Computer-Aided Civil and Infrastructure Engineering 21 (2006) 561â572. Comparison of Two Evolutionary Algorithms for Optimization of Bridge Deck Repairs.
Computer-Aided Civil and Infrastructure Engineering 21 (2006) 561–572
Comparison of Two Evolutionary Algorithms for Optimization of Bridge Deck Repairs Hatem Elbehairy, Emad Elbeltagi, Tarek Hegazy∗ & Khaled Soudki Civil Engineering Department, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Abstract: Prioritization of bridge decks for maintenance and repair purposes involves decisions related to project-level repair strategies as well as network-level selection of bridges that bring the highest return on the repair budget. These decisions, however, represent complex optimization problems that traditional optimization techniques are often unable to solve. This article introduces an integrated model for bridge deck repairs with detailed life cycle costs of network-level and project-level decisions. Two evolutionary-based optimization techniques that are capable of handling large-size problems, namely Genetic Algorithms and Shuffled Frog Leaping, are then applied on the model to optimize maintenance and repair decisions. Results of both techniques are compared on case study problems with different numbers of bridges. Based on the results, the benefits of the bridge deck management system are illustrated along with various strategies to improve optimization performance.
1 INTRODUCTION Bridges are vital links in road infrastructure networks and it is important to keep them well maintained despite their difficult operating conditions. Harsh environment, increase in both traffic intensity and heavy transport, and the aging of existing bridges are the main reasons for bridge deterioration. The increase in bridge deterioration and in maintenance costs, in view of the limited funds, have motivated the development of Bridge Management Systems (BMS) (e.g., Pontis, 2001; Hugh, 1995) that support decision makers in allocating the limited funds to top-priority bridges. ∗ To
whom correspondence should be addressed. E-mail: tarek@ engmail.uwaterloo.ca.
While the selection of bridges for prioritization is considered a network-level decision, selection of repair type is considered a project-level decision. Most of the current BMS systems are developed to support either network- or project-level decisions. Dealing with network and project levels separately may lead to neither being optimally decided. According to Thompson et al. (2003), both network-level and project-level decisions are very inter-related, therefore, both levels should be integrated. The work of Johnston and Zia (1983) is a good example of network-level prioritization, where bridges are sorted according to their level of service. The output is a set of candidate bridges for maintenance, repair and rehabilitation (MR&R) activities, without deciding on the repair alternatives or the timing to apply this repair. Also, Jiang (1990) developed a network-level prioritization model based on maximizing the area under the performance curve for bridge networks. On the other hand, project-level decisions mainly focus on selecting the repair methods, estimated costs, and expected improvements. The Finnish project-level BMS (for example, Soderqvist and Veijola, 2000) uses the recommendations from network-level BMS to decide on the repair strategy for individual bridges based on life cycle cost analysis (LCCA). Some of the limited but increasing efforts related to incorporating both network-level and project-level decisions can be found in Liu et al. (1997), Miyamoto (2001), and Hegazy et al. (2004). Thompson et al. (2000a) is an example of a good effort to combine both levels using cost-benefit analysis. However, incorporating projectlevel details into network-level analysis complicates the LCCA and makes traditional optimization tools insufficient to deal with the large formulation involved. This is particularly the case when hundreds of bridges are involved (Jacobs, 1992). For example, if the number of
C 2006 Computer-Aided Civil and Infrastructure Engineering. Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA, and 9600 Garsington Road, Oxford OX4 2DQ, UK.
Elbehairy et al.
bridges is N, the planning horizon is T years, and the number of repair alternatives is R, then the number of possible combinations is RN∗T (Nunoo and Mrawira, 2004). Therefore, the complexity of the problem grows exponentially with the increase in number of bridges, making traditional optimization techniques not able to handle such a huge problem. To handle such a large scale optimization problem, the Genetic Algorithms technique was introduced by many researchers as a non-traditional optimization tool for BMS. Miyamoto et al. (2001), for example, developed the Japan-BMS to detect the deterioration of bridge components and determine the appropriate repair method. In this system, GA technique was used to search for the optimum maintenance plan. Also, Liu et al. (1997) developed a BMS to optimize the selection of a maintenance strategy using GAs. Despite the increasing trends in developing integrated BMS, existing research still has some drawbacks related to modeling complexity, which the present research is attempting to overcome. In addition, new evolutionary techniques, other than GAs, have recently been introduced (e.g., Shuffled Frog Leaping). With all evolutionary techniques being heuristic in nature and targeting nearoptimum solutions, this study attempts to investigate and compare the suitability of two powerful evolutionary techniques to the bridge deck repair optimization problem. Many BMSs have dealt specifically with bridge decks, which is the most deteriorated component of bridges. The project-level BMS of Markow (1994) suggested a life-cycle cost approach to decide on the time and methods of improving bridge decks. At the network-level, Jacobs (1992) used an integer programming approach to optimize long-term bridge deck rehabilitation and replacement plans. At the combined level, Dogaki et al. (2000) presented an optimization model for maintenance planning of reinforced concrete bridge decks using GAs. This model considers selecting the repair method and allows multiple bridge visits during the planning horizon. In this article, a simplified framework for a bridge deck management system (BDMS) that considers both network-level and project-level decisions, as well as user, work zone, and traffic control costs is presented. To deal with the primary challenge of large problem size, two evolutionary-based algorithms: Genetic Algorithms (GAs) and Shuffled Frog Leaping (SFL) are introduced and various experiments are conducted to maximize their effectiveness. A comparison between the GAs and the SFL is then presented for case studies with a different number of bridges to investigate their suitability for infrastructure management applications.
2 INTEGRATED BRIDGE DECK MANAGEMENT SYSTEM The components of a general BMS that incorporates both project-level and network-level decisions include the following (Figure 1):
r Detailed BMS models (time-dependent deteriorar r
tion, repair cost, and repair-dependent improvement). BMS constraints (industry, governmental, political, user defined constraints, project, network, and execution constraints such as deadline, resources, etc.) BMS decision support module (user interface, condition assessment, and LCC optimization).
2.1 BMS models 2.1.1 Deterioration model. A BMS requires a deterioration model that estimates the future decline in bridge condition so that an appropriate rehabilitation strategy can be decided (Sobanjo, 1997). In this research, one of the most common models, the Markovian deterioration model, is used to predict future bridge conditions (Jiang et al., 1988). The Markov deterioration model assumes that bridge condition state takes a value from 0 to 9 (according to FHWA ratings) (FHWA, 1998). Each row of the matrix has two values that represent the probability of the deck to remain in its condition state and the probability of moving to the next lower condition state; the summation of both probabilities equals 1 (i.e., P11 + P12 = 1 (Equation (1)) (Jiang et al., 1988). An important assumption for the probability transition matrix is that the transition to the next lower state is dependent only on current state, and is done in 1 year intervals. Condition state 9
P=
Condition state
562
⎡
9
8
7
6
5
4
P11
P12 P22
0 P23
0 0
0 0
0 0
0
P33
0
0
0
0
P34 P44
0 0
0 0
0 0
P45 P55 0
0 P56 P66
0
0
0
0
0
8 ⎢ ⎢ 0 ⎢ 7 ⎢ 0 ⎢ 6 ⎢ ⎢ 0 ⎢ 5 ⎢ 0 ⎢ 4 ⎢ ⎣ 0
3
0
3
⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ P67 ⎥ ⎦ P77 (1)
For bridge decks, the present study uses the probability transition matrices proposed by Jiang et al. (1988) for different bridge deck materials, highway types, and age ranges.
Two evolutionary algorithms for optimization of bridge decks
563
Fig. 1. Components of a bridge management system.
2.1.2 Cost model. In this article, three repair options are used for bridge decks. According to Seo (1994), repair costs can be estimated as a percentage of initial (or total replacement) cost: light repair, medium repair, and extensive (full replacement) repair. Light repairs are intended to restore the deck surface and include patching, sealing, and cleaning of debris. Medium repairs, on the other hand, involve strengthening or increasing bridge deck thickness, and, as such, may require closure of the bridge to traffic. The third repair option is the extensive repair or deck replacement and requires a complete closure of the bridge to traffic. In the present model, the repair costs associated with the three repair options are estimated to be 28.5%, 65%, and 100%, respectively (as suggested by Seo, 1994). Also in the present model, rough estimates for other cost elements such as user costs, work zone costs, and traffic control costs are included in the life cycle cost optimization. 2.1.3 Improvement model. The impact of each repair option on the condition of a bridge deck is important to be analyzed. As represented by Seo (1994), estimated repair improvements are shown in Table 1. For example, to raise the bridge deck condition from 3 to 5, a medium
Table 1 Impact of repair option on bridge deck condition Condition rating before repair Condition rating Condition Rating after Repair
3, 4 5, 6 7, 8
3, 4
5, 6
7, 8
Light Medium Extensive
– Light Medium
– – Light
repair should be selected while to raise it to condition 7, extensive repair should be selected. 2.2 BMS constraints As shown in Figure 1, several constraints should be taken into account in a BMS including: available technology, governmental, political, user, project, and network constraints. A general BMS should be able to consider all practical constraints imposed on a bridge not only at the project and network levels but also at the user, government, and municipality levels as well. In addition, an important aspect that is overlooked in the existing BMS is to consider several execution constraints such as the order of execution of individual bridges, deadline, resource
Elbehairy et al.
564
(labor, equipment and materials) limits, scheduling of in-house workforce versus employed contractors, and time/cost control. These are important aspects that, while being beyond the scope of this article, are important to consider in the design of a general BMS. 2.3 BMS decision support 2.3.1 Bridge condition assessment. Several rating systems have been developed to asses the condition rating of bridges. Condition rating used in this article was developed by the FHWA (1998), which uses a scale from 0 to 9 for bridge elements, where 9 corresponds to the best condition (like new). It is assumed that bridges are serviceable until the rating reduces to a value of 3 (non-serviceable). Condition ratings are used to describe the existing condition of a bridge. It is considered as the most important phase on which subsequent decisions are based. Bridge condition is determined based on detailed assessment of the deck and all other bridge components, which is a large research area that is outside the scope of this article. 2.3.2 Life-cycle optimization. Having defined the present condition of a network of bridges with the deterioration model, repair alternatives, and improvement model, the proposed BMS uses evolutionary-based optimization techniques to determine an optimum priority list of bridges and their repair methods. The procedure searches for the optimal repair types and the times to perform these repairs, while satisfying the budget and condition rating constraints. As an optimization problem, an objective function is constructed by summing up the present values of the annual cost of repairs for all bridges, user, work zone, and traffic control costs (Equation (2)). The objective function is to minimize the total life cycle cost (TLCC), while maintaining acceptable bridge condition: N T 1 Min TLCC = (Cti + UCti ) (1 + r )t t=1 i=1
N + (WZi + TCi ) (2) i=1
where, Cti is the repair cost of bridge i at time t, r the discount rate, T the number of years, N the number of bridges, UCti the user cost of bridge i at time t, WZi the work zone cost for bridge i, and TCi is the traffic control cost for bridge i. Both user and work zone costs are subject to further extensive research with background materials provided in Thompson et al. (2000b) and Najafi and Soares (2001). Added to the objective function, the proposed BDMS accounts for the following constraints:
Fig. 2. Solution structure.
1. Yearly N repair costs should be ≤ yearly budget limits; i=1 Ct ≤ (AB)t , 2. Condition rating for individual bridges ≥ 3 (or N pre-defined user desirable value) condi=1 ≥ min cond; 3. Network overall condition rating is ≥ pre-defined user desirable value; 4. Repair method used in a specific year for a specific bridge = user forced value; and 5. The number of repair visits to a certain bridge can be constrained to a user-desirable maximum number. With the above defined variables, the number of variables becomes N × T. A solution for the problem is structured as a string of elements equal to the number of variables as shown in Figure 2. Each element can take an integer value from 0 to 3 corresponding to one of the repair options (0 = nothing, 1 = light repair, 2 = medium repair, and 3 = extensive repair).
3 EVOLUTIONARY ALGORITHMS The difficulties associated with using mathematical optimization on large scale problems have contributed to the development of alternative solutions. Linear programming and dynamic programming techniques, for example, often fail (or reach local optimum) in solving NP-hard problems with large numbers of variables and non-linear objective functions (Lovbjerg, 2002). To overcome these problems, researchers have proposed evolutionary-based algorithms for searching near-optimum solutions. Evolutionary Algorithms (EAs) are stochastic search methods that mimic the metaphor of natural biological evolution and/or the social behavior of species. The behavior of such species is guided by learning, adaptation, and evolution (Lovbjerg, 2002). To mimic the efficient behavior of these species, various researchers have developed computational systems that seek faster
Two evolutionary algorithms for optimization of bridge decks
and more robust solutions to solve complex optimization problems. The first evolutionary-based technique introduced in the literature were the GAs (Holland, 1975; Goldberg, 1989; Adeli and Hung, 1995). In an attempt to reduce processing time and improve the quality of solutions, particularly to avoid local optima, other EAs have been introduced during the past 10 years, including various GA improvements (e.g., Adeli and Cheng, 1993) and a recently developed technique: shuffled frog-leaping (Eusuff and Lansey, 2003). In general, EAs share a common approach for their application to a given problem. The problem usually requires some representation to suit each method, then, the evolutionary search algorithm is applied iteratively to arrive at optimum or near-optimum solution. Elbeltagi et al. (2005) compared the performance of five evolutionary algorithms for solving general optimization problems and reported the powerful performance of the GAs and SFL techniques. A brief description of these two algorithms is presented in the following subsections. 3.1 GAs GAs are inspired by the biological systems improved fitness through evolution (Holland, 1975). A solution to a given problem is represented in the form of a string (Figure 2), called a “chromosome,” consisting of a set of elements, called a “genes,” that hold a set of values for the optimization variables (Goldberg, 1989). GAs technique is to work with a random population of solutions (chromosomes). The fitness of each chromosome is determined by evaluating it against an objective function (Equation (2)). To simulate the natural “survival of the fittest” process, the best chromosomes exchange information, through crossover or mutation, to produce offspring chromosomes. The offspring solutions are then evaluated and used to evolve the population if they provide better solutions than their parents. Usually, the process is continued for a large number of generations to obtain a best-fit (near-optimum) solution. Crossover among parent chromosomes is a common natural process and traditionally is given a rate that ranges between 0.6 and 1.0. In crossover, the exchange of parents’ information produces an offspring, as shown in Figure 3. To avoid stagnation in the evolutionary process, a mutation rate (< 0.1) is usually used (Goldberg, 1989). The GA used in this article is a steady-state (an offspring replaces the worst chromosome only if is better than it) and real coded (the variables are represented in real numbers) algorithm. Based on this discussion, the main parameters used in the GA procedure are: population
565
Fig. 3. Crossover operation to generate offspring.
size, number of generations, crossover rate, and mutation rate. 3.2 SFL algorithm The SFL is another heuristic search algorithm. It attempts to balance between a wide scan of a large solution space and also a deep search of promising locations for a global optimum. As such, in the SFL, the population consists of a set of frogs (solutions) each having the same solution structure as in the GA technique (Figure 2). The whole population of frogs is then partitioned into subsets referred to as memeplexes. The different memeplexes are considered as different cultures of frogs that are located at different places in the solution space (i.e., global search). Each culture of frogs performs a deep local search. Within each memeplex, the individual frogs hold information that can be influenced by the information of their frogs within their memeplex, and evolve through a process of change of information among frogs from different memeplexes. After a defined number of evolutionary steps, information is passed among memeplexes in a shuffling process (Eusuff and Lansey, 2003). The local search and the shuffling processes (global relocation) continue until a defined convergence criterion is satisfied (Eusuff and Lansey, 2003). As explained, the SFL formulation places emphasis on both global and local search strategies, which is one of its major advantages. As shown in Figure 4a, the SFL algorithm starts with an initial population of “P” frogs created randomly. Frog i is represented as X i = (xi1 , xi2 , . . . , xiS ); where S represents the number of variables. Afterwards, the frogs are sorted in a descending order according to their fitness. Then, the entire population is divided into m memeplexes, each containing n frogs (i.e., P = m × n). In this process, the first frog goes to the first memeplex, the second frog goes to the second memeplex, frog m goes to the m memeplex, and frog m + 1 goes to the first memeplex, etc. Within each local memeplex (Figure 4b), the frogs with the best and the worst fitness are identified as X b and X w ,
Elbehairy et al.
566
Fig. 4. Flowchart for Shuffled Frog Leaping (SFL) algorithm.
respectively. Also, the frog with the global best fitness (the overall best frog) is identified as X g . Then, an evolutionary process is applied to improve only the frog with the worst fitness (not all frogs) in each cycle. Accordingly, each frog updates its position to catch up with the best frog as follows: Change in frog position (Di ) = rand( ) · (Xb − Xw ) (3) New position Xw = Current position Xw + Di ; Dmax ≥ Di ≥ −Dmax
(4)
Where rand( ) is a random number between 0 and 1, and Dmax is the maximum allowed change in frog’s position. If this process produces a better solution, it replaces the worst frog. Otherwise, the calculations in Equations (3) and (4) are repeated with respect to the global best frog (i.e., X g replaces X b ). If no improvement becomes possible in this case, then a new solution is randomly generated to replace the worst frog. The calculations then continue for a specific number of iterations (Eusuff and Lansey, 2003). Accordingly, the main
parameters of SFL are: population size (P), number of memeplexes, number of generations for each memeplex before shuffling, number of shuffling iterations, and maximum step size. More discussions about the SFL and its variations can be found in Elbeltagi et al. (2006).
4 BDMS PROTOTYPE AND IMPLEMENTATION The proposed BDMS was implemented on a commercial spreadsheet program (Microsoft Excel). The developed application includes different forms and worksheets. The data of a network of bridges are input to the BDMS as shown in Figure 5. For each bridge, inputs are: construction year, initial cost, deck type (steel or concrete), highway type (interstate or other), average daily traffic (ADT), width, length, and inspected condition (current condition). The user can force a desirable repair decision by defining the repair type and year. This provides the flexibility to cater for technical/environmental/political constraints on using a specific repair type in a certain year. Also, one of the flexible options in the proposed
Two evolutionary algorithms for optimization of bridge decks
567
Fig. 5. Main worksheet showing user inputs.
model is that the number of repair visits to a certain bridge can be constrained to a user desired maximum number, for practicality reasons to reduce traffic disruption (Figure 5). Once the bridges’ data are input, the user starts the optimization by defining the optimization constraints as shown in the user form of Figure 6: Networkrelated constraints (e.g., minimum overall rating for the network = 6.5 and minimum rating for individual bridges = 4.5), and organization-related constraints (e.g., yearly budget limit, yearly discount rate, and the percentages of full replacement associated with the different repair options). These inputs are then fixed during the optimization process; however, the user can change these values and re-optimize to examine the sensitivity of the results to budget limits, for example. Once the constraints are set, the evolutionary process starts and continues until the stop criterion is reached. A sample output for a network of 50 bridges is shown in Figure 7. Figure 7a shows the cell for the TLCC, which is linked by formulas to all the other parts of the model (TLCC reached a minimum of $27,435,000). Figure 7b shows the cells associated with the variables of the model (five variables for each bridge). The values inside these cells represent the repair decisions (0, 1, 2, or 3) for each year of the 5-year planning horizon. Also, the deck condition ratings before and after the repair are shown in Figures 7c and d, respectively. For example, bridge 2 which had high condition rating before the optimization (7.08) is not selected for repair (decisions are zeroes in all years). Accordingly, its condition will deteriorate in the following years (Figure 7d). Another example of bridges that started with low condition ratings is seen in bridges 1 and 9 (both were 4.5 before optimization).
Fig. 6. Optimization constraints for a network of 50 bridges.
Accordingly, the cheapest repair strategies are shown in Figure 7b, with multiple repairs along the planning horizon (i.e., no constraint on the number of visits was used). Figure 7e shows the repair costs associated with the repair decisions. For example, the repair decision for bridge 1 at year 1 is to perform medium repair, therefore the repair cost is estimated to be $200,000 (based on the percentages defined in Figure 6). The allowable yearly budget and the resulting annual repair costs are
Elbehairy et al.
568
Fig. 7. Sample output for a bridge network.
shown in the top of Figure 7. It should be noted that the resulting annual repair costs are less than the budget limits. The overall network condition was 6.52, which is greater than the constraint value (6.5, Figure 6). Similarly, all individual bridges show a rating in each year that is higher than the constraint used (4.5, Figure 6). As shown in Figure 7, the developed BDMS is transparent and has many flexible and practical features. The user can manually input a repair decision and instantaneously the model presents the implications in terms of cost, bridge condition, and the overall network condition.
5 EXPERIMENTATION WITH EVOLUTIONARY ALGORITHMS Using the VBA Macro programming language of Microsoft Excel, the GA and SFL algorithms were coded and integrated with the developed BDMS. Since the developed GA algorithm will be a baseline for judging the performance of the SFL, it was tested against two commercial GAs software systems (Evolver, 1996; Gene Hunter, 1999), which are Microsoft Excel add-on programs. Experimenting with both systems showed that the developed GA performs equally well. The benefit of the developed GA code as opposed to the commercial programs, however, lies in its flexibility to accommodate
any adjustments to the algorithm to suit the problem at hand, as mentioned later. Initial experiments using the developed GA and SFL algorithms were then conducted on a network of 50 bridges to have a feel of their performance when minimizing the 5-year TLCC. The results of these experiments were unsatisfactory and the following observations were made on the performance of both algorithms: - The large number of variables involved even in a 50-bridge network took a lot of time (around 1 hour) to show improvements. This is because the genes of all population members had randomly generated values that represent random repairs (with associated costs). As such, all starting solutions in the population were excessively violating the budget constraints and it takes many evolutionary cycles to converge; - Although the TLCC (sum of yearly expenditures) was reduced during the evolutionary process and is getting closer to the desired total budget limit, the yearly distribution of the expenditures violates the yearly budget limit; - The best obtained solutions still violated the minimum desirable condition (4.5) for some of the individual bridges; and - The SFL was noticeably better in performance than the GAs. This is due to its use of a step term in
Two evolutionary algorithms for optimization of bridge decks
569
Equation (4) to adjust and refine solutions (i.e., deeper local search). This is opposed to the GAs where the crossover exchanges large portions of the parent chromosome to cause slower refinement to solutions, particularly with a small mutation rate. Due to this poor performance, various additional experiments were used to improve the performance of both algorithms on this typical infrastructure problem. These experiments were structured as follows, and discussed in the following paragraphs: - Examining other objective functions and deciding the most proper one to use with the five-year optimization model; - Developing a pre-processing function to avoid violating the minimum desirable condition of each bridge; - Examining the effect of changing the percentage of non-zero values in the initial solutions; - Determining the best values for the parameters of each algorithm; and - Investigating the use of a year-by-year optimization strategy. Experimenting with different objective functions: In addition to minimizing the TLCC, two other objective functions were experimented with (constraints were modified accordingly) on the 50-bridge network: (a) minimize the budget error, which is the sum of the absolute difference between each year’s repair cost and allowed budget (to force the optimization to avoid violating the yearly budget), and (b) maximize the overall network condition rating (to get the maximum network improvement for the money spent). The best performance obtained in terms of processing time and solution quality was by minimizing the total budget error (Case A). Maximizing the overall network condition consistently over allocated repair money to earlier years, thus violating the yearly budget limit and resulting in unfeasible solutions. Pre-processing of solutions: Once a suitable objective function was selected, several experiments were then carried out using different numbers of bridges: 50, 100, 200, and 400 to compare the performance of both algorithms. With the increase in the number of bridges, the number of variables increased. One of the difficult constraints to meet, due to the random nature of the evolutionary process, is that each bridge needs to maintain a condition that is above the desirable limit of 4.5 in all years. To overcome this problem, a pre-processing function was added to the GA and SFL code. The pre-processing
Fig. 8. Effect of non-zeroes in the initial chromosomes and frogs.
function is simple and does not change the process; rather, it may make a minor adjustment (if necessary) to some of the no-repair (i.e., zero) values of any solution generated during the evolutionary process (population members, offspring members, and frogs). The function checks the consequent conditions associated with the initial values of a solution, then forces a repair (changes the zero value to a 1) to any bridge that has a condition lower than the minimum acceptable level at any year. This process ensures that repair funds are first spent on the must-repair bridges first, then randomly on the rest of bridges. It also ensures that this constraint is not violated during the optimization. Adjusting initial solutions: Because bridge management problems involve strict budget limits, only a small proportion of the bridges is expected to receive repairs. As such, the final solution string is expected to have a lot of zeroes, which is not the case in the totally random manner in which initial solutions (population members and frogs) are generated. Thus, a simple approach to speed the evolutionary process is by generating each initial solution with all zeros except a percentage of random non-zero values. This percentage can be roughly estimated as the ratio of available budge to the total repair need. To examine the validity of this approach and to determine the most suitable percentage for each algorithm, different percentages of non-zero values (10%, 20%, 40%, 55%, 70%, and 85%) were introduced in the initial population and frogs and were experimented with on the 50-bridge network. Based on the results shown in Figure 8, 70% and 20% of non-zero values were the best percentages for the GA and SFL, respectively.
Elbehairy et al.
570
Table 2 Results of optimization experiments No. of bridges 50 100 200 400 50 100 200 400 50 100 200 400
Budget limit
Total repair cost
Sum of budget errors
Network condition
Results of GAs—one Five-Year Optimization (minimize budget error) $30,000,000 $29,910,000 $356,666 6.13 $60,000,000 $59,651,111 $784,444 6.11 $120,000,000 $119,844,444 $571,111 6.08 $240,000,000 $239,712,000 $1,156,000 6.08 Results of SFL—one Five-Year Optimization (minimize budget error) $30,000,000 $30,026,667 $173,333 6.11 $60,000,000 $60,263,333 $543,333 6.10 $120,000,000 $119,960,000 $480,000 6.08 $240,000,000 $239,980,000 $826,666 6.07 Results of GAs—five Year-by-Year Optimizations (maximize condition) $30,000,000 $29,633,333 $366,666 6.84 $60,000,000 $59,733,333 $266,666 6.82 $120,000,000 $119,926,667 $73,333 6.71 $240,000,000 $239,860,000 $140,000 6.57
Further experiments on networks of 50, 100, 200, and 400 bridges were carried out and the results are shown in Figure 9. Ten trial runs were performed for each case and all experiments took place on a 1.8 MHz laptop PC machine. The performance of the two algorithms is compared using the minimum error objective function. Determining best parameter values: After initial experimentation, the best GAs parameters were set as: crossover probability = 0.8, mutation probability = 0.08, population size = 100, and number of generations = 500. This is added to the use of the solution pre-processing procedure and a 70% nonzero value for the population members. Also, the criterion to stop the optimization was set to no improvement in the objective function for 10 consecutive generation cycles (10 × 500 generations). For the SFL algorithm, on the other hand, initial parameter settings were those suggested in the literature (Elbeltagi et al., 2005) and found to work efficiently for the problem at hand: population size = 200 frogs, memeplexes = 20, and iterations = 10 per memeplex. One parameter, maximum step size, however, was reduced based on experimentation from 2 to 1 to improve the SFL performance. This is added to the use of the solution pre-processing procedure and a 20% non-zero value for the frogs. The SFL criterion to stop the optimization was also set to no improvement in the objective function for 10 consecutive shuffling cycles (10 × 20 × 10 generations). Based on these best settings for the
Optimization time (minute) 35 39 64 108 12 23 37 74 25 25 25 50
two algorithms, Figure 10, and the top two sections of Table 2, show the best performance achieved by both the GA and the SFL with different numbers of bridges. The SFL showed very close and slightly better performance than the GAs. Examining a year-by-year optimization: The results in Table 2 clearly show that the network overall condition slightly reduces with the problem size. With the desire to maximize the return on the repair dollar, the 5-year formulation, as mentioned before, was not suitable with an objective function of maximizing the network condition. An alternative optimization strategy was therefore used to consider each year individually in a 5 year-byyear consecutive optimization. Considering the first year, for example, only the variables in that year (the first column of Figure 7b) are involved, with the objective function being to maximize the overall network condition, and only one constraint that the sum of repair costs for that year is within this year’s budget limit. As such, each of the 5 optimizations is much smaller in size and can logically maximize the network’s return on the repair investment made every year. Applying this year-by-year strategy was then experimented with on the different size networks with the pre-processing procedure of fully random values of the population members. The results of this strategy show a substantial improvement in network condition (third section of Table 2). Also, no noticeable difference was observed between the results of GA and SFL.
Two evolutionary algorithms for optimization of bridge decks
571
Fig. 10. Best performance of GA and SFL algorithms.
Fig. 9. Performance of Genetic Algorithms (GA) and SFL using non-zeros in initial solutions.
6 DISCUSSION OF RESULTS Comparing the results of the various network sizes is made possible by the fact that the 50, 100, 200, and 400-bridge networks are multiples of a single 10-bridge network. As such, the optimization experiments on the larger networks had a defined solution against which the optimization performance is judged. The results of using the GAs and the SFL algorithms shown in Figure 9 show that each algorithm can perform better, given the proper setup for its parameters. Also, the best results in Figure 10 and Table 2 show that the two algorithms are consistent in producing results that correspond to the objective function used. Both algorithms could allocate the repair funds efficiently (the total repair cost in the third column of Table 2 is very close to the budget limit in the second column). Based on the experiments made, the best optimization strategy for this typical infrastructure problem is a year-by-year strategy. This strategy could determine an overall network condition of 6.84 for the 50-bridge network (Table 2), which is higher than comparable values obtained by any trials of the single 5-year optimization strategy (best was 6.13). This strategy, coupled with the use of the pre-processing function, also worked
well for the larger networks with additional processing time. As shown by the slight degradation of the overall network condition as the network size increases (from 6.84 for 50 bridges to 6.57 for 400 bridges), problem size still represents a challenge for optimizing bridge maintenance and repair decisions. The complexity of the problem substantially increases as the number of bridges increases and the solution space becomes too large. For example, in the case of 400 bridges, the number of possible solutions is 42000 , which is extremely large. The problem is also expected to become even much larger if the model is expanded to the case of multiple bridge components (e.g., deck, substructure, and superstructure). As such, some strategies may need to be applied to reduce the number of bridges to include in the optimization. One approach is to sort the bridges according to their current condition, and optimize the decisions only for the set of bridges with a condition below a threshold value (for example, less than 6). This is expected to reduce the number of bridges for the optimization process and accordingly improve the quality of solutions.
7 CONCLUDING REMARKS A flexible and transparent BDMS that integrates both project-level and network-level decisions was developed. Two evolutionary techniques, GAs and SFL were used in the BDMS for optimizing maintenance and repair activities for bridge decks. Ten trial runs with different numbers of bridges were experimented with to evaluate the performance of both the GAs and the SFL. For the problem at hand, the results of the experiments showed that both techniques are equally suitable for dealing with the problem at hand. The key issue is to determine the set of parameters that optimize performance. Also, based on the experiments made and the
Elbehairy et al.
572
various approaches used to improve the optimization performance, the best optimization strategy for this typical infrastructure problem is a year-by-year optimization strategy, coupled with the use of a pre-processing function to allocate repair funds first to critical bridges.
REFERENCES Adeli, H. & Cheng, N.-T. (1993), Integrated genetic algorithm for optimization of space structures, Journal of Aerospace Engineering, ASCE, 6(4), 315–28. Adeli, H. & Hung, S. L. (1995), Machine Learning—Neural Networks, Genetic Algorithms, and Fuzzy Sets, John Wiley & Sons, New York. Dogaki, M., Furuta, H., Tsukiyama, I. & Frangopol, D. (2000), Optimal Maintenance Planning of Reinforced Concrete Decks on Highway Network, Life Cycle Cost Analysis and Design of Civil Infrastructure, ASCE, 107–19. Elbeltagi, E., Hegazy, T. & Grierson, D. (2005), Comparison among five evolutionary-based optimization algorithms, Journal of Advanced Engineering Informatics, Elsevier Science, 19(1), 43–53. Elbeltagi, E., Hegazy, T. & Grierson, D. (2006), A modified shuffled-frog-leaping optimization algorithm: Applications to project management, Structure and Infrastructure Engineering, Taylor and Francis, Accepted for Publication. Eusuff, M. M. & Lansey, E. (2003), Optimization of water distribution network design using the shuffled frog leaping algorithm, Journal of Water Resources Planning and Management, ASCE, 129(3), 210–25. Federal Highway Administration. (1998), Recording and Coding Guide for the Structure Inventory and Appraisal of the Nation’s Bridges, Report FHWA-ED-89-044, U.S. Department of Transportation, Washington, DC. Goldberg, D. (1989), Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, New York. Hegazy, T., Elbeltagi, E. & El-Behairy, H. (2004), Bridge deck management system with integrated life cycle cost optimization, in Transportation Research Record, vol. 1866, TRB, National Research Council, Washington, DC, pp. 44–50. Holland, J. (1975), Adaptation in Natural and Artificial System, The University of Michigan Press, Ann Arbor, MI. Hugh, H. (1995), BRIDGIT deterioration models, in Transportation Research Record, vol. 1490, TRB, National Research Council, Washington, DC, pp. 19–22. Jacobs, T. J. (1992), Optimal long-term scheduling of bridge deck replacement and rehabilitation, Journal of Transportation Engineering, ASCE, 118(2), 312–22. Jiang, Y. (1990), The development of performance prediction and optimization models for bridge management systems, Ph.D. thesis, Purdue University.
Jiang, Y., Saito, M. & Sinha, K. (1988), Bridge performance prediction model using Markov chain, in Transportation Research Record, vol. 1180, TRB, National Research Council, Washington, DC, pp. 25–32. Johnston, D. & Zia, P. (1983), A Level-of-Service System for Bridge Evaluation, Report prepared for the North Carolina Department of Transportation, North Carolina State University, USA. Liu, C., Hammmad, A. & Itoh, Y. (1997), Maintenance strategy optimization of bridge decks using genetic algorithm, Journal of Transportation Engineering, ASCE, 123(2), 91–100. Lovbjerg, M. (2002), Improving particle swarm optimization by hybridization of stochastic search heuristics and selforganized criticality, Masters Thesis, Department of Computer Science, Aarhus Universitet, Denmark. Markow, M. J. (1994), Optimal rehabilitation times for concrete bridge decks, in Transportation Research Record, vol. 1394, TRB, National Research Council, Washington, DC, pp. 79–89. Miyamoto, A. (2001), Development of a bridge management system (J-BMS) in Japan, Life Cycle Cost Analysis and Design of Civil Infrastructure, ASCE, 179–221. Miyamoto, A., Kawamura, K. & Nakamura, H. (2001), Development of a bridge management system for existing bridges, Advances in Engineering Software, Elsevier, 32(11), 821–33. Najafi, F. & Soares, R. (2001), User costs at the work zone, Canadian Journal of Civil Engineering, 28, 747–51. Nunoo, C. & Mrawira, D. (2004), Shuffled complex evolution algorithms in infrastructure works programming, Journal of Computing in Civil Engineering, ASCE, 18(3), 257–66. Pontis, (2001), Technical Manual Report, Version 4.0, FHWA, U.S. Department of Transportation. Seo, J. H. (1994), A Lagrangean relaxation and network approach to large-scale optimization for bridge management systems, Ph.D. thesis, Texas A&M University. Sobanjo, J. O. (1997), A neural network approach to modeling bridge deterioration, in Proceeding of the 4th Congress on Computing in Civil Engineering, ASCE, Philadelphia, PA, pp. 623–26. Soderqvist, M.-K. & Veijola, M. (2000), Finnish project level bridge management system, in Transportation Research Circular, vol. 498, TRB, National Research Council, Washington, DC, pp. 1–7. Thompson, P., Merlo, T., Kerr, B., Cheethan, A. & Ellis, R. (2000a), The new Ontario bridge management system, in Transportation Research Circular, vol. 498, TRB, National Research Council, Washington, DC, pp. F-6/1–F-6/15. Thompson, P., Soares, R., Choung, H., Najafi, F. & Kerr, R. (2000b), User cost model for bridge management system, in Transportation Research Record, vol. 1697, TRB, National Research Council, Washington, DC, pp. 6–13. Thompson, P., Sobanjo, J. & Kerr, R. (2003), Florida DOT project-level bridge management models, Journal of Bridge Engineering, ASCE, 8(6), 345–52.
