Time-Cost-Quality Trade-Off Analysis for Highway Construction

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Time-Cost-Quality Trade-Off Analysis for HighwayConstruction

Khaled El-Rayes, M.ASCE,1 and Amr Kandil, A.M.ASCE2

Abstract: Many departments of transportation have recently started to utilize innovative contracting methods that provide newfor improving construction quality. These emerging contracts place an increasing pressure on decision makers in the constructto search for an optimal resource utilization plan that minimizes construction cost and time while maximizing its quality. Thpresents a multiobjective optimization model that supports decision makers in performing this challenging task. The model is dtransform the traditionaltwo-dimensional time-costtradeoff analysis to an advancedthree-dimensional time-cost-qualitytrade-off analysis. The model is developed as a multiobjective genetic algorithm to provide the capability of quantifying and considering qconstruction optimization. An application example is analyzed to illustrate the use of the model and demonstrate its capagenerating and visualizing optimal tradeoffs among construction time, cost, and quality.

DOI: 10.1061/~ASCE!0733-9364~2005!131:4~477!

CE Database subject headings: Information technology~IT!; Algorithms; Highway construction; Cost control; ContractConstruction industry.

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Introduction

Existing networks of highways and roads are aging and derating at a fast pace that requires immediate rehabilitation e~ASCE 2002!. Public expenditures have increased in recent yin order to renew and revitalize these aging networks. In 2total expenditures on highway capital outlay and maintenprojects reached a volume of $87 billion~FHWA 2000!, and theFederal Highway Administration estimates that an average obillion per year is needed over the next 20 years to improvenation’s highways and bridges~U.S. DOT 2000!. To maximizelong-term return on this public investment, government agenhave recently started utilizing new types of contracting metthat are designed to achieve multiple project objectives, incluminimizing construction cost and duration, while maximizingquality.

In recent years, many departments of transportation~DOTs! invarious states have started to apply new highway contramethods, including:~1! Bidding on cost/time~i.e., A+B method!to encourage competition among contractors to minimize prduration ~Herbsman 1995; El-Rayes 2001!; ~2! incentive/disincentive contract clauses that provide financial incentive

1Assistant Professor, Dept. of Civil and Environmental EngineeUniv. of Illinois at Urbana–Champaign, Urbana, IL 61801. [email protected]

2PhD Candidate, Dept. of Civil and Enviromental EngineerUniv. of Illinois at Urbana–Champaign, Urbana, IL 61801. [email protected]

Note. Discussion open until September 1, 2005. Separate discumust be submitted for individual papers. To extend the closing daone month, a written request must be filed with the ASCE ManaEditor. The manuscript for this paper was submitted for review andsible publication on June 19, 2003; approved on June 18, 2004.paper is part of theJournal of Construction Engineering and Management, Vol. 131, No. 4, April 1, 2005. ©ASCE, ISSN 0733-9364/2005
477–486/$25.00.
JOURNAL OF CONSTRUCTIO

J. Constr. Eng. Manage. 2

reduce construction duration~Jaraiedi et al. 1995!; ~3! nighttimeconstruction that seeks to cut service disruption and projectby requiring contractors to work during off-peak nighttime ho~Ellis and Amos 1996; El-Rayes and Hyari 2002, 2004!; ~4! war-ranty contracting that attempts to improve construction qualitmaking contractors liable for the performance of the facility aproject completion~Anderson and Russell 2001; ENR 2002!; and~5! multiparameter contracts that provide incentives to contrato improve quality performance~Anderson and Russell 200!.These new and emerging contracts place an increasing pron decision makers in the construction industry to search fooptimal/near-optimal resource utilization plan that minimizesstruction cost and time while maximizing its quality. This crenew and pressing needs for advanced resource utilization mthat are capable of optimizing the multiple and conflicting obtives of construction time, cost, and quality, as shown in Fig

Significant research advancements have been made in thof optimizing construction resource utilization. This led to a nber of optimization models that were developed using a variemethods, including linear programming, integer programmdynamic programming, and genetic algorithms~GAs!. Thesemodels can be classified according to their optimization otives into models that attempted to:~1! Minimize project timeand/or improve resource utilization~Easa 1989; Chan et al. 199Hegazy 1999; Gomar et al. 2002!; ~2! minimize time and cost fononrepetitive construction using time-cost trade-off ana~Burns et al. 1996; Feng et al. 1997; Li and Love 1997; Maxet al. 1998; Li et al. 1999; Feng et al. 2000!; and ~3! minimizetime and/or cost for repetitive construction~Moselhi and ElRayes 1993; Senouci and Eldin 1996; Adeli and Karim 1El-Rayes 2001; El-Rayes and Moselhi 2001; Hegazy and Er2001; Hegazy and Wassef 2001; Leu and Hwang 2001!. While theabove research studies have provided significant contributiothe area of optimizing construction resource utilization, therebeen little or no reported research focusing on multiobje
models for optimizing construction time, cost, and quality.
N ENGINEERING AND MANAGEMENT © ASCE / APRIL 2005 / 477

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Objective

The objective of this paper is to present the development oadvanced multiobjective optimization model that supports mmizing construction time and cost, while maximizing its quaThe model is formulated to transform the traditionaltwo-dimensional time-costtrade-off analysis to an advancedthree-dimensional time-cost-qualitytrade-off analysis, as shown in F1. This transformation provides planners and decision makehighway construction with a much-needed model that is capof: ~1! Generating optimal/near-optimal resource utilization pfor highway construction projects that optimize construction tcost, and quality;~2! considering and quantifying quality in costruction optimization; and~3! visualizing three-dimensional reresentation of the trade-offs among project time, cost, and quin order to support decision makers in evaluating the impavarious resource utilization plans on project performance.optimization model is developed in three main stages: Modemulation, quantifying construction quality, and model implemtation. These three development stages are described in motail in the following sections.

Model Formulation

The primary purpose of this development stage is to formulrobust optimization model that supports advanced thdimensional time-cost-quality trade-off analysis. To this endpresent model is formulated in two major steps:~1! Determiningthe major decision variables in this resource utilization prob

Fig. 1. Impact of new contracting methods on decision makinconstruction

Table 1. Feasible Resource Utilization Options for Concrete Paving

Resourceutilizationoptionn

Resource composition

Material~MPa! Crew

Ov

1 31 A

2 31 B

3 34 A

4 34 B

5 31 A

6 31 B

7 34 A

8 34 B

478 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT


-

and~2! formulating the three major objectives of optimizing cstruction time, cost, and quality in a robust optimization mod

Decision Variables

For each construction activity in the project, the present moddesigned to consider all relevant decision variables that mayan impact on project time, cost, or quality. This includes:~1!Construction methodsmd, which indicates the availability of diferent types of materials and/or methods that can be utilized~2!crew formationsfd, which represents feasible sizes and confirations for construction crews; and~3! crew overtime policyspd,which represents available overtime hours and nighttime shiforder to control the complexity of the optimization model,present model combines these three major decision variablesingle decision variable called resource utilizationsnd, as shownin Fig. 2.

For example, feasible resource utilization options for perfoing the concrete paving activity in a highway project may incthe utilization of: ~1! A construction materialof concrete thaprovides a strength of 31 MPas4,500 psid or 34 MPas5,000 psid;~2! a crew formation Athat consists of one paving machine,grader, and a labor force of one foreman, three laborersequipment operators, and one cement finisher, or crewB that hasthe same composition except that it has a larger and moreerful concrete paving machine~RSMeans 2000!; and~3! anover-time policyof 0 or 4 h of overtime. Various combinations of thavailable options can be represented by different resource ution options snd, as shown in Table 1. Each of these feasresource utilization options has an expected daily productioncost rate, and quality performance; and accordingly it leads

Fig. 2. Time-cost-quality trade-off optimization model

Performance

Costs$/m2d

Productivitysm2/dayd

Quality~%!

32.44 2,090 90

37.36 2,510 90

33.54 2,090 96

38.57 2,510 96

38.86 2,613 88

44.71 3,135 88

33.65 2,613 94

46.28 3,135 94

ertime~h!

0

0

0

0

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© ASCE / APRIL 2005

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unique time, cost, and quality for this activity, as shown in Ta1.

Similarly, each of the remaining activities in the projecthave a similar set of feasible resource utilization options thabe used to construct the activity. The major challenge confroconstruction planners in this problem is to select an optimasource utilization option, from the available set of feasible anativessn=1 to Nd, for each activitysi =1 to ld in the project, ashown in Fig. 2. The possible combinations of these alternacreate a large search space ofNl, where each solution in this sparepresents a possible resource utilization option for deliverinproject. For example, a small-size project that includes 20 aties and 5 possible resource utilization options for each accreates a search space of approximately 95 trillion~i.e., 520! pos-sible solutions. The present model is designed to help plannthe challenging task of searching this large solution space into identify optimal resource utilization plans that achieve multproject objectives~see Fig. 2!.

Optimization Objectives

The present optimization model is formulated in order to prothe capability of minimizing construction time and cost, whmaximizing its quality. The model is also designed to quanand measure the impact of various resource utilization decion performance in each of the identified three project objec~see Fig. 2!. To this end, the model incorporates three majorjective functions as shown in the following three equationenable the evaluation of project performance in constructioncost, and quality, respectively.

Minimize project time =oi=1

l

Tin s1d

where Tin=duration of activitysid on the critical path using re

source utilizationsnd. In this model, project duration is estimausing newly developed algorithms for scheduling the construand renewal of highways~El-Rayes 2001; El-Rayes and Mose2001!.

Minimize project cost =oi=1

1

fsMin + Di

n 3 Rind + sBi

ndg s2d

whereMin=material cost of activitysid using resource utilizatio

snd; Din=duration of activity sid using resource utilizationsnd;

Rin=daily cost rate in $/day of resource utilizationsnd in activity

sid; Bin=subcontractor lump sum cost for resource utilizationsnd

Table 2. Quantifying Construction Quality

Quality indicatorskd

Base coarse

Gradation Drainage

Quality performancesQj ,kn d 85% 88%

Weight of indicatorswtj ,kd 0.2 0.4

Activity quality performance 88.2%

Weight of activity swtid 0.2

Project quality performance

in activity i, if any.



Maximize project quality =oi=1

l

wtiok=1

K

wti,k 3 Qi,kn s3d

where Qi,kn =performance of quality indicatorskd in activity sid

using resource utilizationsnd; wti,k=weight of quality indicatoskd compared to other indicators in activitysid; andwti =weight ofactivity sid compared to other activities in the project. In tmodel, a new and practical approach is utilized to measurequantify construction quality using Eq.~3!. This adopted approach is explained in more detail in the following section.

Quantifying Construction Quality

Estimating and quantifying the impact of a given resourcezation optionsnd on the quality of a construction activity aultimately the entire project is a more challenging task thandicting its impact on construction cost and duration. This caattributed to two major challenges:~1! The difficulty in measurinand quantifying the impact of each resource utilization optiothe quality of the activity being considered; and~2! the complexity of aggregating quality performance at the activity leveprovide an overall quality at the project level. For examplethe concrete paving activity shown in Table 1, estimating thafirst resource utilization optionsn=1d is expected to produceoverall quality of 90% is more challenging than predictingthe expected cost and production rate is 32.44 $ /m2 and2,090 m2/day, respectively. In addition, aggregating these quperformances at the activity level to estimate a quality pemance at the project level~see Table 2! is a more complex tasthan that required for construction cost and duration. In ordovercome these two major challenges, the present model incrates a new objective function for optimizing construction quaThe following two sections provide a brief description of hownewly formulated function@Eq. ~3!# is designed to overcome tabove two major challenges in quantifying and consideringstruction quality in the optimization process.

Measuring Quality Performance

In order to facilitate the measurement and quantification ofstruction quality, the incorporated quality objective function@seeEq. ~3!# enables the consideration of a number of measuquality indicators for each activity in the project. These indicahave been investigated and identified in recent studies thataimed at developing quality-based contractor prequalificationtems~Anderson and Russell 2001; Minchin and Smith 2001!. The

Activity sid

Concrete pavement

CompactionCompressive

strengthFlexuralstrength

Ridequality

90% 94% 91% 87%

0.4 0.3 0.3 0.4

90.3%

0.8

89.9%

identified quality indicators were derived from performance-


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based models that correlate the long-term performance of thproduct of each activity to its quality indicators. For examTables 2 and 3 illustrate a sample of these quality indicatorthe concrete paving activity that includes compressive streflexural strength, and ride quality.

Quality indicators should be selected in a way that allpractical and objective measurement of performance in eacdicator as shown in Table 3. For example, performance inpressive strength and ride quality indicators can be easily tand measured using a standard test for concrete comprstrength and pavement ride quality test, respectively~ASTM2003a, b!. For each resource utilization optionsnd, the results othese quality performance tests can be easily obtained andin a similar way to existing methods that collect and analyzeand time performances from previous projects. In fact, mDOTs are currently gathering and storing these types of datatheir ongoing projects using a variety of forms, such as theDaily Paving Record~Form BC 2531! and the Resident Engneer’s Daily Report~Form CEM-4501! used by the Illinois DOTand CalTrans, respectively~IDOT 2002; Caltrans 2001!. This col-lected and stored data can be statistically analyzed in ordestimate the construction time, cost, and quality that can bpected from employing resource utilizationsnd in activity sid forthe project being planned as shown in Tables 1 and 2.

It should be noted that test results for the selected quindicators are often expressed in different units of measureAs such, they need to be transformed to a unified system ofsurement that can be consistently used to evaluate performadifferent quality indicators. In this model, the results of quatests in different indicators are transformed to a common uvalue that ranges from 0 to 100%~Meredith et al. 1985! in orderto represent the degree of satisfaction of quality performaneach indicator as shown in Table 2.

Aggregation of Quality Performance

The formulated quality objective function@Eq. ~3!# enables thaggregation of the estimated quality for all the considered aties to provide an overall quality at the project level usinsimple weighted approach. For each activity being evaluatequality, this method requires planners to identify two typeweights: ~1! Weight of the activityswtid to represent the impo

Table 3. Quality Indicators~Anderson and Russell 2001; Minchin aSmith 2001!

Construction activity Possible construction quality indicators

Concrete pavement W/C ratio, consolidation/density, air contentthickness, compressive strength, flexuralstrength, ride quality

Bituminous pavement Compaction density, asphalt content, gradasurface smoothness, thickness, aggregate quvoid ratio, skid resistance

Bridge deck Consolidation/density, rebar cover, W/C ratiodensity, curing, air content, strength

Structural concrete Consolidation/density, rebar cover, W/C ratidensity, curing, air content, strength

Base course Aggregate quality, drainage, gradation,thickness, compaction/density, moisture conte

Embankment Compaction/density, moisture content, matequality, uniformity, drainage

Note: W/C=water/concrete.

tance and contribution of the quality of this activity to the overall



quality of the project; and~2! weight of quality indicatorkswti,kdin activity i to indicate the relative importance of this indicatoothers being used to measure the quality of the activity. Thestypes of weights are used to estimate the overall quality pemance at the project level, as shown in Eq.~3! and Table 2. Thillustrated method in Table 2 can be applied to additionalstruction activities using other measurable quality indicatorslar to those suggested by Anderson and Russell~2001!, andMinchin and Smith~2001!, as shown in Table 3.

Model Implementation

In order to support decision makers in their search for opttrade-offs between time, cost, and quality, the present modimplemented as a multiobjective GA. GAs are search and opzation tools that assist decision makers in identifying optimanear-optimal solutions for problems with large search space.are inspired by the mechanics of evolution and they adopsurvival of the fittest and the structured exchange of geneticterials among population members over successive generatia basic mechanism for the search process~Goldberg 1989!. Assuch, the present model is implemented in three major phase~1!Initialization phase that generates an initial set ofS possible solutions for this resource utilization problem;~2! fitness evaluatiophase that calculates the cost, time, and quality of each gensolution; and~3! population generation phase that seeks toprove the fitness of solutions over successive generationsdetailed computation procedure in these three phases is expin the following sections and is shown in Fig. 3.

Phase 1: Initialization

The main objective of this phase is to initialize the optimizaprocedure in the present model, using the following two msteps.1. Read project and GA parameters needed to initialize

search process. The project parameters include:~1! Projectsize; ~2! activity precedence information; and~3! availableresource utilization options for each activity and theirpected impact on the activity cost, duration, and quality slar to those shown in Table 1. The required GA paramefor this initialization phase include:~1! String size;~2! num-ber of generations;~3! population size;~4! mutation rate; an~5! crossover rate. The string size is determined bymodel, considering the total number of construction activincluded in the analyzed project as shown in Fig. 2.number of generationssGd and population sizesSd are identified based on the selected string size in order to improvquality of the solution. Similarly, the mutation and crossorates are determined considering the population size anmethod of selection employed by the algorithm~Reed et al.unpublished 2002!.

2. Generate random solutionsss=1 to Sd for the initial population P1 in the first generationsg=1d. These solutions reprsent an initial set of resource utilization options that caused to construct each and every activity in the projecshown in Fig. 2. This initial set of possible solutions is tevolved in the following two phases in order to generateof optimal resource utilization options for each activity inproject that establishes an optimal trade-off among cons
tion time, cost, and quality.

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Phase 2: Fitness Functions Evaluation

The main purpose of this phase is to evaluate the cost, timequality of each possible solutionssd in generationsgd in order todetermine the fitness of the solution. This fitness determinelikelihood of survival and reproduction of each solution inlowing generations. As such, this phase evaluates the threetified fitness functions for each solution using the following thsteps.1. Calculate project duration~Timeg

s for solution ssd in generation sgd, which consists of the total duration of all the actties on the critical path as shown in Eq.~1!. In this modelproject duration is estimated using newly developed arithms for integrating the scheduling of repetitive and nopetitive activities commonly encountered in the construcand renewal of highways~El-Rayes 2001, El-Rayes aMoselhi 2001!.

s

Fig. 3. Mod

2. Calculate project costsCostgd for solution ssd in generation



sgd, which consists of material, crew, and subcontractor cas shown earlier in Eq.~2!.

3. Calculate project qualitysQualitygsd for solution ssd in gen-

erationsgd as explained earlier in Eq.~3!.

Phase 3: Population Generation

The purpose of this phase is to create three types of populateach of the considered generations:~1! Parent;~2! child: and~3!combined. For each generationsgd, a parent populationsPgd isused to generate a child populationsCgd in a similar manner tthat used in traditional GAs~Goldberg 1989!. The purpose ogenerating this child population is to introduce a new set of stions by rearranging and randomly changing parts of the soluof the parent population. This child population can then be cbined with the parent population to create an expanded s

lementation
el imp
possible solutions that forms the combined populationsNgd for


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generationsgd. This combined populationsNgd is used to facilitatthe comparison among the initial solutions in the parent poption and those generated in the child population. The besttions in this combined population regardless of their originretained and passed to the following generation as a parentlation ~Deb 2001; Deb et al. 2001; Zitzler et al. 2001!. The com-putational procedure in this phase is implemented in the follosteps.1. Calculate Pareto optimal rank and crowding distance

each solution~s=1 to S! in the parent populationsPgd. First,this is done by ranking the solutions in the populationcording to their Pareto optimal domination of other solutiowhere a solution is identified as dominant if it is better tall other solutions in all of the considered optimizationjectives simultaneously. For example, solution S1 in Fig. 4can be considered a Pareto optimal solution that domi~i.e., provides higher quality and lower cost and durathan! all other solutions that may fall within the highlightcube. Second, this step calculates the crowding distaneach solution, which represents the closeness of neighbsolutions to the solution considered. The crowding distavalues help the algorithm spread the obtained solutionsa wider Pareto optimal front instead of converging to pothat cover only a small part of the tradeoff surface~Deb et al2001!.

2. Create a new child populationsCgd using the GA operationof selection, crossover, and mutation. The selection operchooses the individuals that will go through the reproducprocess, by favoring those with higher optimal rankswider crowding distances. The crossover operation, onother hand, crosses each pair of the selected individualrandomly determined point and swaps the variables codthe strings at this point, resulting in two new individuals. Tmutation operation randomly changes the value of one ovariables in the string to induce innovation and to prepremature convergence to local optima~Goldberg 1989!. Thefitness of the generated child population is then analusing the earlier described steps of Phase 2 in order to othe values of project cost, duration, and quality for eachlution as shown in Fig. 3.

3. Combine child populationsCgd and parent populationsPgd toform a new combined populationsNgd of size 2S. This com-bined population acts as a vehicle for the elitism, where gsolutions of the initial parent population are passed on tofollowing generation to avoid the loss of good solutions o

Fig. 4. Pareto optimal solution

they are found~Deb et al. 2001!.



4. Calculate Pareto optimal rank and crowding distanceeach solution~s=1 to 2S! of the newly created combinpopulationsNgd. This step performs the same operationStep 1 of this phase on the new combined populationsNgd.

5. Sort the new combined populationsNgd using the nichecomparison rule. This sorting rule selects solutionshigher Pareto optimal ranks and breaks ties betweentions with the same rank by favoring solutions with higcrowding distances.

6. Keep the topS solutions from the combined populationsNgdto form the parent populationsPg+1d of the next generatioThis parent population is then returned to Step 1 of this pfor generating a new child population as shown in FigThis iterative execution of the second and third phases omodel continues until the specified number of generaare completed.

Application Example

A project example is analyzed in order to illustrate the use opresent optimization model and demonstrate its capabilitiesexample consists of 18 construction activities, where eachnumber of possible resource utilization options that can be usconstruct the activity as shown in Table 4. The example was onally introduced by Feng et al.~1997! to illustrate constructiotime-cost trade-off analysis, and therefore it did not includeon construction quality. In order to highlight the capabilitiesthe present model, the data of the original example are expato illustrate the impact of each resource utilization option onstruction quality in addition to its time and cost, as showTable 4. For this simple example, there is an average of 3.source utilization options to construct each of the 18 activitieshown in Table 4, which produces more than 3.6 billion~i.e.,3.418! possible combinations for delivering the entire projEach of these possible combinations leads to a unique impaproject performance, and the main challenge here is to searclarge search space to find solutions that establish an optimadelicate balance among construction time, cost, and quality.

The present optimization model was used to search thisspace of possible solutions using the developed multiobjeoptimization model. The model was able to significantly redthis large space by precluding dominated solutions in the susive generations of the GA, using the Pareto optimality princshown in Fig. 4. This led to the selection of 305 Pareto opt~i.e., nondominated! solutions for this application example. Eaof these solutions identifies an optimal resource utilization opfor each of the 18 construction activities and, accordingly, itvides a unique and optimal tradeoff among project time, costquality. A sample of these solutions and their impact on properformance is shown in Table 5. The entire set of the idenPareto optimal solutions are represented graphically using athree-dimensional surface as shown in Fig. 5. This thdimensional representation of the identified solutions can beto visualize the trade-offs among project time, cost, and qualorder to support decision makers in evaluating the impact ofous resource utilization plans on project performance. Ttrade-offs can also be visualized in two-dimensional slicesillustrates trade-offs between two objectives at a time as shoFigs. 6 and 7.

In order to validate the results provided by the present mthey are compared to those reported in the literature for the
application example~Feng et al. 1997!. The comparison confirms

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Table 4. Available Resource Utilization Options

Activitysid

Resourceoption

sndDuration~days!

Cost~dollars!

Activityweight

~%!swtid

Quality indicatork=1 Quality indicatork=2 Quality indicatork=3

Indicatorweightswti,kd

Qualityperformance

sQi,kn d


Qualityperformance

sQi,kn d


Qualityperformance

sQi,kn d

1 1 14 2,400 3 50 100 30 96 20 98

2 15 2,150 90 89 89

3 16 1,900 86 77 84

4 21 1,500 75 72 73

5 24 1,200 63 60 65

2 1 15 3,000 5 40 98 40 94 20 99

2 18 2,400 87 94 95

3 20 1,800 81 92 85

4 23 1,500 77 72 70

5 25 1,000 60 66 59

3 1 15 4,500 8 70 100 15 97 15 98

2 22 4,000 80 82 81

3 33 3,200 62 60 63

4 1 12 45,000 11 50 99 35 95 15 94

2 16 35,000 74 71 76

3 20 30,000 59 63 64

5 1 22 20,000 10 60 100 20 97 20 99

2 24 17,500 93 89 89

3 28 15,000 77 71 72

4 30 10,000 61 64 61

6 1 14 40,000 11 50 95 25 95 25 100

2 18 32,000 76 74 79

3 24 18,000 59 62 68

7 1 9 30,000 10 30 97 30 99 40 93

2 15 24,000 70 73 71

3 18 22,000 61 62 67

8 1 14 220 1 100 95 0 NA 0 NA

2 15 215 83

3 16 200 75

4 21 208 68

5 24 120 61

9 1 15 300 1 50 100 50 99 0 0

2 18 240 97 92 0

3 20 180 81 88 0

4 23 150 71 75 0

5 25 100 63 64 0

10 1 15 450 1 60 94 40 97 0 0

2 22 400 79 83 0

3 33 320 63 69 0

11 1 12 450 2 70 96 30 95 0 0

2 16 350 72 75 0

3 20 300 61 66 0

12 1 22 2,000 3 50 99 35 98 15 95

2 24 1,750 89 85 87

3 28 1,500 70 71 79

4 30 1,000 62 61 63

JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / APRIL 2005 / 483

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that the present model is capable of generating the sameoptimal solutions as those reported by Feng et al.~1997! for thetime-cost trade-off analysis~see Fig. 7!. Furthermore, the presemodel provides the added capability of considering and quaning quality and visualizing trade-offs among project time, cand quality for the analyzed example~see Fig. 5!.

Conclusion

A multiobjective optimization model was developed to transfthe traditionaltwo-dimensional time-costtrade-off analysis to aadvancedthree-dimensional time-cost-qualitytrade-off analysisThe model is designed to search for optimal resource utilizplans that minimize construction time and cost while maximiits quality. The optimization model is developed in three mstages:~1! Model formulation stage that incorporates all madecision variables and optimization objectives;~2! quality quantification stage that formulates new functions to enable the

Table 4. ~Continued.!

Activitysid

Resourceoption

sndDuration~days!

Cost~dollars!

Activityweight

~%!swtid

Quali

Indicatoweighswti,kd

13 1 14 4,000 7 40

2 18 3,200

3 24 1,800

14 1 9 3,000 6 80

2 15 2,400

3 18 2,200

15 1 16 3,500 7 70

16 1 20 3,000 3 30

2 22 2,000

3 24 1,750

4 28 1,500

5 30 1,000

17 1 14 4,000 6 70

2 18 3,200

3 24 1,800

18 1 9 3,000 5 30

2 15 2,400

3 18 2,200

Note: NA=not available.

Table 5. Sample Pareto Optimal/Nondominated Solutions

SolutionOptimal resource

utilization optionsa

Project performance

Time~days!

Cost~dollars!

Quality~%!

1 $1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1% 104 166,320 95

2 $1,5,3,3,4,3,3,5,1,1,3,1,3,2,1,5,1,1% 114 105,470 71

3 $2,3,1,1,2,3,1,1,1,1,1,1,1,1,1,3,1,1% 115 141,620 90

4 $2,5,1,3,4,2,3,3,1,1,1,1,3,1,1,5,1,1% 109 121,350 77

5 $1,5,1,3,4,3,3,5,1,1,2,1,3,2,1,5,3,1% 124 104,620 72aEach solution is represented by a vector that indicates the iden
resource utilization option for each of the 18 activities.


fsideration of construction quality in this optimization probleand ~3! model implementation stage that implements a muljective GA for highway construction to enable the simultaneoptimization of construction time, cost, and quality. An appltion example is analyzed to illustrate the use of the modeldemonstrate its capabilities in considering quality in the opzation process and in developing optimal trade-offs amongstruction time, cost, and quality. These new capabilities shprove useful to decision makers in highway construction anhabilitation projects, especially those who are involved intypes of contracts that demand high-quality performance.

catork=1 Quality indicatork=2 Quality indicatork=3

Qualityerformance

sQi,kn d


Qualityperformance

sQi,kn d


Qualityperformance

sQi,kn d

99 40 96 20 97

73 71 76

60 62 63

100 10 95 10 98

79 82 81

63 67 66

100 30 98 0 NA

97 30 96 40 98

89 85 87

81 79 78

72 73 74

67 60 62

98 20 97 10 99

73 75 72

62 65 61

98 45 99 25 94

75 77 71

63 66 67

Fig. 5. Time-cost-quality trade-off surface

ty indi

rt p


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Acknowledgment

The writers gratefully acknowledge the financial support provby the National Science Foundation for this research prunder the CAREER Award CMS-0238470.

Notation

The following symbols are used in this paper:Bi

n 5 subcontractor lump sum cost for resource utilizatiosnd in activity sid, if any;

Cg 5 child population in generationsgd;Di

n 5 duration of activitysid using resource utilizationsnd;

Min 5 material cost of activitysid using resource utilization

snd;Ng 5 combined population in generationsgd;Pg 5 parent population in generationsgd;

Qi,kn 5 performance of quality indicatorskd in activity sid

using resource utilizationsnd;Ri

n 5 daily cost rate in $/day of resource utilizationsnd in

Fig. 7. Time-cost trade-off analysis

Fig. 6. Cost-quality trade-off analysis

activity sid;



Tin 5 duration of activitysid on the critical path using

resource utilizationsnd;wti 5 weight of activity sid compared to other activities in

the project; andwti,k 5 weight of quality indicatorskd compared to other

indicators in activitysid.

Subscripts and Superscripts

g 5 genetic algorithm generation~from g=1 to G!;i 5 construction activity~from i =1 to I!;k 5 quality indicator~from k=1 to K!;n 5 resource utilization optionsn=1 to Nd; ands 5 solution ss=1 to Sd.

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