect to yij on Y. A minimax value may be said to be stable if it is constituted by such strategy sets that no unilateral change can better the utility of either player. Consequent to the holding of the three aforementioned requirements, while at some strategy profile of both players, they receive a payoff that is equal to both the minimax value and the maxmin value. In a finite, twoplayer zero-sum game, such a strategy profile will be a Nash Equilibrium (NE). It follows that ∀ Y ∈ ℜn, ufoe(X*, Y*) ≥ ufoe(X*, Y) where ui represents the payoff function. Hence there exists a stable minimax value, as there is no incentive for either player to change strategies at the minimax value. Further, all NE in any finite, two-player zero-sum game have the same payoff vector (Nash, 1951). Thus, the value of the objective function ∑ xi j ci j ðxi j ; yi j Þ will be unique, even though the decision variables ij

might not be. Proposition. There exists a stable and unique minimax value for the formulation presented in Eqs. (1) to (5) when the cost function is as in Eq. 1(6). 0 1α   x i j ci j xi j ; yi j ¼ T i j þ β@K A

ð6Þ

ij

yi j

ij

X and Y will remain convex. As convexity also is retained over summation when the coefficients are positive, it may be seen that ∑ xi j ci j ij

ðxi j ; yi j Þ is convex on X and Y separately. Hence, the weaker conditions of quasi-convexity/concavity with respect to X and Y also holds. The three conditions in the extended version of the classical minimax theorem hold for the formulation in Eqs. (1) to (5) with the cost function given in Eq. (6). Consequently, it can be concluded that there exists a stable and unique minimax value for this formulation.∎ Also, as the three sufficient conditions described in the beginning of this section hold for the formulation with BPR function as the cost function, min max f ðx; yÞ ¼ max min f ðx; yÞ. y∈Y

x∈X

x∈X

y∈Y

The above proposition proves that the network operational cost at the critical state of capacity is unique. In other words, given a network and a traffic demand, we have a unique critical-state-operational-cost on the network. The assurance on its existence, and more importantly on its uniqueness, qualifies it to be a benchmark of network performance under capacity uncertainty. 3.3. Computing the solution using two-space genetic algorithm The minimax optimization problem formulated in Section 3.1 is solved using a two-space genetic algorithm developed by Herrmann (1999a). This algorithm has been used to solve many minimax optimization problems in engineering, including parallel machine scheduling problem (Herrmann, 1999a), demand stochastic transportation network design problem (Herrmann, 1999b), and damping controller design (Jiang et al., 2002). This method belongs to the class of coevolutionary genetic algorithms. The method is suitable for the class of problems in which the condition min max f ðx; yÞ ¼ max min f ðx; yÞ y∈Y

x∈X

where Tij is free-flow travel time on link ij, and α and β are constants. Proof. The objective function ∑ xi j ci j ðxi j ; yi j Þ; xi j ∈X; ij

yi j ∈Y; has X

and Y as non-empty subsets of Euclidean space, as they have elements that are the link flow values and link capacity degradation respectively in them. Since the link flow values are bounded between zero and an upper bound of maximum physically possible capacity, X is bounded. Since the link flow values are bounded between zero and an upper bound of maximum physically possible capacity, X is bounded. The elements of X can have the values at these bounds, namely zero and the capacity, which makes X a closed space. It follows that X is a compact space. The decision variable yij are lower-bounded at one; and an upper bound is exogenously input to the problem by the analyst,

1 This is the BPR cost function — popularly used for analysis of urban transportation networks — modified to incorporate degradability of capacity.

x∈X

y∈Y

holds (Jensen, 2004). As discussed in the closure of Section 3.2, this condition holds for the problem being discussed in this paper. As compared to conventional genetic algorithm that generates one population of solutions, applies genetic rules, and evolves through generations to a set of most fit individuals at the solution, the two

Fig. 2. Test Network 1. Sheffi (1985).

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Fig. 3. Convergence of two-space GA on Test Network 1.

space algorithm maintains two populations: One population represents solutions, and the other, scenarios. Fitness of an individual in one population is evaluated by associating it with each individual in the other population. The populations evolve simultaneously, and they converge to a robust solution and its worst-case scenario respectively. For our formulation, the decision variables' set is divided into two populations based on their respective interest (i.e. to minimize or to maximize) on the objective function: a population X that have flow vectors as individuals in it, and a population Y that have capacity degradation vectors as individuals in it. Each individual in the X population is a vector of destination specific flow values. Similarly, that of Y population is a vector of degradations of capacity, on each link. Individuals in the initial populations of X and Y are randomly generated within the feasible region, i.e. values for fsij and yij are found such that the constraints given in Eqs. (2) to (5) are not violated. For each individual of either population, the objective function value is calculated with respect to each individual of the other population, i.e. the fitness function value of each flow vector in X is evaluated by considering it together with each degradation vector in Y The fitness function is defined for both the populations using Eq. (1)—after substituting the cost function in Eq. (5) to it—such that the fitness function penalizes higher values of STT for individuals in X and rewards higher values of STT for individuals in Y. This fitness function decides the parents in the present generation on whom the genetic operations are performed to create the next generation of fijs and yij. This process is continued till the stopping criterion is reached, i.e., objective function value, STT, converges, for both the populations. A flowchart of the two-space genetic algorithm is presented in Fig. 1. Since the problem involves constrained optimization involving continuous variables, the genetic algorithm may not be able to converge efficiently on huge networks (Michalewicz, 1995). However, solving the present formulation for a big network is not the focus of this paper. There is no absolute assurance that genetic algorithm will converge to a global optimum in all cases. It is known that genetic algorithm has the potential to converge to global optimum, if run for enough generations. To quote Greenhalgh and Marshall (2000), “by running the genetic algorithm for long enough we can guarantee convergence to the

Table 1 Result from Test Network 1. Link

Start-end nodes (i-j)

Flow (xij)

Percentage degradation

Total link travel time (xij . cij)

STT⁎

1 2 3 4 5

1-2 1-3 3-2 2-4 3-4

3.00 3.00 1.00 4.00 2.00

75.00 75.00 91.67 66.67 83.33

15.45 15.45 5.15 20.60 10.30

66.95

global optimum with any pre-specified level of confidence”. In their paper, they have obtained a much lower bound for the number of iterations necessary to do this. 4. Resilience using critical state network cost Quantification of network resilience is important, because of the various uncertainties that real-life networks are prone to. Östh et al. (2015) presents a wholesome perspective of the philosophy of spatial economic resilience. A recent definition of network resilience calls it “the ability of the network to provide and maintain an acceptable level of service in the face of various faults and challenges to normal operation” (Sterbenz et al., 2010). Resilience is a measure of fault tolerance on networks that are susceptible to such faults. There are instances in the literature where the term robustness is used to represent this idea of fault tolerance (Duan & Lu, 2013; Matisziwet al., 2012). Due to the enormous size that most real-life networks bear, network resilience has been addressed in terms of the topologic parameters in graph theory (Newman, 2003). For instance, network resilience used to be popularly measured as the expected number of node pairs which can communicate (Colbourn, 1987). Another graph theoretic interpretation of network resilience yields a probabilistic measure on the maximum failures to disconnection (Najjar & Gaudiot, 1990). Reggiani et al. (2002) has explored the significance of resilience in socioeconomic systems, and has defined resilience in a perspective similar to what is presented in this paper, i.e. in presence of exogenous perturbations. An unrepaired fault in a network can induce both direct and indirect costs. Indirect costs are incurred outside the network (say, the loss incurred in a manufacturing firm due to a fault in the power network), while direct cost is the increment in operational cost of the network itself (for example, the excess pumping power consumed due to a fault in the water distribution network). While quantifying network resilience, direct cost—incurred directly on the primary functionality of the network—has to be accounted. Graph theoretic measures fail to do it, as they are principally based on topology. Urban road networks are prone to many challenges in its day-today operations. Quantification of resilience is hence of interest for urban and regional planners and policymakers. The resilience of urban road networks has been studied by Murray-Tuite (2006) and Ip and Wang (2011). Murray-Tuite (2006) uses simulation to examine the influence of different vehicle routing schemes on various qualitative dimensions of resilience; they being: adaptability, mobility, safety, and the ability to recover quickly. Giving ten different dimensions of resilience, the author opines that no widely accepted measure of resilience existed for transportation systems. Ip and Wang (2011) evaluates resilience of a city as the weighted average

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113

Fig. 4. Test Network 2. Harker & Friesz (1984).

number of reliable passageways with all other cities in the network. Further, they calculate the networks resilience as the weighted sum of resilience of all cities. However, neither of them focuses on uncertainty in roadway capacity even though it is a primary challenge that warrants resilience analysis. Even though a standardized definition does not exist yet, transportation network resilience is not totally a new concept. A basic definition of transportation network resilience in the infrastructural perspective given by Ta et al. (2009) reads: “the ability of the network, given its capacity to supply lane miles, to facilitate the movement of goods under capacity-constrained conditions that are due to a disruption”. In the perspective of network topology, Levinson (2012) studies the connectivity related resilience of different metropolitan areas in the United States of America. A recent review by Reggiani (2013) discusses different methodological considerations of network resilience in the perspective of transport security. Serulle et al. (2011) suggested an index of transportation network resilience, and opined that the base resiliency of a transportation network has two parts: infrastructure resilience and user resilience. While user resilience deals with travelers' perception and cost, infrastructural resilience takes into account primarily road available capacity and alternate path accessibility. It can be seen that this paper focuses on the infrastructural component of transportation network resilience. For a given demand, critical state network cost indicates the unique operable bound on network performance, beyond which the network cannot facilitate the demand. Hence it can be used to measure the resilience of the network. Network resilience can be interpreted as the

difference between best operational cost and the critical operational cost; and can be normalized relative to the critical operational cost as both these values are unique at a given demand for a network. The index for network resilience (ρ) can be formally written as: ρ ¼ ðSTT –SO‐STTÞ=STT :

ð7Þ

Note that the best operational cost is denoted as SO-STT, which stand for system optimal travel time. It can be determined easily by a simple minimization problem (Sheffi, 1985). A network that takes-in greater magnitudes of disruption (and thus operate at low capacity realizations) and yet caters the traffic demand will have a higher value in the numerator of Eq. (7). As the numerator increases, ρ tends to become one. Conversely, if the demand and the network parameters are such that the network cannot take in any disruption at all, then STT⁎ will be equal to SO-STT and the value of ρ remains zero. Conversely, a value of ρ close to zero indicates that the difference in STT⁎ and SO-STT is small, and the network cannot endure much degradation if it has to satisfy the demand. Such a network is not resilient since it will become inoperable with a little degradation. On the other hand a value of ρ close to 1 indicates that the network can withstand significant degradation before becoming inoperable. It should be noted that the resilience index depends on the current demand between OD pairs, since both STT⁎ and SO-STT are functions of current demand. A comparable notion of resilience has been presented by Matisziw et al. (2012). They use the term robustness and resilience more or less

Fig. 5. Convergence of two-space GA on Test Network 2.

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Table 2 Results from Test Network 2. Link

Start-End nodes (i-j)

Flow (xij)

Percentage degradation

STT⁎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1-2 1-3 2-1 2-3 2-4 3-1 3-2 3-5 4-2 4-5 4-6 5-3 5-4 5-6 6-4 6-5

4.72 5.28 4.98 0.00 5.00 0.02 0.28 5.00 4.99 0.01 7.01 0.01 2.01 2.99 4.99 0.01

69 65 67 99 67 99 98 67 67 99 53 99 87 80 67 99

520.3

synonymously. They report from their analysis that network resilience can be highly sensitive to flow. They too use a bound of various possible network states, but do not incorporate routing (traffic assignment) in their base model.

Table 3 Parameters of Test Network 3. Ukkusuri, Mathew, & Waller (2007). Link

Start-End nodes

Length (mi)

No. of lanes

Kij

Free flow speed

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1-5 1-12 4-5 4-9 5-6 5-9 6-7 6-10 7-8 7-11 8-2 9-10 9-13 10-11 11-2 11-3 12-6 12-8 13-3

4.08 5.25 5.25 7 1.75 5.25 2.91 7.58 2.92 5.25 5.25 5.83 5.25 3.5 5.25 4.67 4.08 8.17 6.42

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200 2.200

35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35

This limits the capacity degradation to 99% on the first network, and to 90% on the bigger networks. This was done to decrease computational complexity on bigger networks.

5. Numerical examples on minimax 5.1. Test Network 1 To illustrate the minimax formulation and the two-space genetic algorithm, we present results on four networks—small, medium and large sized—which are popularly used in the transportation network analysis literature. The following parameters have been chosen for the genetic algorithm. The population size or the number of parent solutions is 20, the mutation function distribution is Gaussian, and the crossover function is scattered, based on a random binary vector. Of the twenty individuals of a new generation, two are elite children, fourteen are crossed-over, and four are mutated. While the value of mij is taken as 100 for the first two examples, it is taken as 10 for the bigger networks.

Test Network 1 shown in Fig. 2 is an example commonly used to intuitively comprehend transportation network problems. The network has one OD pair from node 1 to node 4 with demand equal to 6. The free flow travel time is 5 time units and Kij is 12 on all links. The multiplicative coefficient (β) and the power coefficient (α) of the cost function in Eq. (6) are respectively 0.15 and 1. The two-space genetic algorithm converged quickly, as the network is simple. The convergence is shown in Fig. 3. The results from Test Network 1 are given in Table 1.

Fig. 6. Test Network 3. Nguyen & Dupuis (1984).

B.K. Bhavathrathan, G.R. Patil / Computers, Environment and Urban Systems 54 (2015) 108–118 Table 4 OD Demand Matrix for Test Network 3.

115

Table 5 Results from Test Network 3.

From/To

2

3

Link

Start-End nodes (i-j)

Flow (xij)

Percentage degradation (%)

STT⁎

1 4

660 412.5

495 495

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1-5 1-12 4-5 4-9 5-6 5-9 6-7 6-10 7-8 7-11 8-2 9-10 9-13 10-11 11-2 11-3 12-6 12-8 13-3

466 689 309 598 483 293 507 235 122 385 552 424 467 658 520 523 259 430 467

77 69 70 73 78 51 77 54 80 82 59 81 78 70 76 76 88 80 38

73317

5.2. Test Network 2 Test Network 2 is shown in Fig. 4. The network has 6 nodes, 16 links and 2 OD pairs (1 to 6 and 6 to 1). The demand between 1 to 6 is 10 and that between 6 to 1 is 5. The free flow travel time is 1 time-unit, and the capacity is 15 on all links. In this example, α and β are 10 and 1 respectively. The convergence of the genetic algorithm on Test Network 2 is shown in Fig. 5 and the results are shown in Table 2. 5.3. Test Network 3 Test Network 3 is shown in Fig. 6 and the network parameters are given in Table 3. The OD demand matrix is in Table 4. α and β are 4 and 0.15 respectively. Convergence of STT is shown in Fig. 7. Two reasons could have contributed for the quicker convergence on Test Network 3, even though the size of the network is larger than Test Network 2. One, Test Network 3 has links in only one direction, and hence lesser number of paths are available; also, the domain of search-space is lower (1 to 10 against the 1 to 100 in previous example) for population Y. Results after performing the test on Test Network 3 is given in Table 5. 5.4. Test network 4 In the fourth example, we employ the topology and parameters of a network that relates to the transportation network at Sioux Falls, the largest city in the US state of South Dakota. The network has 24 nodes and 76 links. To eliminate computational complexity, we discard all demands other than that from Zone 1 to Zone 24 and vice-versa. The motivation behind selecting the Sioux Falls network is that it is a widely cited benchmark for transportation network analysis. The Sioux Falls network is shown in Fig. 8 and the network parameters are given in Table A.1 in Appendix A. A demand of 1500 from zone 1 to 24 and vice-versa were considered. The convergence of the two-space genetic algorithm is shown in Fig. 9. The results from the Test Network 4 are given in Table A.2 in Appendix A. The percentage degradation shown in against each link would result in the critical capacity state of the network. However, it must be noted that even though the minimax equilibrium value

(STT*) is unique as demonstrated in the Proposition, the strategies (X*, Y*) need not be. This property is well-known in minimax optimization, and many similar results can be found in the literature (Bell, 2000; Didinsky et al., 1993; Heikkinen, 1999; Luss, 1999; Salmon, 1968) 5.5. Resilience on test networks The SO-STTs are determined for the four networks in numerical examples and the resulting resilience values of these networks are presented in Table 6. Test Network 1 is the least resilient among the four, followed by Test Network 3. In the perspective of topology, Test Network 4 network appears to be more resilient than Test Network 2. On a closer look, Test Network 2 could take in more degradation—with half of the links taking in more than 80% degradation, and 6 of them taking in 99%—at the critical state. Hence at the demand levels considered, Test Network 2 turns out to be the most resilient. The results indicate that network resilience so calculated captures the effect of both road available capacity and alternate path accessibility, thereby justifying the definition of transportation network's infrastructure resilience. The measure can be used to evaluate different networks and alternatives, and to propose strategies that improve network resilience. 6. Summary and conclusions The paper introduces a taxonomy of critical capacity state in networks subject to capacity uncertainty. A methodology is outlined to

Fig. 7. Convergence of two-space GA on Test Network 3.

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B.K. Bhavathrathan, G.R. Patil / Computers, Environment and Urban Systems 54 (2015) 108–118 Table 6 Resilience calculated using network cost at critical capacity state.

Fig. 8. Test Network 4. LeBlanc et al. (1975).

model critical capacity state on transportation networks. The methodology envisages a game between two players, playing for and against the network, and finds the minimax value of their game. The theoretic contribution in this paper is the methodology to find the critical network operation cost, and the assertion that it will be unique. However, similar to minimax problems in other area, there can be more than one flow and capacity combination, which results in the critical state. The sufficient conditions on the formulation that

Network

STT⁎

SO-STT

Network resilience (ρ)

Test Network 1 Test Network 2 Test Network 3 Test Network 4

67 520 73317 84778

60 112 63761 45014

0.097 0.785 0.130 0.469

would impart this uniqueness have been detailed. The formulation has been exemplified on networks where the minimax optimization problem is solved using a two-stage genetic algorithm. The performance at the critical state of transportation networks can be useful in comparing alternatives and analyzing effectiveness of the system. The practical contribution of this paper—in a planner's perspective—is that it outlines a methodology to determine the resilience of a transportation network. We introduce a definition of network resilience based on the critical operation cost; for instance, an option which has a higher difference between critical operation cost and best operation cost can be considered more resilient than another. Network resilience has been established on the example networks, as an illustration on the applicability. This value of resilience can be employed by a planner in the benefit-cost analyses to prioritize among different alternatives. Similarly, in an operations-management perspective, while deciding on which all links are to be repaired, the resilience values can be incorporated as one of the different parameters, by employing the decremented capacity realizations in the formulation. Capacity realizations are constrained on an expected minimum level in this work. However, a planner may improve the realistic flavor by defining distribution functions on link capacity degradations, if such data can be obtained. To guarantee the convergence to global optimum, the genetic algorithm has to be performed for enough number of generations, which imposes a computational challenge when the network size increases. The core issue that poses this challenge is the issue of constraint handling. Constraint handling is a hot research topic in evolutionary algorithms; thus finding a specific superior method for finding critical state on large networks is an open research problem. Theoretically, it may be confidently stated that the solutions obtained in this paper are close to the global optima, as the number of generations remain unconstrained. Rather, the stopping criterion is a stability check on the equilibrium to ensure that the optimum has been reached. Majority of world's cities still face inordinate uncertainty in facility capacity, due to errors in estimation and hindrances in realization. Network resilience, if incorporated as a design criterion, has the potential to narrow down the errors due to suboptimal design resulting from capacity uncertainty.

Fig. 9. Convergence of two-space genetic algorithm on Test Network 4.

B.K. Bhavathrathan, G.R. Patil / Computers, Environment and Urban Systems 54 (2015) 108–118

Appendix A

Table A.2 (continued)

Table A.1 Parameters of the Test Network 4. LeBlanc et al. (1975). Link

Start-end nodes

Kij

Travel time

Link

Start-End nodes

Kij

Travel Time

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

1-2 1-3 2-1 2-6 3-1 3-4 3-12 4-3 4-5 4-11 5-4 5-6 5-9 6-2 6-5 6-8 7-8 7-18 8-6 8-7 8-9 8-16 9-5 9-8 9-10 10-9 10-11 10-15 10-16 10-17 11-4 11-10 11-12 11-14 12-3 12-11 12-13 13-12

25900.2 23403.47 25900.2 4958.18 23403.47 17110.52 23403.47 17110.52 17782.79 4908.83 17782.79 4948 10000 4958.18 4948 4898.59 7841.81 23403.47 4898.59 7841.81 5050.19 5045.82 10000 5050.19 13915.79 13915.79 10000 13512 4854.92 4993.51 4908.83 10000 4908.83 4876.51 23403.47 4908.83 25900.2 25900.2

6 4 6 5 4 4 4 4 2 6 2 4 5 5 4 2 3 2 2 3 10 5 5 10 3 3 5 6 4 8 6 5 6 4 4 6 3 3

39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

13-24 14-11 14-15 14-23 15-10 15-14 15-19 15-22 16-8 16-10 16-17 16-18 17-10 17-16 17-19 18-7 18-16 18-20 19-15 19-17 19-20 20-18 20-19 20-21 20-22 21-20 21-22 21-24 22-15 22-20 22-21 22-23 23-14 23-22 23-24 24-13 24-21 24-23

5091.26 4876.51 5127.53 4924.79 13512 5127.53 14564.75 9599.18 5045.82 4854.92 5229.91 19679.9 4993.51 5229.91 4823.95 23403.47 19679.9 23403.47 14564.75 4823.95 5002.61 23403.47 5002.61 5059.91 5075.7 5059.91 5229.91 4885.36 9599.18 5075.7 5229.91 5000 4924.79 5000 5078.51 5091.26 4885.36 5078.51

4 4 5 4 6 5 3 3 5 4 2 3 8 2 2 2 3 4 3 2 4 4 4 6 5 6 2 3 3 5 2 4 4 4 2 4 3 2

Table A.2 Results from Test Network 4. Link

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Start-end nodes (i-j) 1-2 1-3 2-1 2-6 3-1 3-4 3-12 4-3 4-5 4-11 5-4 5-6 5-9 6-2 6-5 6-8 7-8 7-18 8-6 8-7 8-9 8-16 9-5 9-8

117

Flow (xij)

Percentage degradation (%)

Link

Start-End nodes (i-j)

Flow (xij)

Percentage degradation (%)

543.3 956.7 554.1 543.3 945.9 334.7 622.1 351.6 172.1 307.6 193.8 183.6 217 554.1 189.9 353.4 188.3 240.4 370.5 172.6 56 179.5 232.4 47.2

89.2 76.7 88.1 88.2 87.2 78.3 75 75.6 82.5 89.8 83.3 82.1 89.8 88.5 23.1 83.1 77.3 76.7 89.9 83.1 78.3 89.2 83.9 88.5

39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

13-24 14-11 14-15 14-23 15-10 15-14 15-19 15-22 16-8 16-10 16-17 16-18 17-10 17-16 17-19 18-7 18-16 18-20 19-15 19-17 19-20 20-18 20-19 20-21

557.7 572.7 252.6 66.6 306.2 234.5 94.7 145.1 220.9 189.8 65 69.7 206.3 84.3 70.5 151.9 256.1 228.4 321.9 175.4 141.6 168.4 359.7 188.4

88.5 41.2 85.7 90 89.8 84.4 84.1 83.3 23.1 89.7 71.4 89.5 85.1 73 88.4 89.7 75.6 77.3 78.3 89.2 88.5 87 58.3 89.6

Link

Start-end nodes (i-j)

Flow (xij)

Percentage degradation (%)

Link

Start-End nodes (i-j)

Flow (xij)

Percentage degradation (%)

25 26 27 28 29 30 31 32 33 34 35 36 37 38

9-10 10-9 10-11 10-15 10-16 10-17 11-4 11-10 11-12 11-14 12-3 12-11 12-13 13-12

218.2 224.8 135.3 240.3 52.3 95.4 302.8 146.2 90.2 259.4 594.3 103 572.7 557.7

89.6 89.4 89.6 65.5 82.1 85.3 89.6 88 87.8 54.5 73.7 61.5 85.1 89.2

63 64 65 66 67 68 69 70 71 72 73 74 75 76

20-22 21-20 21-22 21-24 22-15 22-20 22-21 22-23 23-14 23-22 23-24 24-13 24-21 24-23

274.5 170.4 315.5 178.6 456.2 212.9 187.1 181.7 164.9 271.4 176.8 471.1 557.7 494.1

89.7 89.2 89.1 89.4 81.8 89.8 81.5 89.1 89.8 89.5 89.7 89 89.6 89

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