A Multi-year Pavement Maintenance Program Using

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A multi-year pavement maintenance program using astochastic simulation-based genetic algorithm approach

Piya Chootinan a, Anthony Chen a,*, Matthew R. Horrocks b, Doyt Bolling c

a Department of Civil and Environmental Engineering, Utah State University, 4110 Old Main Hill, Logan, UT 84322-4110, USAb Horrocks Engineers, American Fork, UT 84003, USA

c Utah Local Technology Assistance Program, Utah State University, Logan, UT 84322-4110, USA

Received 19 April 2005; accepted 15 December 2005

Abstract

The objective of this paper is to introduce a multi-year pavement maintenance programming methodology that canexplicitly account for uncertainty in pavement deterioration. This is accomplished with the development of a simula-tion-based genetic algorithm (GA) approach that is capable of planning the maintenance activities over a multi-year plan-ning period. A stochastic simulation is used to simulate the uncertainty of future pavement conditions based on thecalibrated deterioration model while GA is used to handle the combinatorial nature of the network-level pavement main-tenance programming. The effects of the uncertainty of pavement deterioration on the maintenance program are investi-

gated using a case study. The results show that programming the maintenance activities using only the expected pavementconditions is likely to underestimate the required maintenance budget and overestimate the performance of pavementnetwork. 2005 Elsevier Ltd. All rights reserved.

Keywords: Pavement maintenance programming; Simulation-optimization; Genetic algorithms

1. Introduction

One of the major requirements of pavement management system (PMS) is the ability to develop a multi-year pavement maintenance program for the entire road network under the jurisdiction of the highwayagency. The development of a multi-year pavement maintenance program is highly dependent on the abilityto estimate future pavement conditions (Zimmerman, 1995). Unfortunately, the accuracy of the predictedpavement conditions (as well as other types of infrastructure) could be influenced by the choice of predictionmodel (Durango and Madanat, 2002) as well as the accuracy of inputs of the deterioration model (e.g., futuretraffic load, weather condition, etc.). As a result, the predicted pavement conditions could be subject tosubstantial uncertainty. Because the uncertainty of the predicted pavement conditions contributes to the

0965-8564/$ - see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.tra.2005.12.003

* Corresponding author. Tel.: +1 435 7977109; fax: +1 435 7971185.E-mail address: [email protected] (A. Chen).

Transportation Research Part A 40 (2006) 725743

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reliability of the pavement maintenance plan, these uncertainties (e.g., inputs and prediction model) should becarefully considered when making pavement maintenance decisions (Ben-Akiva et al., 1993). Durango andMadanat (2002) suggested using a family of models to represent the physical deterioration of infrastructure(e.g., pavement). Each model contributes to the predicted value of pavement condition differently accordingto the belief (weight or probability) of decision makers that the model would be a good representation of the

deterioration process at each stage of pavement condition. This belief can be updated when the additionalinformation regarding the actual pavement deterioration becomes available.Although a large number of sophisticated deterioration models have been developed during the past several

decades, the Markov transition probability matrix (TPM) approach appears to be the only model explicitlydeveloped to address the uncertainty issue of the predicted pavement condition. In the Markov process, a pave-ment deteriorates by moving from one state to another state according to the state transition probability spec-ified for each category of pavement condition and type of maintenance undertaken. As opposed to thedeterministic deterioration models, the Markov TPM approach forecasts the distribution of future pavementconditions (i.e., the proportion of pavements in each group of pavement condition) rather than the future con-dition of an individual pavement section (e.g., a single numerical value). Therefore, it is usually applied for thenetwork-level management process. Additional information such as subjective opinions (or rules) of the high-way agency is generally required to implement the TPM approach at the project level (Robert et al., 2002). In

addition, the requirement of comprehensive historical performance data for developing the reliable TPMsappears to limit its application in programming maintenance activities (Wang et al., 1994; Li et al., 1997). Toreduce the amount of data required, Li et al. (1996) suggested using a Monte Carlo simulation (MCS) combinedwith a deterministic deterioration model to develop TPMs for a variety of pavement categories and/or individualpavement sections. In fact, the idea of using MCS allows the stochastic behavior (uncertainty) of the pavementdeterioration to be incorporated explicitly in the modeling process of PMS without the employment of TPMs.

Another challenge in developing a long-term maintenance plan at a network level is the ability to identifyproper maintenance activities for individual pavement sections at the project level that are consistent withthe network-level recommendation (Chan et al., 1994; Mbwana and Turnquist, 1996). This type of problemis usually formulated using integer variables to represent maintenance activities selected for individual pave-ment sections (i.e., integer program). It is also known as the combinatorial problem, which is very difficult to

solve due to a vast solution space (i.e., the number of possible solutions increases exponentially as the problemsize increases). Chan et al. (1994) introduced the genetic algorithm (GA), which is one of the evolutionary com-puting techniques that have shown considerable success in solving a number of complex large-scale problems inmany disciplines, to handle this difficulty in the pavement maintenance programming. In this paper, we presenta simulation-optimization framework for programming the maintenance activities over a multi-year planningperiod. The combination of stochastic simulation and GA allows the development of a project-based network-level maintenance plan that can explicitly take into account the uncertainty of future pavement conditions in thedecision-making process. In other words, the risk that the maintenance plan would fail to fulfill the requiredpavement standard is explicitly considered when selecting maintenance activities. The stochastic simulationis used to simulate the uncertainty of future pavement conditions based on the calibrated deterioration model(for the evaluation of objective function and constraints of the mathematical program) while GA searches for agood solution (i.e., pavement maintenance plan) for a given funding level and a required pavement standard.

The organization of this paper is as follows. Section 2 presents the development of mathematical formula-tions of two pavement maintenance programs, which aim to minimize the maintenance cost and to maximizethe pavement performance, respectively. The stochastic formulations in which the uncertainty of future pave-ment conditions is explicitly modeled are presented along with their deterministic counterparts. In Section 3,the solution procedure, which is the stochastic simulation-based GA approach, is described. Section 4 presentsthe numerical results of a case study conducted using the proposed framework. Finally, the findings and con-clusions of the study are reported and discussed in Section 5.

2. Methodology

The problem of pavement maintenance programming is one of maintaining the serviceability of the entire

pavement network with the available funds and resources. To fulfill these requirements, highway agencies can

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utilize a variety of maintenance goals, such as maximizing network performance, maximizing the cost-effec-tiveness of maintenance activities, minimizing road user cost, minimizing the present worth of the total main-tenance cost, etc., with a certain set of constraints (e.g., budget, pavement standard, manpower, equipment,etc.). The development of a multi-year maintenance plan for the entire network requires a systematic approachto optimally select the appropriate maintenance activities. In addition, the prediction of future pavement con-

ditions and the impacts of maintenance activities on the deterioration of pavement are also very crucial. Forconvenience, the notation used in this study is provided in Table 1.

2.1. Pavement performance prediction and associated uncertainties

Pavement performance prediction models are required to perform a long-term maintenance programming.These prediction models assist highway agencies to determine what has happened to pavements over the pastyears and to predict what may happen in the future (Peterson, 1987) so that the maintenance works could beperformed cost-effectively. To simplify the pavement condition analysis and the communication to the higher-level (administrative) management, the composite performance indices representing the overall pavement con-dition are often used. In general, the composite performance indices are a function of pavement type, trafficloading, age of pavement, severity level of individual distress, etc. (Haas and Hudson, 1982).

In this study, present serviceability rating (PSR), developed by Lee et al. (1993) for the highway pavementmonitoring system (HPMS), is used. However, it should be noted that the framework developed in this studycould be applied using any deterioration models with a similar structure. PSR is a surface-condition ratingscheme based on a numeric scale between 0 and 5 (where 4.5 is always used in practice), 0 indicating extremelypoor condition and 5 indicating a perfect pavement.

The condition of pavement section s at any time period t, as given in Eq. (1), is a function of the initialpavement condition after construction (P0), pavement structural number (STRst), age of pavement (Yst),and the cumulative 18-kip axle loadings on pavement section s at year t(Dst). The calibrated parameters(i.e., a, b, c, and d) of this model for different types of pavement can be found in the study of Lee et al.(1993). The factor AF adjusts the deterioration rate of a pavement in a particular climate zone and functionalgroup.

PSRst P0 AF a STRbst Ycst D

dst. 1

Table 1Mathematical formulation nomenclature

Variable Description

xst Decision variable, type of treatment applied to pavement section s at year tT Number of years in the planning horizonS Total number of pavement segmentsc(xst) Unit cost function related to type of treatmentP(xst) Condition of pavement section s after applied treatment at year tr Discount rateBt Budget constraint at year tLs Length of pavement segment sWs Width of pavement segment sPt Required network performance standard at year tP0 Maximum possible pavement condition ratingqst(xst, Yst) Actual improvement in pavement condition as a function of treatment typeAF Adjustment factor regarding climate zoneSTRst Structural number of segment s at year tYst Age of pavement at year t since the initial construction or last major rehabilitationDst Cumulative 18-kip equivalent single-axel loads, EASL (millions) carried by the pavement segment s in year ta, b, c, d Pavement performance coefficients for pavement performance prediction modelPr() Probability of eventxst Predicted (expected) traffic load on segment s at year t

est Prediction error of traffic load on segment s at year t

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As different maintenance activities are implemented, the performance of pavement is affected differently.Routine maintenance activities (e.g., chip seal) minimize the effects of deterioration and provide minorimprovements in pavement condition (i.e., PSR), some types of treatment, such as minor rehabilitations(e.g., thin overlay) may add additional strength to the pavement (i.e., increase the thickness of pavement)and provide a new riding service. The major rehabilitations or reconstruction will reset the pavement to the ini-

tial stage (i.e., zero cumulative load, age of zero). Accordingly, the characteristics of pavement not only changeover time, but they are also affected by the maintenance activities applied. Mathematically speaking, pavementcharacteristics (e.g., STRst, Yst, Dst, etc.) become a function of the maintenance activity undertaken (xst).

In this study, the uncertainty of predicted pavement conditions is assumed to be associated with the pre-diction error (est) of future traffic loads. Although other sources of uncertainty are present, when predictingfuture pavement conditions, it is assumed here that the uncertainties of future traffic loads are several magni-tudes larger than other sources of uncertainty. The accumulative traffic load on pavement section s at year t ismodeled as a random variable and can be expressed as:

~Dste ~Dst1e xst est; 2

where xst is the predicted (expected) traffic load on pavement section s at year t, and est is the prediction error,which could be drawn from any probability distribution (e.g., a normal distribution). It should be noted that

the effect of prediction error could be accumulated over time. Since the predicted annual traffic load is a ran-dom variable, the predicted pavement performance (Eq. (3)), which takes the effect of maintenance activityinto account, becomes a random variable as well. Here, it is assumed that the pavement structural number(STRst) remains unchanged regardless of the maintenance activity.

PSRstxst; e P0 AF a STRb

st Yc

stxst ~D

d

stxst; e

Xti1

qsixsi; Ysi; 3

where qsi is the actual improvement in the condition of pavement section s at year i, as a result of the treatmentoption applied (xsi) (Eq. (4)). Since the maintenance activity cannot improve the condition of pavement higherthan P0 (the upper limit), the actual increment is the minimum between the effect of the treatment option (seeTable 2, ~qsi) and the maximal possible increment (i.e., the difference between P0 and the pavement condition

before applying treatment).

qsixsi; Ysi Min P0 PSRsi; e; ~qsixsi; Ysif g. 4

Because a pavement performance is a function of traffic load, environment, age, and previous maintenanceactivities, the improvement in the condition of pavement section s at year i ~qsi regarding the same mainte-nance activity is not consistent between different ages of pavement (Al-Suleiman et al., 1991). For instance, aroutine maintenance could be very effective when applied within the first few years of a new pavement seg-ment, but its effectiveness reduces when pavements approach their design life. A reconstruction always returnsthe pavement to the initial condition regardless the existing pavement condition.

2.2. Deterministic formulations

For comparison purposes, the deterministic models will be presented first and followed by their stochasticcounterparts. In this paper, two maintenance goals commonly used in the network-level pavement mainte-

Table 2Available treatment type and effect on pavement condition (Utah LTAP, 2004)

Code Treatment type Unit cost ($/m2) Age of pavement applicable (< year)

20 19 16 13 10 7 4 1

0 0.00 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0001 Routine maintenance 0.20 0.000 0.000 0.000 0.225 0.225 0.450 0.450 0.4502 Surface treatment 0.74 0.000 0.450 0.675 1.125 1.575 1.575 1.575 1.5753 Overlay 4.67 0.000 0.900 1.575 1.800 1.800 1.800 1.800 1.800

4 Major rehabilitation 7.74 4.500 4.500 4.500 4.500 4.500 4.500 4.500 4.500



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higher pavement performance is more preferable. In this study, both objectives are given a weight (e.g., pri-ority) and combined into a single objective value as shown in Eq. (7).

Maximize : w1 ~Z1 w2 ~Z2;

subject to : Eqs. (5b)(5h),7

where w1 and w2 are weights (priorities) given to the performance of pavement and the maintenance cost sav-ing. ~Z1 and ~Z2 are respectively, the standardized versions of Eqs. (5a) and (6). The standardization (e.g., nor-malized between 0 and 1) is required because different objectives are usually defined in incomparable scales(e.g., money and time). In addition, the direct usage of the original objective values may cause the dominationof one objective over another. The original objective values can be normalized as follows.

~Z1 Z1 Z

min1

Zmax1 Z

min1

; 8

~Z2 Z

max2 Z2

Zmax2 Z

min2

; 9

where Zmaxi

and Zmini

are respectively, the maximal and minimal possible values of objective i.

2.3. Stochastic formulations

In the preceding section, pavements are assumed to deteriorate deterministically (i.e., no uncertaintyinvolved in the predicted pavement condition). By assuming that the prediction error (est) of future trafficloads follows a known probability distribution, it can be incorporated into the existing pavement performancemodels (Eqs. (5e)(5h)) to account for the uncertainty of pavement deterioration as follows.

2.3.1. Expected performance maximization

Maximize: Z3 E XT

t1

Htx; e" #; 10asubject to:

XSs1

cxstLsWs 6 Bt; 8t; 10b

Pr Pxst; eP Pt P ast; 8s; t; 10c

xst 2 f0; 1; . . . ;mg; 8s; t; 10d

where Pxst; e P0 AF a STRb

st Yc

stxst ~D

d

stxst; e

Xti1

qsixsi; Ysi; 8s; t; 10e

~Dstxst; e xst est; if xst m

~Dst1 xst est; otherwise

&; 8s; t; 10f

Ystxst 1; if xst mYst1 1; otherwise

&; 8s; t; 10g

Htx; e Min8s

Pxst; ef g; 8t. 10h

As a result of the uncertainty of future traffic loads, the future pavement conditions become random vari-ables. In addition, all relations associated with pavement performance (e.g., Eqs. (5a), (5c), (5e), etc.) becomeprobabilistic functions. Since the value of these functions is not a single numerical value but a probability dis-tribution, the expectation and/or the probability of a certain event are usually used to formulate the stochasticoptimization problem (Liu, 1999). Now, the first problem considered in the previous section becomes one ofmaximizing the expected value of pavement performance (Eq. (10a)) for a given maintenance budget. Whileaccounting for several possibilities of future pavement conditions, the conditions of individual pavements

maintained in each year must satisfy the required standard with a certain confidence level (ast)a chance

http://-/?-http://-/?-


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constraint. In other words, the probability measure of the event {P(xst, e)P Pt} is at least ast, which can bereferred to as the confidence that the requirement (e.g., constraint) will not be violated. The remaining deter-ministic formulations can be modified to account for the uncertainty of future pavement conditions in thesame manner.

2.3.2. Maintenance cost minimization

Minimize: Z2 XTt1

1 rt1

XSs1

cxstLsWs;

subject to: Eqs. (10b)(10g).

11

2.3.3. Expected performance maximization and cost minimization

Maximize: w1 ~Z3 w2 ~Z2;

subject to: Eqs. (10b)(10h),12

where~Z

3 is determined similarly to~Z

1 shown in Eq. (8).As shown above, these formulations involve non-linear functions, non-differentiable functions, step func-tions, and integer variables. Although the step functions can be generalized to the linear forms, the transfor-mation will require additional variables, which will increase the problem size. In addition, the pavementperformance model (deterioration model) is certainly non-linear. Moreover, when the stochastic elementsare incorporated into the formulation, these characteristics all together are incompatible with the traditionaloptimization techniques. This motivates the usage of GA for solving pavement maintenance programs pro-posed in this study.

3. Simulation-optimization framework

3.1. Stochastic simulation

A simulation-based GA procedure that combines the stochastic simulation and the GA to solve the sto-chastic programs is developed in this study. Traditionally, simulation is the process of replicating reality basedon a set of assumptions and conceived models of reality (Ang and Tang, 1984). To handle the uncertainty offuture pavement conditions, the stochastic simulation is used to simulate the uncertainty of future traffic loadsused in the prediction model based on a probability distribution with the predefined mean and standard devi-ation. Latin hypercube sampling (LHS) technique, one of the stratified sampling techniques that has shown tooutperform the simple Monte Carlo sampling technique (McKay, 1988), is employed in this study. LHS par-titions the distribution of future traffic loads into several equal intervals according to the number of samplesrequired. Only one random variate is sampled from each interval. This sampling technique significantlyreduces the number of samples yet delivers a reasonable level of accuracy. LHS is incorporated into theGA to evaluate the values of objective function and constraints corresponding to the maintenance decisionunder a stochastic environment (i.e., pavement deterioration). That is, the objective function and constraintsof the stochastic formulations presented earlier are evaluated several times (i.e., the number of samples) usingthe simulation.

3.2. Genetic algorithm (GA)

GA is an evolutionary computing technique that, in principal, mimics the mechanism of natural selectionprocess. According to Goldberg (1989), GA differs from the classical, calculus-based optimization techniquesin the following ways: (i) instead of using a point-to-point search method, as in the traditional optimizationtechniques, GA simultaneously searches from a population of points, known as chromosomes, to explore the

solution space; (ii) GA uses probabilistic transition rules (for its operators) as a guide to search the solution



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space with likely improvement; (iii) GA can work with continuous and discrete parameters, differentiable andnon-differentiable functions, uni-modal and multi-modal functions, as well as convex and non-convex feasibleregions.

3.2.1. Representation of decision variables

One important aspect of applying GA to any problem is the representation of the decision variables in thegenetic fashion or as a chromosome. For the pavement maintenance problem studied here, the chromosome iscoded as a series ofT-year maintenance activities for all pavement segments S, as shown in Fig. 1. As depictedin Fig. 1, each chromosome consists of a series of maintenance actions represented by numerical values:0,1,2, . . . , m (see Table 2 for treatment options). For each pavement section s, there are T genes representingthe maintenance treatments for T years. For example, the first T genes represent the maintenance treatmentsfor the first pavement section over Tyears; the second set ofTgenes is for the second pavement section, and soon.

3.2.2. Constraint handlingBecause the traditional GA operators are blind to constraints of an optimization problem, the special-pur-

pose constraint-handling methods are usually required to ensure the feasibility of solutions. As with any opti-mization procedure, improper constraint handling will result in a considerable amount of effort wasted inevaluating infeasible solutions. To ensure that GA performs the search effectively, the budget constraintand the pavement performance constraint are handled by the special constraint-handling procedure. This pro-cedure utilizes a combination of the penalty and repair methods. The penalty method converts a constrainedproblem into an unconstrained problem by including a penalty value in the objective function (Goldberg,1989). The repair method attempts to repair the infeasible solutions by using a special solution mapping toensure the feasibility of the solutions (Liu, 1999).

For the pavement maintenance problem studied here, the repair method is applied first by attempting toobtain a feasible or near-feasible solution. Under the repair method, both budget and pavement performanceconstraints are examined simultaneously to determine the positions of genes in the chromosome that causethe solution to be infeasible. The positions of the bad genes indicate when and where (i.e., whichpavement segment and at what year) the budget and/or pavement performance constraints are violated. Oncethese bad genes are identified, the repair method attempts to fix these genes with the most cost-effective treat-ment. This process is repeated several times until the solution is feasible or the repair is impossible. If the solu-tion is not feasible, the penalty method penalizes the fitness according to the degree of the constraintviolations.

3.3. Stochastic simulation-based GA procedure

The simulation-based GA procedure for solving the pavement maintenance program is displayed in Fig. 2

and is summarized as follows:

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20

r

aey

ht9

ra

ey

ht01

. . . .

r

aey

ts1

raeydn2

raey

dr3

r

aey

ht8

r

aey

ht8

r

aey

ht9

ra

ey

ht01

. . . .

Xi = type of treatment {0,1,2,3,4}

r

aey

ts1

raeydn2

raey

dr3

1st Section 2nd Section

Fig. 1. Representation of decision variables.



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1. Initialize a set of maintenance solutions (N

pop solutions) according to the pavement inventory data andpavement performance model.

2. Set N= 1 (the counter for number of solutions) and S= 1 (the counter for number of sample points). Eachmaintenance solution will be evaluated based on the objective function, and checked for the feasibility Smaxtimes by the stochastic simulation. If the maintenance solution violates the stochastic constraints, con-straint-handling procedure is applied.

3. Rank the maintenance solutions based on their fitness to the problem. Update solutions (maintenanceplans) by GA operators, reproduction, crossover and mutation in order to obtain a new set of maintenance

solutions.

No

No

No

No

+1

S= S+1

1

3

2

4

Yes

1

S=1

N_ pop

Yes

Yes

Yes

N=N+1

S= S+1

1

3

2

54

Pavement Inventory

Initial pavement condition

Initial traffic loads

Growth Rate

Generate Initial Maintenance

SolutionsUpdate Maintenance Strategies

by GA OperatorsReproduction

Crossover

Mutation

N= 1

S=1

Generate Random Traffic

Loading

Evaluate Maintenance Strategies

Objective function

Constraints

Collect Statistical Inferences S> Smax

Stochastic

Constraints met?

Constraint

HandlingN>N_

Solution and Probabilistic

SummaryStopping

criteria met?

Fig. 2. Flow diagram for a stochastic simulation-based GA for PMS.



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4. Repeat steps 2 and 3 until stopping criterion is met (e.g., the maximum number of generations).5. Report the solution and its probabilistic summary.

4. Case study

4.1. General information

To illustrate the feasibility of the proposed method, a case study was conducted. The study network consistsof two routes of flexible pavement. Both routes consist of a 31.450 lane-miles (50.621 lane-kilometers) of flex-ible pavement with three control sections, one control section on the first route and the remaining two controlsections on the second route. This study aggregates the pavement inventory data into approximately 3281-feet(1000-m) intervals across both roadways. As a result, there are 53 pavement segments for the case study (seepavement inventory data in Chootinan, 2001). In this case study, age of pavement segments ranges from 3years to 9 years with an average pavement age of 4.75 years. Initial PSR values range from 2.49 to 3.48 witha network average PSR of 3.16.

A 10-year maintenance plan (year 20002009) is programmed in order to maintain the pavement condition

above a PSR value of 2.5 with an assumed annual budget of US$ 61,410. Present worth calculations are con-ducted using an arbitrary discount rate of 12%. The available treatment options, as shown in Table 2, are: 0no treatment, 1routine maintenance, 2surface treatment, 3minor rehabilitation, 4major rehabilita-tion. The prediction of future traffic loadings utilizes an annual growth rate of 5%. Prediction error of thefuture traffic loadings is simulated using a normal distribution and is assumed to increase proportionally withtime. In addition, a confidence level of 90% (ast = 0.90) is set to ensure that stochastic constraints are fulfilledto at least 90%.

4.2. Selection of simulation-based GA parameters

Although a large population pool in GA has a higher probability of obtaining a better solution, it might

be impractical in a simulation-based framework since the number of function evaluations per generation isnot simply the number of chromosomes in the population pool. The number of function evaluations isessentially the number of chromosomes times the sample size used in the stochastic simulation. Conse-quently, it is important to size the population pool such that it is reasonably small yet allows for reason-ably good solutions. The following settings of GA are used in this study: the population size is 32.Crossover probability is 0.50 (uniform crossover) while mutation probability is 0.01 (random mutation).The readers may refer to Goldberg (1989) for the detail description of GA operators. The initial popula-tion is randomly generated. The roulette wheel (proportional) selection and the half-replacement strategyare used for reproduction. Chromosomes in each generation are ranked based on their fitness (penalizedobjective value) and divided into two parts. Only the chromosomes in the top half (i.e., better solutions)are eligible for reproduction. Chromosomes in the bottom half will be replaced by the offspring generated

by crossover and mutation. The genetic search will be performed until the maximum number ofgenerations (50,000 generations) is reached, In the case that the infeasible solution cannot success-fully be repaired; a very high penalty will be applied. The sample size of the stochastic simulation is100.

4.3. Numerical results

Multi-year pavement maintenance plans were developed using both the deterministic and stochastic formu-lations. All three objectives, which are the performance maximization, maintenance cost minimization, as wellas the bi-objective model, are considered in both formulations. Comparison of the maintenance plans devel-oped by these two formulations is used to investigate the effects of uncertainty on pavement deterioration

under different maintenance objectives.



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4.3.1. Pavement performance maximization

Under the pavement performance maximization objective, the developed maintenance programs attempt tofully utilize the available maintenance budget such that the serviceability level of the lowest (minimum) con-dition among all pavement sections will be as high as possible. Fig. 3 displays the convergence curves mea-sured in terms of the summation of minimum pavement segment PSR values over the 10-year planning

period for both the deterministic and stochastic formulations. As can be seen from the figure, the deterministicand stochastic approaches converge to the solutions (i.e., maintenance plans) with the objective values of 37.23and 36.53, respectively.

In order to evaluate the performance of the maintenance plans developed under the deterministic and sto-chastic formulations, the pavement performance curve and pavement deterioration curve of both maintenanceplans were displayed in Fig. 4. The pavement performance curve displays the network average PSR valuesbefore treatment and after treatment for each year of the 10-year maintenance period. To illustrate the sto-chastic nature of pavement deterioration, a 90% confidence level is also included for both pavement-perfor-mance and pavement-deterioration curves. The evaluation of performance curves showed that bothmaintenance plans (deterministic and stochastic approaches) steadily improve the average network-perfor-mance throughout the duration of the 10-year planning period. In nearly every year of the planning period,the stochastic formulation generates a lower average network-performance than that of the deterministic for-mulation. This indicates that the uncertainty increases the predicted pavement deterioration and results in alower-than-expected pavement performance. This is confirmed by evaluating the allocation of maintenancebudget in Fig. 5 and the maintenance activities summarized in Tables 3 and 4.

From Fig. 5, it can be seen that the stochastic formulation utilizes a slightly larger budget than the deter-ministic formulation for nearly every year of the planning period. When accounting for the uncertainty ofpavement deterioration, pavement sections are expected to deteriorate at a higher rate. As a result, a higherlevel of treatments (e.g., treatment Type 2) is generally required to maintain the required level of pavementstandard. In Tables 3 and 4, it can be seen that a higher percentage of pavement area is being treated by treat-ment Type 2 in the stochastic formulation compared to the deterministic formulation. The increase in pave-ment area being treated by treatment Type 2 results in a maintenance cost difference of US$ 6807 and anetwork average PSR difference of approximately 0.01. Under this situation, the stochastic formulation sug-

gests that a 1.1% increase in the maintenance cost would be required to sustain a comparable pavement

27.0

28.5

30.0

31.5

33.0

34.5

36.0

37.5

39.0

0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000

Number of Generations

CumulativePSR

Deterministic Approach (37.23)

Stochastic Approach (36.53)

Fig. 3. Convergence curve of pavement performance maximization.



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condition to the deterministic formulation. In general, the following observations can be made about the per-formance maximization results:

1. The maintenance plans recommended by both deterministic and stochastic formulations are composed pri-marily of treatment types 1 and 2. However, the results in Tables 3 and 4 indicate that, by accountingfor the uncertainty in the stochastic formulation, a larger number of pavement sections are required toundergo a higher level of treatment (treatment Type 2).

2. Higher maintenance cost required for the stochastic formulation results in an increase of US$ 216 per lane-mile of roadway compared to the deterministic formulation. This indicates that the deterioration of pave-ment is underestimated in the deterministic case, which results in an underestimation of the maintenance

cost.

1.500

2.000

2.500

3.000

3.500

4.000

4.500

1999 2001 2003 2005 2007 2009

Year

NetworkAveragePSR

Deterioration Lower Deterioration Mean

Deterioration Upper Performance Lower

Performance Mean Performance Upper

Deterministic Performance

StochasticPerformance Curve

DeterministicPerformance Curve

Fig. 4. Performance of pavement under performance maximization.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Year

AnnualMaintenan

ceCost($M)

Deterministic

Stochastic

Budget

US$ 61,410

Fig. 5. Maintenance cost allocation under pavement performance maximization.



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4.3.2. Maintenance cost minimization

For the cost minimization objective, the maintenance plans developed under both deterministic and sto-chastic formulations aim to minimize the present worth of the total maintenance cost spent throughout theplanning period. The deterministic and stochastic formulations recommended the maintenance plans withthe total maintenance costs (present value) of US$ 138,946 and US$ 159,615, respectively (see Fig. 6). Asexpected, the stochastic formulation requires a higher maintenance cost compared to that of the deterministicformulation in order to maintain the same level of pavement standard. To gain more insight on the increase ofmaintenance cost, the budget allocation and average network-performance curves displayed respectively inFigs. 7 and 8 were investigated.

Similar to the results shown in the previous section, a larger amount of maintenance cost is required for thestochastic formulation. Evaluation of the performance and deterioration curves reveals that the stochastic for-mulation maintains a higher average network PSR. Although this is counterintuitive, the treatment allocationscheme recommended by the stochastic formulation in Table 6 creates a higher level of pavement perfor-mance. Because the pavement conditions in the stochastic formulation deteriorate at a higher rate than thedeterministic formulation, a larger pavement area requires treatment, and, in some cases, a higher level oftreatment. This results in a 13.6% increase in maintenance cost in the stochastic formulation to maintainthe same level of pavement standard. Investigation of the treatment scheme summarized in Tables 5 and 6

indicates that the deterministic formulation treated a larger area of pavement while the stochastic formulation

Table 3Summary of maintenance activities under performance maximization (deterministic case)

Treatment option Area of treatments (m2)

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

0 (Do-nothing) 130,648 157,493 107,994 110,390 141,694 95,567 162,499 128,662 115,591 50,176

1 (Routine) 204,534 166,510 233,313 229,901 191,619 251,746 161,033 205,967 226,696 312,1312 (Surface treatment) 27,125 38,304 21,000 22,016 28,994 14,994 38,775 27,678 20,020 03 (Minor rehabilitation) 0 0 0 0 0 0 0 0 0 04 (Major rehabilitation) 0 0 0 0 0 0 0 0 0 0% of total area treated 63.94 56.53 70.19 69.53 60.89 73.62 55.15 64.49 68.10 86.15

Maintenance cost (US$) 60,182 60,948 61,324 61,402 59,022 60,520 60,219 60,871 59,302 61,337Network ave. PSR before

treatment3.16 3.33 3.51 3.68 3.84 3.95 4.08 4.14 4.14 4.14

Network ave. PSR aftertreatment

3.52 3.70 3.86 4.02 4.13 4.26 4.31 4.32 4.31 4.32

Table 4

Summary of maintenance activities under performance maximization (stochastic case)


2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

0 (Do-nothing) 194,347 142,606 107,675 149,676 124,521 108,424 146,648 127,477 121,968 109,5411 (Routine) 115,871 186,396 234,332 176,329 210,941 232,988 180,659 207,082 214,448 231,5772 (Surface treatment) 52,089 33,305 20,300 36,302 26,845 20,895 35,000 27,748 25,891 21,1893 (Minor rehabilitation) 0 0 0 0 0 0 0 0 0 04 (Major rehabilitation) 0 0 0 0 0 0 0 0 0 0% of total area treated 46.36 60.64 70.28 58.69 65.63 70.07 59.52 64.82 66.34 69.77


treatment3.16 3.31 3.48 3.66 3.82 3.98 4.11 4.15 4.14 4.13


3.50 3.67 3.84 4.00 4.16 4.29 4.32 4.31 4.30 4.29

Standard deviation of PSR 0.00 0.020 0.024 0.027 0.029 0.028 0.024 0.026 0.029 0.031



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treated less area of pavement with a higher level of treatment. A budget increase of US$ 28,206 from the deter-ministic approach to the stochastic approach results in a PSR value increase of 0.08.

In general, the following observations can be made about the cost minimization results:

1. The deterministic formulation underestimates the pavement deterioration resulting in the reduction ofmaintenance cost throughout the planning period. This underestimation of maintenance cost translatesto a cost difference of US$ 897 per lane-mile of roadway.

2. Maintenance activities recommended by the stochastic formulation require a larger amount of maintenancecost. This indicates that the deterministic formulation underestimates the pavement deterioration, which

leads to the underestimation of the required maintenance cost.

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000


TotalMaintenance

Cost($M)

Stochastic Approach (US$ 159,615)

Deterministic Approach (US$ 138,946)

Fig. 6. Convergence curve of maintenance cost minimization.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Year

AnnualMaintena

nceCost($M)

DeterministicStochastic

Budget

US$ 61,410

Fig. 7. Budget allocation under maintenance cost minimization.



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3. On average, the stochastic formulation is likely to recommend a higher level of treatment in order to main-tain the minimum pavement condition at the PSR value of 2.5. The uncertainty of pavement deteriorationconsidered in the stochastic formulation led to 96% more of Type 2 treatment being applied when com-pared to the deterministic formulation.

4.3.3. Bi-objective model

Unlike the single-objective models, the bi-objective model attempts to optimize both pavement perfor-mance and maintenance cost simultaneously. Although a wide range of weight combinations could be usedto represent different levels of importance put toward these two objectives, equal weights for both objectives(w1 and w2 are 0.50) are adopted in this study to illustrate the effects of simultaneous optimization of twoobjectives to develop a compromise maintenance plan. Fig. 9 displays the convergence curves for both deter-ministic and stochastic formulations. From Fig. 9, the deterministic and stochastic formulations recom-

mended the maintenance plans with the total maintenance costs of US$ 284,595 and US$ 287,257,

1.500

1.700

1.900

2.100

2.300

2.500

2.700

2.900

3.100

3.300

3.500

1999 2001 2003 2005 2007 2009

Year

NetworkAveragePSR

Deterioration LowerDeterioration Mean

Deterioration UpperPerformance Lower

Performance MeanPerformance Upper


Fig. 8. Performance of pavement under maintenance cost minimization.

Table 5Summary of maintenance activities under cost minimization (deterministic case)


2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

0 (Do-nothing) 263,607 284,626 249,396 233,894 246,981 233,830 288,324 304,584 294,435 318,3461 (Routine) 98,700 60,865 105,911 1 21,413 108,242 118,777 73,983 57,723 61,852 43,9612 (Surface treatment) 0 16,816 7000 7000 7084 9700 0 0 6020 03 (Minor rehabilitation) 0 0 0 0 0 0 0 0 0 04 (Major rehabilitation) 0 0 0 0 0 0 0 0 0 0% of total area treated 27.24 21.44 31.16 35.44 31.83 35.46 20.42 15.93 18.73 12.13

Maintenance cost (US$) 19,396 24,353 25,971 29,018 26,491 30,489 14,539 11,343 16,591 8639Network ave. PSR before

treatment3.16 3.10 3.05 3.02 3.03 3.01 3.01 2.92 2.78 2.66


3.29 3.24 3.20 3.21 3.18 3.18 3.09 2.95 2.83 2.69



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respectively. The network performances delivered by these two maintenance plans are 34.80 and 34.22,respectively.

In order to evaluate the performances of the bi-objective maintenance plans developed by the deterministicand stochastic formulations, the pavement performance curve and pavement deterioration curve in Fig. 10were developed. The pavement performance curve displays the average network PSR value before treatmentand after treatment for each year of the 10-year maintenance period. Due to the stochastic nature of pavementdeterioration, the 90% confidence level is also included for both the pavement performance and pavementdeterioration curves.

From Fig. 10, it can be seen that the bi-objective models exhibit similar traits as the single-objective coun-terparts. That is, the first several years of the planning period have steadily increasing pavement performancevalues. This is similar to the pavement performance curves of the pavement performance maximization objec-tive. As the bi-objective model also attempts to satisfy the cost minimization objective, after the sixth year,when the pavement network reaches its maximum condition, the average network performance level starts

to decrease. This is similar to the pavement performance curves of the cost minimization objective. Similar

0.27

0.29

0.31

0.33

0.35

0.37

0.39

TotalMaintenanceCost($M)...

0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000


26.4

28.0

29.6

31.2

32.8

34.4

36.0

CumulativePSR

Maintenance Cost (Deterministic)- US$ 284,595

Maintenance Cost (Stochastic) -US$ 287,257

Pavement Condition (Stochastic) -34.22

Pavement Condition (Deterministic) -34.80

Fig. 9. Convergence curve of bi-objective model.

Table 6Summary of maintenance activities under cost minimization (stochastic case)


2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

0 (Do-nothing) 272,699 253,808 256,794 280,085 253,849 240,723 300,700 327,762 305,614 331,192

1 (Routine) 67,908 85,779 95,522 75,222 101,458 98,210 55,223 34,545 49,819 31,1152 (Surface treatment) 21,700 22,720 9991 7000 7000 23,374 6384 0 6874 03 (Minor rehabilitation) 0 0 0 0 0 0 0 0 0 04 (Major rehabilitation) 0 0 0 0 0 0 0 0 0 0% of total area treated 24.73 29.95 29.12 22.69 29.94 33.56 17.00 9.53 15.65 8.59


treatment3.16 3.15 3.16 3.13 3.09 3.07 3.10 3.01 2.87 2.75


3.33 3.35 3.32 3.27 3.24 3.27 3.19 3.04 2.93 2.78




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to the pavement performance curves, the bi-objective budget allocation of Fig. 11 mimics the budget allocationof the performance maximization objective in the first five years and the cost minimization objective in the lastfive years.

Similar to the results of single-objective models, a larger amount of maintenance cost is required for thestochastic formulation. Investigation of the treatment scheme summarized in Tables 7 and 8 indicates thatthe deterministic formulation treated a larger area of pavement, where as the stochastic formulation treated

less area of pavement with a higher level of treatment. A maintenance cost increase of US$ 1308 from thedeterministic formulation to the stochastic formulation is required to sustain comparable PSR values. In gen-eral, the following observations can be made about the bi-objective results:

1.500

2.000

2.500

3.000

3.500

4.000

4.500

1999 2001 2003 2005 2007 2009

Year

NetworkAveragePSR

Deterioration Lower Deterioration Mean

Deterioration Upper Performance Lower

Performance Mean Performance Upper


Fig. 10. Performance of pavement under bi-objective model.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Year

AnnualMaintenanc

eCost($M)...

Deterministic

Stochastic

Budget

US$ 61,410

Fig. 11. Budget allocation under bi-objective model.



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1. The deterministic formulation underestimates the pavement deterioration resulting in a reduced mainte-nance cost throughout the planning period. This underestimation of maintenance cost translates to a costdifference of US$ 42 per lane-mile of roadway.

2. Maintenance activities of the stochastic formulation require a larger amount of maintenance cost. This indi-cates that the deterministic formulation underestimates the pavement deterioration; thus, underestimatesthe required maintenance cost.

3. On average, the stochastic formulation applies a higher level of treatment in order to maintain comparablepavement performance values. This led to 9.0% more of Type 2 treatment being applied in the stochasticformulation when compared to the deterministic formulation.

5. Summary and conclusion

In order to address the growing concerns of pavement management at the network level, accounting foruncertainty in pavement maintenance programming is essential. Based on the results of this study, it is evidentthat the pavement maintenance programming models using a deterministic formulation for the developmentof pavement maintenance plan underestimate the level of pavement deterioration in future years. A directresult of this is the underestimation of the required maintenance cost and the overestimation of the expectedpavement performance. By accounting for uncertainty, in the form of prediction error, multi-year pavement

maintenance plans can more accurately predict the future needs of the pavement network.

Table 7Summary of maintenance activities under bi-objective model (deterministic case)


2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

0 (Do-nothing) 170,240 162,132 188,336 186,209 126,764 154,737 193,889 259,813 309,660 362,307

1 (Routine) 151,222 171,706 143,596 142,238 214,634 193,696 161,418 94,500 52,647 02 (Surface treatment) 40,845 28,469 30,375 33,860 20,909 13,874 7000 7994 0 03 (Minor rehabilitation) 0 0 0 0 0 0 0 0 0 04 (Major rehabilitation) 0 0 0 0 0 0 0 0 0 0% of total area treated 53.01 55.25 48.02 48.60 65.01 57.29 46.48 28.29 14.53 0.00


treatment3.16 3.32 3.46 3.55 3.66 3.81 3.89 3.91 3.82 3.68


3.50 3.64 3.73 3.84 3.99 4.07 4.08 3.99 3.85 3.68

Table 8

Summary of maintenance activities under bi-objective model (stochastic case)Treatment option Area of treatments (m2)

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

0 (Do-nothing) 194,584 172,340 181,524 148,083 151,963 152,529 227,774 246,176 315,411 362,3071 (Routine) 118,429 154,764 156,010 186,343 173,540 183,948 134,533 116,131 46,896 02 (Surface treatment) 49,294 35,203 24,773 27,881 36,804 25,830 0 0 0 03 (Minor rehabilitation) 0 0 0 0 0 0 0 0 0 04 (Major rehabilitation) 0 0 0 0 0 0 0 0 0 0% of total area treated 46.29 52.43 49.90 59.13 58.06 57.90 37.13 32.05 12.94 0.00


treatment3.16 3.28 3.42 3.55 3.68 3.84 3.96 3.92 3.83 3.68

Network ave. PSR aftertreatment 3.47 3.61 3.73 3.86 4.02 4.13 4.10 4.00 3.85 3.68




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Through the development of the simulation-optimization framework proposed in this paper, PMS candevelop multi-year pavement maintenance programs that can account for uncertainty in the prediction pro-cess. While accounting for uncertainty, a wide range of PMS objectives can be implemented into the frame-work while utilizing existing deterioration models. This allows for simple adoption of these methods sinceexisting PMS procedures do not need to change. In addition to the needs of PMS, decision makers are allowed

to limit the probability or risk that the selected maintenance plan would fail to maintain the required levels ofperformance. This allows for the generation of maintenance plan alternatives that account for various degreesof risk.

Acknowledgement

This research is supported by the Community/University Research Initiative (CURI) grant from the Stateof Utah.

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