Multiobjective Optimization of Bridge Deck Rehabilitation Using a Genetic Algorithm

Microcomputers in Civil Engineering12 (1997) 431–443

Multiobjective Optimization of Bridge DeckRehabilitation Using a Genetic Algorithm

Chunlu Liu, Amin Hammad

Department of Civil Engineering, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan

&

Yoshito Itoh∗

Center for Integrated Research in Science and Engineering, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan

Abstract: The solution methods of multiobjective optimiza-tion have undergone constant development over the past threedecades. However, the methods available to date are notparticularly robust. Because of the complicated relationshipbetween the rehabilitation cost and deterioration degree ofinfrastructure systems, it is difficult to find a near-optimalsolution using common optimization methods. Since geneticalgorithms work with a population of points, they can capturea number of solutions simultaneously and easily incorporatethe concept of Pareto optimality. In this paper a simple geneticalgorithm with two additional techniques, Pareto optimalityranking and fitness sharing, is implemented for the deck reha-bilitation plan of network-level bridges, aiming to minimizethe total rehabilitation cost and deterioration degree. Thisapproach is illustrated by a simple example and then appliedto a practical bridge system with a large number of bridges.

1 INTRODUCTION

Rehabilitation planning for infrastructure systems representsa resource-intensive activity. From a mathematical perspec-tive, the determination of efficient maintenance plans is asignificant challenge because the problem contains many in-herent interactions among management objectives, system

∗ To whom correspondence should be addressed.

elements, planning period, and so on. Previous research hasmodeled the bridge maintenance problem using a mixed-integer mathematical model to optimally schedule long-termbridge deck rehabilitation and replacement activities.13 Dy-namic programming has been applied to determine the pri-ority of pavement sections for replacement.7 Appropriatemarkovian transition matrices were developed for specificbridge elements and used to determine the probability dis-tribution for the condition assessment of each element.3,14,21

Cesare et al.4 proposed a methodology that requires a stochas-tic model of the deterioration of bridge elements, a reliabil-ity analysis of individual bridges, and an efficient approachfor the selection of repairs within the entire bridge popula-tion. Sobanjo et al.23 presented a decision-support system forselecting the best set of strategies for bridge rehabilitationand replacement on a highway network using uncertaintyapproach. In most of the previous research, cost optimiza-tion has been considered by optimizing a related objectivefunction. However, most of the optimization problems in in-frastructure maintenance have multiple objectives that mayconflict with each other. An example of conflicting objectivesis the minimization of cost and minimization of risk.

The bridge deck is an important part of the bridge that isdirectly subjected to cyclic loading and harsh environmentalconditions. Concrete decks are more widely built than steeldecks. For instance, 269 of 287 bridges in Nagoya City (thethird largest city in Japan) have concrete decks. Accordingto the newest version of the specifications of the Japan Road

© 1997Microcomputers in Civil Engineering. Published by Blackwell Publishers, 350 Main Street, Malden, MA 02148, USA,and 108 Cowley Road, Oxford OX4 1JF, UK.

432 C. Liu, A. Hammad & Y. Itoh

Association,24 the truck load and the minimum concrete deckthickness are 245 kN (25 tf) and 18 cm, respectively. How-ever, a large number of the existing decks have been designedwith smaller thicknesses of 14, 16, or 17 cm according toprevious specifications to support lower truck loads limitedto 196 kN (20 tf). The problem has been worsened by thesharp increase in traffic volume and the large percentage ofoverloaded trucks passing on the bridge decks. The prevail-ing damages to concrete bridge decks are concrete cracking,concrete exfoliation, corrosion of reinforcing bars, and partialdropping of the concrete of the deck where it is in tension.10,22

Another case less frequently seen is the breaking of reinforc-ing bars due to fatigue. In addition, a number of problemsof other bridge elements can be traced back to the deck dur-ing the bridge inspection process.22 For example, corrosionand unexpected deflection or distortion of deck-supportingelements are clues that a deck may not be performing its pro-tective function. Corrosion of supporting elements indicatesthat water may be passing through the deck and attackingthe structure beneath. Deflection or distortion of supportingelements can indicate that unanticipated movements are oc-curring in the bridge decks. For these reasons, the inspection,assessment, and maintenance of concrete bridge decks havebeen given special interest.17,18

Multiobjective optimization extends optimization theoryby permitting multiple objectives to be optimized simulta-neously. In contrast with single-objective optimization prob-lems, there may not exist a single solution that is optimal withrespect to all objectives of the multiobjective optimizationproblem. Instead, there is a set of solutions that is superiorto the rest of the solutions in the search space consideringall objectives, and no solution in this set is absolutely betterthan the other solutions. This set is called thePareto optimalset. In other words, the improvement of any single objectivein one solution of a Pareto optimal set will result in a nega-tive impact on at least one other objective. Several methodsfor generating the Pareto optimal set of a multiobjective op-timization problem have been proposed, such as weightingobjectives, constraint approach, goal programming, and min-imax approach.15,19 The basis of these methods is the trans-formation of the multiobjective optimization problem into asingle-objective optimization problem by combining multi-ple objectives into a single objective or transforming someobjectives into constraints. Then this single-objective opti-mization problem is solved using some optimization tech-nique. In such cases, the optimal solution obtained is highlysensitive to the input data of the problem, such as objectives,variables, and constraints. In addition, the multiobjective op-timization methods available to date are not particularly ro-bust, and none of them performs equally well on a broad classof problems.15

A genetic algorithm(GA) is a search algorithm based onartificial evolution.11 The basic idea behind GAs is to gener-ate a pool of solutions (population), represented by a string

structure (coding structure). Then, in a manner similar to thenatural genetic operators of selection, crossover, and muta-tion, copying, swapping, and modifying of partial strings areapplied to improve the solutions. These genetic operators ofselection, crossover, and mutation on populations of potentialsolutions are the essence of GAs. The selection operator isused to generate the new population of the next generation.The selection probability of a string surviving in the next gen-eration depends on the objective function (i.e., fitness func-tion) value of this solution and the selection method such asproportionate selection.9 The crossover operator exchangesthe string bits after the crossover point between two selectedparent strings to form two offspring strings. Crossover oc-curs with some specified probability. The crossover point israndomly selected along the strings. The mutation operatoroccurs with some specified probability for each bit in thestrings that have undergone crossover.

GAs have been proved to be one of the most effectiveand robust optimization techniques for single-objective opti-mization problems in previous research.9,16 GAs are differentfrom the traditional optimization procedures in the followingaspects9: (1) During the search, GAs will be working with agroup of possible solutions. This means that GAs can pro-cess a large number of bridge rehabilitation plans at the sametime. In contrast, the traditional optimization procedures usu-ally work with only one solution. (2) GAs work with a codingof the parameter set and not with the parameters themselves,thus allowing use of a wide variety of parameters as deci-sion variables. Because the alternatives that can be selectedas bridge maintenance strategy are determinate and finite, thealternative optimization of network-level bridges is a discreteproblem and appropriate for the coding approach of GAs.(3) GAs do not require gradient information of the objectivefunction, which could be very difficult to obtain in bridgerehabilitation plan optimization problems. And (4) GAs im-prove the search process in an adaptive manner using prob-abilistic transition rules, which guide a search easily towarda set of rehabilitation plans with likely improvement. Thesedifferences make the GAs more robust for the optimizationproblem of bridge deck rehabilitation plans at a network level.

Since GAs work with a population of points, they can cap-ture a number of solutions simultaneously and easily incor-porate the concept of Pareto optimality in their evolutionaryprocess. The first practical GA for multiobjective optimiza-tion was developed by Schaffer20 and is called thevector-evaluated genetic algorithm(VEGA). One of the problemswith VEGA is its bias for some solutions at the extremities ofthe Pareto optimal set. Goldberg9 suggested a nondominatedsorting procedure to overcome this weakness of VEGA. Healso suggested that this procedure should be used in conjunc-tion with some technique for maintaining the Pareto optimalset distribution over a larger region. Fonseca and Fleming8

implemented these two suggestions and called a simple GAwith these two suggestions amultiobjective optimization ge-

Multiobjective Optimization of Bridge Deck Rehabilitation 433

Table 1Deterioration conditions

Deterioration Deterioration Deterioration conditionlevel degree D description(1) (2) (3)

I 0.8 – 1.0 Potentially hazardousII 0.6 – 0.8 Obvious deterioration and may

need detailed inspectionIII 0.4 – 0.6 Aggravating deterioration and

may need further investigationIV 0.2 – 0.4 Minor deteriorationV 0.0 – 0.2 Like new

netic algorithm (MOGA). Horn et al.12 and Srinivas andDeb25 also examined these two suggestions through severalexamples of mathematical functions. Because both the pre-ceding suggestions are related to the selection operator only,MOGA differs from a simple GA only in the way the selectionoperator works. In this research, a MOGA is implemented toset up and refine the Pareto optimal set for the rehabilita-tion plan of concrete bridge decks. The optimization aimsat minimizing the total rehabilitation cost and the averagedeterioration degree weighted by the bridge deck area.

2 MULTIOBJECTIVE REHABILITATION PLAN OFBRIDGE DECKS

2.1 Deterioration levels and maintenance methods

According to the results of inspection, the conditions of bridgedecks are assessed by some organizations in Japan accordingto one of the following five deterioration levels, as shown inTable 1:level I: deterioration is serious and is affecting the ser-viceability and traffic safety;level II: deterioration is obvious,and a detailed inspection may be necessary;level III: bridgedeck satisfies the functional and structural requirements, butdeterioration is aggravating, and further investigation is nec-essary;level IV: minor deterioration exists and needs to be puton record, but further investigation is unnecessary; andlevelV: the bridge deck is like new. The deterioration degree is anindex to quantify the deterioration condition of a bridge deckand has been adopted in much previous research.7,16 In thisstudy, each deterioration level is quantified by a deteriorationdegree that varies between 0 and 1, as shown in column 2 ofTable 1. For deterioration levels I, II, III, IV, and V from theinspection data, the values of deterioration degrees are takenas 0.9, 0.7, 0.5, 0.3, and 0.1, respectively.

Researchers and agencies have given different classifica-tions and definitions to maintenance methods such as routinemaintenance, repair, and rehabilitation.10,16 For simplicity,rehabilitation is assumed to be the only possible maintenance

method in this paper. Rehabilitation implies fairly major re-construction of the deck and a large maintenance effort andsometimes closure of the bridge to traffic.1,17,22 Attachingadditional longitudinal girders or steel plates and increasingthe cover thickness of the reinforcement are common reha-bilitation methods. Rehabilitation can be a measure to allowtime to develop plans for deck replacement, and its effect isquantified in the deterioration model in this research by ex-tending the life of the rehabilitated bridge deck, as will beexplained later.

2.2 Deterioration model of concrete bridge decks

There exist a large number of factors influencing the dete-rioration process of concrete bridge decks such as thicknessof the deck, structural type, materials properties, drainagesystem, girder spacing, construction method, age, traffic vol-ume, environmental factors, and so on. However, it is noteasy to represent all these factors in the mathematical formu-lation of the deterioration model. Many deterioration modelswere developed through analyses, tests, and statistical stud-ies. Although some stochastic models such as the Markovchain method have been developed to represent the transi-tion probability from one condition level to another,3,14,20

these models are difficult to adopt at present due to the lackof inspection data. In this research, a nonlinear deteriorationmodel is adopted from refs. 17 and 18, in which all the pre-ceding factors are classified into two categories dependingon whether they have a close relationship with the age of thebridge or not. Two comprehensive parameters,αi andβi , rep-resenting these two categories of bridgei , respectively, areused in this model, as shown in Eq. (1):

d(t, i ) = 1

1+ eαi−βi×t(1)

whered(t, i ) is the predicted deck deterioration degree ofbridgei at aget . This nonlinear deterioration model is con-sistent with observations of bridge deck deterioration overtime.2,17,18 Furthermore, it is a bounded function, ensuringthat the deterioration degree of the deck lies between 0 and 1,which is very suitable for representing the deterioration con-dition of a bridge deck.17,18 Because of the lack of inspectiondata,αi is determined by assuming a value for the initial de-teriorationd(0, i ). The parameterβi is related to the age ofthe bridge and is calculated for each bridge using inspectiondata. The determination ofαi andβi will be discussed indetail later in this paper.

2.3 Objective functions

Two objective functions, total rehabilitation cost and weightedaverage deterioration degree, are to be minimized. The yearlydeck rehabilitation cost of a bridge is calculated using thedeck area and the unit cost of the rehabilitation. The total


costC of a bridge system over the rehabilitation planningperiod is determined by

C =N∑

i=1

T∑t=1

[(1+ r )−t × c× s(i )× n(i, t)] (2)

where N is the number of bridges,T is the length of theplanning period,r represents the discount rate that is assumedto be constant during the planning period,c is the unit areacost of rehabilitation,s(i ) is the deck area of bridgei , and thevalue ofn(i, t) is 1 if a rehabilitation activity is performedon the deck of bridgei at yeart or it is 0 for the case of norehabilitation activity. The rehabilitation cost is calculated atthe beginning of the planning period without considering thepossible changes in unit costs due to inflation.

The second objective function, average deterioration de-greeD over the planning period weighted by the deck areaof each bridge, is formulated as follows:

D =N∑

i=1

T∑t=1

d(t, i )× s(i )

T × S(3)

whereS is the sum of deck areas of all bridges.

3 OPTIMIZATION OF BRIDGE DECKREHABILITATION USING MOGA

The flowchart illustrating the multiobjective optimization ge-netic algorithm (MOGA) implemented in the present studyis shown in Fig. 1. The three modules numbered in this figureare the main processes. These are (1) production of the ini-tial generation and establishment of an initial Pareto optimalset, (2) application of MOGA techniques of Pareto optimalityranking and fitness sharing, and (3) reproduction by selection,crossover, and mutation operators and revision of the Paretooptimal set. These modules are explained in the followingsections. For each generation, MOGA first determines thefitness function of each population individual of the previ-ous generation and then generates strings by selecting twoparents on the basis of their fitness and reproducing them bycrossover and mutation until the whole population is recre-ated. Finally, MOGA decodes and evaluates the strings ofthis new generation and revises the Pareto optimal set of theprevious generation. This procedure is repeated many timesuntil one of the following termination criteria is satisfied: (1)the maximum generation number is reached or (2) the con-vergence index is sufficiently small. Termination criterion 1is necessary to prevent a run with excessively long time. Ter-mination criterion 2 is an important criterion to check theconvergence of the optimization procedure, as will be shownin the numerical example of the next section. In the numer-ical examples, the convergence of the preceding procedureis checked for several cases with different parametric valuesusing a suggested convergence index.

Fig. 1.Flowchart of MOGA.

3.1 Coding and decision variables

An essential feature of GAs is the coding of the decisionvariables that describe the problem. In this research, the re-habilitation actions are used directly to code the GA strings.The string bits of “doing nothing” and “undertaking reha-bilitation action” are defined by the binary values 0 and 1,respectively. As shown in Fig. 2, the string of a rehabilitationplan consists of many substrings representing all bridges ina given order. The string length is the sum of all substrings’lengths. Every substring represents the rehabilitation strategyof a bridge. In a substring, every string bit from left to rightrepresents the rehabilitation action at 1-year intervals fromthe beginning to the end of the planning period.

The Pareto optimal set is first established at the initial gen-eration and then revised with the increase in the generationnumber. For an arbitrary generation, each strategy that can-not be dominated will be added to the Pareto optimal set.The process of selecting the Pareto optimal set is shown inFig. 3. A rehabilitation plani will be added to the Paretooptimal set if there is no other planj that is better than theplani considering both the deterioration degree (det) and the


Fig. 2.An example of the coding structure and crossover and mutation operators.

Fig. 3.Process of selecting the Pareto optimal set.

rehabilitation cost (cost). In this case, the plani is called anondominated individualof the Pareto optimal set.

3.2 Pareto optimality ranking and fitness sharing

Pareto optimality ranking suggested by Goldberg9 is a rank-based fitness assignment method that takes into considerationeach of the different optimization objectives. To illustrate thismethod, an example of a ranked population of 20 rehabili-tation plans, plotted according to rehabilitation cost versusaverage deterioration degree, is shown in Fig. 4. The super-scripts of these individuals are the rank number, and theirsubscripts represent indices. First, all individuals in the cur-rent population are compared, and the nondominated individ-uals are identified and assigned a rank of 1, which is also thePareto optimal set of this population. Then these individualsof rank 1 are set apart, and the remaining individuals are com-pared to select a new nondominated set with a rank of 2. Thisprocess continues until the entire population is ranked. Fi-nally, the fitness function value of each individual is assignedaccording to its rank, as shown in Eq. (4):

fit(i ) = 1

rank(i )(4)

wherefit(i ) andrank(i ) are the fitness function and the ranknumber of individuali , respectively.

Fitness sharing aims to divide the population into subpopu-lations of similar individuals. In a multiobjective optimizationproblem, fitness sharing is useful in stabilizing the multiplesubpopulations that arise along with the Pareto optimal setand preventing excessive competition among distant popula-tion members. For the present research, the cost, one of thetwo objective functions, is divided into several intervals. Eachplan is assigned to an interval, thus forming subpopulationsclasses of solutions. Theshared fitness functionof a plani istaken as its fitness function divided by the number of plansbelonging to its class, as follows:

sharedfit(i ) = f i t (i )

num(i )(5)


Fig. 4.Pareto optimality ranking and class of population individuals.

wherei represents any plan in the population,sharedfit(i ) isthe shared fitness function of plani , andnum(i ) is the numberof plans in the class to which individuali belongs. The sharedfitness function of each plan replaces its fitness function asthe selection criterion.

3.3 Selection, crossover, and mutation

In this research, the fitness proportionate selection is adoptedas the selection method.9 With this method, the rehabilita-tion plans with shared fitness function values that are equalto or greater than the average shared fitness function in thepopulation will survive and be selected to generate the newpopulation individuals of the next generation, while the re-habilitation plans with smaller values will be eliminated inthe selection procedure. As shown in Fig. 2, crossover is in-troduced within every substring corresponding to one bridge,and the number of the crossover points is the same as the num-ber of bridges. This multipoint crossover affects every bridgewith the same probability and accelerates the optimizationprocess. Similarly, the bitwise complement mutation oper-ator changes one binary value to the opposite within everysubstring such as 0 to 1 or 1 to 0. The details of multipointcrossover and multipoint mutation can be found in Liu et al.16

3.4 Performance measure

Performance measure aims to quantify the distribution ofpopulation individuals for any one generation. Deb andGoldberg6 used the chi-square-like deviation measure, whichrepresents the fitness function (phenotypic measure). Collins

and Jefferson5 described another method of performance mea-sure, which focused on the distributions of the genotype alle-les (genotypic measure). In this study, the performance mea-sure at each generation is calculated following the formermethod as

Performance measure=√√√√ q∑

i=1

[m(i )− µ

µ

]2

(6)

wherem(i ) is the actual number of plans in the classi , µ isthe average number of plans in each class, andq is the numberof classes. The smaller the performance measure, the moreuniform is the population distribution. Therefore, there existsa higher possibility for the bridge engineers to find a solutionlocating any given objective function range.

4 APPLICATIONS AND DISCUSSION

In this section, two examples are used to demonstrate thecapability of MOGA of optimizing the rehabilitation plan ofbridge decks. First, an example with six bridges is used to ex-amine the MOGA optimization approach. Then this approachis applied to the rehabilitation plan of the whole bridge systemof Nagoya City. In order to compare several solutions, onlythe maximum generation number is used as the terminatingcondition.


Table 2Bridge deck data used in numerical Example 1

Bridge Number1 2 3 4 5 6

(1) (2) (3) (4) (5) (6) (7)

Length (m) 39.3 27.8 18.4 87.4 26.2 27.0Width (m) 23.7 8.50 7.0 26.0 7.0 12.0Construction year 1957 1961 1968 1971 1979 1982Deterioration degree 0.5 0.5 0.3 0.3 0.1 0.1

at 1992αi 3.892 3.892 3.892 3.892 3.892 3.892βi 0.111 0.126 0.127 0.145 0.130 0.169

4.1 Numerical Example 1

For the purpose of examining the multiobjective optimizationapproach discussed above, a small example with six bridgesin Nagoya City is prepared. Using this example, the calcu-lation results can be checked manually. The data of thesebridges are obtained from the Bridge Inspection Database ofNagoya City, as shown in Table 2. From Eq. (1), the param-eterαi is equal toβi × t + ln[1/d(0, i ) − 1], which is de-termined according to the initial deterioration degreed(0, i )of bridge i at age 0. Estimating this degree is difficult, andit should be different for each bridge deck, since the data ofonly one inspection are available at present. For the purposeof simplicity, a constant value of 0.02 is assumed for the ini-tial deterioration degree for all bridge decks in this research.From this assumption, the parameterαi of all bridge decksbecomes 3.892, as shown in Table 2. The deterioration degreed(t, i ) of each bridge decki in 1992 is used to determine theparameterβi according to{αi − ln[1/d(t, i )− 1]}/t , and theresults are shown in Table 2. The parameters’ values of thedeterioration model of each bridge deck can be adjusted whenmore inspection data are available. According to the presentvalues of these parameters, the deterioration degrees of mostbridge decks will reach 0.98 at an average age of about 60years, which is the design service life of most bridges. Takingbridge 2 as an example, Fig. 5 shows the inspection data anddeterioration process from age 0 to age 70 years assumingthat no rehabilitation is performed.

In the following calculations, the objective functions ofEqs. (2) and (3) are used. It is also assumed that the reha-bilitation can extend the service life of a bridge deck by 10years,22 and its cost is 20,000 yen/m2 (about US$200 perm2). The planning period of deck rehabilitation is taken as 5years in accordance with the rehabilitation planning of otherinfrastructures in Japan. The discount rate is assumed to be1.75% per year during the planning period. This exampleexamines the MOGA process where the GA parameters arebased on the suggestion in ref. 9, as given in column 2 ofTable 3. A moderate population size of 300, a high crossover

Fig. 5. An example of the change in the deterioration degree ofbridge decks.

Table 3GA parameters used in numerical examples

Example1 Example2(1) (2) (3)

Maximum generations 50 100Population size 300 500String length 30 760Probability of crossover 0.8 0.8Probability of mutation 0.01 0.01Number of classes 1, 3, 5 5

probability of 80%, and a low mutation probability of 1% areadopted. These parameters affect the evolutionary process ofthe population of rehabilitation plans and the Pareto optimalset.

The MOGA optimization procedure has been programmedin Fortran. The execution time per run on a SUN SPARC-Station II is only a few minutes. This example is solved byseveral runs of the program. Although the specific results arenot completely identical because of the randomness involvedin GAs, these results are very similar. Given the number of


Fig. 6.Population distribution of MOGA with fitness sharing at generations 0, 10, and 50 for Example 1.

classes as 5, the population distributions of one run at gener-ations 0, 10, and 50 are shown in Fig. 6. It is found that withthe increase in the generation number, the rehabilitation costand deterioration degree of most solutions decrease, and mostsolutions approach the Pareto optimal solutions. The conver-gence of the optimization process can be checked accordingto the change in a convergence index with the generationnumber, as shown in Fig. 7. The convergence index of gen-erationi (i > 0, and generation 0 is the initial generation)is obtained by comparing the individuals of the Pareto opti-

mal set with the Pareto optimal set of the previously adjacentgeneration, as shown in the following equation:

Convergence index= 1

N

N∑n=1

|Di (n)− Di−1(m)|Di (n)

(7)

whereN is the number of individuals in the Pareto optimal setof generationi , Di (n) is the deterioration degree of individ-ual n in the Pareto optimal set of generationi , andDi−1(m)is the deterioration degree of individualm in the Pareto opti-mal set of generationi −1, which has the smallest difference


Fig. 7.Convergence of MOGA with fitness sharing for Example 1.

Fig. 8. Population distribution of VEGA at generation 50 forExample 1.

with individual n from the rehabilitation cost of view. Ac-cording to this figure, it can be found that the convergenceindex is high and fluctuates before generation 30 and thengradually decreases to reach the value of 0.01 at generation40. After generation 40, the convergence index is near 0.003,which means that the Pareto sets of two adjacent generationsare almost same. This convergence index also can be usedas a termination criterion within the optimization process,together with the predefined generation number.

In order to check the effect of Pareto optimality ranking,the preceding MOGA results are compared with results fromVEGA. In VEGA, the selection is performed by consideringthe rehabilitation cost only, one of the two objective func-tions. Figure 8 shows the population distributions accordingto VEGA at the final generation (generation 50). By visual

Fig. 9. Population distribution of MOGA without fitness sharingat generation 50 for Example 1.

comparison of Figs. 6 and 8, MOGA gives better distributionof population plans than VEGA. For instance, in the Paretooptimal set of VEGA, there is no plan whose rehabilitationcost is greater than 250 million yen (about US$2.5 million).However, MOGA can provide a lot of plans satisfying thisrequirement.

The effect of fitness sharing on the optimization results isalso studied by comparing the results at generation 50 withand without fitness sharing, as shown in Fig. 6c and Fig. 9,respectively. In the case without fitness sharing, only oneclass is available. From this comparison, the case with fitnesssharing gives better distribution than the case without fit-ness sharing. Furthermore, this comparison can be done usingthe performance measures, as shown in Fig. 10. Initially, thefour runs of the three methods start with a high-performance


Fig. 10.Comparison by performance measure for Example 1.

measure because the initial population is spread over thesearch space with fewer individuals in some subpopulations.VEGA’s increasing performance measure with the increase inthe generation number indicates its poor distributing ability.The initial decrease is due to the convergence of the popula-tion toward the nondominated region. Up to generation 50,the performance measures of MOGA with classes of 3 and5 are better than the performance measures of VEGA andMOGA without fitness sharing. This means that MOGA canprovide the decision maker with a greater number of alterna-tive solutions at each generation.

One ideal rehabilitation plan can be selected from thePareto optimal set at the final generation (generation 50),as shown in Fig. 6c, according to particular requirements.For example, if a large budget for the deck rehabilitationof these bridges is available, a suggested solution is with arehabilitation cost of 193.5 million yen (about US$1.9 mil-lion) and an average deterioration degree of 0.06. On theother hand, the average deterioration degree is about 0.42 ifno rehabilitation action is taken. A solution with a moderaterehabilitation cost of 88.06 million yen (about US$0.9 mil-lion) and a moderate deterioration degree of 0.16 also can befound from the preceding Pareto optimal set. Furthermore,the bridge engineer also can apply some multiobjective op-timization method such as weighting objectives, constraintapproach, or minimax approach, as mentioned before, to se-lect one satisfactory solution from the final Pareto optimalset and considering other available information.

4.2 Numerical example 2

Because of the lack of data for many bridges, only 152 ofthe 269 bridges with concrete decks from the Bridge Inspec-

tion Database of Nagoya City are studied. These bridges havethe following characteristics: They have concrete decks, theirages are less than 50 years, their lengths vary from 15.1 m to531.8 m, their widths vary from 5 m to 49.2 m, and the inspec-tion data of the deck deterioration conditions are available.These bridge decks were constructed or replaced between1944 and 1991. The initial deterioration levels of bridge decksare assessed aslevel III (11 bridges),level IV(9 bridges), andlevel V (132 bridges). The ages of bridge decks in 1992 andthe number of bridges at each deterioration condition areshown in Fig. 11.

A sensitivity analysis is done to check the effects of thesimple GA parameters, including population size, crossoverprobability, and mutation probability, taking the maximumgeneration number of 100 and the class number of 5. It isfound that the Pareto optimal set will improve with large pop-ulation size, high crossover probability, and moderate muta-tion probability. (Taking the mutation probabilityPm as anexample, Fig. 12 shows its effect on the Pareto optimal set. Ifthe mutation probability is too high, such as 10%, or too low,such as 0.01%, the set of Pareto optima is far from the bestset found whenPm = 1%.) Therefore, a mutation probabilityof about 1% is suitable for this application. The populationsize and the crossover probability are taken here as 500 and80%, respectively. All GA parameters are shown in column3 of Table 3.

Figure 13 shows the population distributions at several gen-erations. It is clear that the solutions are improved with in-crease in the generation number. This means that the MOGAoptimization process is applicable for a bridge system with alarge number of bridges. The set of Pareto optima at gener-ation 100 shows the tradeoff between the rehabilitation costand the deterioration degree. This tradeoff shows the pos-sible rehabilitation plans. The convergence is checked usingthe convergence index, as explained before. The convergenceindex is less than 0.01 after generation 90. One satisfactorysolution can be selected by comparing all solutions in thePareto optimal set according to some additional information.For example, from Fig. 13, if the rehabilitation budget is4000 or 5000 million yen (about US$40 or US$50 million),the possible minimum deterioration degree is 0.09 or 0.07,respectively. This value is relatively small, because the ini-tial deterioration degrees of most bridge decks (132 of 152bridges) are equal to 0.1.

It is notable that the number of possible combinations inthis problem is 2760, and it is impossible to get the entireset of Pareto optimal solutions through full random search.This example is also solved using the random search methodwith a given number. Figure 14 shows a comparison betweenthe random search results with a population size of 50,000individuals and the Pareto optimal set generated by MOGA atgeneration 100 (after considering less than 5000 individuals),which indicates that the MOGA gives far better results.


Fig. 11.Bridge deck data for Example 2.

Fig. 12.Effect of mutation probability for Example 2.

5 CONCLUDING REMARKS

Although there exist a number of classic multiobjective opti-mization techniques, they require detailed problem informa-tion. In this research, a multiobjective optimization geneticalgorithm was used to solve the rehabilitation optimizationproblem of bridge decks. The results suggest that a multiob-jective optimization genetic algorithm can be used success-fully to find the Pareto optimal set, simultaneously satisfy-ing the minimum cost and minimum deterioration degree ofbridge decks. The following conclusions can be stated:

1. A multiobjective optimization genetic algorithm wassuccessful in optimizing bridge deck rehabilitationplans. The optimization approach has been tested andapplied to two examples. It was found that this approachcan find a satisfactory Pareto optimal set in a short cal-culation time.

2. Pareto optimality ranking and fitness sharing were twonecessary techniques to modify the fitness function ofeach population plan, by which a uniform distributionof the population plans evolved with increase in thegeneration number.


Fig. 13.Population distribution at several generations for Example 2.

Fig. 14.Comparison between MOGA and random search for Example 2.

3. The Pareto optimal set at the final generation illustratedthe tradeoff between the rehabilitation cost and dete-rioration degree. This tradeoff provided the decisionmaker with several candidate solutions. According to

this tradeoff, the decision maker can work out his or herrehabilitation plan considering both rehabilitation costand deterioration degree.


ACKNOWLEDGMENT

We would like to thank the Bridge Office of Nagoya CityCivil Bureau for providing data on the bridges of NagoyaCity.

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Multiobjective Optimization of Bridge Deck Rehabilitation Using a Genetic Algorithm

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