hybrid multiobjective optimization model for regional pavement-preservation resource allocation

28

10 years). Although efforts are being made toward developing aperformance-based long-term repair solution, many state DOTs, withlimited or inconsistent historical records of pavement conditiondata, still use a needs-based budgeting process that does not requirethe incorporation of pavement performance prediction models.Accordingly, budget requests and allocations are developed basedon identified annual or biannual pavement preservation needs fromall component maintenance districts. A typical procedure to identifythe pavement preservation needs involves, in sequence, collectingpavement condition data, prescreening the “in need” pavement sec-tions, calculating the relevant costs and benefits, and prioritizing orranking identified sections (1).

Disadvantages accompanying this procedure when used forstatewide short-term preservation budgeting may include

1. A tendency of some of the individual districts to exaggeratetheir true needs for preservation and

2. A resulting statewide budget request that is often not linkeddirectly to the optimal benefit of the overall pavement network.

The allocation of funding across districts is a challenging task: witha tight budget, it is not possible to satisfy the preservation needs fromall competing districts. A typical method of funds allocation under thiscircumstance is to rely on certain fixed criteria (e.g., total road lengthand traffic volume) that are applied to all districts. As the service con-ditions of their respective road networks are unlikely uniform, differ-ent districts tend to have different preservation needs and goals. Thus,the use of fixed criteria, though convenient and easy to apply from thestandpoint of the central administration, would not lead to optimalusage of funds and resources. Applying a common fund allocationformula to all cannot meet the various needs and goals of different dis-tricts (2). Therefore, an allocation method capable of negotiating andbalancing within the competing component districts in a formal ordocumented way and with necessary optimization considerationswould be helpful in conducting the needs-based budgeting process.

OBJECTIVE

The paper explores an alternative method for the central administra-tion to set short-term pavement preservation budgeting under a widerinformation context, linking budget allocation to multiple criteria andperformance targets through structured procedure and interactivecommunication. The result is a practical decision support model thatenables the central administration in a decentralized state DOT

• To identify optimal maintenance actions and budget alloca-tions across the component districts that are consistent with agencyneeds and resource limitations and

Hybrid Multiobjective Optimization Modelfor Regional Pavement-PreservationResource Allocation

Zheng Wu, Gerardo W. Flintsch, and Tanveer Chowdhury

Because of a lack of reliable performance prediction models, many statedepartments of transportation (DOTs) use a needs-based budgetingprocess, namely, annual budget requests. Allocations of funding acrossmaintenance activities and districts are developed on the basis of pave-ment maintenance needs derived from pavement inventory and annual orbiannual condition data. This allocation of funding across maintenanceactivities and districts is challenging and often involves negotiation andbalancing. A decision support model is proposed for the optimization ofshort-term pavement preservation budgeting based on two proven oper-ations research techniques: goal programming for handling multipleobjectives and an analytic hierarchy process for priority setting undermultiple criteria. The model simultaneously considers two incommensu-rable and conflicting objectives: maximization of the preservation effec-tiveness in terms of extended service life and minimization of the totalpreservation cost. Application of the model is demonstrated with a short-term pavement preservation budgeting problem for a decentralized stateDOT with nine maintenance districts. The illustrative example reveals thatthe proposed model is practical for supporting needs-based budgeting.

The development trend for the pavement budgeting process within astate department of transportation (DOT) is to progress from histori-cal allocations (how much funding has already been allocated to eachdistrict, residency, or area headquarters by its preceding authorities),to needs-based allocations, and finally to performance-based alloca-tions. How far a state DOT can progress in this direction relies onseveral key factors including, but not limited to, availability and reli-ability of pavement performance prediction models (network level orproject level), appropriate decision policies, and an efficient organi-zational structure that promotes effective communication betweenupper- and lower-level maintenance entities in the department.

Statewide pavement repair planning often involves programmingdecisions to develop budgets and allocate financial resources over theentire network, with the planning period being either short-term (1 to2 years), medium-term (up to 10 years), or long-term (more than

Z. Wu, Charles E. Via, Jr., Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. Currentaffiliation: MACTEC Engineering and Consulting, Inc., Suite A, 12104 Indian CreekCourt, Beltsville, MD 20705. G. W. Flintsch, Charles E. Via, Jr., Department ofCivil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. T. Chowdhury, Asset Management Division,Virginia Department of Transportation, 6600 West Broad Street, Richmond, VA 23230. Corresponding author: G. W. Flintsch, [email protected].

Transportation Research Record: Journal of the Transportation Research Board,No. 2084, Transportation Research Board of the National Academies, Washington,D.C., 2008, pp. 28–37.DOI: 10.3141/2084-04

Wu, Flintsch, and Chowdhury 29

• To understand the trade-off between the preservation cost andthe associated network benefit.

The implementation of the model in an illustrative example showsits potential application for supporting needs-based budgeting.

BACKGROUND

Goal programming is a popular multiobjective method that employsa minimum-distance notion of best, meaning that an ideal solutionwould minimize the weighted sum of deviations of all objective func-tions from their respective goals. The weight assigned to the deviationof each objective function from its respective goal plays a key role inthe final optimization result. Accordingly, goal programming can becategorized into two types: nonpreemptive and preemptive (3, 4).

The analytic hierarchy process (AHP) is a theory of measurementfor dealing with quantifiable or intangible criteria that has found richapplications in decision theory and conflict resolution, among otherfields. This method is based on the principle that in decision making,experience and knowledge are at least as valuable as the data used tosupport the decision (5). The AHP is designed for subjective evalu-ation, providing a vector of weight expressing the relative impor-tance of a set of alternatives on the basis of multiple criteria. Decisionapplications of the AHP are carried out in three main phases (6, 7):hierarchic design that decomposes the problem complexity, pairwisecomparisons that allow the systematic determination of the intensi-ties of interrelationships of various decision factors, and synthesis ofinformation that weights the lower-level priorities with higher-levelpriorities to derive the composite priorities.

METHODOLOGY

Pavement preservation budgeting occurs at the programming level,which develops budget and allocates resources over the entire network.This type of analysis does not require the extent of detail necessary forthe life-cycle cost analysis and design of individual projects. There-fore, a macroscopic optimization model in which decision variablesare related to the lengths or proportions of pavement classes insteadof individual pavement sections is generally recommended to reduceunnecessary model complexity (8). The proposed optimizationmodel aims to address simultaneously the issues of short-term budgetrequests and funds allocation that would have been handled separatelyin a traditional needs-based budgeting process. Applicability of themodel will be most appropriate to maintenance-decentralized stateDOTs that do not currently possess extensive historical records ofpavement condition data to develop reliable pavement performanceprediction models.

Model Formulation

The proposed model uses goal programming for handling multipleobjectives and the AHP for establishing the relative importance ofmultiple criteria within some of the optimization constraints. Morespecifically, the model formulation consists of the following steps.

Determination of Decision Variables

Common pavement management practices in state DOTs at the pro-gramming level include evaluating pavement conditions with discretecondition states based on a certain performance index (e.g., PavementCondition Index 90–100 means pavement is in excellent condition),

classifying various treatment methods into maintenance categories(e.g., preventive maintenance includes minor patching, crack sealing,and surface treatment), and assigning an average cost to each categorybased on historical maintenance cost data for simplicity.

In a maintenance-decentralized state DOT, the central administra-tion responsible for pavement preservation budgeting would expectthe outcome from a needs-based budgeting process to include theamount of funds allocated to each of the component districts and therelevant benefits proposed to the overall network. Therefore, the rec-ommended decision variable in the proposed model is “length (lane-miles) of pavement in district i in condition state j that will be treatedby some treatment method from maintenance category k,” whichleads directly to the quantification of both costs and benefits.

Selection of Objective Functions

The objective(s) to be considered in an optimization formulationshould reflect the true concerns of engineers and planners managingthe pavement network. Modeling the pavement management problemhas been typically addressed with two different approaches (8): thefirst approach aims to maximize the preservation benefit subjected tobudget constraints, while the second approach aims to minimizethe total preservation cost subjected to certain pavement conditionimprovement constraints. The main reason for cost minimization andbenefit maximization being considered separately is due to their inher-ent conflict under limited resources. However, optimizing only one ofthe objectives may not provide the most appropriate solution sinceboth of them are major concerns for the pavement managementagency. Therefore, a multiobjective optimization model, as proposedin this paper, is established to address both objectives simultaneouslyand to provide insights into the trade-off.

Many parameters characterize the preservation benefits, includingpreservation treatment effectiveness (the area between before- andafter-treatment pavement performance curves multiplied by trafficand pavement length) and average increase of pavement conditionscore (9, 10). In general, the selection of the parameters to use ismainly determined by the availability and reliability of relevant data.In this paper, the total expected age gain (extended service life withunit “year-lane-mile”) for the network is selected to characterize thepreservation benefit. The main reasons for the selection are two:

1. The expected service life for a repair action is readily available,either estimated from engineers’ experience or obtained by lookingup in maintenance or design files and

2. Expected service life is a simple but effective long-term mea-sure of pavement condition improvement, especially for agencies withno access to pavement performance prediction models (8).

In summary, two incommensurable and conflicting optimiza-tion objectives, maximization of the preservation benefit in termsof extended service life and minimization of the total preservationcost, are considered in the proposed model, as shown in Equations 1and 2, respectively.

where

Z1 = total network gain of extended service life (year-lane-mi), Z2 = total preservation cost (US$),

minimize Z x Cijk kk

s

j

n

i

m

2111

2====

∑∑∑ ( )

maximize Z x Aijk kk

s

j

n

i

m

1111

1====

∑∑∑ ( )

xijk = length (lane mi) of pavement in district i in condition state jthat will be treated by some treatment method from mainte-nance category k,

Ak = expected average age (year) associated with the kth mainte-nance category,

Ck = average repair cost associated with the kth maintenancecategory,

m = number of districts,n = number of condition state (the larger the value, the worse the

condition), ands = number of maintenance category.

Setting Up Constraints

Constraints to be considered in state-level pavement preservation pro-gramming should reflect the resource limitations and performancetargets of the agency. As the proposed model aims to generate theoptimal budget request and fund allocations simultaneously, variouspossible sets of constraints from the central administration, thedistricts, and the decision variables themselves may be considered.

Most state DOTs are under intense scrutiny to set up performancetargets and report on how well those targets are met for pavements andother infrastructures. Therefore, the performance targets or goals needto be understood by the legislation. In this regard, a macroscopic quan-tifiable target is in general preferred. For example, the performance tar-get for pavements in Virginia is “no more than 18 percent of interstateand primary pavements are rated as deficient (condition state beingpoor and very poor based on critical condition index)” (11). The per-formance target that specifies the agency requirements on the overallcondition of pavement network can be expressed as in Equation 3.

where

Lij = length (lane mi) of pavement in district i and conditionstate j,

T = performance target at the state level, usually in decimalform, and

u, v = integer values—a subset of (1, n)—for various conditionstates.

The second set of constraints is the cost constraint set, which rep-resents the preservation cost, either in total or for each district, asso-ciated with the repair plan of the entire pavement network. Morespecifically, the total preservation cost is required to be less than orequal to the allowable total network budget B, as provided in Equa-tion 4. In addition, to ensure a relatively equitable budget allocationacross districts, the ratio of the respective preservation cost betweenany two districts is required to be proportional to the respective lengthof “maintenance needed” pavement but within a certain percent-age range (±a%), as provided in Equation 5. The setting of percent-age range a% is flexible and requires an interactive communicationbetween the central administration and the districts.

x C Bijk kk

s

j

n

i

m

===∑∑∑ ≤

111

4( )

L x

L

Tij ijk

k

s

j u

v

i

m

ijj

n

i

m

−( )≤===

==

∑∑∑

∑∑11

11

3( )

where Mi and Mu = length (lane mi) of “maintenance needed”pavement in districts i and u, respectively.

The third set of constraints is related to the funding allocationacross districts. The application of certain fixed criteria (e.g., based ontotal road length under its jurisdiction) to all component districts maynot lead to optimal usage of funds. Practically, more than one crite-rion (quantifiable or intangible) often needs to be considered beforefinal allocations are made. AHP, in this regard, provides a formal ordocumented way to help differentiate funds allocation across districtsbased on multiple criteria. Given any district, the extent of pave-ment network condition improvement directly relates to the amountof funding it receives. Therefore, a set of constraints can be set upthrough AHP (as in Equation 6) that imposes variable system condi-tion improvement across districts to reflect the relative importance foreach district in receiving funds:

where wi equals the AHP-determined weights across districts thatreflect the relative importance for each district in receiving funds, andsum of wi over all districts equals to 100%.

The fourth set of constraints represents the different districts’preservation needs, as provided in Equations 7 and 8. For example,some districts may want to achieve within their network a conditionstate above the state performance target, while other districts may onlybe able to handle certain amount of pavement length in a year due tomanpower and/or facility restrictions.

where Ti is the preservation requirements at the district level, usu-ally in decimal form, and Q is the percentage amount, which speci-fies that the sum of pavement length selected for treatment in eachdistrict, regardless of condition state and maintenance category,does not exceed a certain percentage of the total pavement length inthat district. Q is determined based on factors such as pavementmaterials availability and the maintenance capacity of the regionalpaving industry.

The fifth set of constraints are the limitation constraints placingbound limits on the decision variables as indicated by Equations 9 and10. Equation 9 assures that the sum of pavement length within a givendistrict and condition states to be treated does not exceed the corre-sponding total pavement length. Equation 10 is the nonnegativityconstraints placed on all decision variables.

x i m j n k sijk ≥ ∀ = = =0 1 2 1 2 1 2, , . . . , ; , , . . . , ; , , . . . , (110)

x L i m j nijkk

s

ij=

∑ ≤ ∀ = =1

1 2 1 2 9, , . . . , ; , , . . . , ( )

x Q L i mijkk

s

j

n

ijj

n

== =∑∑ ∑≤ ∀ =

11 1

1 2 8, , . . . , ( )

L x

LT i m

ij ijkk

s

j u

v

ijj

i

−( )≤ ∀ ===

∑∑∑1 1 2 7, , . . . , ( )

1 1

11 11

11wx A

wx A

iijk k

k

s

j

n

ii jk k

k

s

j

=== +

+( )==

∑∑ ∑nn

i m∑ ∀ = −1 2 1 6, , . . . , ( )

1 11

11

−( ) ≤ ==

==

∑∑

∑∑a

M

M

x C

x C

i

u

ijk kk

s

j

n

ujk kk

s

j

n% ≤≤ +( ) ∀ =1 1 2 5a

M

Mi u mi

u

% , , , . . . , ( )

30 Transportation Research Record 2084


Model Solution

As the two objectives addressed in the optimization formulationare assumed to be of similar importance to the decision maker,nonpreemptive goal programming is therefore used to solve themultiple-objective optimization problem. The process consists ofthe following two steps:

1. Solve through a payoff table two individual maximization(minimization) problems in Equations 1 and 2 subject to the same setof constraints to find the respective optimal solution, thus obtain-ing a range of values [Z1lb

, Z1max] for objective Z1 and [Z2min, Z2ub] for

objective Z2.2. For the maximization (minimization) problem, the upper bound

(lower bound) of the range value of the ideal solution will be consid-ered as the relevant goal. As the goals often represent different unitsand have different orders of magnitude, to avoid comparing “applesto oranges” (12) and also to avoid unacceptable deviations from goals,the final integrated objective function, Equation 11, takes the form ofa linear scale function with the range of [0, 1] for each goal:

subject to Equations 3–10

where Z1, Z2 are defined in Equations 1 and 2, and d–u, d

+u (u = 1, 2)

are positive and negative deviations, respectively, of uth objectivefrom its goal.

IMPLEMENTATION

The model was applied to determine the optimal pavement preser-vation budgeting across districts for the Interstate and primary flex-ible pavement network. Features of the network are adopted fromdocumentation and state-of-the-practice pavement management inVirginia (11, 13, 14).

d d uu u− + ≥ =, , ( )0 1 2 14

Z d d Z2 2 2 2 13+ − =− +min

( )

Z d d Z1 1 1 1 12+ − =− +max

( )

minimize Zd d

Z Z

d d

Zlb u

=+( )−

++( )+ − + −

1 1

1 1

2 2

2max bbZ− 2

11min

( )

Inputs

A composite index called the critical condition index (CCI), on a scaleof 0 to 100, was employed to evaluate the condition of the pavements.A CCI of 100 represents a pavement in perfect condition, and a 0 rep-resents pavements in the worst possible condition (15). A CCI-basedbreakdown of Interstate and primary flexible pavements by conditionstates and districts in 2006 is given in Table 1.

At the programming level, it is feasible to categorize the varioustreatment methods into categories and assign an average cost to eachcategory on the basis of historical maintenance cost data. Differenttreatment methods under a particular treatment category could be usedto correct the pavement sections that have reached a selected thresh-old, and the specific type of treatment to choose for a given pavementsection is up to each individual district. The maintenance activities,corresponding costs, and expected service life are listed in Table 2.

Assumptions

Because of the difficulty in obtaining accurate data for some of themodel inputs, it was necessary to use some engineering judgmentin this example. However, this does not reduce the realism of themodel application. The following assumptions are only for illustra-tion purpose in the paper; different agencies may modify theseassumptions based on their respective network condition and agencyrequirements:

1. System condition improvement across districts will be deter-mined on the basis of multiple criteria to reflect the relative importancefor each district in receiving funds.

2. In Table 1, 14.6% of total state lane miles are rated as deficient(pavements classified as poor and very poor), which is within theestablished state-level target to keep the percentage of deficient pave-ments below 18% (11). However, a target of 12% is assumed in thispaper, so that the objective of minimization of total preservation costmakes more sense in this example.

3. According to the FY 2006–2007 annual budget report, the totalpreservation cost for the Interstate and primary flexible pavement net-work is around $185 million (16), which is assumed to be the allow-able total network budget in this example.

4. The districts that currently do not meet state performance target(Districts 1, 2, 4, 6, 8, and 9 in this case) should be improved to meetthat target. In addition, to ensure a relatively rational and equitablebudget allocation, each district will be allocated at least 50% of

TABLE 1 Breakdown of Interstate and Primary Flexible Pavements by Condition States and District, 2006

IS and PS System

Condition States District 1 District 2 District 3 District 4 District 5 District 6 District 7 District 8 District 9

Excellent (CCI ≥ 90) 399.7 421.0 878.2 936.3 572.3 516.8 638.5 988.2 498.8

Good (CCI: 70–89) 1,664.1 1,537.5 956.6 1,678.8 866.5 928.8 880.5 1,255.9 826.4

Fair (CCI: 60–69) 789.0 552.0 618.8 928.2 401.8 570.4 340.1 638.2 466.3

Poor (CCI: 50–59) 330.7 442.8 183.8 431.8 99.4 282.8 196.1 377.5 273.3

Very poor (CCI ≤ 49) 261.8 165.3 64.9 60.5 89.3 139.0 31.3 171.6 104.1

Sum lane miles for each district 3,445.3 3,118.6 2,702.3 4,035.6 2,029.3 2,437.8 2,086.6 3,431.4 2,168.9

Percentage of deficient 17.2 19.5 9.2 12.2 9.3 17.3 10.9 16.0 17.4(poor and very poor )

the budget based only on its percentage of deficient pavementover the entire network.

5. There are no maintenance needs for “excellent” pavements.Pavements in condition j ( j 2–5 corresponding to condition “good–very poor,” respectively) can only select, if to be treated, one con-dition from the maintenance category k (k 2–5 corresponding tocategory PM–RC, respectively) that is equal to or higher than j. Forexample, if pavements in fair condition are to be treated, then onlymaintenance categories CM, RM, and RC are candidate choicessince DN and PM do not help address effectively the problems in faircondition from the perspective of pavement life-cycle performance.

6. Due to limitations on annual asphalt production and pavingequipment availability and to avoid excessive variation in pavingquantities from year to year, it is assumed that no more than 30% ofeach district’s network can be maintained every year and no more thanhalf of the 30% can be maintained with preventive maintenance.

Multiple Criteria Consideration in Funds Allocation

The criteria adopted to help allocate funds across districts are preferredto be concrete and objective. In this example, three criteria are chosento help differentiate current year funds allocation across districts:

1. Need: “district current year preservation needs,” which can beobtained on the basis of the assessment of current year conditions byaggregating prioritized maintenance strategies proposed by componentresidencies;

2. Usage: “district Interstate and primary system vehicle miles,”which can be obtained by multiplying the total system mileage by theaverage daily traffic the system serves; and

3. History: “previous budget allocation per maintenance needmile across districts,” which can be obtained by dividing the allo-cated budget to each district by the maintenance-need mileage in thatdistrict.

The AHP is used here to systematically consider these three crite-ria. As the extent of pavement network condition improvement isdirectly linked to the funding allocated, the outcome from AHP willbe weighted associated with system condition improvement for alldistricts to reflect their relative importance in receiving funds. ExpertChoice, a decision support software package, can be used to constructthe AHP model and perform the underlying mathematics (17). Alter-natively, this may also be done by using Microsoft Excel. A detaileddescription of the procedure is provided in the following sections.

Hierarchy Construction

A three-level hierarchy is considered (Figure 1). Level 1 representsthe overall goal (to determine the ranking of importance across dis-tricts in receiving the funding). Level 2 contains three criteria thatinfluence the goal. At Level 3, there are nine elements (the districts)that influence Level 2.

Comparisons and Consistency Check

In comparing the importance of any two elements at same level,the two primary questions asked of the decision maker would be(a) “Which attribute is more important or has greater influence onthe attribute one level higher in the hierarchy?” and (b) “What is theintensity of that importance according to a predetermined scale,where 1, 3, 5, 7, 9 represent ‘equal importance,’ ‘moderate impor-tance,’ ‘strong importance,’ ‘very strong importance,’ and ‘extremeimportance,’ respectively, and 2, 4, 6, 8 represent the intermediatevalues between adjacent scale values?” (6). It is important that con-sistency be maintained during the pairwise comparisons, althoughAHP does allow for reasonable deviations in consistent comparisonand does not require the consistency to follow exact mathematicalproportions (7 ). The detailed pairwise comparisons procedure isshown in Figure 2.


TABLE 2 Maintenance Actions for Interstate and Primary Flexible Pavements

Activity Cost Expected LifeCategory Maintenance Activity ($/lane mile)a (years)b

Do nothing (DN)

Preventive maintenance (PM)

Corrective maintenance (CM)

Restorative maintenance (RM)

Rehabilitation–reconstruction (RC)

NOTE: N/A = not applicable.aAverage value of most likely cost for Interstate and primary pavements, calculated on the basis of a lane width of 12 ft and only material costs included.bUse average value.

N/A

1. Minor patching (<5% of pavement area; surface patching; depth 2 in.)2. Crack sealing3. Surface treatment (chip seal, slurry seal, latex)

1. Moderate patching (<10% of pavement area; partial depth patching; depth 6 in.)

2. Partial depth patching (<10% of pavement area; depth 4–6 in.) and surface treatment

3. Partial depth patching (<10% of pavement area; depth 4–6 in.) and thin (≤ 2 in.) AC overlay

4. ≤ 2 in. milling and ≤ 2 in. AC overlay

1. Heavy patching (<20% of pavement area; full depth patching; depth 12 in.)2. ≤ 4 in. milling and replace with ≤ 4 in. AC overlay3. Full-depth patching (<20% of pavement area; full-depth patching;

depth 9–12 in.) and 4 in. AC overlay

1. Mill, break and seat and 9–12 in. AC overlay2. Reconstruction

N/A

$4,547

$51,867

$133,261

$362,043

N/A

2–5Average 3

7–10Average 8

8–12Average 10

15+Average 20


Synthesis of Information

Once priorities on weights are established for the entire pairwisecomparison matrices, the process of synthesis can proceed. Lower-level priorities are weighted by higher-level priorities until the bot-tom level is reached. At this stage, the composite priorities (i.e., theoverall relative weights of the alternatives, where these weights sumto one) are calculated and can be used as rank scales for overall rela-tive importance. Take District 1 as an example; the relevant weightw1= 0.309 � (0.180) + 0.581 � (0.145) + 0.110 � (0.110) = 0.152.Similarly, the weights for District 2 to District 9 are 0.176, 0.048,0.147, 0.052, 0.116, 0.048, 0.137, and 0.124, respectively.

Model Customization

In this example, two objective functions were considered simulta-neously: (a) Objective Z1: maximization of the preservation benefitin terms of total system age gain (extended service life in “year-lane-miles”) and (b) Objective Z2: minimization of the total preser-vation cost in “dollars.” In summary, the complete formulation for thecase study is constructed as follows:

subject to

x C M

L

L

B iijk kk jj

ijj

ijji

==

=

==

∑∑∑

∑∑≥ ∀ =

5

2

54

5

4

5

1

911 2 9 19, , . . . , ( )

x C Bijk kk jji ===∑∑∑ ≤

5

2

5

1

9

18( )

L x

L

ijji

ijkk jji

ijji

== ===

==

∑∑ ∑∑∑

∑

−4

5

1

9 5

4

5

1

9

1

5

11

917

∑≤ T ( )

minimize Z x Cijk kk jji

2

5

2

5

1

9

16====

∑∑∑ ( )

maximize Z x Aijk kk jji

1

5

2

5

1

9

15====

∑∑∑ ( )

where

xijk = decision variable, length (lane mi) of pavement in dis-trict i in condition state j that will be treated with sometreatment method from maintenance category k;

Ak = expected average age (years) associated with the kthmaintenance category (use average value in third col-umn of Table 2);

Ck = average repair cost associated with kth maintenancecategory (use data in second column of Table 2);

Lij = length (lane mi) of pavement in district i and conditionstate j;

T = established state-level target to keep percentage ofdeficient pavements below a certain percentage (in thisexample, 12%);

B = allowable total network budget (in this example,$185 million);

x i j kijk ≥ ∀ = = =0 1 2 9 2 5 2 5 27, , . . . , ; , . . . , ; , . . . , ( ))

x L i jijkk

ij=

∑ ≤ ∀ = =2

5

1 2 9 2 5 26, , . . . , ; , . . . , ( )

xx x x x x

ijkk

ij ijkk

ij ijkk

ij

=

= =∑∑ ∑

=− + − +

2

5 23

5

34

5

44 5

3

1 2 9 2 5 25

−

∀ = =

x

i j

ij

, , . . . , ; , . . . , ( )

x k j i j kijk = ∀ < = = =0 1 2 9 2 5 2, , , . . . , ; , . . . , ; , . . . , 55 24( )

xQ

L iijj

ijj

22

5

1

5

21 2 9 23

= =∑ ∑≤ ∀ = , , . . . , ( )

x Q L iijkk jj

ijj== =

∑∑ ∑≤ ∀ =5

2

5

1

5

1 2 9 22, , . . . , ( )

L x

L

P iij

jijk

k jj

ijj

= ==

=

∑ ∑∑

∑

−≤ ∀ =4

5 5

4

5

1

51 2 4 6, , , ,, , ( )8 9 21

1 15

2

5

11

5

2wx A

wx A

iijk k

k jj ii jk k

k jj

=== +

+( )==

∑∑ ∑55

1 2 8 20∑ ∀ =i , , . . . , ( )

Level 1Goal

Level 2Criteria

Level 3Alternatives

District4

District5

District6

District7

District8

District9

District1

District2

District3

District Interstate andprimary system vehicle miles

District current-yearpreservation needs

Previous budget allocationper maintenance-need mileacross districts

Ranking of districts

FIGURE 1 Hierarchy for analyzing sample network.

wi = AHP-determined weights associated with average sys-tem condition improvement for all districts that reflecttheir relative importance in receiving funds given theabove three criteria, as given in Figure 2; and

M, P, Q = district-level requirements (in this example, M = 50%,P = 12%, and Q = 30%).

Equation 17 is the performance target that specifies the agencyrequirements on the overall condition of pavement network. Equation18 is the cost constraint that is required to be less than or equal to theallowable total network budget. Equation 19 is the cost constraint thatensures a relatively fair budget allocation. The selection of percentageM often requires an interactive communication between the central


FIGURE 2 Comparison matrices for analyzing sample network.

Goal: Ranking of districts

Criteria: C1 = District Interstate and primary system vehicle miles

C2 = District current-year M&R need

C3 = Previous budget allocation per maintenance-need mile across districts Alternatives: Z1, Z2, ... Z9 = District 1, District 2, ... District 9, respectively

0.309 0.581 0.110

Consistency check:

Maximum eigen value = 3.004

Consistency ratio = 0.36% (less than 10%, accepted)

Level 3 (Alternatives) Pairwise comparison with respect to the criterion C1

0.180

0.122

0.079

0.270

0.046

0.062

0.041

0.157

0.043

Weight Vector

Z1

Z2

Z3

Z4

Z5

Z6

Z7

Z8

Z9

Consistency check:


Consistency ratio

= [(9.242 - 9) / 8] / 1.45

= 2.1% (less than 10%, accepted).

Z1

Z2

Z3

Z4

Z5

Z6

Z7

Z8

Z9

1

1/2

1/3

2

1/4

1/3

1/4

1

1/4

2

1

1/2

3

1/3

1/2

1/3

1

1/3

3

2

1

5

1/2

1/2

1/2

3

1/3

1/2

1/3

1/5

1

1/5

1/3

1/5

1/2

1/4

4

3

2

5

1

1

1

3

1

3

2

2

3

1

1

1/ 2

3

1/2

4

3

2

5

1

2

1

4

1

1

1

1/3

2

1/3

1/3

1/4

1

1/3

4

Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9

3

3

4

1

2

1

3

1


0.145

0.207

0.034

0.061

0.039

0.159

0.055

0.137

0.163

Weight Vector Z1

Z2

Z3

Z4

Z5

Z6

Z7

Z8

Z9

Consistency check:


Consistency ratio

= [(9.117 - 9) / 8] / 1.45


Z1

Z2

Z3

Z4

Z5

Z6

Z7

Z8

Z9

1

2

1/5

1/3

1/4

1

1/2

1

1

1/2

1

1/6

1/3

1/5

1

1/4

1/2

1

5

6

1

2

1

4

2

4

4

3

3

1/2

1

1/2

3

1

2

3

4

5

1

2

1

4

1

3

5

1

1

1/4

1/3

1/4

1

1/3

1

1

2

4

1/2

1

1

3

1

3

3

1

2

1/4

1/2

1/3

1

1/3

1

1

1

1

1/4

1/3

1/5

1

1/3

1

1


0.110

0.157

0.038

0.257

0.141

0.041

0.028

0.082

0.146

Weight Vector

Z1

Z2

Z3

Z4

Z5

Z6

Z7

Z8

Z9

Consistency check:


Consistency ratio

= [(9.225 - 9) / 8] / 1.45


Z1

Z2

Z3

Z4

Z5

Z6

Z7

Z8

Z9

1

2

1/3

3

1

1/4

1/5

1

2

1/2

1

1/5

2

1

1/4

1/5

1/2

1

3

5

1

5

4

1

1/2

3

4

1/3

1/2

1/5

1

1/2

1/5

1/6

1/4

1/2

1

1

1/4

2

1

1/4

1/5

1/2

1

4

4

1

5

4

1

1/2

2

3

6

5

2

6

5

2

1

3

4

1

2

1/3

4

2

1/2

1/3

1

2

1/2

1

1/4

2

1

1/3

1/4

1/2

1

Level 2 (Criteria) Pairwise comparison with respect to the goal

1 1/2 3 2 1 5 1/3 1/5 1

Weight Vector C1 C2 C3 C1 C2 C3

C1 C2 C3


administration and the districts. Equation 20 is the set of constraintsthat imposes variable system condition improvement across dis-tricts to reflect the relative importance for each district in receiv-ing funds. These constraints force the model to allocate resourcesproportionally to the importance of each district.

Equations 21–23 are constraints that represent the different dis-tricts’ preservation needs. Equation 21 requires the districts that cur-rently do not meet the state performance target to improve and meetthat target. Equation 22 controls that the sum of pavement lengthselected for treatment in district i, regardless of condition state andmaintenance category, does not exceed a certain percentage of thepavement network in that district. Equation 23 limits the total lengthof pavement that can be maintained with preventive maintenance ineach district.

Finally, Equations 24–27 are the limiting constraints that placelimits on the decision variables. Equations 24 and 25 assure thatpavements in condition j can only select, if to be treated, one condi-tion from maintenance category k that is equal to or higher than j.Equation 26 assures that the sum of pavement length within a givendistrict and condition state to be treated does not exceed the totalpavement length within that district and condition state. Equation 27shows the nonnegativity constraints placed on all decision variables.

Results

As both objective functions and constraints in the optimization for-mulation are linear, a global optimum solution is obtained by usingthe optimization software LINGO 10.0 (18) (Figure 3). The distribu-tion of the preservation budget across districts and the breakdown ofpavements that would receive treatments from different maintenancecategories are also presented in this figure. As expected, the solu-tion emphasized preventive maintenance. This is because preventivemaintenance provides the most improvement in average expected ser-vice life per dollar invested. For example, the dollars invested per unitgain of extended life ($/year-lane-mile) for preventive maintenanceare approximately 1,342 (4,547/3); comparatively, the same unit forcorrective maintenance, restorative maintenance, and rehabilitationor reconstruction is around 6,483 (51,867/8), 13,326 (133,261/10),

and 18,102 (362,043/20), respectively, an obvious advantage thatfavors the selection of preventive maintenance. The solution alsoprogrammed enough corrective maintenance, restorative mainte-nance, and rehabilitation or reconstruction to reach the establishedperformance targets and different districts’ preservation requirements.

Figure 4 shows a comparison of pavement network conditionsacross districts among different scenarios: the current year condition,the projected conditions for the optimized (“Model”) preservationplan, the “Max. Benefit” repair plan, and the “Min. Cost” repair plan.As shown, there are some marked differences in current conditionsamong the districts, ranging from around 9% of deficiency lane-milesin District 3 to about 20% in District 2. All three optimization solu-tions lead to a more comparable distribution among districts, improv-ing the overall condition of the network by concentrating on thosecomponent districts that have a significantly high percentage of defi-cient pavements. While there is no obvious difference in terms of per-centage of deficient pavements, the respective “benefit” and “cost”behind the three optimization scenarios are noticeably different. Thegain in extended system service life (year-lane-mile) and total preser-vation cost ($ million) for the “optimized” repair plan are 21,293 and148.6, respectively. Comparatively, they are 13,288 and 137.2 for the“minimize total preservation cost” repair plan, and 28,025 and 185 forthe “maximize preservation benefit” repair plan. By considering both“maximize preservation benefit” and “minimize total preservationcost,” the “model” repair plan reaches a compromise solution:

• An 8.3% increase in preservation cost that results in a 60.2%increase in system age gain with respect to the “minimize totalpreservation cost” plan, and

• A savings of 19.7% ($36.4 million) in preservation cost withonly a 24.0% decrease in system service life units compared with the“maximize preservation benefit” plan.

DISCUSSION OF PROPOSED MODEL

The proposed global network optimization model is simple to applyfor multilevel pavement management with minimal data requirements.Objectives and constraints can be established on the basis of specified

20.29 $M

20.64 $M

5.68 $M

19.41 $M

4.71 $M

12.29 $M

6.21 $M

33.33 $M 26.05 $M

0.0

200.0

400.0

600.0

800.0

Len

gth

(la

ne

mile

)

Preventive Maintenance Corrective Maintenance Restorative Maintenance Rehabilitation/Reconstruction

Rehabilitation/Reconstruction 0.0 0.0 14.0 0.0 8.9 0.0 12.4 0.0 0.0

Restorative Maintenance 179.1 233.9 0.0 8.0 0.0 129.3 0.0 137.3 117.1

Corrective Maintenance 0.0 0.7 0.0 165.9 2.0 10.1 0.0 0.0 61.7

Preventive Maintenance 482.1 467.8 247.1 574.2 304.4 365.7 257.8 514.7 325.3

District 1 District 2 District 3 District 4 District 5 District 6 District 7 District 8 District 9

FIGURE 3 Global optimal solution and recommended preservation plan.

agency requirements and analysis processes can be done automati-cally by computer. However, the analyst should be cautious withthe data reliability and definition of the constraints. While the for-mer is a requirement that holds for any pavement managementmodel, the latter is more oriented toward the problem or situationunder consideration. The solution tends to be very sensitive to con-straints related with district-level preservation requirements. Forexample, constraints in Equations 21∼23 reflect the different strate-gies (e.g., “worst first,” “preserving the existing”) that variousdistricts may choose; an improper setting of the relevant values maylead to infeasible solutions. It is therefore recommended that theanalyst consult with district or residency engineers before settingcombined constraints.

As the nonpreemptive goal programming method attempts to min-imize the weighted sum of deviations of all objective functions fromtheir respective goals, appropriate determination of the weights asso-ciated with deviations is therefore very important. Although thecommon practice is to approximately use equal weights, as in thisexample, techniques capable of setting the weights in a more objec-tive way are encouraged due to their benefits in helping improve theaccountability and objectivity of the output solutions. It is not uncom-mon for the AHP user to feel uncertain about some of the pairwisecomparisons. Therefore, sensitivity analysis is a good supplement totest the robustness of the solutions. In some cases, where the mitiga-tion of potential bias or system error is necessary for a single decisionmaker context, group decision making can be used. For example, thiscan be accomplished by establishing a maintenance program leader-ship group and integrating through certain methods the group AHPoutputs as the final input for the optimization model. The most chal-lenging part in using AHP is setting up the comparison matrices. Asthis method is based on the principle that in decision making experi-ence and knowledge are at least as valuable as the quantitative data, itis expected that experience and knowledge will help compensate forthe possible uncertainty in making comparisons.

Application of the model on a yearly basis (i.e., a new run everyyear) allows the determination of the optimum preservation actions tobe implemented in the immediate future. However, by consideringonly the system service life gain of each individual year instead of thetotal planning horizon as the long-term measure of pavement condi-tion improvement, such optimum preservation actions sometimes can

be short-sighted, as they do not take into account the future evo-lution of the system, which truly depends on reliable pavementperformance predictions. Many state DOTs have realized this andare working toward a performance-based instead of a needs-basedbudgeting process.

Finally, the model cannot replace the decision makers in the plan-ning process as it does not fully capture all the variables involved.The optimum decisions recommended by the model will help deci-sion makers make good decisions, but it is, of course, not enough toguarantee them.

SUMMARY AND CONCLUSIONS

This paper proposes a decision-support model for the optimization ofshort-term pavement preservation needs-based budgets based on twoproven operations research techniques: goal programming for han-dling multiple objectives and AHP for priority setting under multiplecriteria. The model simultaneously considers two incommensurableand conflicting objectives: maximization of the preservation effec-tiveness in terms of extended service life and minimization of the totalpreservation cost. AHP is applied to capture the relative importanceof each district for receiving funds.

The implementation of the proposed model in an illustrative exam-ple shows that the approach is practical, flexible, and easy to imple-ment. Objectives and constraints can be established based on specifiedagency requirements, and analysis processes can be done automati-cally by computer. While the example focuses on the Interstate andprimary flexible pavement network, the approach proposed is applic-able to any transportation infrastructure network (e.g., bridge networkor secondary pavement network) at the state, district, or local levelwhere allocation across subnetworks under multiple criteria and mul-tiple objectives is an important consideration in the decision-makingprocess.

REFERENCES

1. AASHTO. Pavement Management Guide. American Association of StateHighway and Transportation Officials, Washington, D.C., 2001.


Current condition Projected condition _'Model' repair plan

Projected condition _'Max.Benefit' repair plan Projected condition _'Min.Cost' repair plan

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

District 1 District 2 District 3 District 4 District 5 District 6 District 7 District 8 District 9 Statewide

Def

icie

ncy

pav

emen

t %

state-level performance target

FIGURE 4 Comparison of different optimization scenarios.


2. Chan, W. T., T. F. Fwa, and J. Y. Tan. Optimal Fund-Allocation Analy-sis for Multidistrict Highway Agencies. Journal of Infrastructure Systems,Vol. 9, No. 4, 2003, pp. 167–175.

3. Ignizio, J. P. Goal Programming and Extensions. Lexington Books, Lex-ington, Mass., 1976.

4. Charnes, A., and W. W. Cooper. Goal Programming and Multiple Objec-tive Optimization, Part 1. European Journal of Operational Research,Vol. 1, 1977, pp. 39–54.

5. Vargas, L. G. An Overview of the Analytic Hierarchy Process and ItsApplications. European Journal of Operational Research, Vol. 48, No. 1,1990, pp. 2–8.

6. Saaty, T. L. The Analytic Hierarchy Process. McGraw–Hill, New York,1980.

7. Saaty, T. L. The Analytic Hierarchy Process: Planning, Priority Setting,Resource Allocation. McGraw–Hill International Book Co., New York,1980.

8. Abaza, K. A. Expected Performance of Pavement Repair Works in aGlobal Network Optimization Model. Journal of Infrastructure Systems,Vol. 13, No. 2, 2007, pp. 124–134.

9. Haas, R., W. R. Hudson, and J. Zaniewski. Modern Pavement Manage-ment. Krieger Publishing Company, Malabar, Fla., 1994.

10. Hudson, W. R., R. Haas, and W. Uddin. Infrastructure Management.McGraw–Hill, New York, 1997.

11. Virginia Department of Transportation. Asset Management Methodol-ogy: Appropriation Act Item 444 A (Special Session I, 2006). Asset Man-agement Division, VDOT, Richmond, Va., 2006.

12. Romero, C., C. Sutcliffe, J. Board, and P. Cheshire. Naive Weighting inNon-Preemptive Goal Programming. Viewpoint and Reply. Journal ofthe Operational Research Society, Vol. 36, 1985, pp. 647–649.

13. Asset Management Division. State of the Pavement—2006, Interstate andPrimary Flexible Pavements. Pavement Management Program, VirginiaDepartment of Transportation, Richmond, Va., 2006.

14. Asset Management Division. NBB3: Documentation and State of thePractice. Systems Management Group, Virginia Department of Trans-portation, Richmond, Va., 2005.

15. McGhee, K. H. Development and Implementation of Pavement ConditionIndices for the Virginia Department of Transportation (Phase I): Flexi-ble Pavements. Richmond, Va., 2002. www.virginiadot.org/business/resources/asd-rfp124roATTCH2.pdf. Accessed July 13, 2006.

16. VDOT Annual Budget, June 2007. Financial Planning Division, VirginiaDepartment of Transportation, Richmond, Va., 2007. www.virginiadot.org/business/resources/asd-rfp124roATTCH2.pdf. Accessed July 19,2007.

17. Expert Choice, Inc. www.expertchoice.com/markets/index.html#AHP.Accessed July 16, 2007.

18. Schrage, L. Optimization Modeling with LINGO, 6th ed. LINDO SystemsInc., Chicago, Ill., 2006.

The Pavement Management Systems Committee sponsored publication of thispaper.

hybrid multiobjective optimization model for regional pavement-preservation resource allocation

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