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2004

A Multi-Objective Decision-Making Frameworkfor Transportation InvestmentsMashrur ChowdhuryClemson University, mac@clemson.edu

Pulin TanClemson University

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A Multi-Objective Decision-Making Framework for Transportation Investments Author(s): Mashrur Chowdhury and Pulin Tan Source: Journal of the Transportation Research Forum, Vol. 43, No. 1 (Spring 2004), pp. 91-104 Published by: Transportation Research Forum

Stable URL: http://www.trforum.org/journal

The Transportation Research Forum, founded in 1958, is an independent, nonprofit organization of transportation professionals who conduct, use, and benefit from research. Its purpose is to provide an impartial meeting ground for carriers, shippers, government officials, consultants, university researchers, suppliers, and others seeking exchange of information and ideas related to both passenger and freight transportation. More information on the Transportation Research Forum can be found on the Web at www.trforum.org.

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This paper presents a framework based on multi-objective optimization that can be used togenerate and analyze the most desirable transportation investment options based on their ob-jectives and constraints. The framework, which is based on the surrogate worth trade-offanalysis, could be applied to both discrete or continuous decision-problem scenarios. In adiscrete problem, a pre-defined set of alternatives is available, whereas continuous problemsare not characterized by a pre-defined set of alternatives. This framework was applied with thedata generated for a Capital Beltway Corridor investment study. The multi-objective decision-making framework was found to be adaptable to this typical investment case study.

by Mashrur Chowdhury and Pulin Tan

A Multi-Objective Decision-MakingFramework for TransportationInvestments

Transportation infrastructure decisions thatoptimize available resources and providemaximum benefits hold tremendous value tothe transportation community. Decisionmakers attempt to reach their goals with well-timed and cost effective decisions that investlimited available resources according to futureneeds. Thus, it becomes increasinglyimportant for decision makers to use objectivetools to make proper investment choices.

Most decision-making scenarios in thetransportation field are complex and includemultiple and often conflicting objectives.These objectives are sometimes difficult tomeasure in monetary units alone, so traditionaleconomic methods such as benefit-costanalysis may not be sufficient. If a policymaker has to decide between several mutuallyexclusive projects, benefit-cost analysis(CBA) is a useful economic tool forcomparing projects and deciding which oneis optimal. However, benefit-cost analysisrequires a common unit of measurement, andthe most typically used common unit is money.Therefore, all costs and benefits have to beexpressed in monetary values, including theones that are not so easily measurable such as

air quality or safety. Ideally, all valuation ofbenefits should be measured based on directlyobserved behavior in the market.Unfortunately, it is not always possible toobtain market price for all benefits. Monetaryvaluation of benefits for benefit cost analysismay be difficult to do and unreliable in itsresult for many decision scenarios. Multi-objective analysis enables evaluation of thealternatives without the need to convert theobjectives into monetary units. Additionally,multi-objective analysis is suitable fordecision scenarios with multiple objectiveswhere there is no single optimum solution.Multi-objective analysis provides a set of bestsolutions from a large set of available options,and provides an objective framework toeliminate a large number of possible optionsfrom any further consideration. At the sametime, the framework provides an acceptableconfidence bound where most desirablesolutions are included in the set of bestsolutions.

Trading off labor costs versusenvironmental impacts is a challenge for alltransportation projects. This and several othertradeoff scenarios comprise the multitude of

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criteria that a project must serve. To effectivelyreach a decision, a practitioner must utilize adecision-making framework.

A typical decision problem intransportation investment could becontinuous, discrete, or a combination of both.In a discrete problem, a predefined set ofalternatives is available. Many transportationinvestment projects have this characteristic,as project alternatives are selected from anumber of possible alternatives (such as theselection of a project site from several possiblesites). Continuous problems are notcharacterized by a pre-defined set ofalternatives. Instead, for a continuous problem(such as miles of a particular road that needto be reconstructed), a mathematical modelincluding decision variables, constraints, andmultiple objective functions must beformulated to generate alternatives. Thedecision alternatives are not pre-defined in acontinuous problem while a finite numberexist in discrete problems.

Several previous studies applied differentmulti-objective methods to develop tools formaking decisions in transportation-relatedissues, mostly for discrete problems. Many ofthese applications have been limited to autility-based approach, rather thanoptimization-based approach, where variousweights are assigned to different decisioncriteria. This type of decision-makingapproach was used in a study to identify thecritical highway safety needs of specialpopulation groups (Dissanayake et al., 2002).In another utility-based approach, a softwareprogram was developed based on assigningweights to multiple objectives (NCHRP2001). This utility-based approach could beused to evaluate transportation investmentdecisions on the basis of multiple goals,objectives, and measures.

The optimization-based approach is moreobjective than the utility-based approach formulti-objective decision analysis. The utility-based approach is greatly reliant upon decisionmakers’ input and criteria weighting, whenfinal output and project selection could beinfluenced by personnel changes among thedecision makers. The utility-based approachincludes the decision maker’s input as a part

of developing the output. Input is soughtthrough a set of questionnaires on the relativeimportance of selected measures ofeffectiveness (MOEs).

Haimes et al., applied a powerfuloptimization-based approach called theSurrogate Worth Trade-off (SWT) method fordecision making in water resource systems(Haimes et al., 1975). The SWT method isused to generate surrogate worth functions.The method is composed of severalconsecutive phases in two major steps. Thefirst step generates the non-dominatedsolutions using the constraint (ε ) method. Ina non-dominated solution, any improvementof one objective can be achieved only at thedegradation of the other. The constraintmethod is applied by optimizing any one ofthe objectives from n number of objectiveswhile all of the other objectives (n-1) areconstrained to some value (ε). The solutionto the problem largely depends on the chosenεk vector, which is chosen between theminimum and maximum values of the kth

objective function.The second step in the SWT method,

known as the interactive process, includesdirect interaction between the analyst and thedecision maker. It can guide the decisionmaker to develop tradeoffs among objectivesfrom the set of feasible solutions generatedearlier to find the preferred solution. In manysituations, transportation projects considerboth economic factors and financial return onthe investment and other factors such asquality of life and preservation of theenvironment. However, even though thegeneration of non-dominated solutions meetsthose criteria, decision makers need to maketradeoffs among such criteria.

In a study funded by the National ScienceFoundation, researchers developed aframework for evaluating the safety ofalternative automobile designs in terms of thelikelihood of crash occurrence and severityof likely injury (Haimes et al., 1994). Theresearchers used a multi-objective decisionanalysis approach called “Partitioned Multi-objective Risk Method” to develop theframework for evaluating the vehicle-basedcrash avoidance and worthiness technologies

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based on the expected and worst-caseoutcomes. This methodology permittedevaluation based on unconditional expectedevents as well as worst-case outcomes. Theproposed framework included the SWTmethod to assess the preference of the decisionmaker/design engineer for competing designalternatives by interviewing him or her andcommunicating the possible outcomes andcorresponding trade-offs.

A study by Chowdhury et al. (2000),“Multi-Objective Methodology for HighwaySafety Resource Allocation,” presented anInteractive Multi-Objective ResourceAllocation (IMRA) tool to help decisionmakers minimize the frequency and severityof vehicular crashes by selectingcountermeasures and allocating resourcesoptimally among various competing highways.This methodology also illustrated the tradeoffbetween various decision options and how toset priorities for a variety of potential crashcountermeasures. The main objective of thisresearch was to develop a tool that would aidin optimal resource allocation to improvehighway safety. The IMRA tool supportedinteraction between the analyst and thedecision maker that would help the decisionmaker select the best among various options.

An optimization-based approach, whichaddresses a combination of discrete andcontinuous problems for transportationinvestment analysis would be applicable tomany investment decisions faced by pubicagencies. A case study that demonstrates theapplication of a multi-objective framework toactual transportation investment scenarios willprovide motivation for real-worldapplications.

OBJECTIVES

The general objective of the researchdocumented in this paper was to develop amulti-objective decision support system thatwould aid the decision-making process in a

more systematic way. Specific objectives arelisted below:1. Develop a decision-making framework

for selecting between multiple competingalternatives, continuous and/or discretedecision problems, while maximizing thedesired and minimizing the undesiredattributes.

2. Demonstrate the practical application ofthe methodology to actual transportationprojects.The proposed framework was applied

with the data generated in a Capital BeltwayCorridor transportation investment analysis(Maryland State Highway Administration2001). The detailed case study follows adiscussion of the framework.

FRAMEWORK OF THE MULTI-OBJECTIVE METHODOLOGY

The multi-objective framework includes atotal of five steps. These steps are describedbelow.

Step 1. Identify Objectives

The first activity is to identify the objectivesto be measured. An objective is a statementabout the desired state of the system underconsideration. Objectives should be specificand cover the main goal or need, such asminimizing delay time, minimizing cost, andmaximizing safety. Although some projectshave a single objective, most transportationprojects have multiple objectives. The analystshould consider all of the objectives in thisprocess.

Step 2. Select Measures of Effectiveness

In this step, the objectives to be measured aredefined. The effectiveness of each projectalternative is measured according to theperformance of these alternatives on all of theobjectives specified in Step 1.

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Step 3. Formulate a Mathematical Model

The mathematical model is expressed as amathematical function that represents theproblem. The model is used to generate thevalue of decision variables and maximize orminimize the objective function subject to thespecified constraints. If there are n-relateddecisions to be made, they are represented asdecision variables (x1,x2,…,xn) whoserespective values are to be determined.

The appropriate measures ofeffectiveness, such as cost and travel time, arethen expressed as a mathematical function ofthese decision variables. This function iscalled the objective function. For example, inthe scenario given above the objective is tominimize cost and the decision variable (xi)is number of miles of road to build in area i,where i = 1, 2, and 3. The identification of thedecision variable leads to essential answersto the questions the decision-maker is seeking.The constant value (Ci) in this case may bethe cost per mile of road built in area i, wherei = 1, 2, and 3. So, the objective functionwould be total cost, C = C1x1 + C2x2 + C3x3.Any restrictions on the values that can beassigned to these decision variables are alsoexpressed mathematically, typically by meansof inequality or equality. This mathematicalrestriction is called a constraint function.

A constraint function can restrict orreduce the number of alternatives. Commonconstraints in a transportation project arefunding, right of way, and sometimestechnology. For example, a region may haveestablished a monetary limit per fiscal yearfor pavement programs, which is a budgetconstraint. A highway project in a metropolitancity has a right-of-way constraint to buildingnew access, and an ITS project usually has atechnological constraint to meet thespecifications.

The decision variable (xi), objectivefunction (Zj), and constraint function are usedto represent the decision making problems bytransforming them into a mathematical model.The decision variable is used to differentiatethe mathematical model between a continuousand discrete problem. In a continuous

problem, the decision variables will becontinuous, such as the case where decisionvariable xij represents miles of pavement typei in area j that will be built.

In a discrete problem, the decision makersimply decides which projects are to bechosen. The mathematical model for this typeof problem uses the following decisionvariables:

Each xi is a binary variable, which has valueof 0 or 1. Binary variables are important inmathematical models because they representa “yes” or “no” decision. In this case, a yes/no decision means project i is or is notselected. After solving the mathematicalmodel, the result will show that xA = 1 whenproject A is selected. However if the resultshows xA = 0, this means Project A is notselected. An example is provided below whereminimizing cost is the objective, subject to aset of constraints such as travel time andemissions that are less than some amount ornumber,εK.

The mathematical model can beexpressed as follows:

where, xi = 0 or 1. εT and εE are constants,which are given or acceptable limit of traveltime and emissions, respectively.

(3) Subject to,

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1 if a project i is selected, i = 1,2,…,m (1) xi = 0 if project i is not selected

(2) Minimize Cost, Z = ∑i=1

m xi * Ci,

Ci= cost of project i

∑i=1

m xi * Ti ≤ εT,

Ti= travel time of project i

∑i=1

m xi * Ei ≤ εE,

Ei= emissions from project i

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Step 4. Generate Non-Dominated SolutionsUsing Surrogate Worth Tradeoff Method(SWT)

The SWT method is a multi-objective methodused to generate a set of solutions and providea technique that incorporates the decisionmaker’s preferences in choosing the optimalsolution. The following tasks are performedin this step:

Construct a Payoff Table

A payoff table (Table 1) consists of allobjective values, when each objective isoptimized subject to constraints. The first rowin the table shows that the result Z1(X

1)represents the objective values for the firstoptimization run, X1, optimizing objective Z1.This process (optimization run) is repeated fora number of times equal to the number ofobjectives, Z1, Z2,.., Zp. For example, whenthe total number of objectives are three (Z1,Z2, and Z3), the optimization should be runthree times to construct a payoff table. XP

refers to the number of optimization runs foreach Z. The maximum (max) and minimum(min) refer to the optimization runs with thehighest and lowest values of Z.

The purpose of developing a payoff tableis to help formulate the constraint model inthe next task by determining the lower andupper bounds for the constraint e value, such

The primary objective is (Zh) where the hth

objective is chosen arbitrarily for followingoptimization model:

(6) Maximize Zh (X1, X2,…,Xn)

(7) Subject to:

as lower and upper bounds of cost or traveltime constraints.

Transform a Multi-Objective Probleminto a Single Objective Problem

This task involves considering one objectiveas primary and transforming other objectivesas constraints. The general form of a multi-objective problem with p objectives and mconstraints is shown below transformed intoa constraint model (Cohon, 1978).

(4) Maximize or minimize Z1(X1, X2,…,Xn),Z2 (X1, X2,…,Xn),…,Zp(X1, X2,…,Xn)

(5) subject to: c1 (X1, X2,...,Xn) < 0,c2 (X1, X2,...,Xn) < 0,...,cm (X1, X2,...,Xn) < 0Xj > 0, j = 1,2,...,n

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c1 (X1, X2,…,Xn) ≤ 0, c2 (X1, X2,…,Xn) ≤ 0, …,cm (X1, X2,…,Xn) ≤ 0 Zk (X1, X2,…,Xn) ≤ εk k = 1,2,…,h-1, h+1,…,p

Xj ≥ 0, j = 1,2,…,n

Z1(Xk) Z2(X

k) & ZP(X

k)

X1 Z1(X

1) Z2(X1) & Zp(X

1)

X2 Z1(X

2) Z2(X2) & Zp(X

2)

. . . & .

. . . & .

. . . & .

Xp Z1(X

p) Z2(Xp) & Zp(X

p)

Max Maximum value of Z1(Xk) Maximum value of Z2(X

k) Maximum value of ZP(Xk)

Min Minimum value of Z1(Xk) Minimum value of Z2(X

k) Minimum value of ZP(Xk)

Table 1: Payoff Table

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Choose the Different Values of εk from theRange of Minimum and Maximum Values forEach Objective (identified in Step 1)

The minimum and maximum values (for eachcolumn representing Z1(X

k), Z2(Xk),…, Zp(X

k)in the payoff table) are derived from the payofftable. Feasible solutions to the constraintmodel will exist when εk is chosen betweenthe minimum and maximum limit. Theselection of constraint values, εk, between theminimum and maximum limit, ensure thatfeasible solutions to the constraint problemcould be generated. Each solution of theconstraint model, with a selected combinationof εk values between the minimum andmaximum limit, will produce a non-dominatedsolution when all the objective constraints arebinding.

Solve the Constraint Model for EveryCombination of Values for the εk

The mathematical models (constraint model)with every combination of constraint values,εk, are solved in this task to generate a set ofnon-dominated solutions. The model may besolved mathematically or by usingcommercially-available optimization softwarepackages.

Step 5. Choose the Preferred Solution

The analyst presents the set of non-dominatedsolutions to the decision maker. The decisionmaker can choose a solution from the set ofnon-dominated solutions presented in thisstep.

AN EXAMPLE APPLICATION OFTHE MULTI-OBJECTIVEFRAMEOWRK

The multi-objective decision-making frame-work discussed above was applied to theCapital Beltway Project in the metropolitanWashington, DC, area to demonstrate thepractical application of the methodology toactual transportation projects. The data thatwere used for this example were based on a

Capital Beltway Corridor Study (MarylandDepartment of Transportation 2001).

Project Description (MarylandDepartment of Transportation 2001)

The Capital Beltway corridor is located in themetropolitan Washington, DC, area. It is theonly circumferential route in the area,connecting many radial routes. A studyconducted by the Maryland State HighwayAdministration found that the projected highincrease in travel demand within the Beltwaycorridor in the year 2025 requires that bothHigh Occupancy Vehicle (HOV) lanes and railtransit will be needed to handle the projectedtraffic. This Maryland study recommendedthat both HOV lane and rail transit alternativesbe considered. HOV lanes and rail transitwould perform different functions. It wouldserve different markets within the region andcorridor. HOV lanes are added to concurrentlanes by adding one lane in each direction,providing commuters who are willing tocarpool or take a bus with one lane on theBeltway that operates without too muchcongestion. It was concluded that even whenrail transit is available, a large percentage oftotal trips in the corridor would be made byautomobile. It was further concluded that theHOV lanes would help to improve travelconditions for HOV users. The MarylandDepartment of Transportation conductedseparate impact studies for the HOV lane andthe rail transit alternatives for the study area.The HOV lane corridor was divided into fivesegments. Rail transit was divided into sixdifferent alignments (P1, P2, P3, P4, P5, P6),in which P1, P2, and P3 were aligned forheavy rail and P4, P5, and P6 for light railtransit. Figure 1 shows a map of the study area.

Decision Problem

In this study case, the decision-makingproblem was solved using the decisionframework presented in the previous section.The best solution was found by combiningboth HOV lane and rail transit options anddeciding how many miles of HOV lane were

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Figure 1: Capital Beltway Corridor Case Study Area

(Source: Maryland Department of Transportation 2001)

needed for each segment. Additionally, analignment was to be selected with thecorresponding light or heavy rail transit.

Problem Solving

Step 1. The identification of objectives is thefirst step in the decision-making process. Theobjectives for the Capital Beltway Corridorproject were as follows (Maryland Departmentof Transportation 2001):

• Support regional mobility and addresstravel demand,

• Minimize incremental costs whilemaximizing transportation capacity, and

• Improve accessibility to existing andplanned economic development areas andregional activity centers.

Step 2. Measure of Effectiveness used fivedifferent criteria to evaluate each alternative.

• Total costs• Annual ridership

• Daily new ridership• Public support, and• Economic development.

Each HOV segment and transit alternativewas evaluated and measured for theireffectiveness. “Annual ridership” representscurrent ridership without any improvementsand “daily new ridership” represents theincrease in ridership because ofimprovements. “Public support” wasmeasured based on comments received fromthe public. “Public support” scores rangedfrom “0” to “5,” respectively, representingpublic opposed to the improvements (i.e., 0)to high public support (i.e., 5). “Economicdevelopment” was measured by forecastingwhether the improvement will facilitateeconomic development by connecting majorresidential and/or employment activitycenters. “Economic development” scoresranged from “0” to “5,” respectively,representing the improvements will have noeffect on economic development (i.e., 0) to

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Table 2: Measures of Effectiveness for HOV Lanes

SEGMENT1 2 3 4 5

* scores fro

the improvements will greatly facilitateeconomic development by connecting majorresidential and/or employment activity centers(i.e., 5). The measures of effectiveness areshown in Tables 2 and 3. These values wereobtained for this case study from a MarylandDepartment of Transportation Capital BeltwayCorridor Transportation study. Ridershipvalues in Tables 2 and 3 are approximationsas they are based on the assumption that theyare not affected by the HOV length or type oftransit selected.

Each of the criteria is measured in termsof cost, annual ridership, and daily newridership. Public support and economicdevelopment are scored between 0 and 5(negative to positive impact). This case studyis basically a combination of a continuous(HOV) and a discrete (rail transit) problem.

The goal is to generate how many HOV milesneed to be built and decide which rail transitalignment should be chosen.

The rail transit alternatives are based onalignment and rail transit type (light and heavyrail):P1 = heavy rail transit alternative with

alignment 1P2 = heavy rail transit alternative with

alignment 2P3 = heavy rail transit alternative with

alignment 3P4 = light rail transit alternative with

alignment 4P5 = light rail transit alternative with

alignment 5P6 = light rail transit alternative with

alignment 6

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Step 3. Develop a MathematicalFormulation. There are two types of decisionvariables in this model. One, which iscontinuous, is for the HOV alternative,representing how many miles of road need tobe built for each segment. The HOValternative is divided into five differentsegments:Xi = miles of road in segment i,

i = 1,2 …,5

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1, if a project j is selected, Pj = j = 1,2,…,6

0, if a project is not selected.

The second, which is discrete, is for rail transitrepresenting a “yes” or “no” decision (1 = yesand 0 = no) based on six different alignments(Pj with j = 1,2,…,6) and type of rail transit;heavy rail (P1, P2, P3) or light rail (P4, P5,P6).

Table 3: Measures of Effectiveness for Rail TransitTOTAL COST (IN MILLI ONS OF $)

ALIGNMENT

SEGMENT P1 P2 P3 P4 P5 P6

1 $2,136 $1,659 $2,136 $786 $857 $786

2 $3,766 $3,019 $3,066 $1,254 $1,257 $1,430

3 $2,599 $2,599 $2,599 $766 $630 $630

4 $1,418 $1,418 $1,418 $470 $423 $423

ANNUAL RIDERSHIP

ALIGNMENT

SEGMENT P1 P2 P3 P4 P5 P6

1 5,935,403 4,382,770 5,228,753 3,507,785 5,376,469 3,492,959

2 33,810,886 40,724,948 28,604,913 53,625,924 30,363,296 39,989,544

3 11,133,474 10,531,544 10,789,939 10,251,554 15,392,336 12,122,800

4 4,522,762 4,338,398 4,317,583 6,005,887 9,396,844 8,627,636

DAILY NEW RIDERSHIP

ALIGNMENT

SEGMENT P1 P2 P3 P4 P5 P6

1 10,028 7,401 8,830 5,925 9,085 5,900

2 30,847 32,054 26,090 42,332 27,696 31,822

3 10,156 9,606 9,840 9,360 14,036 11,063

4 7,636 7,325 7,291 10,148 15,870 14,578

PUBLIC SUPPORT

ALIGNMENT

SEGMENT P1 P2 P3 P4 P5 P6

1 4.00 5.00 4.00 4.00 4.00 4.00

2 4.00 4.33 4.00 4.00 4.00 4.33

3 4.00 4.00 4.00 4.00 4.00 4.00

4 5.00 5.00 5.00 4.00 5.00 5.00

ECONOMIC DEVELOPMENT

ALIGNMENT

SEGMENT P1 P2 P3 P4 P5 P6

1 5.00 5.00 5.00 5.00 4.00 5.00

2 4.00 4.67 4.00 4.33 4.50 4.33

3 4.00 4.00 4.00 4.50 4.00 4.00

4 4.00 4.00 4.00 4.00 4.00 4.00

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Ci = total cost per mile for HOV in segment iCj = total cost for rail transit alternative jAi = annual ridership per mile for HOV insegment iAj = annual ridership for rail transit alternativejDi = daily new ridership per mile for HOV insegment iDj = daily new ridership for rail transitalternative jSi = score of public support per mile for HOVin segment i

Z1 Z2 Z3 Z4 Z5

Min Cost X1 $3,167,000,000 60,528,944.40 66,686.90 17.00 16.50

Max AnnualRidership X

2 $4,497,745,000 79,832,327.60 89,523.10 26.00 32.83

Max Daily NewRidership X

3 $4,497,745,000 79,832,327.60 89,523.10 26.00 32.83

Max Public Support X4 $9,916,745,000 66,418,837.10 78,142.70 28.33 32.67

Max Economic Dev. X5 $4,497,745,000 79,832,327.60 89,523.10 26.00 32.83

Minimum

Value $3,167,000,000 60,528,944.40 66,686.90 17.00 16.50

Maximum

Value $9,916,745,000 79,832,327.60 89,523.10 28.33 32.83

Table 4: Payoff Table of Capital Beltway Corridor

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Five objectives are considered based on theMOEs selected earlier,1. Minimize Total Cost,

Z1 = ∑i=1

5 Xi * Ci + ∑

j=1

6 Pj * Cj

2. Maximize Annual Ridership,

Z2 = ∑i=1

5 Xi * Ai + ∑

j=1

6 Pj * Aj

3. Maximize Daily New Ridership,

Z3 = ∑i=1

5 Xi * Di + ∑

j=1

6 Pj * Dj

4. Maximize Public Support,

Z4 = ∑i=1

5 Xi * Si + ∑

j=1

6 Pj * Sj

5. Maximize Economic Development,

Z5 = ∑i=1

5 Xi * Ei + ∑

j=1

6 Pj * Ej

Sj = score of public support for rail transitalternative jEi = score of economic development per milefor HOV in segment iEj = score of economic development for railtransit alternative j

Constraints:X1 < 4.0 milesX2 < 2.6 milesX3 < 8.6 milesX4 < 8.3 milesX5 < 16.6 milesP1 + P2 + P3 + P4 + P5 + P6 = 1Xi > 0

There is a restriction for Pj (j = 1,2,…,6)where the sum of Pj should equal 1,representing mutually exclusive alternatives(i.e., only one rail transit alternative needs tobe chosen). In addition, the total HOV mileageshould be less than or equal to the totalsegment length.

Step 4. Generate a non-dominated solutionusing the SWT, including the following tasks:

Construct a Payoff Table. The first task wasto construct a payoff table (Table 4) byoptimizing each of the five objectivesseparately (cost, annual ridership, daily newridership, public support, and economicdevelopment) to obtain maximum or minimumvalues.

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Formulate a Constraint Model. The secondtask is to transform a multi-objective probleminto a single objective problem using theconstraint method. In this case, the maximizedpublic support objective (Z4) was chosen asa primary objective and all other objectives(Z1, Z2, Z3, and Z5) are transformed asconstraints, as shown below:

Constraints:

X1 < 4.0 milesX2 < 2.6 milesX3 < 8.6 milesX4 < 8.3 milesX5 < 16.6 milesP1 + P2 + P3 + P4 + P5 + P6 = 1Xi > 0

Choose Constraint Values. The L values(Table 5) are chosen arbitrarily from differentvalues of the range of minimum and maximumfor objectives 1, 2, 3, and 5 from the payofftable (Table 4). By choosing L between theminimum and maximum values, the feasiblesolutions for the above constraint problem canbe generated. Each solution of the constraintmodel, with a selected combination of L valuesbetween the minimum and maximum limit,will produce a non-dominated solution whenall the objective constraints are binding.

Table 5: Constraint Values of Capital Beltway Corridor

Constraint Selected Constraint Values

L1 $5,416,915,000.00 $7,666,830,000.00 $9,916,745,000.00

L2 66,963,405.47 73,397,866.53 79,832,327.60

L3 74,298.97 81,911.03 89,523.10

L5 21.94 27.39 32.83

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(8) Maximize Public Support,

Z4 = ∑i=1

5 Xi * Si + ∑

j=1

6 Pj * Sj

(9) Z1 = ∑i=1

5 Xi * Ci + ∑

j=1

6 Pj * Cj ≤ L1

(10) Z2 = ∑i=1

5 Xi * Ai + ∑

j=1

6 Pj * Aj ≤ L2

(11) Z3 = ∑i=1

5 Xi * Di + ∑

j=1

6 Pj * Dj ≤ L3

(12) Z5 = ∑i=1

5 Xi * Ei + ∑

j=1

6 Pj * Ej ≤ L5

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Step 5. The final stage of this decisionframework is to select the preferred solutionfrom the set of alternatives. The analyst couldpresent the solved objective function valuesin a graphical format to the decision maker todemonstrate the relative trade-off for choosingbetween competing alternatives (Y1,Y2,…,Y7). This will help the decision makerselect an alternative from the optimal set ofalternatives generated through solving thedecision model.

CONCLUSIONS

Most decision making in transportationagencies involves multiple objectives thatoften conflict and cannot be measured inmonetary units. This makes the use oftraditional investment analysis tools, such asbenefit-cost analysis, difficult. This studypresented a multi-objective framework thatcould be applied under different decisionscenarios in transportation investmentprocesses. Instead of transforming all different

Derive Solutions from the Model. The finaltask was to solve the constraint problem bymaximizing the public support score subjectto all constraints for every combination ofvalues for L1, L2, L3, and L5. The model wassolved for 80 runs with each run including adifferent combination of objective constraintvalues. The optimization process shows thatPj = 1 when project J is selected. If the resultshows Pj = 0, project J is not selected. Sevenproject alternatives (Y1, Y2, Y3, Y4, Y5, Y6,and Y7) were generated as combinations ofHOV and rail transit projects by solving theconstraint model presented in this step. Thecombinations are shown in Table 6.

In Table 6, Y1 represents the result of theoptimization where HOV decision variablevalues of X1, X2, X3, X4, and X5 equaled 4,2.6, 8.6, 8.3, and 16.6 miles, respectively andthe rail transit decision variable values of P1,P2, P3, P4, P5, and P6 equaled 0, 1, 0, 0, 0,and 0, respectively.

Based on the decision variable valuesgenerated in Table 6, the objective values (Z1,Z2, Z3, Z4, and Z5) for each alternative areshown in Table 7. To show the relativeimportance of its objective values, these valuesare transformed (as shown in Table 8) into a0-to-1 (Zp*) scale using the equation below.This helps demonstrate their relativeimportance or utility, which could becommunicated to the decision maker in agraphical format.

For example, the relative importance value ofobjective 1 (Z1*) for alternative Y1 is:

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(13) Zp* = {Zp - Zp(min)}/

{Zp(max) – Zp(min)}

(14) Z1* = {7666830000 - 5416915000}/ {9294426400 – 5416915000} = 0.58

Table 6: Generated Alternatives and Associated Decision Variables from theSolved Constraint Model

HOVProject

Selection X1 X2 X3 X4 X5

Rail

Transit

Y1 4.00 2.60 8.60 8.30 16.60 P2

Y2 2.01 2.60 0.00 0.00 0.00 P2

Y3 4.00 2.60 8.60 2.94 0.00 P2

Y4 4.00 2.60 2.06 0.00 16.60 P2

Y5 4.00 2.60 8.60 4.28 0.00 P5

Y6 2.65 2.60 0.00 0.00 0.00 P5

Y7 4.00 2.60 8.60 8.30 16.60 P6

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The application of the proposedframework in the Capital Beltway Corridorinvestment study demonstrated the suitabilityof the methodology. The framework presentedin this study could be considered by publicagencies as an alternative or complement totraditional economic analysis and integratedwith agency funding processes andmanagement systems. The proposed decision-analysis framework, which is general in nature,could also be applied to other transportationareas, such as aviation, rail, and water.

Table 8. Objective Value Scale of Capital Beltway CorridorObjectiveProject

Alternative Z1 (min) Z2(max) Z3(max) Z4(max) Z5(max)

Y1 0.58 1.00 1.00 1.00 1.00

Y2 0.58 0.00 0.00 0.03 0.00

Y3 0.58 0.51 0.67 0.54 0.51

Y4 1.00 0.38 0.38 0.56 0.51

Y5 0.00 0.56 0.91 0.51 0.51

Y6 0.00 0.07 0.26 0.00 0.00

Y7 0.58 1.00 1.00 1.00 1.00

Table 7: Objective Value of Generated Solution of Capital Beltway CorridorObjectiveProject

Alternative Z1(min cost) Z2(max annual ridership) Z3(max daily new ridership) Z4(max public support) Z5(max economic development)

Y1 7,666,830,000 68,183,307.3 81,035.6 27.9 32.5

Y2 7,666,830,000 60,954,471.3 61,376.5 21.1 21.9

Y3 7,666,830,000 64,630,147.7 74,572.9 24.7 27.4

Y4 9,294,426,400 63,682,935.7 68,904.4 24.8 27.4

Y5 5,416,915,000 64,989,685.3 79,309.7 24.4 27.4

Y6 5,416,915,000 61,433,500.7 66,446.4 20.8 21.9

Y7 7,666,830,000 68,183,307.3 81,035.6 27.9 32.5

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project alternatives or objectives intomonetary values, these alternatives orobjectives can be approached on an equalbasis in their own measures of effectiveness,either in monetary or non-monetary terms.The proposed framework permits objectivedecision analysis for any transportationinvestment.

The proposed framework addresses bothdiscrete or continuous decision problems anda combination of the two. This makes theproposed framework applicable to a widerange of decisions that are required intransportation investment scenarios.

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References:

Chowdhury, Mashrur, Nicholas Garber, and Duan Li. “Multi-objective Methodology for HighwaySafety Resource Allocation.” American Society of Civil Engineers Journal on Infrastructure Systems,6(4), (2000): 138-144.

Cohon, Jared L. Multi-objective Programming and Planning, Academic Press, London, 1978.

Dissanayake, Sunanda, J. Lu, X. Chu, and P. Turner. “Use of Multi-criteria Decision Makingto Identify the Critical Highway Safety Needs of Special Population Groups.” TransportationResearch Record, No. 1693, (2002): 13-17.

Haimes, Y., J. Eisele, M. Chowdhury, P. Kuczzminski, R. Schwing, N. Garber and D. Li,Improvement of Highway Safety Through Optimal Vehicle Design: Fault-Tree and Multi-objective Analysis, National Science Foundation (NSF) Project SES–9308247 Report, 1994.

Haimes, Y.Y., W. Hall and H. Freedman. Multi-objective Optimization in Water Resources Systems,Elsevier Scientific, New York, 1975.

Maryland Department of Transportation, Capital Beltway Corridor Transportation Study, Maryland,2001.

National Cooperative Highway Research Program (NCHRP). Development of a Computer Modelfor Multimodal, Multicriteria Transportation Investment Analysis, Research Results Digest,Number 258, 2001.

Mashrur A. Chowdhury is an assistant professor in the Department of Civil and EnvironmentalEngineering and Engineering Mechanics at the University of Dayton. Chowdhury received his Ph.D. inCivil Engineering from the University of Virginia. He is a registered professional engineer in Ohio andthe District of Columbia.

Pulin Tan was a graduate research assistant in the Department of Civil and Environmental Engineeringand Engineering Mechanics and the University of Dayton. She received her M.S. in Civil Engineeringfrom the University of Dayton in May 2003.

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