Investigating the Sensitivity of Road Network Resilience to Demand Variability

1 Roadways, the links of transportation networks, are susceptible to a variety of degradations. 2 Some degradation patterns are severe; they make the network incapable of meeting origin3 destination (OD) demands. Most others are non-severe; they increase the network operational 4 cost while OD demands remain met. In other words, the network structure is resilient to such 5 degradations. It is important for planners to quantify this resilience of a network. We have, in a 6 previous study, framed a minimax optimization problem to find the network operational cost— 7 the minimax value—which is irreducible even by re-assignment. The relative difference of the 8 minimax value and the best possible value is interpreted as the network’s resilience. This 9 resilience measure is demand-specific. To generalize the resilience measure, its sensitivity to 10 change in demand should be studied. As it is difficult to analytically establish the sensitivity of 11 resilience to demand, we perform computational experiments on different network topologies to 12 bring out a relationship between network resilience and demand. We observe that the resilience13 demand curve, irrespective of the difference in network size or network topology, follows a trend 14 very much comparable to negative exponential curve. With this result, one need to compute the 15 demand-specific resilience values at only a given number of demand levels to attain generality. 16 Finally, we introduce a generalized index of resilience (GIR) incorporating the demand-spread. 17 We compare the GIR to traditional network indicators and find that it is better in modeling 18 topological and dynamic properties of a network. 19 Patil and Bhavathrathan 3 INTRODUCTION 20 Urban roads—the links of an urban transportation network—are susceptible to a variety of 21 degradations due to traffic incidents, flooding, storming, snowing, road space reallocations, road 22 space infiltrations, etc. When each link of a network is degradable, there exist many degraded 23 states for the network. It is computational-intense to analyze system performance at each of 24 them. Researchers have tried to tackle this problem in exact and approximate ways. The exact 25 methods may lead to computationally complex problems (1). One has to perform traffic 26 assignment for all the possible component state, while component states enumeration itself bears 27 the curse of dimensionality. The approximate methods focus on a subset of the degraded states, 28 trying to contain a large fraction from them and attempt to obtain bounds on the resultant 29 network indices (2). Most of the approximate methods are in the perspective of network 30 reliability. However, as there is uncertainty arising from two sides: the supply as well as the 31 demand, to cover a large fraction of component uncertainty and a large fraction of demand 32 uncertainty is again complex. Thus, it is important to identify a unique upper operable bound of 33 degraded states, so that the network’s spatial and topological structure can be evaluated based on 34 its performance at this boundary. 35 Some degraded states are severe; they make the network incapable of meeting origin36 destination demands. Most others are non-severe; they increase the network operational cost 37 while origin-destination (OD) demands remain met. The network structure is usually resilient to 38 such degradations. The authors have reported the existence of a unique upper-bound on network 39 operational cost beyond which no traffic re-assignment can bring the cost down, in a previous 40 study (3). The authors determined the unique upper-bound by formulating a minimax 41 optimization problem and solving it using a coevolutionary algorithm. This unique upper42 bound—called the worst operable case (WOC)— is compared with the best possible network43 cost at zero degradation—which is also unique—to measure the resilience of the network 44 structure to degradations. The unique worst and best bounds of network operability have been 45 similarly interpreted (4) while considering stochastic degradations too. 46 This measure of resilience is demand specific; for a given OD demand matrix, the 47 network has a resilience value between zero and one. The value changes with change in demand. 48 Hence, a single value of resilience at some demand scenario cannot be attributed as a property of 49 the network structure. In this paper we extend our earlier work and present a generalized index 50 that accounts for the variation of resilience with a change in demand. It would be costly to 51 calculate it for different demand levels. We present results that can avoid this burden. 52 This paper is organized as follows. In the next section, an overview is presented on 53 transportation network resilience. Subsequent section reviews the methodologyto find out of 54 transportation network resilience. The following section is devoted to the analysis of network 55 resilience at different demand levels. We then present the generalized index incorporating 56 demand variation, after which comparisons are made of the improved index with different 57 existing network indictors. Finally, we draw some conclusion. 58 TRANSPORTATION NETWORK RESILIENCE 59 A recent definition of resilience relating to the internet reads: “the ability of the network to 60 provide and maintain an acceptable level of service in the face of various faults and challenges to 61 normal operation” (5). Road transportation networks are also prone to many challenges in its 62 day-to-day operations. It is thus important that network resilience emerge as a design 63 consideration. Network resilience is not as widely studied in the field of transportation, as it is in 64 certain other fields that involve network analysis, like the internet. However, recently, resilience 65 study on freight transportation networks has attracted researchers’ interest (6, 7, 8, 9, 10, 11). 66 Patil and Bhavathrathan 4 Resilience of a system, as a research issue, has classically developed with studies pertaining to 67 ecology. Over time, many different definitions of systemic resilience have come into existence. 68 Fundamentally, they can be classified into two. One talks in the perspective of system properties 69 near some stable equilibrium (12). The other—the more popular—in the perspective of a 70 maximum agitation the system can take in before getting displaced from one state to another 71 (13). Which among the two philosophies suit better for a road transportation system is arguable, 72 as is the case in any field. In view of road capacity degradation that are non-severe in nature, we 73 opine that one has to account for the maximum agitation that the transportation network can 74 absorb before undergoing a state shift from operability to inoperability. In this direction, the 75 authors have presented a methodology to find the maximum operational cost that a network can 76 bear before changing from operability to inoperability, in a previous work (3). 77 Though in its emergent phase, resilience measures in transportation networks has had 78 different interpretations. Various network resilience measures were proposed like load capacity 79 (14), weighted average number of passageways between nodes (15), vital paths (16), maximum 80 number of node failures the network can tolerate (17), network loading (18), and network-state 81 dependent accessibility variations (19). Recently, a detailed review has been done by Regianni 82 (20), which discusses different methodological considerations of network resilience in the 83 perspective of transport security. However, the sensitivity of road network resilience to demand 84 variability is under-researched, specifically on fragile networks that are subject to multiple 85 simultaneous disruptions. This is the research gap that this paper attempts to address. 86 METHODOLOGY 87 We first outline the methodology presented by the authors in a previous work (3), to determine 88 maximum capacity degradation that the network can take-in before being inoperable. By 89 inoperability, it is meant that the OD demand will not be met. The methodology uses a minimax 90 optimization formulation which minimizes the network cost taking flow vector as the decision 91 variable, while maximizing it taking capacity degradation vector as the decision variable. The 92 formulation, solved using a coevolutionary algorithm called two-space genetic algorithm, gives a 93 unique value of network cost. The formulation is included hereunder for reference. 94 On a traffic network represented as a digraph G , where N is the set of nodes and A is the 95 set of links; R denotes the set of origins and S denotes the set of destinations, D is the degraded 96 capacities, and F is the flows on each link of the set A, the worst-operable-case system travel 97 time (WOC-STT) that cannot be minimized even by the best possible re-assignment can be 98 found by solving the program: 99     ij ij ij ij ij y x y x c x STT WOC ij ij , . max min (1) 100 Subject to: 101 102 s i i q f f i i A ij A ji is s ji s ij           , (2) 103    s ij s ij ij x f (3) 104   ij y K x ij ij ij    0 (4) 105 ij m y ij ij    1 (5) 106 Patil and Bhavathrathan 5 where q is the demand between the origin r and destination s ( rR , sS ); ij is a directed 107 link between node i and j, ijA; xij is the flow on link ij; s ij f is the flow specific to destination s 108 on link ij; Kij is the capacity on link ij before degradation; yij is the divisive factor such that 1/ yij 109 is the available proportion of capacity after degradation;   i i A A , set of outgoing links and the 110 set of incoming links, respectively, at node i; mij is the upper bound on divisive factor that the 111 analyst inputs;   ij ij ij y x c , is the link cost function such that: 112