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Structural Safety 67 (2017) 96–104

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Structural Safety journal homepage: www.elsevier.com/locate/strusafe

Bridge network maintenance prioritization under budget constraint Weili Zhang a, Naiyu Wang b,⇑ a b

Department of Industrial and Systems Engineering, University of Oklahoma, USA School of Civil Engineering and Environmental Science, University of Oklahoma, USA

a r t i c l e

i n f o

Article history: Received 28 March 2017 Accepted 1 May 2017

Keywords: Transportation network Travel time Optimal maintenance scheduling Prioritization Optimization

a b s t r a c t This study develops a decision model to assist bridge authorities in determining a preferred maintenance prioritization schedule for a degraded bridge network in a community that optimizes the performance of transportation systems within budgetary constraints at a regional scale. The study utilizes network analysis methods, structural reliability principles and meta-heuristic optimization algorithms to integrate individual descriptive parameters such as bridge capacity rating, condition rating, traffic demand, and location of the bridge, into global objective functions that define the overall network performance and maintenance cost. The performance of the network is measured in terms of travel time between all possible origin-destination (O-D) pairs. In addition to the global budgetary constraint, the optimization is also conditioned on local constraints imposed on traffic flow by insufficient load carrying capacity of deficient bridges. Uncertainties in traffic demands, vehicle weights and maintenance costs are also considered in the problem formulation. Two project priority indices are introduced – the static priority index (SPI), defined as a function of the difference in network travel time between block running (with reduced load carrying capacity before repair) and smooth running (design-level load carrying capacity after repair) of a bridge, and the dynamic priority index (DPI) defined as the likelihood of a bridge being selected for repair when the budget is fixed and the uncertainties governing the performance of the transportation network are considered. Finally, this decision model is illustrated with a hypothetical network with 160 bridges. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Highway bridges deteriorate in service as a result of a wide variety of events (e.g. floods, heavy truck traffic, aggressive environmental conditions, industrial action, and inadequate maintenance), making bridges the vulnerable links in transportation networks. According to the Federal Highway Administration’s 2013 National Bridge Inventory Database (NBI), approximately 25 percent of the nation’s bridges are either structurally deficient or functionally obsolete, causing significant social and economic impact to communities. Resources allocated to the maintenance of transportation networks in the United States (and societies worldwide) typically are limited, and seldom are sufficient to maintain in-service performance levels required for the infrastructure system. As stated in the 2013 ASCE Infrastructure Report Card, every year over $12 billion has been spent on the maintenance and rehabilitation of the nation’s bridges, while the annual investment that would be necessary to improve the current condition of existing highway bridges has been estimated to be $20 billion. Bridge managers are facing ever-increasing challenges in prioritizing ⇑ Corresponding author. E-mail address: [email protected] (N. Wang). http://dx.doi.org/10.1016/j.strusafe.2017.05.001 0167-4730/Ó 2017 Elsevier Ltd. All rights reserved.

expenditures to maintain safety and functionality of deteriorating bridge systems. A decision-making framework that maximizes the functionality of a regional transportation system while ensuring that individual bridges conform to the minimum safety requirements stipulated by Association of State Highway and Transportation Officials (AASHTO) [2] is essential. The study described in this paper is aimed at developing a decision model for bridge network management and project prioritization that enables the operational performance of a transportation system to be optimized, given the safety requirements mandated by AASHTO and the inevitable budgetary constraints imposed by limited resources. The study utilizes modern network analysis methods, structural reliability principles and meta-heuristic optimization algorithms to integrate individual descriptive parameters, such as bridge capacity rating, condition rating, traffic demand, and location of the bridge in the network, in global objective functions that define the network performance and maintenance cost. The network performance is measured in terms of travel time, computed based on the traffic demand within the network between all possible origin/destination (O-D) pairs. In addition to the overall network budgetary constraint, this optimization is also conditioned on the local traffic flow constraints due to insufficient load carrying capacity of structurally

W. Zhang, N. Wang / Structural Safety 67 (2017) 96–104

deficient bridges that has been load posted. Uncertainties in traffic demands, vehicle weights and bridge maintenance costs are considered in the problem formulation. The rest of the paper is organized as follows. Firstly, we review related literature and present the highlight of this study. Secondly, we summarize the bridge safety criteria (bridge condition rating and capacity rating) and introduce the bridge network efficiency measure (travel time) used in this study. A complete mathematical formulation of the decision optimization is then presented and two project prioritization mechanisms are introduced. The application of the overall developed methodology is illustrated using a hypothetical transportation network with 160 bridges, followed by concluding remarks. 2. Background Finite resources for renewal/replacement of highway bridges within a transportation system must be distributed strategically among all bridges to optimize the performance of the system as a whole. Bridge maintenance programs in the past, however, usually have been developed to optimize the life-cycle cost (LCC) of individual bridges without considering the interaction between these bridges in making the transportation system as a whole functional. In reality, the condition of one bridge, e.g. deterioration, failure, maintenance priority and the timing of expenditure, may very well affect the performance and maintenance scheduling of neighboring bridges. Adopting a system perspective to assess the role of bridges serving a community will add an additional dimension to conventional bridge maintenance decisions by placing such decisions in the overall context of enhancing reliability and functionality of the entire transportation system. In the past decade, several research studies have investigated bridge maintenance strategies considering bridges collectively as integral parts of a network. Frangopol and his co-researches have made number of important contributions to the bridge network maintenance planning. Liu and Frangopol [30–32] introduced a bridge reliability importance factor that relates individual bridge reliability to the reliability of the bridge network, and proposed a comprehensive mathematical model for evaluating the overall performance of a bridge network based on probabilistic analyses of network connectivity, user satisfaction, and structural reliability of the critical bridges in the network. Their later study [33] optimized bridge network maintenance based on a multi-objective approach using genetic algorithms. A thorough review of their work can be found in Frangopol [20]. Orcesi and Cremona [40] proposed a bridge network management approach using visual inspection data and Markov chains. These studies have based maintenance and project prioritization decisions on the LCC analysis over the projected service life of the bridges. Although LCC analysis is the most mature and broadly understood decision method [19,35,10,4,17,47,39], problems exist in applying LCC analysis to bridge network maintenance. First, there are large uncertainties associated with LCC analysis due to a lack of supporting databases on cost estimation, especially when deterioration and natural hazards and their consequences are considered. Second, most decision makers for highway infrastructure are government officials, who are driven by public perceptions of their political performance and seldom have the motivation to make decisions based on the analysis that optimizes the outcome for a time span of 50–75 years [14]. Third, available budgets for bridge network maintenance vary from year to year, depending on social, economic and political factors, which often impose constraints on bridge maintenance-related decision making. Rather than using LCC as a criterion for decision, several studies have attempted to obtain optimal bridge maintenance strategies by maximizing the operational performance of a transportation

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network. Current network performance indicators can be divided into two categories. The first one is network topology-based. Connectivity reliability (also known as reachability reliability) is defined as the probability that there exists at least one path between O-D pairs of interest and has been proved as a #Pcomplete problem [43]. Liu and Frangopol [30,33] applied genetic algorithm to maximize the connectivity reliability between a single O-D pair and simultaneously minimize associated maintenance cost. Bocchini and Frangopol [8] and Hu and Madanat [24] later expended the model to multiple O-D pairs. When a network is disrupted by extreme natural hazards (earthquakes, floods, etc.) and its links fail in unfavorable configurations, the network becomes disconnected and the connectivity reliability is often used to evaluate post-disaster network performance. Even when fully connected, however, a network may still fail to provide an adequate level of service to the local community. In such cases, the second type of performance metrics that is network functionality-based becomes more appropriate. Flow capacity reliability and travel time reliability often have been proposed to assess the efficiency of transportation network functionality. Chen et al. [11,12] defined the flow capacity reliability as the probability that the network can serve a certain travel demand using a user equilibrium model, and investigated its sensitivity to the capacity variations of individual links. Nojma [37] presented a prioritization method using maximum flow as the network performance metric to find Birnbaum’s importance measure for each network component. Sanso and Milot [45] defined the transportation network performance in terms of its ability to transport passengers from their origins to destinations in a reasonable amount of time. Other previous studies on capacity reliability [15,26] and travel time reliability [6,7,5] provided useful tools to analyze network traffic equilibrium, but in these studies the links (bridges) of the network were modeled as either fully functional or completely closed. In reality, however, many structurally deficient bridges are in neither status; rather, they continue to operate with a reduced load (flow) capacity imposed by lane closures or load posting limits. In this paper, our objective is to inform decision-making at point-in-time regarding project prioritization that maximizes the operational performance of a transportation system measured in terms of travel time under budgetary constraints. The major contributions of the paper include: (a) the formulation of the bridge maintenance optimization that integrates bridge safety, operation efficiency measured by network travel time, uncertainty of the network, maintenance cost and budgetary constraints; (b) the modeling of local constraints imposed by reduced load capacity of deficient bridges in a transportation system, realistically reflecting the operational status of degraded bridge networks; (c) the application of a metaheuristics method (binary particle swarm optimization algorithm) to provide solutions to the mixed integer optimizations problem formulated for large networks; and (d) the introduction and comparison of the two priority indices - static priority index (SPI) and dynamic priority index (DPI). 3. Bridge network safety and functionality 3.1. Safety criteria for individual bridges – Bridge condition rating and capacity rating Safety is the first priority among all bridge performance objectives. In current engineering practice, bridge condition rating (on the scale of 0–9), assigned according to National Bridge Inspection Standard (NBIS) (summarized in Table 1), is widely used in bridge condition assessment in the US and is an overall measure of the physical condition of highway bridges. Bridge engineers assign condition ratings to existing bridges based on inspection data, traffic survey and highway types. The NBIS stipulates that a bridge

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W. Zhang, N. Wang / Structural Safety 67 (2017) 96–104

Table 1 Bridge condition rating (CR) criteria [36]. Condition Rating

Description

9 8 7 6

EXCELLENT CONDITION VERY GOOD CONDITION-No problems noted. GOOD CONDITION-Some minor problems. SATISFACTORY CONDITION-Structural elements show some minor deterioration. FAIR CONDITION-All primary structural elements are sound but may have minor Section loss, cracking, spalling or scour. POOR CONDITION-Advanced section loss, deterioration, spalling or scour. SERIOUS CONDITION-Loss of section, deterioration, spalling or scour have seriously affected primary structural components. Local failures are possible. Fatigue cracks in steel or shear cracks in concrete. CRITICAL CONDITION-Advanced deterioration of primary structural elements. Fatigue cracks in steel or shear cracks in concrete may be present or scour may have removed substructure support. Unless closely monitored, it may be necessary to close the bridge until corrective action is taken. IMMINENT FAILURE CONDITION-Major deterioration or section loss present in critical structural components structure stability. Bridge is closed to traffic but corrective action may put back in light service. FAILED CONDITION-Out of service. Beyond repair.

5 4 3

2

1

0

with a condition rating less or equal to 4 must be repaired, replaced or closed to operation. In Europe, a similar system (known as Bridge Management in Europe) uses a five-point scale for condition rating. Bridges in good physical condition could still pose a threat to the functioning of the transportation system if they were designed according to archaic standards because they may not have adequate load carrying capacity for modern traffic demands. Therefore, in addition to condition rating, the AASHTO Manual for Bridge Evaluation AASHTO [1] utilizes the bridge live load capacity rating factor (RF), calculated using bridge design strength minus dead load effect, and divided by the live load effect, to reflect a bridge’s live load capacity with respect to its traffic demand. If the resulting RF  1.0, the bridge is deemed to have adequate load carrying capacity, while if RF < 1, the bridge is required to be strengthened, replaced or posted. If the bridge is posted, vehicles that are heavier than the weight limit suggested by the bridge’s RF are restricted from passing the bridge. Accordingly, either a low condition rating or a low capacity rating factor can trigger maintenance activities for a given bridge. If closure of a bridge due to either severe deterioration (i.e. condition rating  4) or posting (i.e. RF < 1) creates intolerable adverse impacts on the performance of the transportation system, the bridge should be scheduled for renewal or replacement. The priority of renewal/replacement projects should be based on the level of impact that each individual bridge’s operational status (e.g. fully functional, posting, or closure) has on the efficiency of the transportation system as a whole, which is reflected in the network performance metric, the travel time efficiency for this study, as discussed next. 3.2. Bridge network functionality measure – travel time efficiency In addition to the safety concerns regarding individual bridges, it is important that appropriate system performance measure is considered in order to improve the performance efficiency of the transportation network as a whole. As discussed previously, wellaccepted metrics to quantify the efficiency of a network performance include connectivity, flow capacity and travel time. While connectivity is often used to evaluate the performance of a disrupted network (e.g. after a natural hazard) and flow capacity is

appropriate for congested networks, in this study we use travel time to evaluate network performance as it is an appropriate system performance metric where the major source of disruption is traffic detour due to inadequate load carrying capacity of deteriorated bridges in degraded transportation networks. We define travel time efficiency, RT , as:

RT ¼ E

  /p ðnÞ /ðnÞ

ð1Þ

in which E½ is the mean operator; /ðnÞ denotes the service metric for an existing bridge network in its current condition, computed by summing up the travel time of the fastest paths for all vehicles traveling between all O-D pairs under uncertain parameters (n) which will be defined later in detail; /p ðnÞ is the minimum travel time for the network in its designed (‘‘new”) condition under uncertainty. The value of RT ranges between 0 and 1, and the network achieves its best performance when RT ¼ 1. If a network cannot transfer vehicles from their origins to destinations, /ðnÞ approaches to infinity and RT to 0. Such defined RT is a function of traffic supply/ demand in the network, distance between O-D pairs, the shortest time paths between each O-D pair for vehicles of different weight, and other network properties such as its topology, speed limit on links, and deterioration condition and load capacities of every bridge in the network. The RT measures the efficiency of a transportation network in transferring vehicles from their origins to destinations, and is used herein as a network-level objective for optimizing maintenance schedules for deficient bridges. The resulting decisions will include a selection of bridges that need renewal/ replacement and a priority index for each selected bridges.

4. Mathematical formulation and solution procedure In this section, we formulate the optimization algorithm that will prioritize bridge maintenance project selections at a pointin-time to improve the transportation network performance under limited financial budget. The mathematical challenges involved in the problem solution and the metaheuristics optimization algorithm used in addressing those challenges are also discussed. Finally, we introduce two ranking mechanisms to assist decision makers in prioritizing maintenance projects under budget constraints. 4.1. Travel time optimization We define the network topology G ¼ ðN; AÞ as a set of nodes N and set of arcs A. The nodes represent origins, destinations, and transshipment nodes. The arcs, each denoted by a distinct node pair ði; jÞ where i; j 2 N for i–j, represent all existing roadways and bridges in the transportation system. Let K denote the set of vehicles in the network and index t 2 K identify each vehicle. Let W denote the set of all O-D pairs in G. T tOt Dt is the travel time of vehicle t from its origin, Ot , to its corresponding destination Dt , where ðOt ; Dt Þ 2 W, and can be obtained by modifying the classical shortest path method [3]. The network service value, /, represents the sum of all vehicle travel times and can be calculated as below

/ðGÞ ¼

X X t2K ðOt ;Dt Þ2W

T tOt Dt

ð2Þ

Let dij and v ij respectively denote the distance and speed limit on arc ði; jÞ 2 A, B # A denote the set of network bridges, and lij denote the posted load limit of bridge ði; jÞ 2 B on arc ði; jÞ: Let rti denote the traffic supply/demand type at node i of vehicle t (i.e. rti ¼ 1 if i is the origin of vehicle t; rti ¼ 1 if i is the destination

W. Zhang, N. Wang / Structural Safety 67 (2017) 96–104

of vehicle t; and r ti ¼ 0 if i is a transshipment node of vehicle t). Flow decision variable xtij denotes the flow of vehicle t on arc ði; jÞ and is defined as continuous variable bounded within ½0; 1. For example, xtij ¼ 1 if arc ði; jÞ is on the fastest path of vehicle t from Ot to Dt , and xtij ¼ 0 if otherwise. The total travel time of all vehicles between all O-D pairs can be obtained by solving the following optimization problem:

X X t2K ðOt ;Dt Þ2W

T tOt Dt ¼ min

X X dij t2K ði;jÞ2A

v ij

8 i ¼ Ot > < 1; s:t: xtij  xtji ¼ 1; i ¼ Dt > : fj:ði;jÞ2Ag fj:ðj;iÞ2Ag 0; otherwise X

ð3Þ

xtij

X

8i 2 N; 8t 2 K ð4Þ

0 6 xtij 6 1;

8ði; jÞ 2 A;

8t 2 K:

ð5Þ

Eq. (4) ensures that the inflow and outflow satisfy the traffic supply/demand at node i of vehicle t. A posted load limit of structurally deficient bridges will certainly affect the path choice of many heavy trucks which, in turn, will impact the travel time of those trucks from their origin to destination. These effects are integrated in the calculation of the travel time by prohibiting vehicle t from passing bridge ði; jÞ if the vehicle weight, xt , exceeds the posting limit of that bridge, lij , where p p lij ¼ f ij zij lij ; lij represents the design (as ‘‘new”) load capacity of bridge ði; jÞ; zij denotes the capacity rating factor (RF) of bridge ði; jÞ; and f ij ¼ 0 when bridge condition rating is less or equal to 4 and f ij ¼ 1 if otherwise. It is assumed that the load capacities of roads (arcs without bridges) are large enough to allow all vehicles to pass. It is further assumed that the load capacity of any bridge p that is renewed or replaced will be brought to lij (as ‘‘new” condition). Let yij denote the binary maintenance decision variables, where yij ¼ 1 if bridge ði; jÞ is selected for renewal/replacement and yij ¼ 0 otherwise. The local constraints imposed on traffic flow by posting limits of deficient bridges are:

xt sgnðxtij Þ 6 ð1  yij Þlij þ yij lpij ; 8ði; jÞ 2 B; 8t 2 K

ð6Þ

yij ¼ f0; 1g; 8ði; jÞ 2 B:

ð7Þ

Eq. (6) ensures that the vehicle t 2 K can only travel through bridge ði; jÞ 2 B if and only if its weight is less or equal to the load capacity. The right-side of Eq. (6) updates the load capacity (i.e. RF zij increases to 1) if this bridge is selected for retrofit (yij ¼ 1); otherwise, its load capacity remains at its current level. Let cij denote the renewal cost for bridge ði; jÞ 2 B, which is assumed to be a function of both posting limit and deck area of the bridge and let H denote the total annual budget available for maintenance of the entire community bridge inventory. The cost constraint can then be expressed as:

X

cij yij 6 H

ð8Þ

ði;jÞ2B

Accordingly, the optimal maintenance strategy and project prioritization can be obtained by minimizing the network total travel time [as calculated using Eqs. (2)–(5)] considering local constraints imposed by the reduced load capacity of deficient bridges [as expressed by Eqs. (6) and (7)] within a prescribed network-level (global) budget limit [as expressed by Eq. (8)]. Renew cost (c), vehicle weight (x), and traffic demand (K) are modeled as random vari-

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x , and n K denote the random ables in the optimization process.  nc , n samples of these variables, respectively. The optimization formulation for each sample set is summarized in Table 2. There are two general types of methods to solve the mixed integer programming problem stated in Eqs (2)–(8): exact methods and approximate methods. For example, the branch-and-bound (B&B) method is a commonly used exact method using upper and lower bounds to discard the fruitless branches [9,41,25]. Due to the lack of tight upper and lower bounds in mixed binary programming problems of this kind, this method, however, is inefficient, especially when network is large and complex. Whether the upper and lower bounds are ‘‘tight” depends on the nature of the objective function and its constraints. The most widely used approximate methods for solving mixed binary programming problems are categorized as metaheuristic, including Tabu Search, Simulated Annealing [27], One-pass [21], Greedy and Local Search [34], Particle Swarm Optimization [28] and Genetic Algorithm [23]. Binary Particle Swarm Optimization (BPSO) is a binary version of PSO, which was designed to solve discrete optimization problems [29] and has been proved to perform better than Genetic Algorithms in several research studies [22,46,44,13,42,16]. In BPSO, each state variable is modeled as a particle; all possible positions of the particles define the feasible solution space; BPSO iteratively updates the positions of all ‘‘particles” to search for a better solution according to its mathematical formulations. For our problem, each ‘‘particle” is a vector of maintenance decision variables. The maintenance cost, vehicle weight, and traffic demand (c, x, and K) are considered as random variables in optimization, making the travel time efficiency [as defined in Eq. (1) and computed in Eqs. (2)–(8)] uncertain in nature. Monte Carlo Simulation (MCS) is employed to generate random sample set of these variables (i.e.  nc ,  nx , and  nK ) in the optimization process. This optimization process is summarized in the flow chart presented in Fig. 1. 4.2. Project priority indices Due to budget consideration and limited available resources, it is usually the case that only a portion of deficient bridges can be scheduled for renewal or replacement. Bridges that have a larger impact on the overall network performance should be prioritized for such activities. With the optimization model formulated, we introduce two type of priority indices for individual bridge project ranking - static priority index (SPI) and dynamic priority index (DPI). Both indices take the advantage of the analysis result from the optimization model but with different perspectives. The static priority index, SPI, is defined as a function of the difference in network travel time efficiency between block running (with reduced load carrying capacity before repair) and smooth running (with design-level load carrying capacity after repair) of the bridge considered, and can be calculated as:

SPIij ¼

E½/0   E½/ij  E½/0   minfE½/ij g

ð9Þ

ði;jÞ2B

where E½ is the mean operator; /0 is the service value of the network without any renewal/replacement activities (i.e. the network is in its as-is condition); /ij represents service value of the network when only bridge ði; jÞ 2 B is selected for renewal/replacement; minði;jÞ2B fE½/ij g is the minimum of all /ij . The priority index defined by Eq. (9) reflects the net (or ‘‘absolute”) impact of the renewal of bridge ði; jÞ on the network efficiency. SPIij will take on values between 0 to 1, with 1 being the top priority. The value of /ij is obtained by solving the proposed optimization problem with fix yij ¼ 1 for the bridge of interest and yij ¼ 0 for all the other bridges. The budget constraint is relaxed to infinity for computing SPI for

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W. Zhang, N. Wang / Structural Safety 67 (2017) 96–104

Table 2 Summary of the optimization formulation. Description

Equation P

Global objective: (minimizing total travel time) Global constraint: (network flow equilibrium)

t2Kð nk Þ ðOt :Dt Þ2W

ði;jÞ2B

P

t2Kð nk Þ ði;jÞ2A

dij

(3)

t

v ij xij

i ¼ Ot nk Þ i ¼ Dt ; 8i 2 N; 8t 2 Kð otherwise

(4)

(8)

ij

0 6 xtij 6 1; 8ði; jÞ 2 A;

(6)

8t 2 Kðnk Þ

(5)

yij ¼ f0; 1g; 8ði; jÞ 2 B

velocities, particle best fitness value (pbest), and particle best

Take the positions of particles into optimization system and compute the fitness value by solving the linear programming

No

Keep the previous pbest

No

Keep the previous gbest

Yes Update pbest with current positions

Is current fitness value better than gbest?

¼ min

P

xt ðnx Þsgnðxtij Þ 6 ð1  yij Þlij þ yij lpij ; 8ði; jÞ 2 B; 8t 2 Kðnk Þ

A bridge network with property variables generated using Monte Carlo simulation

Is current fitness value better than pbest?

T tOt Dt

8 < 1; P t P t xij  xji ¼ 1; : ði;jÞ2A ðj;iÞ2A 0; P  cij ðnc Þy 6 H

Global constraint: (budgetary limit) Local constraint: (reduced live load capacity of deficient bridges) Decision variable 1: (continuous flow variable to reflect path choices) Decision variable 2: (binary variable to reflect maintenance decisions)

Initialize the population best fitness value (gbest) to infinity and best particle as

Eq. No.

P

Yes

(7)

where S is the total sample size used in MCS; s denotes the sample ID. yijjs are binary variables, where yijjs ¼ 1 if bridge ði; jÞ is selected for renewal/replacement in the optimization with the sample set s and the fixed budget H, and yijjs ¼ 0 otherwise; /s = service value of the network resulting from the optimization with the sample set s and the fixed budget H. Note that we do not impose any renewal/replacement decisions on any network bridge in computing for DPI, and only solve the optimization problem for all S set of random samples. In contrast to the SPI, DPI not only depends on the characteristics of the network, but also on the available budget and which bridges are selected under this budget; in this sense, it is a dynamic, situation-specific bridge importance measure which ensures that the selected bridges collectively maximize the network performance for a given budgetary limit. When decision makers have a clearly determined budget limit to work with, the DPI is likely to result in bridge selections and prioritizations that can improve the network performance beyond what SPI would provide. Comparison of these two ranking mechanisms is further illustrated next through an example. 5. Numerical application

Update gbest with current positions

No

All particles are computed? Yes

Meet the stop criteria?

Yes

End and output the gbest

No

Fig. 1. Flow chart of binary particle swarm optimization.

each bridge in the network. We refer this index as static because it can be viewed as an ‘‘absolute” importance measure of each bridge within the network in term of its impact on network functionality. In contrast, the dynamic priority index, DPI, is defined as a function of the likelihood of a bridge being selected for repair for a given maintenance budget when the uncertainties in the transportation network are considered. It can be calculated as:

DPIijjH

 9 8 S  > X >  > > > > y / > ijjs s  > <  = s¼1  ¼ H> S > X  > > > > > /s  > : ; s¼1

ð10Þ

To illustrate the application of the proposed methodology and priority ranking measures, a hypothetical bridge network is generated, in which the nodes represent origins, destinations, and major road intersections, and links are roads, with or without bridges. There are 600 roads in the network, 160 of which contain a bridge on their road path. We divide the network into nine equal-area regions and randomly select one node from each region to represent the regional business hub with assumed mean traffic supply/demand as listed in Table 3 which is randomly generated on a uniform distribution on [400, 3000], for the purpose of illustration. In reality, the O-D matrix can be estimated from daily traffic measurements collected by state department of transportation or local road/bridge owners, or from a model simulation of traffic patterns based on assumed users travel routines. Table 3 Mean traffic demand/supply at O-D nodes. Node (City)

Mean Traffic supply (O)

Mean Traffic demand (D)

3 11 19 28 30 35 37 41 49

2000 1700 1000 600 4500 800 1000 1200 1600

1400 3000 1500 900 3600 400 700 1800 1100

Total

14400

14400

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A capacity rating factor less than 1.0 is assumed for 22% of the bridges randomly selected in the network; these bridges would be posted if not renewed or replaced (e.g., 19% bridges in Georgia are structurally deficient; 28% bridges in Oklahoma are so classified). In addition, a condition rating less than or equal to 4 (as described in Table 1) is assigned to another 3% of the bridges which, if no maintenance activities were performed, would be closed due to severe deterioration. Accordingly, 40 out of the 160 network bridges are the candidates for renewal if budget for network maintenance is sufficient. We assume that the remaining bridges can meet network traffic demands without any maintenance actions. Distributions used to generate the values of network parameters (i.e., capacity rating factor, speed limits on roads, traffic supply/ demand, vehicle weights and bridge capacity ratings) are reported in Table 4. It must be noted that the capacity rating factor (RF) and speed limit (v ) are only sampled once from their distributions in order to generate the network characteristics for analysis, and these are then treated as deterministic in the optimization process; on the other hand, the traffic supply/demand (K), vehicle weight (x), and renewal cost (c), are modeled as normal random variables in the optimization procedure through MCS. In particular, the mean of traffic supply/demand are summarized in Table 3; the mean weight of the vehicles in the network is 65kips [38]; and the mean renewal cost is positively proportional to the capacity RF and deck area of the bridge [18]. In total, 14,458 random variables (including supply and demand of 9 nodes, the weights of 14,400 vehicles and the renewal cost of the 40 candidate bridges for retrofit) are sampled in the optimization procedure summarized in Table 2. The BPSO method is advantageous for this large virtual network optimization problem, as discussed previously. Before using the BPSO method to solve this problem, we first applied both BPSO and B&B methods to three smaller bridge networks consisting of only 5, 8 and 10 nodes, respectively, with ten sets of network parameters for each of three network as tabulated in Appendix A. The number of particles used in the BPSO is 20 and the maximum iteration is 100. An early termination criterion is employed which stops the search when the objective value is not improved for 10 consecutive iterations. In all cases, the BPSO yields the same optimal solution as the exact B&B method, with less computing time (detailed comparison can be found in the table in Appendix A). 5.1. Project selection and prioritization Upon the validation, we applied the BPSO algorithm to the much larger hypothetical network, utilizing a sample size of 10,000 in MCS to account for the uncertainties in maintenance cost, vehicle weight and traffic supply/demand for each O-D pair. We consider the sample size of 10,000 sufficient since our bridge ranking mechanisms, both SPI and DPI, are based on the mean estimates of the metric. It was found that the mean service value for the ‘‘as-designed” network (all bridges are ‘‘new” without deterioration or weight posting), /p , and that for the ‘‘as-is” network (degraded network without maintenance), /0 , are 125,926 h and 288,541 h, respectively. That is, the network travel time in its cur-

rent ‘‘as-is” condition is 2.29 times larger than that of the network when it was newly built, corresponding to a RT ¼ 0:44. The optimization results indicate that with the 150 cost unit, on average 13 bridges can be selected for renewal and the mean total travel time of the network, /, decreases to 1:63/p , corresponding to a RT ¼ 0:61. The coefficient of variation (COV) of / is approximately 9%, which is greater than the COV of the traffic supply/demand (6%) and smaller than that of vehicle weights (17%) or renewal costs (15%). This seemingly small COV is explained by the following: 1) about 56% vehicles do not travel through the 40 candidate bridges (out of the total 1200 directed arcs of the network); in addition, the weights of approximately 50% vehicles are well below the average bridge load capacity; therefore, the uncertainties associated these vehicles do not have significant impact on the uncertainty of the metric; and more importantly, 2) for each MC sample set, the BPSO always searches for the optimal bridge sets for retrofit, which always yields the maximum value of the metric; therefore the COV of 9% describes the uncertainty of the maximum (extreme) value distribution of the metric resulted from optimization, which is usually much smaller than the COV of the parent distribution of the metric (calculated from random selections of bridges without optimization process). We acknowledge that this COV is conditional on the specific network characteristics assumed in this study, which may not be generalized, however the analysis produce itself is capable of realistic estimation of the metric when real network data is provided. The cumulative distribution function (CDF) of the normalized travel time is presented in Fig. 2. To further investigate the relation between the budget limit and the service metric, multiple budget values are tested with the same network. Fig. 3 illustrates that the normalized service value decreases as the available budget for maintenance increases. As the budget increases to a certain level, approximately 400 cost units in this case, the service metric of the network became 1:0/p (RT ¼ 1), meaning that the network performance recovers to its ‘‘as-new” condition. At this point, the budget is no longer a constraint for network maintenance activities and all 40 deficient bridges can be scheduled for renewal/replacement, as illustrated in Fig. 4. The increase in the number of selected bridges with the increase in the budget shown in Fig. 4 is not simply due to adding more bridges to the selected group associated with a lower budget. As shown in Fig. 5, the likelihood of some bridges being selected, such as bridge (31,42), increases as H increases. Some other bridges that are critical to network performance are always selected, regardless of the budget limit, such as bridge (23,31). Finally, for bridges such as bridge (4,7), the likelihood of being selected decreases initially because a neighboring bridge on an alternative road path might become a more effective candidate for improving the network performance as budget limit increases; however, when the budget become essentially sufficient, they would be selected again. The DPI, as a function of the likelihood of being selected [as defined in Eq. (10)], can reflect the dynamics of the selection process encapsulated in the formulation of the optimization. As an example, Fig. 6 shows the comparison between DPI and SPI for selected bridges for a fixed budget H ¼ 150.

Table 4 Statistics of the network attributes. Parameter

Symbol

Distribution

Mean

Coefficient of variance (COV)

Capacity rating factor Speed limit Traffic supply/demand

RF

Discrete [0.4, 0.5, 0.6, 0.7, 0.8, 0.9] Uniform [50, 60, 70, 80] mph Normal

0.65 65 Table 3

0.20 0.13 0.06

Normal

65 kips

0.17

Normal

16.6 unitsa

0.15

Vehicle weight Renewal cost a

v

rð nK Þ wð nx Þ c Þ cðn

The mean cost assumed to be a function of the deck area and RF of the bridge. The unit is hypothetical for ranking purpose.

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Fig. 2. CDF of the normalized post-maintenance network travel time (with H = 150 units).

Fig. 3. Normalized mean network travel time as a function of budget.

Fig. 5. The likelihood of bridge being selected for renewal as budget increases.

Fig. 6. Comparison of SPI and DPI (H = 150 units).

Apparently, the DPI and SPI result in different project selections and priority rankings. While the maintenance strategy using the SPI resulted in a mean normalized travel time of 1.946, the strategy using the DPI led to a normalized mean travel time of 1.630, representing a 32% improvement over the SPI for a H of 150. 6. Conclusions

Fig. 4. Number of bridges selected for renewal as a function of budget.

This paper presented a framework for optimizing bridge maintenance decisions under budget constraints, which integrates network traffic demand, bridge condition ratings, bridge capacity ratings, and network characteristics (e.g. topology, vehicle speed limit, etc.). The following concluding remarks can be drawn from the study: (a) weight posting of deficient bridges collectively impact the operational performance of a transportation network significantly and must be taken into account in making maintenance decisions to maximize the network performance; (b) BPSO is efficient for solving mixed binary programming problems involved in optimizing maintenance schedules of large transportation networks; and (c) the dynamic priority index (DPI) is a more effective ranking mechanism than the static priority index (SPI) when the available budget for network maintenance is fixed.

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Appendix A. Validation of BPSO on small network instances

Table A Comparison of BPSO and B&B on small network instances. Instances for testing

Number of nodes

Number of vehicles

Number of arcs

/B&B  /BPSO

CPU time of B&B

CPU time of BPSO

1 2 3 4 5 6 7 8 9 10

5

2247 1754 1591 2384 2623 4574 4226 2684 5822 4649

4 10 5 5 8 9 6 5 10 6

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

2.00E03 2.00E03 2.00E03 2.00E03 2.00E03 2.00E03 2.00E03 2.00E03 9.99E04 9.99E04

1.43E06 1.71E06 1.43E06 2.00E06 1.71E06 1.71E06 2.28E06 1.71E06 1.71E06 1.71E06

1 2 3 4 5 6 7 8 9 10

8

1719 6581 2295 4193 6440 3378 2259 2180 4785 2072

9 28 16 12 12 10 21 11 17 22

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.02 11.94 0.03 0.07 0.99 5.47 0.10 0.02 10.97 7.96

0.77 2.11 0.94 0.94 1.46 1.74 1.05 0.87 1.24 1.40

1 2 3 4 5 6 7 8 9 10

10

5075 2197 2210 2637 6424 3089 5456 2313 4677 1792

44 44 17 17 32 44 20 17 43 35

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.08 45.56 1.21 0.69 305.99 639.11 14.15 0.14 62.17 20.49

1.07 3.33 1.27 1.27 2.33 2.77 1.44 1.27 1.88 2.21

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