New Methodology for Transportation Investment Decisions with ...

New Methodology for Transportation Investment Decisions with Consideration of Project Interdependencies Zongzhi Li, Hemanshu Kaul, Sanjiv Kapoor, Eirini Veliou, Bei Zhou, and Sang Hyuk Lee analytical steps to ensure achieving the most cost-effective and holistic decision outcomes. For project evaluation, benefits of a project implemented on a link or node (i.e., segment or intersection) of a transportation network are computed as net reductions in agency costs and user costs in the facility service life cycle. Agency cost items primarily include costs of constructing, preserving, and maintaining physical facilities, and user cost items are associated with vehicle operations, travel time, crashes, and emissions (often treated as externalities). Through project evaluation, all economically feasible candidate projects could be identified. Because of budget constraints, not all economically feasible projects would be selected for actual implementation. Thus, project selection aims to identify the best subcollection of economically feasible projects in order to yield maximized overall benefits for the available budgets. This study introduces a new methodology for project evaluation and selection that explicitly addresses issues of networkwide impacts of a project and interdependencies of multiple projects. Specifically, the network impacts of a project refer to the project networkwide benefits measured by total reductions in agency costs and user costs in the facility service life cycle resulting from traffic redistribution in the network extending beyond the physical range of the project after project implementation. The project interdependency is defined as the difference in project networkwide benefits from implementing multiple projects at the same time and the sum of project networkwide benefits resulting from separately implementing the same set of projects one at a time. After a brief review of pertinent literature on project evaluation and selection, a methodology is proposed for applying the multicommodity minimum-cost network (MMCN) model for estimating networkwide benefits of implementing a single project or implementing multiple projects and the hypergraph Knapsack model for finding the best subcollection of multiple interdependent projects under budget constraints. Then the proposed methodology is applied in a computational study.

A new methodology for transportation project evaluation and selection explicitly addresses issues of networkwide impacts of a project and interdependencies of multiple projects. The methodology consists of two key elements: a multicommodity minimum-cost network model to estimate networkwide benefits of a single project or multiple interdependent projects and a hypergraph Knapsack model to identify the best subcollection of projects to yield maximized overall benefits. The proposed methodology is applied in a computational study with data on roadway network design, geometrics, travel demand, and traffic operations as well as the scope and costs of six major projects proposed for investment in the Loop area of Chicago, Illinois. The study reveals that the overall benefits with consideration of project networkwide impacts and their inter dependencies are significantly lower by 38% to 64% as compared with those benefits established without consideration of project interdependencies, and the rate of increase in overall benefits follows a diminishing trend with higher budget levels; these findings suggest that an optimal level of investment could be established. The proposed methodology may be refined by developing a decision rule to identify a reasonably small number of subcollections of projects guaranteed to contain the optimal solution so that it could practically support large-scale transportation investment decisions.

The steady growth in travel demand and the increase in truck loads in the past several decades, coupled with a much slower pace of transportation system capacity expansion, have led to accelerated deterioration of physical highway facilities such as pavements, bridges, and travel safety hardware, including traffic signs, signals, lighting, pavement markings, and guardrails, and escalated concerns regarding vehicle operating costs, traffic congestion, crashes, and vehicle emissions. Meanwhile, the recession and decreased revenues have led in large part to limited budgets. Consequently, pressure has intensified to address growing transportation needs with limited budgets. Specific to transportation investment decision making, project evaluation, and project selection are two critical

Related Work

Z. Li, E. Veliou, B. Zhou, and S. H. Lee, Department of Civil, Architectural, and Environmental Engineering, 3201 South Dearborn Street; H. Kaul, Department of Applied Mathematics, 10 West 32nd Street; and S. Kapoor, Department of Computer Science, 10 West 31st Street, Illinois Institute of Technology, Chicago, IL 60616. Corresponding author: Z. Li, [email protected].

In current practice, various methods and models have been developed for project evaluation and selection. For estimating project benefits, the life-cycle cost analysis method has been widely used to compute net reductions in agency costs and user costs after project implementation for different types of transportation facilities such as pavements and bridges. For instance, FHWA made a concerted effort for use of life-cycle cost analysis in highway pavement design

Transportation Research Record: Journal of the Transportation Research Board, No. 2285, Transportation Research Board of the National Academies, Washington, D.C., 2012, pp. 36–46. DOI: 10.3141/2285-05 36

Li, Kaul, Kapoor, Veliou, Zhou, and Lee

(1). Wilde et al. introduced a life-cycle cost analysis framework for rigid pavement design (2). Abaza developed an optimal life-cycle cost analysis model for flexible pavements (3). Labi and Sinha and Peshkin et al. studied systematic preventive maintenance and the optimum timing strategies to achieve minimum pavement life-cycle cost (4, 5). Purvis et al. performed life-cycle cost analysis of bridge deck protection and rehabilitation (6). Mohammadi et al. introduced the concept of life-cycle cost analysis for highway bridge planning and design (7). Hawk developed bridge life-cycle cost analysis software for evaluating bridge-related investments (8). In recent years, researchers have begun to utilize the risk-based life-cycle cost analysis approach to establish highway project benefits. For instance, Tighe performed a probabilistic life-cycle cost analysis of pavement projects by incorporating mean, variance, and probability distribution for typical construction variables, such as pavement structural thickness and costs (9). Reigle and Zaniewski incorporated risk consideration into the pavement life-cycle cost analysis model (10). Setunge et al. developed a methodology for risk-based life-cycle cost analysis of alternative rehabilitation treatments for highway bridges with Monte Carlo simulation (11), and Li and Madanu introduced a method for highway project lifecycle cost analysis by considering mixed cases of certainty, risk, and uncertainty (12). As for project selection, optimization models have been developed for prioritizing investment projects with the objective of maximizing total benefits by using such techniques as integer programming (13–15), goal or compromise programming (16, 17), and multiobjective optimization (18–22). In addition, a few researchers introduced stochastic models to address issues of data reliability and predictability of pavement, bridge, safety, and mobility performance models as well as uncertainty of traffic volume, project cost, and discount rate by using simulation (23–26) and Markovian decision processes and dynamic programming (27–31). Two limitations are found in the existing analysis methods and models. First, localized project impacts in terms of total reductions in agency costs and user costs within the physical range of the project are typically assessed. All too often, local changes in the transportation network can lead to agglomerative changes in its global behavior that extend beyond the project physical range. For instance, expanding the capacity of a highway link typically improves traffic operations of the link and may also result in better (or sometimes worse) traffic conditions elsewhere; this change leads to much larger or smaller overall networkwide benefits. Second, existing models for prioritizing transportation projects are generally formulated as the 0-1 integer Knapsack problem (32). Such a formulation only addresses cases in which multiple investment projects are fully independent of each other. In fact, the overall benefits of multiple projects may be greater than, equal to, or smaller than the sum of individually estimated benefits. The lack of a rigorous methodology for addressing the issues of networkwide impacts of a single project and interdependencies of multiple projects will likely produce biased investment decisions.

Proposed Methodology General The benefits to a transportation network of project implementation could be computed as the total reduction in agency costs (construction, rehabilitation, and maintenance) and user costs (vehicle opera-

37

tions, travel time, crashes, and emissions) in the facility service life cycle. The individual agency cost and user cost items associated with a roadway link could be estimated as a function of link-specific traffic volume and vehicle composition. From the perspective of network traffic assignment, the MMCN model could be used to determine traffic volumes and vehicle compositions for individual roadway links before and after project implementation. Networkwide agency benefits could be computed as the difference in agency costs for all network links in the facility service life cycle that could be estimated by link-based traffic volumes and vehicle compositions before and after project implementation. User benefits of each user cost item could be computed as the change in consumer surplus by using link-based traffic volumes and vehicle compositions before and after project implementation. The networkwide agency benefits and user benefits for all user cost items could be aggregated into the overall networkwide benefits of the project. To facilitate cross-comparisons of projects associated with different types of physical facilities that maintain different useful service life cycles, the overall networkwide benefits could be expressed in equivalent uniform annualized amounts. Without loss of generality, this study illustrates the proposed methodology by only computing networkwide travel time savings, which are typically the dominant portion of project-related networkwide benefits. First, the link-based travel time for each link is estimated as a function of the link-specific traffic volume according to the function proposed by the Bureau of Public Roads, whose role is currently performed by FHWA, in the following specification: 4   v( )   C(i, j )=l = TT(i , j )=l ,0 × 1 + 0.15 ×  i, j =l    c(i, j )=l   

(1)

where C(i,j)=l = actual travel time of link l between node i and node j per vehicle, TT(i,j)=l,0 = base free-flow travel time for link l, v(i,j)=l = volume on link l, and c(i,j)=l = capacity for link l. Next, the MMCN model is implemented to determine the linkspecific traffic volumes for the entire network before and after project implementation by assigning origin–destination (O-D) travel demand aimed at minimizing networkwide total travel time. The networkwide total travel time savings after project implementation could be calculated as project networkwide benefits. The difference in project networkwide benefits from implementing multiple projects at the same time and the sum of project networkwide benefits resulting from separately implementing the same set of projects one at a time could be used as a measure to assess project interdependency levels. Finally, the hypergraph Knapsack model is implemented for selecting multiple interdependent projects to achieve maximized overall networkwide benefits.

MMCN Model for Estimating Networkwide Benefits of Projects A transportation network can be modeled by a graph G = (V, E), where the vertex set V corresponds to the nodes connected by network links represented by edges in E. Each edge e has an upper bound, u(e), on its traffic capacity. Also, there are vertices S ⊆ V,

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Transportation Research Record 2285

the origins of travel, and vertices T ⊆ V, the destinations of travel. For each (s, t) ∈ S × T, there is traffic demand f(s, t) > 0 for a specific class of vehicles traveling from each O-D pair from s to t that is regarded as a distinct commodity flow. This formulation enables the modeling of traffic flows as multicommodity flows as follows (33–35): K

minimize ∑

L

∑ ( )

Cij i xijk

(2)

k =1 i, j = l =1

Hypergraph Knapsack Model for Selecting Multiple Projects

subject to L

∑ ( )

i, j = l =1

K

∑x

k ij

an ,(i, j )=l i xijk = bnk

(3)

= uij

(4)

k =1

0 ≤ xijk ≤ uijk

the objective of minimizing the total cost of all links. Then travel times can be computed by applying link-based traffic volumes and vehicle compositions. Finally, the networkwide project benefits can be estimated by aggregating travel time-saving benefits of individual links between the modified transportation network G(i(o)) and the original network G by using the concept of consumer surplus; more details may be found elsewhere (12).

(5)

where K = total number of O-D pairs; L = total number of links; Cij = travel time of link l between node i and node j per vehicle; x ijk = flow on link l between node i and node j coming from kth O-D pair; an,(i,j)=l = node-link incident matrix element, where an,(i,j) = 1 for row i, −1 for row j, and 0 otherwise; b kn = f k if inflow of f k at node n, −f k if outflow of f k at node n, and 0 otherwise; u ijk = capacity of flow on link l between node i and node j coming from kth O-D pair; uij = capacity of flow on link l between node i and node j; f k = flow coming from kth O-D pair; i, j = node i and node j, i, j = 1, 2, . . . , N; (i, j) = link l between node i and node j, (i, j) = l = 1, 2, . . . , L; and k = kth O-D pair, k = 1, 2, . . . , K. As Equation 2, the objective function of the model essentially helps assign trips to network links such that the networkwide total travel time is minimized. Equation 3 is the flow conservation constraint that ensures for each node that the total of inflows would equal the total of outflows. Equation 4 is the link capacity constraint to ensure that assigned traffic on each link does not exceed available capacity. Equation 5 sets the lower and upper bounds of assigned traffic coming from a specific O-D pair. Given a portfolio of N economically feasible projects labeled [N] and the cost of each project cj ( j = 1, . . . , N), the objective of the MMCN model is to minimize the total travel time after a subcollection of N economically feasible projects has been implemented. Let sets Ai (i = 1, 2, . . . , C N1 + C N2 + . . . + C NN) be all possible subcollections of the N projects and let B be a given budget level. For each i ∈ [C N1 + C N2 + . . . + C NN ], let G(i) denote the modified network obtained from the original network G by implementing the investment projects from a specific set Ai, the ith subcollection of projects. For each set Ai, the reassigned link-specific traffic volumes are determined by solving the MMCN problem on network G(i) with

Basic Model Formulation In the 0-1 integer Knapsack problem formulation for networkwide project selection, the objective is to select a subcollection of projects to maximize total benefits under annual budget constraints. The 0-1 value of a decision variable implies rejection or selection of a proposed economically feasible project. A basic optimization model without project networkwide impacts and interdependency considerations is formulated as maximize a1( Local ) i x1 + a2( Local ) i x 2 + . . . + aN ( Local ) i x N

(6)

subject to c1kt i x1 + c2 kt i x 2 + . . . + cNkt i x N ≤ Bkt x1 , x 2 , . . . , x N =

0 1

(7) (8)

where a1(Local), a2(Local), . . . , aN(Local) = localized benefits of individual projects 1, 2, . . . , and N, x1, x2, . . . , xN = decision variables (x1, x2, . . . , xN = 0/1), cikt = costs of project i using budget from management program k in year t, i = 1, 2, . . . , N, k = 1, 2, . . . , K, and t = 1, 2, . . . , M. As Equation 6, the objective function of the model essentially helps select a subcollection of projects to achieve maximized total benefits. Equation 7 gives budget constraints by management program and by project implementation year. The 0-1 constraints for the decision variables (Equation 8) are used for rejection or selection of individual projects.

Model with Project Networkwide Impacts and Interdependency Considerations Since project networkwide impacts are estimated as project network wide benefits of total travel time savings between the original network and the modified transportation network after project implemen tation, the interdependencies of multiple projects can be assessed as the difference in networkwide benefits of implementing multiple projects at the same time and the sum of networkwide benefits obtained by separately implementing individual projects one at a


39

time. That is, the interdependencies in the overall networkwide benefits of multiple projects are computed as ∆aij = ( −1)

2 +1

[a − (a ij

i( Network )

+ a j( Network ) )]

 aijk − ( ai( Network ) + a j( Network ) + ak( Network ) )  3+1 ∆aijk = ( −1)     + ( ∆aij + ∆aik + ∆ ajk ) ∆a1,2 , . . . , N a1,2 , . . . , N  a1,2 , . . . , N − ( a1( Neetwork ) + a2( Network ) + . . . + aN ( Network ) )    N +1  = ( −1)  + ( ∆a1,2 + ∆a1,3 + . . . + ∆a( N −1), N )    − ( ∆a1,2 ,3 + ∆a1,2 ,4 + . . . + ∆a( N − 2),( N −1), N ) + . . . 

(9)

where a1,2, . . . , N(Network) = networkwide benefits of implementing projects 1, 2, . . . , N at same time; a1(Network), a2(Network), . . . , aN(Network) = networkwide benefits of implementing projects 1, 2, . . . , N one at a time; and Δaij, Δaijk, . . . , Δa1,2, . . . , N = interdependencies in overall networkwide benefits of projects i and j, projects i, j, k, . . . , and projects 1, 2, . . . , N. The 0-1 integer hypergraph Knapsack model incorporating project interdependency considerations is formulated as follows:

(10)

subject to N

The purpose of Equations 10, 11, and 16 is the same as that for Equations 6, 7, and 8. For Equation 12, it is required that at most one possible combined implementation scenario be allowed for a project if it is selected for implementation. For Equations 13 through 15, the constraints ensure that x1,2, . . . , N = 1 if and only if x1 = x2, . . . , = xN = 1, indicating that the interdependencies in networkwide benefits of multiple projects will only be considered when all constituent projects have been selected for implementation. Model Solution A heuristic algorithm extended from the heuristic of Volgenant and Zoon (36) and Li et al. (37) by using Lagrange relaxation techniques was developed to solve the hypergraph Knapsack model. X* is denoted the optimal decision vector, s(X′) is the set of projects selected, and s(X″) is the set of projects not selected. In full, the heuristic has the following steps:

N

N

C3   C2  ∑ ∆aij i xij + ∑ ∆aijk i xijk  N ( i , j , k )=1  (i , j )=1  maximize ∑ ai i xi −   N CN i =1    + . . . + ∑ ∆a1,2, . . . , N i x1,2, . . . , N  (1,2, . . . , N )=1  

∑c

Δaij, Δaijk, . . . , Δa1,2, . . . , N = interdependencies in overall networkwide benefits of projects i and j, projects i, j, k, . . . , and projects 1, 2, . . . , N; x1, x2, . . . , xN, xij, xijk, . . . , x1,2, . . . , N = 0/1; cikt = costs of project i with budget from management program k in year t; i = 1, 2, . . . , N; k = 1, 2, . . . , K; t = 1, 2, . . . , M; and Bkt = budget available for program k in year t.

xi ≤ Bkt

Step 0. Initialization and normalize: – Set X* = {0, 0, . . . , 0} (no project selected in the beginning). Hence, s(X′) = ϕ. – Use budget Bkt to perform the following calculations: (a) sort the projects by benefits ai(Network) in descending order, set λkt = 0 for all k, t and xi = 1; (b) normalize cost and budget matrices by setting cikt Bkt

(11)

cikt′ =

xij + xijk + . . . + xij , . . . , N ≤ 1( i = 1, 2, . . . , N )

(12)

for all k, t and Bkt = 1 for all k, t; and (c) compare the sum of normalized costs with normalized budgets

(x

(13)

ikt

i

i =1

i

+ xj ) − 2 i xij ≥ 0

( x1 + x2 + . . . + x N −1 ) − ( N − 1) i x1,2, . . . , ( N −1) ≥ 0

(14)

( x1 + x2 + . . . + x N ) − N i

(15)

x1,2 , . . . , N ≥ 0

x1 , x 2 , . . . , x N , xij , xijk , . . . , x1,2 , . . . , N = 0 1

(16)

where a1, a2, . . . , aN = networkwide benefits of implementing projects 1, 2, . . . , N one at a time;

N

Ckt = ∑ cikt′ i =1

If Ckt ≤ 1 for all k, t, go to Step 4. Otherwise, go to Step 1. Step 1. Determine the most violated constraint k, t: Set C′kt = maximum {Ckt} for all k, t. Step 2. Compute the increase of Lagrange multiplier value λkt: K M   ai( Network ) − ∑ ∑ ( λ kt i cikt′ ) k =1 t =1   θ=  otherwise ∞

K

M



∑ ∑  c′

ikt

k =1 t =1

i

Ckt  ,c > 0 Ckt′  ikt

for all projectt i ∈ s ( X ′ )

40


TABLE 1 O-D Trips for p.m. Peak Period, Internal Zone O-D

1

2

3

4

5

6

7

8

9

10

Total

1 2 3 4 5 6 7 8 9 10 Total

5 16 14 21 20 25 13 11 2 3 130

12 141 50 69 65 81 44 34 9 12 517

13 52 462 71 64 84 45 35 11 14 851

15 62 60 515 72 92 52 42 12 15 937

12 50 43 68 802 73 38 31 10 15 1,142

14 58 53 74 67 1,223 43 36 14 19 1,601

10 49 46 65 51 69 75 29 8 10 412

10 38 38 53 46 55 29 84 7 7 367

3 13 11 18 14 19 10 7 2 2 99

3 14 15 18 18 24 10 7 2 3 114

97 493 792 972 1,219 1,745 359 316 77 100 6,170

– Select project i ∈ s(X′), which has the minimum θi and let θ′i = min{θ1, θ2, . . . , θi, . . .}. Step 3. Increase λkt by θ′i (Ckt /C′kt) and reset xi the value zero: Let

the proposed methodology for estimating networkwide benefits of interdependent transportation projects proposed for possible implementation in this area and identifying the best subcollection of projects for implementation primarily under budget constraints.

C   λ kt = λ kt +  θi′ i kt   Ckt′ 

Data Collection and Processing

and Ckt = Ckt − c′ikt for all k, t. Reset xi = 0 for project i ∈ s(X′) and shift project i from s(X′) to s(X′′). – If Ckt ≤ 1 for all k, t, go to Step 4. Otherwise, go to Step 1. Step 4. Improve the solution: For the feasible solution obtained in Step 3, check whether the projects with decision variable values of zero can have the value 1 without violating constraints Ckt ≤ 1 (k = 1, 2, . . . , K, and t = 1, 2, . . . , M). When this is the case, choose the project with the highest benefits and add it to the selected project list. Repeat this step until no project with zero variable value can be found and stop. Update the set of projects selected, s(X′) = {i | for all xi = 1}, and this establishes an improved solution.

Data on road network design, geometrics, travel demand, traffic operations, and scope and costs of projects proposed for possible implementation are needed for the computational study. Travel demand data were obtained from the Chicago Metropolitan Agency for Planning, which were stored in a cmap zones file containing information on hourly travel demand for a typical day for 1,961 traffic analysis zones for the entire Chicago metropolitan area. For the current study, the O-D travel demand relevant to the Chicago Loop was extracted from the overall travel demand file. Tables 1 and 2 present the O-D trips established for the study area for the most heavily traveled p.m. peak period, which account for approximately 16.7% of the daily trips. The Chicago Loop road transportation network is composed of 486 expressway, arterial, and collector links and 205 nodes. Google Earth photo images were used to accurately create link–node connectivity for through and left- and right-turning movements. Details of travel lane widths and speed limits were also obtained to help determine the link capacities and base free-flow travel times. Further, the inflow and outflow nodes as the sources and sinks of the 100 internal-internal O-D pairs and 81 external-external O-D pairs

Computational Study The financial district portion of the Chicago, Illinois, central business district—the Chicago Loop area bounded by Wacker Drive, Roosevelt Road, and South Lakeshore Drive—was selected to apply

TABLE 2 O-D Trips for p.m. Peak Period, External Zone O-D

E1

E2

E3

E4

E5

E6

E7

E8

E9

Total

E1 E2 E3 E4 E5 E6 E7 E8 E9 Total

0 2 5 2 1 1 0 1 1 13

3 0 25 4 3 2 1 6 10 54

7 30 0 13 9 6 3 10 16 94

1 3 7 0 2 1 1 1 2 18

1 2 4 1 0 1 0 1 2 12

1 1 3 1 1 0 0 0 1 8

0 1 1 0 0 0 0 0 0 2

2 6 11 3 2 1 1 0 3 29

2 10 16 4 3 2 1 3 0 41

17 55 72 28 21 14 7 22 35 271


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TABLE 3 Basic Information on Six Major Investment Projects Project

Name

Scope

1 2 3 4 5 6

Lower Wacker Drive Upper Wacker Drive Interchange Congress Parkway modernization Michigan Avenue resurfacing Lake Shore Drive resurfacing

Congress Parkway to Randolph Street Congress Parkway to Randolph Street Congress Parkway and Chicago River Wells Street to Michigan Avenue Congress Parkway to Roosevelt Road Randolph Street to Roosevelt Road

were identified. When data availability was considered, the analysis period for the computational study was set from 2011 to 2015. Table 3 gives basic information on six major investment projects proposed for possible implementation during this period.

Application of MMCN Model Preparation The zonal travel produces 181 commodity flows, including 100 internal O-D flows and 81 external O-D flows. The MMCN model was first applied to assign the 181 p.m. peak O-D trips to the original network to establish base-case p.m. peak link-specific traffic volume and vehicle composition. The traffic volume was used to compute the base-case p.m. peak networkwide total travel time and vehicle miles of travel for the original network. As a practical matter, the six candidate projects could be implemented separately one at a time or jointly; the implementation could range from two paired to six paired projects all together. As such, the number of modified transportation networks relevant to all project implementation scenarios became C 61 + C 62 + C 63 + C 64 + C 65 + C 66 = 63. For each project implementation scenario, the original roadway network was modified by changing the capacity and base free-flow speed of each link directly affected by one or more projects. The capacity and the base free-flow speed of each affected link were increased by up to 25%. For each project implementation scenario, the MMCN model was executed to assign the 181 p.m. peak O-D trips to the modified network to determine the redistributed p.m. peak link-specific traffic volume and vehicle composition. The redistributed traffic

Cost ($ million) 60 80 60 15 3 6 Total = 224

volumes were utilized to calculate the p.m. peak networkwide total travel time and vehicle miles of travel for the modified network correspondingly. With the networkwide total travel time and vehicle miles of travel determined for the original network and the modified network, the project networkwide benefits were computed as the change in consumer surplus of total travel time before and after project implementation. The networkwide project benefits were assumed to begin after the project construction was completed. If the project implementation scenario involved multiple projects, they were assumed to be implemented simultaneously and the networkwide benefits of multiple interdependent projects would begin after the construction of all projects was completed. To estimate project life-cycle benefits, the first-year benefits were directly estimated and an annual growth rate of 2% in project benefits was considered for future years. A discount rate of 5%, an hourly time value of $9.59, and a useful service life of 15 years for pavement resurfacing projects and 70 years for interchange and elevated road construction projects were used to calculate the project life-cycle benefits expressed in 2010 constant dollar-equivalent uniform annual amounts.

Discussion of Results Tables 4 through 9 summarize the number of network links within the project physical range and number of links involved with traffic reassignments, p.m. peak-hour networkwide total travel time and vehicle miles of travel, and annualized project networkwide benefits for all project implementation scenarios. As shown, the number of network links involved with traffic reassignment is much higher than that affected by project implementation. The ratio of network

TABLE 4 Links Affected by Project Implementation and Traffic Reassignment, Total Travel, and Benefits for Network B-1 (Modified by Implementing a Single Project) Project Implementation Scenario

1 2 3 4 5 6

Links Affected by

p.m. Peak Hour Travel

Project

Hours

Vehicle-Miles

Total Annualized Benefits (in 2010 $)

230.69 227.54 229.39 228.23 229.16 229.91

2,845 2,802 2,842 2,852 2,849 3,112

ai 5,823,110 6,016,722 5,902,871 5,974,515 5,917,095 5,871,426

24 26 17 22 27 13

Reassignment

84 96 84 95 90 90

Note: Original network (a) = 325.41 h and 2,679 vehicle-mi.

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TABLE 5 Links Affected by Project Implementation and Traffic Reassignment, Total Travel, and Benefits for Network B-2 (Modified by Implementing Two Paired Projects) a Project Implementation Scenario 12 13 14 15 16 23 24 25 26 34 35 36 45 46 56

Links Affected by


Annualized Benefits (in 2010 $)

Project

Reassignment

Hours

Vehicle Miles

Total

Interdependencies

42 37 46 51 37 43 48 53 39 38 44 30 42 35 40

98 95 97 96 95 101 113 105 111 99 97 96 103 107 98

224.12 227.94 224.34 225.94 228.84 222.89 222.23 223.04 226.94 224.90 226.47 227.25 223.26 226.60 227.28

2,745 2,840 2,765 2,790 3,118 2,728 2,753 2,745 3,109 2,812 2,842 3,105 2,771 3,131 3,120

aij 6,227,211 5,992,338 6,213,262 6,115,354 5,936,608 6,302,594 6,343,295 6,293,290 6,053,440 6,179,042 6,082,663 6,034,345 6,280,126 6,074,708 6,032,677

bij 6,227,211 5,992,338 6,213,262 6,115,354 5,936,608 6,302,594 6,343,295 6,293,290 6,053,440 6,179,042 6,082,663 6,034,345 6,280,126 6,074,708 6,032,677

Δaij 5,612,621 5,733,643 5,584,363 5,624,852 5,757,928 5,616,998 5,647,942 5,640,527 5,834,707 5,698,343 5,737,303 5,739,951 5,611,484 5,771,233 5,755,844

Note: Original network (a) = 325.41 h and 2,679 vehicle-mi. a Δaij = −[bij − (ai + aj)], aij = (ai + aj) − Δaij

TABLE 6 Links Affected by Project Implementation and Traffic Reassignment, Total Travel, and Benefits for Network B-3 (Modified by Implementing Three Paired Projects) a Project Implementation Scenario 123 124 125 126 134 135 136 145 146 156 234 235 236 245 246 256 345 346 356 456

Links Affected by



Project

Reassignment

Hours

Vehicle Miles

Total

Interdependencies

54 64 69 55 56 63 49 66 59 64 63 70 56 68 61 66 57 50 57 55

165 127 111 116 97 97 102 106 111 108 118 113 117 122 131 124 105 111 109 115

221.70 215.88 221.46 223.25 222.68 225.33 226.62 222.39 223.09 222.79 220.06 220.12 220.77 220.50 221.19 222.90 222.09 223.72 225.52 221.99

2,681 2,579 2,752 3,113 2,765 2,778 3,111 2,772 3,136 3,129 2,746 2,734 3,097 2,765 3,132 3,112 2,757 3,123 3,111 3,135

aijk 12,146,593 12,050,217 12,245,148 11,936,501 12,068,952 12,037,803 11,889,754 12,275,427 11,933,991 11,776,070 12,348,363 12,205,797 11,957,371 12,467,134 12,064,010 12,077,537 12,190,095 12,036,572 12,008,628 12,029,609

bijk 6,375,550 6,733,551 6,390,707 6,280,759 6,315,690 6,152,551 6,073,536 6,333,315 6,290,587 6,308,569 6,476,569 6,472,750 6,433,009 6,449,577 6,407,432 6,301,870 6,351,736 6,251,523 6,141,057 6,357,902

Note: Original network (a) = 325.41 h and 2,679 vehicle-mi. a Δaijk = [bijk − (ai + aj + ak) + (Δaij + Δajk + Δaik)], aijk = (ai + aj + ak) − Δaijk

Δaijk 5,596,109 5,764,130 5,511,780 5,774,757 5,631,543 5,605,273 5,707,652 5,439,293 5,735,060 5,835,562 5,545,745 5,630,891 5,833,647 5,441,199 5,798,652 5,727,706 5,604,386 5,712,239 5,682,764 5,733,427


43

TABLE 7 Links Affected by Project Implementation and Traffic Reassignment, Total Travel, and Benefits for Network B-4 (Modified by Implementing Four Paired Projects) a Project Implementation Scenario 1,234 1,235 1,236 1,245 1,246 1,256 1,345 1,346 1,356 1,456 2,345 2,346 2,356 2,456 3,456

Links Affected by



Project

Reassignment

Hours

Vehicle Miles

Total

Interdependencies

74 81 67 84 77 82 76 69 76 79 83 76 83 81 70

124 118 125 129 134 129 111 116 111 121 127 135 132 139 119

215.77 219.25 220.45 215.44 215.58 220.81 220.65 221.45 221.11 221.19 218.22 219.91 219.90 220.42 221.83

2,752 2,745 3,107 2,767 3,132 3,128 2,766 3,127 3,123 3,146 2,751 3,126 3,108 3,139 3,128

aijkl 18,096,482 18,148,367 17,836,333 18,325,859 17,888,129 17,807,267 18,149,995 17,889,973 17,930,568 17,769,731 18,320,279 17,904,638 17,936,798 18,015,175 17,949,391

bijkl 6,740,100 6,526,476 6,452,650 6,760,472 6,751,934 6,430,592 6,440,502 6,391,007 6,412,297 6,407,370 6,589,901 6,485,747 6,486,475 6,454,421 6,368,048

Δaijkl 5,620,735 5,511,431 5,777,795 5,405,583 5,797,643 5,821,086 5,467,596 5,681,948 5,583,933 5,816,415 5,490,924 5,860,894 5,771,315 5,764,583 5,716,516


Δaijkl = − [bijkl − (ai + aj + ak + al) + (Δaij + Δaik + Δail + Δajk + Δajl + Δakl) − (Δaijk + Δaijl + Δajkl)] aijkl = (ai + aj + ak + al) − Δaijkl

a

TABLE 8 Links Affected by Project Implementation and Traffic Reassignment, Total Travel, and Benefits for Network B-5 (Modified by Implementing Five Paired Projects) a Project Implementation Scenario 12,345 12,346 12,356 12,456 13,456 23,456

Links Affected by



Project

Reassignment

Hours

Vehicle Miles

Total

Interdependencies

94 87 93 97 89 96

134 138 136 139 126 140

214.71 215.21 219.12 214.83 219.66 218.03

2,762 3,127 3,125 3,142 3,141 3,124

aijklm 24,229,201 23,764,967 23,914,335 23,722,404 23,882,916 23,815,684

bijklm 6,805,428 6,775,108 6,534,317 6,798,086 6,500,964 6,601,662

Δaijklm 5,405,111 5,823,675 5,616,889 5,880,463 5,606,100 5,866,944


Δaijklm = [bijklm − (ai + aj + ak + al + am) + (Δaij + Δaik + Δail + Δaim + Δajk + Δajl + Δajm + Δakl + Δakm) − (Δaijk + Δaijl + Δaijm + Δaikl + Δaikm + Δailm + Δajkl + Δajkm + Δajlm + Δaklm) + (Δaijkl + Δaijkm + Δaijlm + Δaiklm + Δajklm)] aijklm = (ai + aj + ak + al + am) − Δaijklm

a

TABLE 9 Links Affected by Project Implementation and Traffic Reassignment, Total Travel, and Benefits for Network B-6 (Modified by Implementing All Six Projects) a Project Implementation Scenario 123,456

Links Affected by



Project

Reassignment

Hours

Vehicle Miles

Total

Interdependencies

108

140

214.44

3,137

a123456 29,767,416

b123456 6,822,276


Δa123456 5,738,323

Δa123456 = − [b123456 − (a1 + a2 + a3 + a4 + a5 + a6) + (Δa12 + Δa13 + Δa14 + Δa15 + Δa16 + Δa23 + Δa24 + Δa25 + Δa26 + Δa34 + Δa35 + Δa36 + Δa45 + Δa46 + Δa56) − (Δa123 + Δa124 + Δa125 + Δa126 + Δa134 + Δa135 + Δa136 + Δa145 + Δa146 + Δa156 + Δa234 + Δa235 + Δa236 + Δa245 + Δa246 + Δa256 + Δa345 + Δa346 + Δa356 + Δa456) + (Δa1234 + Δa1235 + Δa1236 + Δa1245 + Δa1246 + Δa1256 + Δa1345 + Δa1346 + Δa1356 + Δa1456 + Δa2345 + Δa2346 + Δa2356 + Δa2456 + Δa3456) − (Δa12345 + Δa12346 + Δa12356 + Δa12456 + Δa13456 + Δa23456)] a123456 = (a1 + a2 + a3 + a4 + a5 + a6) − Δa123456

a

44


TABLE 10 Annualized Benefits, Costs, Benefit-to-Cost Ratios, and Best Subcollection of Projects Equivalent Uniform Annual Amount ($) Budget Level ($224 Million) (%)

Benefit

Cost

10 20 30 40 50 60 70 80 90 100

6,357,902 6,357,902 6,357,902 12,283,083 12,283,083 12,374,624 18,185,954 18,522,072 18,522,072 18,548,533

1,642,568 1,642,568 1,642,568 4,744,517 4,744,517 5,778,500 7,846,466 8,880,449 8,880,449 11,982,398

links affected by traffic reassignment and project implementation varies from 1.3 to 6.9. This finding intuitively suggests that the network impacts of implementing a single project or multiple projects could extend significantly beyond the physical range of constituent projects. When the p.m. peak travel for the use of the original network and that for the modified network after project implementation were compared, the total hours of travel generally decreased after project implementation. Meanwhile, the total vehicle miles of travel increased slightly. The combined effect of the two reveals a lower level of unit travel time in terms of hours per vehicle mile of travel. The extent of the unit travel time reductions is generally larger when multiple projects are jointly implemented as compared with separate implementation of each project one at a time. For multiproject implementation scenarios, the effect becomes greater with the increase in number of projects involved. Tables 4 through 9 further present the annualized project networkwide benefits for all project implementation scenarios and project interdependencies as differences in project networkwide benefits by implementing multiple projects at the same time and the sum of project networkwide benefits established by separately implementing the same set of projects one at a time. For all joint project implementation scenarios, it was found that the project networkwide benefits achieved by jointly implementing multiple projects are lower than the direct summation of project networkwide benefits obtained from implementing individual projects separately. For instance, the interdependency in networkwide benefits of jointly implementing Projects 1, 2, and 3 is assessed as Δa123/(a1 + a2 + a3) = 5,596,106/(5,823,110 + 6,016,722 + 5,902,871) = 32%. On average, the interdependencies in networkwide benefits of jointly implementing two, three, four, five, and six paired projects are 48%, 32%, 24%, 19%, and 16%, respectively. Application of Hypergraph Knapsack Model The annualized networkwide benefits computed on the basis of results generated from the MMCN model for implementing individual projects one at a time (namely, ai, for i = 1, 2, 3, 4, 5, and 6) and project interdependencies (i.e., Δai, Δaij, Δaijk, Δaijkl, Δaijklm, Δa123456 for i, j, k, l, m = 1, 2, 3, 4, 5, and 6 as well as Δa123456 for simultaneous implementation of all six projects) were used as inputs for application of the hypergraph Knapsack model to find the best subcollection

Benefit-toCost Ratio

Best Subcollection of Projects for Implementation

3.87 3.87 3.87 2.59 2.59 2.14 2.32 2.09 2.09 1.55

4+5+6 4+5+6 4+5+6 156 + 4 156 + 4 2 + 456 156 + 3 + 4 124 + 5 + 6 124 + 5 + 6 14 + 23 + 56

of projects selected for implementation. For the purpose of applying the hypergraph Knapsack model, the solution algorithm developed on the basis of the Lagrangian relaxation technique was coded in the Frontline Solver Xpress V55 software. Table 10 presents the results of project selection at different budget levels ranging from 10% to 100% of $224 million, which are equivalent to the total costs of all six projects. Figure 1 compares overall networkwide benefits achieved from the best subcollection of projects selected with the basic Knapsack model (without consideration of project interdependencies) and the hypergraph Knapsack model (with consideration of project interdependencies). The comparative analysis provides several major observations. First, the overall networkwide benefits with consideration of project interdependencies tend to be significantly lower than the benefits without consideration of project interdependencies. The gap varies by 38% to 64% for different budget levels. Second, the annualized networkwide benefits of the best subcollection of projects with consideration of project interdependencies identified by using the hypergraph model begin to flatten out after a maximum amount of $18.5 million is reached when the annualized budget is maintained at approximately $8.88 million. No additional networkwide benefits of travel time savings are generated with higher levels of investment budgets. However, the overall network benefits of projects selected with the basic Knapsack model without consideration of project interdependencies would still grow along with higher levels of investment budgets. This result appears to be counterintuitive compared with real-world conditions. Currently, the Chicago Loop network maintains a limited capacity and has already accommodated high traffic volumes. There is little room to improve traffic mobility significantly in the area. The foregoing findings provide evidence that the existing methods for project selection such as the basic Knapsack model are likely to considerably overestimate the true networkwide benefits of multiple interdependent projects. Hence, they could not necessarily support truly optimal investment decisions. Study Summary A new methodology was introduced to assist in transportation decision making that consists of two key elements: (a) an MMCN model to establish link-specific traffic volumes and vehicle compositions before and after project implementation that could estimate net-

45

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

0, 00

10

,0 0

00

,0 0

0

0

0 90

0, 00 8, 00

0, 00

0

0 7, 00

0, 00 6, 00

0, 00

0

0 5, 00

0, 00 4, 00

0, 00

0

0 3, 00

0, 00 2, 00

1, 00

0, 00

0

Annual Benefits of Selected Projects (in 2010 Constant Dollars)


Annual Costs of Selected Projects (in 2010 Constant Dollars) Networkwide Benefits_Basic Knapsack Model Networkwide Benefits_Hypergraph Knapsack Model FIGURE 1 Comparison of networkwide benefits of selected projects with and without project interdependency considerations.

workwide benefits of separately implementing individual projects one at a time or jointly implementing multiple interdependent projects and (b) a hypergraph Knapsack model that explicitly considers project interdependencies to identify the best subcollection of projects with maximized overall network benefits under budget constraints. Data on travel demand, network topology, and costs of major investment projects proposed for travel time improvements in the Chicago Loop area were used for methodology application. The following summary is given: • Two limitations were identified in the existing methods for transportation project evaluation and project selection: the methods lacked consideration of the network impacts of a single investment project and of interdependencies of multiple investment projects to achieve truly optimal investment decision outcomes. • Although more complex and data driven, the MMCN model appears to be more reasonable for estimating project networkwide benefits as compared with the existing methods, which mainly consider localized impacts of individual projects and simply sum up individually estimated benefits of multiple projects without consideration of project interdependencies in networkwide impacts. • The hypergraph Knapsack model is more robust than the basic Knapsack model in that it selects the best subcollection of inter dependent projects to yield maximized overall benefits for different levels of budgets. • The computational study revealed a much faster diminishing rate of networkwide benefits when project interdependences are considered. This finding seems to suggest that for a dense urban area with relatively high travel demand the networkwide benefits from travel time savings could only be achieved up to a certain level. Additional benefits may be obtained from reductions in other user cost items such as vehicle operations, crashes, and air emissions. Conclusion The proposed methodology for estimating networkwide benefits of multiple interdependent projects and selecting the best subcollection of projects under budget constraints to yield maximized overall ben-

efits was successfully applied in a computational study. This method shows its potential for practical use to enhance transportation decision making. Although the results generated from the proposed methodology are more reasonable than results from the existing methods, the list of investment projects used for the computational study is quite small. When the list of candidate projects becomes rather large, possible combinations of joint implementation of multiple projects will increase significantly. This result will make it impractical to apply the MMCN model to establish reassigned link-specific traffic volume and vehicle composition exhaustively for all project implementation scenarios and to identify the best subcollection of multiple interdependent projects. In this respect, a decision rule needs to be sought to help identify a reasonably small number of subcollections of projects that are guaranteed to include the optimal solution. When reductions in all agency and user costs in addition to travel time savings are considered as project benefits, the MMCN model application would inevitably be complicated in that individual agency and user cost items could be characterized as nonlinear functions of link-specific traffic volumes and vehicle compositions. Piecewise linear functions may be created according to some decision rules to approximate the nonlinear functions. Some worst-performance analyses are needed to ensure that heuristic solutions are within a small gap of true optimality. Acknowledgments The authors are grateful for the financial support for this research from the Educational and Research Initiative Fund of the Illinois Institute of Technology and for the assistance of transportation agencies in the Chicago metropolitan area in collecting data for the computational study. References 1. Life-Cycle Cost Analysis in Pavement Design: In Search of Better Investment Decisions. FHWA, U.S. Department of Transportation, 1998.

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