Synthesis of Spatial Network Level Metrics with Local
Quantitative Metrics for Prioritizing Transportation Projects
Abstract
Prioritizing and selecting few critical projects from several competing
transportation projects is a multiobjective combinatorial optimization problem (MOCO).
In such problems, transportation planners and managers are interested in visualizing and
analyzing the trade-offs involved. This paper develops methodology for synthesis of
spatial network level metrics with local quantitative metrics for planning and prioritizing
of a large and diverse portfolio of transportation investment projects. The methodology
serves as an aid to planners, managers and engineers to visualize and compare costbenefit trade-offs. The methodology is based on incorporating network level metrics
along with local metrics in formulating generic MOCO algorithm and visualizing
multiobjective trade-offs on a spatial network. A district level case study demonstrates
the use of methodology in trade-offs analysis for short- and long-range transportation
plans. The methodology is adaptable to other areas like water supply management,
manufacturing and service industry where spatial consideration is important.
Key Words: MOCO problem; network visualization; Entropy; transportation planning;
risk analysis.
1. Introduction
Transportation professionals at district and state transportation agencies in the
United States are increasingly pressed with the task of allocating limited public funds
among potential highway improvement projects and the subsequent determination of the
order in which selected projects should be undertaken. There are several projects
competing for these limited funds. Prioritizing highway projects is basically a
Multiobjective combinatorial optimization (MOCO) problem. Over the last decade, many
algorithms have been developed to solve such types of problems. But these mathematical
models have been rarely used by managers and professionals in actual decision making.
The major problem is lack of visualizing and understanding these models and trade-offs
involved. We believe that models should be represented in a clear and informative way.
This paper aims at integrating network level metrics with local quantitative
metrics to better understand the impact of selecting certain projects on the specified
geographical region. The scope of the paper includes: developing a generic MOCO
model for prioritizing and planning transportation projects, developing a spatial network
and network level metrics, identifying useful quantitative metrics for the comparison of
diverse projects, developing an interface for visualizing project portfolio information and
relative trade-offs on the spatial network and presenting a case study with short and longrange plans from the Culpeper district in the state of Virginia.
The organization of this paper is as follows. Section 2 provides background for
generic MOCO formulation of projects planning and prioritizations problem. Section 3
describes the methodology for integrating network level metrics with quantitative
metrics. Section 4 presents the case study and discusses the scalability of the developed
methodology. The last section provides a summary.
2. Background
The purpose of this section is to formulate the projects planning and prioritization
problem as a multiple objective combinatorial optimization problem. Projects planning
and prioritization problem is combinatorial in nature and it involves multiple and
conflicting objectives. In other words, we are interested in studying finite set of
competing projects and combinations and arrangements of these projects that satisfy
specified criteria. Our MOCO formulation is based on the general model as per Ehrgott
and Gandibleaux (2002).
The finite action set A = {a1,…, an} represents n competing transportation
projects, where ai represents a unique project. The power set of A is the set of all subsets
of A and is denoted by P(A). The feasible set of the MOCO problem can be defined as a
subset X such that,
X  P(A)
The problem can be formulated in terms of binary decision variables. A binary decision
variable xi is introduced for every element ai  A. Then, a basic feasible solution S  X
can be represented by a binary vector {x1,…, xn} where
xi = 1
0
ai  S
otherwise
In other words xi =1 indicates that ith project is selected in the final solution and xi=0
suggests that ith project has not been selected in the final solution. A typical
multiobjective problem can have Q objectives or criteria, zj where j = 1,…, Q. Then the
overall problem takes the form of maximizing a set of objective functions:
Maximize {z 1 (S ),..., z Q ( S )}
Examples can be easily constructed to clarify this MOCO formulation. For example,
assume that there are 3 competing projects in a specified region and the decision makers
are considering two criteria for projects prioritization: first, to select a portfolio of
projects that will address higher average daily traffic (ADT) problem is high and second,
to select a portfolio of projects that will address higher crash rates concerns. ADT and
crash rate are two important measures in transportation planning. These two objectives
are not necessarily in conflict, but definitely they will not yield the same portfolio of
projects. The action set, A = {a1, a2, a3} for this particular problem is of order n = 3. Then,
the power set of A will be of order 23= 8 as follows:
P(A) = {, {a1}, {a2}, {a3}, {a1, al2}, {a1, a3}, {a2, a3}, {a1, a2, a3}}
The basic feasible set for this problem will be any subset X of the power set P(A). There
are three binary decision variables x1, x2 and x3 and a basic solution S can be represented
by a binary vector {x1, x2, x3}. One particular solution can be {1, 0, 1}, i.e., in final
portfolio, project a1 and project a3 have been selected and project a2 has not been
included. The two objective functions will be z1(S) and z2(S) representing two different
criteria as mentioned above.
Thus, the overall combinatorial model is:
Maximize {z 1 ( S ), z 2 ( S )}
SX
X  P(A)
A = {a1, a2, a3}
xi = 1
0
ai  S
otherwise
3. Methodology
The purpose of this section is to develop the spatial network level metrics, to
identify useful local quantitative metrics for the comparison of diverse projects and to
implement generic MOCO algorithm and represent multiobjective trade-offs on the
spatial network. The synthesis of spatial network level metrics with quantitative metrics
will greatly enhance understanding and analysis of complex models for transportation
planners and managers. The first step in proposed methodology is to develop a spatial
network, G (N, E) of cities and highways for a geographical region under study where N
is the collection of nodes and subset E of N X N is the collection of arcs. The network is
undirected. The nodes in this specific network represent major cities in the specified
region and arcs represent the highways connecting these major cities. The network can
be a short one consisting of only a few cities or it can be a very detailed one consisting of
large number of cities depending on the nature of analysis a decision maker is interested
in. The second step is to locate the competing projects on the network, so that planners
can have a comprehensive look at the distribution of projects throughout the specified
region.
The next step is to gather quantitative evidences for each candidate project from
a given set of projects being considered for funding.
There can be many local
quantitative metrics based on the criteria to be satisfied while selecting a portfolio. Few
good quantitative metrics are level of service, ADT, crash rate, volume to capacity ratio
and flow rate on each arc in the network. Unemployment rates and environmental issues
in different portions of the network are also important measures. Cost effectiveness of
individual project is another important metric. Then there are some specialized measures
developed by transportation agencies like bridge sufficiency rating, etc. The significance
of each quantitative metric varies according to the criteria selected by planners and
decision makers. For example, if safety is one of the main criteria, then crash rate is the
most important metric. If economic development is a major criterion, then unemployment
rate is a highly significant measure.
Among several measures, we believe that ADT and crash rates on each
individual arc and cost of the project are three meaningful and typical measures for
transportation projects planning and prioritization. The paper mainly uses these three
measures for the case study. But, it is not binding. Similar analysis can be performed
using any of the other local quantitative metrics.
We can apply generic MOCO formulation to the developed spatial network in
which different arcs have different values of quantitative measures. For one criterion,
individual arc crash rates might be more important and for another criterion, arc ADT
might be more significant. Consideration of arc ADTs will give one portfolio of projects
and consideration of arc crash rates will give another. The problem is essentially of which
combination of projects will be satisfy these two criteria. It is generally observed that the
decision makers will try to choose a portfolio that maximizes these local quantitative
measures. With many different local criteria under consideration, there is one more
important criterion, i.e. topology of the network, which is generally neglected by the
decision makers. Fair distribution of the resources among all arcs (highways) is as
important as maximizing the local criteria. The advantage of the methodology is that
decision makers can easily visualize the distribution of the projects throughout the
network. They get good understanding of whether the resources are all concentrated in a
specific portion of the network or whether they are equally distributed. But mere
visualization of the distribution is not sufficient. It can be subjective. One decision maker
might interpret it differently than the other. To solve this problem, we propose other
network level quantitative metrics which can be added as separate criteria along with
local level criteria to be satisfied.
We use a special network complexity index to see how uniformly the resources
are distributed throughout the network. Different indices can be developed for different
network level quantitative metrics. One of such network complexity index can analyze
the distribution of the projects throughout the network. This index is denoted by,
H 1   Pi (ln( Pi )
i
where, Pi 
Ni
N
Here N denotes the total number of projects in the selected portfolio and Ni denotes the
number of projects from the portfolio that are on the ith arc. Higher value of the indicates
more uniform distribution of projects throughout the spatial network and lower value of
index indicates that the distribution of projects is not uniform throughout the network or
the projects are concentrated among few cities or only a few highways in the network. In
general, heavy concentration in any particular area of the network is not advisable unless
there is a good justification for it.
Similarly, another network complexity index can be developed to analyze the distribution
of the funds throughout the network by modifying Pi in the previous equation as:
Pi 
Ci
C
Where C denotes the cumulative cost for all the projects in the selected portfolio and Ci
denotes the total cost of the projects from the portfolio that are on the ith arc. Thus our
goal is to maximize these network level indices along with other local objectives in the
MOCO formulation.
Many solution methods have been developed to solve MOCO problems. These
methods are mainly classified into three categories based on the role of the decision
maker in the portfolio selection process: priori mode, posteriori mode and interactive
mode (Ehrgott and Gandibleaux, 2002). We prefer interactive mode because in this
mode, managers and planners are actively involved in the solution process.
The last step in the methodology is to show the trade-offs graphically. A key
aspect of the methodology is the interface used to present project information on the
network. Graphing and visualizing two objectives trade-off problem in the decision space
or functional space is relatively easier. But when a typical multiobjective problem
consists of three or more objectives, suddenly visualizing and understanding the tradeoffs involved becomes difficult.
The problem essentially turns into visualizing
multivariate data involving several variables simultaneously. Many techniques have been
discusseed to represent trivariate or multivariate data effectively (Cleveland, 1994,
Gower & Digby, 1981, Tufte, 1997). Trivariate data can be represented by bubble plots,
in which values of two variables are indicated by the location of the bubble on a two-
dimensional scatter plot and value of third variable is represented by the size on the
bubble. One important technique to represent multivariate data is scatter plot matrix. A
scatterplot matrix is defined as a square, symmetric table or “matrix” of bivariate
scatterplots (Cleveland, 1994).
We propose a matrix of trivariate bubble plots to visualize and understand the
trade-offs involved in projects planning and prioritization. Our symmetric matrix has m
rows and m columns. Number of rows and columns are equal to the number of nodes
(cities) in the spatial network developed earlier. The intersection of row i and column j
contains a bubble plot showing the total number of projects on a typical arc between two
specific cities. Three criteria can be shown simultaneously on this bubble plot. For
example, a typical matrix will have bubble plots showing average daily traffic on x-axis,
crash rate on y-axis and cost of the project will be indicated by the size of the bubble.
One can develop similar matrix different quantitative metrics and criteria. This matrix
presents a large amount of information to the decision maker in a very efficient manner.
4. Case Study
Following district level case study shows the application of the methodology.
Culpeper district in the state of Virginia is selected for the case study. Table I shows
sample project information collected for this case. Project data include the project ID,
route, starting and end points, ADT, crash rate and cost, of the project.
<< Table 1>>
Specified region consists of 42 competing projects. Thus, the initial action set for
the problem, A = {a1, …, a42} is of order n = 42. The developed spatial network is shown
in Figure 1. The network consists of 14 nodes (cities) and 21 arcs. The number on each
arc represents actual highway number. The network consists of mainly primary highways
in the Culpeper district. Interstate highways are not included in the network since
interstate highway projects come under a different state level interstate highway projects
prioritization program. The methodology is scalable. A similar state-level interstate
highways network can be developed for interstate highway projects prioritization plan.
<< Figure 1>>
Projects are grouped together according to their location in the network and can
be described with corresponding quantitative information such as number of projects,
costs, average cost, ADT, ADT per dollar, etc. As an example, Figure 2 shows the total
number of projects on each arc in the network. Such aggregate statistics can suggest the
network complexity and the potential gross efficiencies of investments in projects on
each individual arc. Another good example of quantitative evidence can be the ratio of
cumulative accident rate and the total project cost.
<< Figure 2>>
After projects are grouped based on their location in the network, analysis can be
done to compare projects within a group. Figure 3 shows an example of such an analysis
for projects planning and prioritization. Shown are the projects on a particular arc.
Coordinate axes are used with average daily traffic and crash rates as measures of crash
exposure and crash intensity. Each of the projects is then represented by a bubble icon,
centered on coordinate values of the project and whose size represents the project cost.
The variations among project costs and the contribution of each project in terms of crash
exposure and crash intensity is shown in the figure.
<< Figure 3>>
The power set of the initial action set is of the order 242. Implementing the
methodology on every element of power set is both unnecessary and impossible. Some of
the solutions in the power set are obviously inferior. It is important to remove such
obvious inferior solutions at this stage to simplify the application methodology. For
example, empty set,  and sets of only one project can not be efficient. Similarly set of all
the projects is also meaningless. As mentioned previously, using interactive mode with
the decision maker, many such inferior solutions can be identified to develop a small set
of meaningful solutions. It is also helpful to know the approximate number of projects the
decision maker is interested in undertaking simultaneously.
After initial analysis, the final set of efficient solutions consists of combinations
of only 20 projects from initial action set of 42 competing projects. These 20 projects
along with other relevant information are shown in Table II.
5. Summary
The developed methodology for synthesis of network level metrics with quantitative
metrics for projects prioritization can serve as a decision aid among the transportation
planners.
6. Acknowledgment
The authors are grateful to the members of the steering committee from Virginia
Transportation Research Council and Virginia Department of Transportation: Chad
Tucker.
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Figure 1. Spatial Network for the Culpeper District
The plains
Culpeper District
Network View
14 cities
{ 17}
Washington
{ 211}
Warrenton
External
{ 29, 25, 17}
{ 28}
{ 522}
Remington
{ 29, 15}
Culpeper
{ 29}
Madison
{ 15}
{ 522}
{ 29}
Stanardsville
Orange
{ 33}
{ 20}
{ 15}
External
{ 29, 240, 250}
Gordonsville
{ 33}
{ 22}
Charlottesville
{208}
{ 22, 208}
{ 15}
{ 20}
Scottsville
Columbia
Louisa
Mineral
Figure 2. An example of comprehensive network-level aggregate information
generated by the developed methodology for projects planning and prioritization
showing total number of projects on each arc
1
Louisa
Gordonsville
Culpeper
Columbia
Charlottesville
2
Madison
2
Orange
5
Warrenton
The plains
1
1
Mineral
3
Remington
2
Scottsville
1
Stanardsville
1
2
3
6
Local
1
1
Washington
10
Figure 3. Example of a simple analysis generated by the developed prototype for
projects planning and prioritization showing project contributions in crash rates
and ADT, and project cost represented by the size of icon
600
500
Crash Rate
400
300
200
100
0
0
20000
40000
ADT
60000
Table I. Sample project information for Culpeper District Case Study
Project ID
#
2070004
2070005
Route
00020
00020
From
RTE 708
RTE 250
ADT
8805
10618
Crash
rate
266.65
393.47
Cost
24826
20378
I-64
To
RTE 53
BARN BRANCH
RTE 250
UNDERPASS
2070010
00029
34938
126.86
10767
2070011
00029
RTE 643
GREENE CL
45407
93346
2070013
00240
RTE 250 W
6692
2070014
00250
14757
204.01
49630
2070015
00250
I-64
RTE 677 OLD
BALLARD
RTE 250 E
RTE 677 OLD
BALLARD
165.05
355.12
14757
320
22119
2070016
00250
22703
253.54
4724
2070017
2070018
2070019
00250
00250
00250
305.76
178.33
210.36
26970
19745
16125
00029
00015
00211
00229
00522
00015
00015
00015
00017
00028
00029
00211
00015
00029
I-64
RTE 22
FLUVANNA CL
WCL
CHARLOTTESVILLE
RTE 686
FAUQUIER CL
RTE 211
RTE 3
RTE 17
RTE 15/29 BYPASS
WARRENTON SCL
RTE 55
PR WILLIAM CL
PR WILLIAM CL
WARRENTON WCL
RTE 250
RTE 33
35950
23526
10558
2070049
2070021
2070023
2070024
2070025
2070028
2070029
2070030
2070031
2070032
2070033
2070034
2070042
2070045
45097
7408
13704
8793
5778
34360
42597
9600
11918
12637
45537
15061
5314
30227
70.21
114.49
301.76
170.15
170.46
94.07
95.13
45.18
261.1
157.4
106.77
54.75
110.92
146.58
15478
26401
4085
14553
24314
21835
45193
4829
5535
45740
61913
24322
12701
34573
2070046
2070050
2070053
2070054
2070055
2070054
2070058
2070059
2070064
2070092
2070074
2070079
2070081
00033
00015
00022
00022
00022
00033
00033
00033
00208
00015
00015
00015
00020
RTE 29/250 BYPASS
ECL
CHARLOTTESVILLE
I-64
RTE 22
RTE 250
UNDERPASS
MADISON CL
RTE 229
RTE 694
ORANGE CL
RTE 15/29 BUS
RTE 17
RTE 15/29 BYPASS
RTE 17 SB
RTE 15/29
RTE 15/29 BUS
CULPEPER CL
RTE 1001
ALBEMARLE CL
SHENANDOAH
PARK ENTRANCE
RTE 617
RTE 33 E
ECL LOUISA
WCL MINERAL
WCL LOUISA
RTE 1005
RTE 22/208
RTE 522
I-64
ORANGE CL
ORANGE NCL
ALBEMARLE CL
RTE 29/250 BYPASS
WCL
CHARLOTTESVILLE
4968
6095
11493
11493
11402
5242
5242
5846
4095
4101
8435
9377
2859
111.86
112.54
495.03
112.82
217.81
97.98
142.64
109.02
147.36
155
120.21
55.69
362.76
12648
25700
2376
14905
968
3091
1470
4329
3801
23985
14482
5797
804
2070088
00015
RTE 211 BUS
RTE 33 BYP
RTE 33
ECL LOUISA
WCL MINERAL
RTE 522
RTE T 669
RTE 22/208
ECL LOUISA
SPOTSYLVANIA CL
RTE 617
CULPEPER CL
MADISON CL
RTE 33W
WINCHESTER
STREET
33940
6.62
6708
28946
2070089
2070091
00015
00211
ALEXANDRIA PIKE
WARRENTON WCL
N1
N2
00029
RTE 29
RTE 33 W
LEE HIGHWAY
SHIRLEY AVENUE
0.5 MN NORTH
RIVANNA RIVER
RTE 15 S
12089
24513
28.34
8.52
3480
4208
57541
11660
438
45.19
151000
11930
Table II. Projects chosen for final analysis
Project ID
#
2070011
2070018
Route
00029
00250
2070017
2070033
00250
00029
From
RTE 643
I-64
ECL
CHARLOTTESVILLE
RTE 15/29 BUS
N1
2070029
2070054
2070053
2070055
00029
00015
00022
00022
00022
RTE 29
RTE 17
ECL LOUISA
RTE 33 E
WCL MINERAL
2070016
2070092
2070032
2070005
2070028
00250
00015
00028
00020
00015
2070049
2070023
2070054
2070024
2070079
2070042
00029
00211
00033
00229
00015
00015
RTE 29/250 BYPASS
I-64
RTE 15/29
RTE 250
RTE 15/29 BUS
RTE 250
UNDERPASS
RTE 229
WCL LOUISA
RTE 694
ORANGE NCL
RTE 1001
Crash
rate
To
GREENE CL
RTE 22
ADT
45407
23526
165.05
178.33
Cost
93346
19745
I-64
PR WILLIAM CL
0.5 MN NORTH
RIVANNA RIVER
RTE 15/29 BYPASS
WCL MINERAL
ECL LOUISA
RTE 522
WCL
CHARLOTTESVILLE
RTE 617
PR WILLIAM CL
BARN BRANCH
RTE 17
WCL
CHARLOTTESVILLE
FAUQUIER CL
RTE T 669
RTE 211
MADISON CL
RTE 250
35950
45537
305.76
106.77
26970
61913
57541
42597
11493
11493
11402
438
95.13
112.82
495.03
217.81
151000
45193
14905
2376
968
22703
4101
12637
10618
34360
253.54
155
157.4
393.47
94.07
4724
23985
45740
20378
21835
45097
13704
5242
8793
9377
5314
70.21
301.76
97.98
170.15
55.69
110.92
15478
4085
3091
14553
5797
12701