Infrastructure Management Decision-Making with Condition
Data Generated by Remote Sensors: A Time Series Framework
Project 04-03
August 2004

Midwest Regional University Transportation Center
College of Engineering
Department of Civil and Environmental Engineering
University of Wisconsin, Madison
Authors: Pablo L. Durango-Cohen and Naveen Tadepalli
Northwestern University, Evanston, IL
Principal Investigator: Pablo L. Durango-Cohen;
Assistant Professor, Department of Civil and Environmental Engineering & Transportation Center,
Northwestern University

 

DISCLAIMER
This research was funded by the Midwest Regional University Transportation Center. The
contents of this report reflect the views of the authors, who are responsible for the facts and the
accuracy of the information presented herein. This document is disseminated under the sponsorship
of the Department of Transportation, University Transportation Centers Program, in the interest
of information exchange. The U.S. Government assumes no liability for the contents or use thereof.
The contents do not necessarily reflect the official views of the Midwest Regional University Trans-
portation Center, the University of Wisconsin, the Wisconsin Department of Transportation, or
the Federal Highway Administration at the time of publication.
The United States Government assumes no liability for its contents or use thereof. This report
does not constitute a standard, specification, or regulation.
The United States Government does not endorse products or manufacturers. Trade and man-
ufacturers names appear in this report only because they are considered essential to the object of
the document.

EXHIBIT B

Technical
Report
Documentation Page



1. Report No.
2. Government Accession No.
3. Recipient's Catalog No.

CFDA 20.701


4. Title and Subtitle
5. Report Date August 31, 2004
Infrastructure Management Decision-Making with Condition Data Generated by Remote Sensors:

A Time-Series Framework
6. Performing Organization Code



7. Author/s Pablo Durango-Cohen and Naveen Tadepalli
8. Performing Organization Report No.
MRUTC 04-03


9. Performing Organization Name and Address
10. Work Unit No. (TRAIS)

Midwest Regional University Transportation Center

University of Wisconsin-Madison
11. Contract or Grant No.
1415 Engineering Drive, Madison, WI 53706
DTRS 99-G-0005


12. Sponsoring Organization Name and Address
13. Type of Report and Period Covered
U.S. Department of Transportation
Research Report [Dates]
Research and Special Programs Administration

400 7th Street, SW
14. Sponsoring Agency Code
Washington, DC 20590-0001

15. Supplementary Notes
Project completed for the Midwest Regional University Transportation Center with support from the Wisconsin Department of
Transportation.
16. Abstract
Recent developments in remote sensing and communications technologies allow agencies to install sensors within infrastructure facilities, such as
pavement segments and bridges in order to collect condition-related data in real-time. In theory, such data can be processed, analyzed and displayed
on-line as a key component for maintenance, and repair decision-making. The reality facing public works agencies that have adopted these
technologies is that vast amounts of data related to the structural and functional condition of infrastructure are accumulated, but not used to address
management needs. The research presented herein, therefore, is to develop methodological tools to support the management of transportation
infrastructure systems given recent developments in facility-condition data collection technologies. In particular, the objectives of this research study
are to develop tools that will allow agencies to process and exploit the data to support IM\&R decision-making, and to provide a framework to
evaluate different strategies for deploying sensing technologies.












17. Key Words
18. Distribution Statement

No restrictions. This report is available through the Transportation Research

Information Services of the National Transportation Library.







19. Security Classification (of this report)
20. Security Classification (of this page)
21. No. Of Pages
22. Price
Unclassified
Unclassified
-0-

Form DOT F 1700.7 (8-72) Reproduction of form and completed page is authorized.

Contents
1
Introduction
5
1.1
Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
Project Description and Outline
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2
Background and Literature Review
8
2.1
Background: Transportation Infrastructure/Asset Management . . . . . . . . . . . .
8
2.2
Motivation: Data Collection Using Remote Sensors and Other Advanced Technologies
9
2.2.1
Using Advanced technologies for Condition Assessment
. . . . . . . . . . . .
10
2.3
Infrastructure Management Decision-Making
. . . . . . . . . . . . . . . . . . . . . .
13
2.3.1
Computational Limitations of the Latent-MDP approach: An Example
. . .
16
3
Model Formulation and Solution
17
3.1
Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.2
Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2.1
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2.2
Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3
Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3.1
Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3.2
State Estimation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4
Application Case Studies
23
4.1
Numerical Example: State-Estimation Problem . . . . . . . . . . . . . . . . . . . . .
23
4.2
State-Estimation using Sensor Data
. . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.3
Empirical Study of the Effect of Uncertainty on Life-Cycle Costs . . . . . . . . . . .
26
4.4
Combining Multiple Technologies for Condition Assessment . . . . . . . . . . . . . .
27
4.4.1
Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
5
Summary and Conclusions
30
A Cost Parameter Generation
31
B Latent Performance and Measurement-Error Models
32
B.1 Estimation of Precisions Associated with Technologies: A Note on Linear Regression
33
3

List of Figures
1
Asset Management Process (Taken from FHWA (1999)) . . . . . . . . . . . . . . . .
9
2
Latent Performance Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . .
15
3
Economic Trade-offs Associated with M&R Investments . . . . . . . . . . . . . . . .
18
4
Updated state-distribution: First moments
. . . . . . . . . . . . . . . . . . . . . . .
24
5
Updated state-distribution: Second moments . . . . . . . . . . . . . . . . . . . . . .
25
6
Updated state-distribution: First moments
. . . . . . . . . . . . . . . . . . . . . . .
26
7
Life-Cycle Costs vs. Deterioration Process Variance
. . . . . . . . . . . . . . . . . .
27
List of Tables
1
Cost Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2
Expected Costs for all Technology Combinations . . . . . . . . . . . . . . . . . . . .
29
3
Discretization and Transformation of PCI Scale . . . . . . . . . . . . . . . . . . . . .
31
4
Agency and User Costs from Madanat and Ben-Akiva (1994) ($/m2) . . . . . . . . .
32
5
Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
6
Parameter Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
7
Precision Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4

1
Introduction
This document is the final report for project 04-03 sponsored by the Midwest Regional University
Transportation Center (MRUTC). The project consists of developing an optimization framework
to provide support for investments in preservation and improvement of transportation infrastruc-
ture facilities that are inspected periodically with sensors or other advanced technologies. In the
remainder of this section, we first state the motivation for our work and the objectives of our study.
We then present an overview of the tasks that we carried out as part of the project and provide an
outline for the report.
1.1
Motivation and Objectives
This research project is motivated by recent developments in remote sensing and communications
technologies that allow public works agencies to install sensors within infrastructure facilities, such
as pavement sections and bridge decks, in order to collect condition-related data. In addition,
a plethora of non-destructive inspection technologies, e.g., video, radar, and laser, have become
commonplace in evaluating and measuring distresses on transportation infrastructure. In theory,
such data should be processed and used as a key component to support maintenance and repair
(M&R) decision-making. The reality facing agencies that have adopted these technologies is that
vast amounts of data are accumulated, but not used to address management needs. The goal of
the research described herein, therefore, is to develop methodological tools to exploit the exten-
sive capabilities of advanced monitoring technologies to support M&R investment decisions. In
particular, the objectives of the study are to develop a framework that allows agencies to:
1. Process condition data efficiently and use them to support M&R decision-making; and
2. Quantify the value of combining different monitoring technologies, which means that frame-
work can be used as a tool to support the development of strategies to deploy advanced
inspection technologies.
1.2
Project Description and Outline
The project consists of five tasks which we describe below. We also provide an outline for the
remainder of the report which roughly is consistent with the tasks.
Task 1: Literature Review
This task consists of conducting an extensive literature review to
identify possible approaches to address the research problem. The challenges involved in developing
optimization models to support M&R investment decisions with condition data generated by sensors
or other (advanced) technologies are related to the potentially vast amounts of data that can be
obtained. These data have to be processed with tools that are both statistically rigorous and
computationally efficient. In addition, the algorithm to solve the underlying optimization problem
5

needs to be tractable. These requirements, for the most part, constitute important limitations
of the available tools to address the research problem that we study. These issues are discussed
extensively in Section 2 of this report. In particular, we present an example that illustrates the
computational shortcomings of the current state-of-the-art approach to address the problem. We
begin Section 2 by putting our research in the context of the "Transportation Asset Management
Framework" presented in The Asset Management Primer (FHWA, 1999). We also review examples
of agencies and initiatives that are collecting condition data using sensors.
Task 2: Model Formulation and Solution
We present an optimization framework to support
M&R investment decisions for transportation infrastructure facilities.
The framework involves
formulating the underlying decision problem as a discrete-time, stochastic optimal control problem
and consists of two components: a state-estimation problem that involves processing vast arrays of
condition data and using them to develop condition forecasts; and an optimization problem whose
solution yields M&R investment policies. Our approach differs from the literature in that both
elements are fully integrated. This, in turn, leads to a framework that is both statistically rigorous
and computationally efficient, i.e., capable of providing effective decision-support. The model is
presented in detail in Section 3 of this report.
Task 3: Application Case Studies
Section 4 of this report presents four application case
studies. The objectives of the empirical studies are as follows:
1. Provide numerical examples to illustrate the methodology presented above to address the
state-estimation problem in the above framework. In particular, we show how the Kalman
Filter processes distress measurements to update the state distribution. For this part of the
study we use both a set of simulated data as well as sensor data. The sensor data was provided
by the Minnesota Road Research Project (MnROAD).
2. Show how the the framework can be used to study the effect of uncertainties in the deteriora-
tion process and in the process of collecting distress measurements on the optimal life-cycle
cost of managing infrastructure facilities.
3. Illustrate how the framework can be used to quantify the value of combining different tech-
nologies for condition assessment.
Initially, our objective was to compare the methodology we developed to state-of-the-art models.
However, early on we determined that the existing framework is inadequate to process data gen-
erated simultaneously by multiple technologies. We use the example presented in Section 2.3.1 to
illustrate the limitations associated with the existing approach.
Task 4: Preparation of Reports and Deliverables
In addition to this report and four
quarterly progress reports delivered earlier, we have prepared (and are preparing) the following
6

materials to disseminate the results of this research effort.
1. A paper submitted for presentation at the Transportation Research Board 84th Annual Meet-
ing to be held January 9­15, 2005 in Washington, D.C. The paper was also submitted for
publication in Transportation Research Record (the journal of the Transportation Research
Board).
2. A paper in preparation to be submitted to a leading journal in the area of transportation
systems analysis.
3. A report for educational purposes that summarizes the results of this effort. The report is
in the form of a powerpoint presentation that has been delivered at institutions, conferences,
and meetings. These include invited presentations at the Third International Symposium
on Infrastructure Management and Financing (Kyoto, Japan ­ September 2003), and at the
Industrial and Systems Engineering Department at Lehigh University (Bethlehem, PA ­ May
2004); contributed presentations at the Annual Meeting of the Institute for Operations Re-
search and Management Sciences (Informs) (Atlanta, GA ­ October 2003). In addition we
prepared a poster for the MRUTC's reception held at the 83rd Annual Meeting of the Trans-
portation Research Board (Washington, D.C. ­ January 2004). The powerpoint presentation
is available directly from the P.I. and can be made available through the MRUTC.
In addition, the P.I. would be happy to participate/teach in a MRUTC-organized workshop to
instruct users and agencies about the work described herein.
Task 5: Exploration of Technology Transfer
We had hoped to use the MRUTC as a liaison
to establish partnerships with public works agencies. Through this effort we established contacts
at the Michigan DOT. Unfortunately, Michigan's efforts to use advanced technologies to monitor
transportation infrastructure are in their infancy. Through our own efforts we obtained pavement
management data from the states of Arizona and Washington. These data, however, were not
generated using advanced technologies. Eventually, we established a partnership with the Min-
nesota DOT through MnROAD. They have provided extensive data and technical assistance and
have expressed interest in partnering with the P.I. to both continue with the research effort and to
disseminate the results. We use data provided by MnROAD in the example presented in Section
4.2.
7

2
Background and Literature Review
This section addresses Task 1 of the project (Literature Review). To put our research in the context
of "Transportation Asset Management", we begin this section by presenting a brief overview of the
broadly accepted and highly regarded framework presented in The Asset Management Primer
(FHWA, 1999). We proceed to motivate the relevance of the research herein, by briefly describing
examples of agencies and initiatives that are currently using remote sensors and other advanced
technologies to inspect infrastructure facilities. To conclude the section, we review the literature on
infrastructure management decision-making. In particular, we focus on the shortcomings associated
with existing methodologies that motivate the need for the research presented in this report.
2.1
Background: Transportation Infrastructure/Asset Management
The nation's transportation infrastructure serves as the backbone of a complex network for sup-
plying goods and services in an increasingly competitive and distributed economy. The quality and
efficiency of this infrastructure, through its ability to provide mobility, and consequently, access
to people, goods, services and resources, impacts quality of life and the continuity of economic
and business growth. Indeed, economists (c.f. Small et al. (1989) and Hulten (1996)) have argued
that investments in the management and efficient use of infrastructure have a greater impact than
investments in additional infrastructure. Consequently, with an aging transportation infrastructure
whose replacement value is estimated at $1 trillion (Kane, 2000), the development of effective and
efficient policies to allocate resources for the construction, operation, preservation and improvement
of such facilities takes on unprecedented social and economic value.
Transportation Asset Management as defined in The Asset Management Guide (FHWA, 1999)
is a strategic approach to manage transportation infrastructure which encompasses a broad array
of business functions, activities and decisions. It provides a systematic framework to support the
resource allocation decisions that are motivated by the trends mentioned in the previous paragraph.
The primer describes the asset management process as follows:
"First, performance expectations, consistent with goals, available resources, and organi-
zational policies, are established and used to guide the analytical process, as well as the
decision-making framework. Second, inventory and performance/condition data are col-
lected and analyzed. This information provides input on future system requirements.
Third, the use of analytical tools and reproducible procedures produces viable cost-
effective strategies for allocating budgets to satisfy agency needs and user requirements,
using performance expectations (condition forecasts) as critical inputs.
Alternative
choices are then evaluated, consistent with long-range plans, policies, and goals. The
entire process is reevaluated annually through performance monitoring and systematic
processes."
8

The process is illustrated in Figure 1.
Goals and Policies
Asset Inventory
Condition Assessment and
Performance Modeling
Budget/
Alternatives Evaluation and
Allocations
Program Optimization
Short and Long Range Plans
(project selection)
Performance Monitoring
Program Implementation

Figure 1: Asset Management Process (Taken from FHWA (1999))
The research described in this report relates to the third step in the process. That is, we have
developed a methodological tool that can exploit the extensive capabilities of advanced inspection
technologies to support investments in M&R of transportation infrastructure. Our work recognizes
that the steps in the asset management process are interconnected, i.e., that fundamental changes
in the condition assessment and performance prediction step (Step 2) warrant improvements in the
optimization models that are used to support M&R investments (Step 3) in order to fully take
advantage of the enhanced capabilities.
2.2
Motivation: Data Collection Using Remote Sensors and Other Advanced
Technologies
The main barrier for the implementation and use of the model presented herein would seem to be
to convince public works agencies to collect condition data using sensors and other advanced tech-
nologies. Here we provide several examples of agencies and initiatives that have deployed advanced
technologies for condition assessment of transportation infrastructure. The purpose is to illustrate
the extent of the potential users for the model we developed. Overall, agencies have adopted these
technologies in the last 10 years. Even though their use has primarily been directed toward experi-
mental infrastructure facilities, it does seem reasonable to assume that technological developments
in areas such as fiber optics, micro-electrical-mechanical systems (MEMS), radar, laser, satellite
imaging, image processing, etc. will increase the availability and cost-effectiveness of using them
for condition assessment of facilities that are in use. In the remainder of this section we proceed to
describe initiatives and agencies that use advanced technologies to monitor/inspect transportation
9

infrastructure.
Prior to discussing the use of advanced technologies for condition assessment, we mention that
such technologies have also been widely used in a slightly different context, to inventory transporta-
tion infrastructure. Satellite imaging, for example, has been used to support planning decisions such
as prioritizing corridors for development, and evaluating overall condition after natural disasters
(floods, earthquakes, etc.). Examples of agencies and initiatives that have been involved in these
efforts include the National Consortium for Remote Sensing in Transportation (NCRST)1 and a
Commercial Remote Sensing Products and Spatial Information Technologies Program2 which is a
partnership between the USDOT and NASA.
2.2.1
Using Advanced technologies for Condition Assessment
Here we describe initiatives and agencies that use advanced technologies to monitor/inspect trans-
portation infrastructure. Inspection/condition assessment in this context refers to the process of
measuring distresses on transportation infrastructure periodically. Distress measurements can be
collected manually or automatically and are comprised of multiple measurements and/or (subjec-
tive) ratings that can be either discrete or continuous. Examples of distresses in pavement man-
agement include roughness, type and extent of cracking, rut depth and profile, extent of surface
patching, and raveling.
The Infrastructure Technology Institute at Northwestern University (ITI)
is recog-
nized as a leader in transferring remote sensing and communications technologies to the inspection
structural elements in infrastructure facilities. For example, the institute has pioneered the devel-
opment and deployment of acoustic, strain, and optical sensors to monitor the growth of cracks in
structural members of steel bridges and other infrastructure facilities. Currently, about 30 facilities
around the country (20 in the Midwest), mostly bridges, are being monitored. The major emphasis
of these projects is to provide continuous remote monitoring. Most of these structures are critical
for safety and need continuous monitoring of the structural fitness and condition of the sub-surface
environment.
The ITI has been successful in developing and applying Acoustic Emission and strain gage
monitoring to steel bridges and Time Domain Reflectometry and Impulse Echo to geotechnical
applications. Relevant projects include: successful deployment strain gages and clinometers to
monitor crack development, in the fracture critical components, of a 70-year-old Michigan Street
Lift Bridge in Sturgeon Bay, Wisconsin (Prine and Fish, 2003). The ITI has also deployed strain
and temperature sensors on the Hoan Bridge, Milwaukee to test if thermally driven stresses would
1http://www.ncgia.ucsb.edu/ncrst
2http://scitech.dot.gov/research/remote
10

induce fatigue cracking in the structure. Apart from these, the ITI has deployed Time Domain
Reflectometers to monitor the crack growth and thus, the structural stability of the rock beneath I-
70 in Souteastern Ohio. Additional information and publications can be obtained from the institutes
website.3
The Minnesota Road Research Project
MnROAD is at the forefront of using advanced mon-
itoring technologies for condition assessment of pavements. They have an extensive network of over
4,572 sensors spread over two pavement segments that run parallel to Interstate 94 near Ostego,
Minnesota. The "mainline" section is 3.5 miles in length and the "low volume" road way consists
of a 2.5 mile closed loop where controlled weight and traffic volume simulate rural road condi-
tions. Static and dynamic sensors record pavement response to traffic loading such as deflections,
strains, stresses, etc. Environmental sensors are used to measure characteristics such as tempera-
ture, precipitation, wind velocity and atmospheric pressure. MnROAD also uses uses probes that
are equipped with lasers, radar and other technologies to measure pavement roughness, cracking,
raveling, and rutting (depth and profile). The main objectives of the project are to evaluate the
effects of heavy vehicles on pavements, evaluating the effects of seasonal changes in paving materi-
als, and to improve the design and performance of low-volume roadways.
MnRoad has successfully completed 10 years in operation. The vast amount of data collected
has been used to validate empirical models for pavement design. Using these models MnROAD
has developed software programs (e.g.: Pavecool, MnPave), which help in designing new pavements
for various climatic, traffic and structural conditions. In addition, MnROAD has tested various
aggregates and crack sealants such as recycled concrete aggregates and carbonate aggregates. For
further details the reader is referred to the MnROAD's website.4
The Smart Road Project
in the state of Virginia is another project which has deployed sen-
sors to monitor the condition of the pavement. This project is 9.6-km connector highway between
Blacksburg and I-81 in southwest Virginia. This project tried to address limitations of test fa-
cilities such as climate control, control of traffic speed and loading and acceleration for loading.
Accordingly, the first 3.2-km stretch is designated as a controlled test facility. This facility allows
for testing of various hypotheses on pavement material performance and characteristics. Using the
"All Weather Testing Facility", pavement materials can be tested in different environmental con-
ditions. The weather conditions are simulated using 76 snow towers which can simulate snowfalls
upto 100 mm/hr and rainfall upto 50 mm/hr. The objectives of the project are to enhance the
methodologies for design and construction of pavements and to evaluate the concepts, technologies
and products of Intelligent Transportation Systems.
3http://www.iti.northwestern.edu/publications
4http://www.mrr.dot.state.mn.us/research/mnresearch.asp
11

Hot-Mix Asphalt Strain Gages, Aggregate Dynamic Strain Gages, Vibrating Wire Static Strain
Gages are used to measure strains in the various layers of the pavement. Pressure cells are deployed
to collect pressures in the various layers of the pavement and Thermocouples are used to measure
the heat flow inside the pavement system. These constitute the dynamic measurement sensors.
The static measurements constitute of environmental data such as temperature, moisture and frost
depth. The construction of this project was recently completed and hence most of the work is in
its infancy. 5
National Consortium for Remote Sensing in Transportation (NCRST)
is at the fore-
front of developing Remote Sensing applications for transportation. The objective of this agency
is to focus on testing and implementation of commercial remote sensing technologies and methods
to meet future transportation requirements.
Many projects are being undertaken to compare and evaluate various techniques and their ef-
fectiveness in pavement management. As a part of these projects, studies were carried out on Laser
Scanning for applications in Construction and Bridge Maintenance. A pilot study conducted by
Iowa Department of Transportation has shown that Ground Laser, which is a very accurate way of
imaging, is useful in developing as-built- 3-D infrastructure data. The study showed that we can
obtain 2-6 mm precision images. But, it was found that this technique was costly by 30 percent
when compared to its competing technique namely, aerial photogrammetry. The investigators be-
lieve that this technique could be made competitive by elevating this scanner on a boom truck and
scanning both sides of the divided roadway (Jaselskis et al., 2003).
Pavement Health Surveys have also been carried out using equipments like hyperspectral sensor,
hand held spectrometers etc. A study is being conducted by University of California Santa Barbara
(UCSB) in joint collaboration with Iowa State University to find a correlation between remotely
sensed parameters (like spectral reflectance) and physical characteristics like rutting and cracking.
The listings of other projects by this agency can be found at the project's website.6
FHWA's Non Destructive Evaluation Validation Center (NDEVC)
is an another agency
which is actively involved in developing and implementing automated pavement and bridge evalua-
tion techniques. The objective of this center is to develop NDE tools and techniques that are both
accurate and efficient. Apart from this, the center also tests the reliability of the NDE technologies
in its laboratories. These laboratories help simulate field conditions. Apart from these, NDEVC
has five decommissioned bridges to evaluate the NDE tools and techniques under realistic environ-
5http://www.cee.vt.edu/program areas/tise/smart/overview.html
6http://www.ncgia.ucsb.edu/ncrst/research.html
12

mental conditions.
High Speed Electromagnetic Roadway Measurement and Evaluation System (HERMES), Ground
Penetrating Radar, Laser Bridge Deflection Measurements, Ultrasonic Stress Measurements, X-ray
computed tomography are some of the tools developed by this center (Washer, 2000).
2.3
Infrastructure Management Decision-Making
In this section, we present an overview of optimization models used to support investment decisions
for M&R of transportation infrastructure. We also discuss the limitations that motivate the need
to develop a framework that can exploit the capabilities of advanced monitoring technologies.
Optimization models to support M&R of transportation infrastructure systems constitute ap-
plications, perhaps the most successful, of the "Equipment Replacement Problem" introduced by
Terborgh (1949) and formulated as a dynamic control problem by Bellman (1955) and Dreyfus
(1960). Friesz and Fernandez (1979) and Golabi et al. (1982) extended the models to support M&R
of transportation infrastructure. State-of-the-art optimization models are formulated as Markov
Decision Processes (c.f. Murakami and Turnquist (1985), Carnahan et al. (1987), and Carnahan
(1988)). Golabi et al. (1982), for example, present a mixed-criteria, constrained, Markov Decision
Process (MDP) for pavement management in the state of Arizona (a network of 12,000 kilometers
of highways). Savings of $14 million were reported in the first year of implementation, and $101
million was forecast for the following four years. The same optimization model drives Pontis (Golabi
and Shepard, 1997), a bridge management system used in over 40 states. The success and impact
of these models is related to the magnitude of investments in M&R of transportation infrastructure
which in the United States is on the order of tens of billions of dollars per year. Recent reviews of
optimization models for transportation infrastructure management are presented in Gendreau and
Soriano (1998) and Durango (2002).
Optimization models to support M&R investments must evaluate both the short and long-term
consequences associated with M&R actions. For this reason, they must incorporate information
about the effect of actions on current and future infrastructure condition. Information about cur-
rent condition is obtained through distress measurements. Distress measurements can be collected
manually or automatically and are comprised of multiple measurements and/or (subjective) ratings
that can be either discrete or continuous. Examples of distresses in pavement management include
roughness, type and extent of cracking, rut depth and profile, extent of surface patching, and rav-
eling. Information about future condition, i.e., condition forecasts, are generated with statistical
deterioration models. A deterioration model relates condition to a set of explanatory variables such
as design characteristics, traffic loading, environmental factors, and history of M&R investments.
Models to support M&R investments based on the MDP framework rely on indices/ratings that
13

combine condition data into a single quantity. Examples include the Concrete Bridge Deck Con-
dition Ratings, the Present Serviceability Index (PSI) and the Pavement Condition Index (PCI)
developed by FHWA (1979), HRB (1962) and Shahin and Kohn (1981), respectively. Unfortu-
nately, these indices lack rigorous justification, have poor explanatory/predictive power, and rely
on predetermined sets of distress measurements which precludes incorporating new ones. In spite
of the computational efficiency of this approach, it is clear that relying on it to process condition
data and to support M&R investments may negate the benefits of using advanced inspection tech-
nologies for condition assessment, and of using standard statistical methods to process condition
data.
Ben-Akiva et al. (1991) introduced the latent performance modeling approach to address the
problems of assessing and forecasting condition when multiple technologies are used to collect con-
dition data. The approach relates distress measurements to the system's current condition through
a measurement-error model. The system's condition is represented by latent/unobservable variables
which capture the ambiguity that exists in defining (and consequently in measuring) a system's
condition. The measurement-error model accounts for uncertainties inherent in the data-collection
process as well as for how different technologies and distress measurements relate to each other.
As discussed by Humplick (1992), the uncertainties in the measurement process can be attributed
to the precision and accuracy of measurement technologies because other biases can be corrected
for. Latent performance models also include a deterioration model that describes the relationship
between a set of explanatory variables and the system's condition, and captures the randomness
inherent in the system's deterioration process.
The latent performance modeling approach is illustrated in Figure 2. The solid arrow represents
the deterioration model and the dashed arrow represents the measurement-error model.
Empirical studies (Ben-Akiva and Ramaswamy, 1993; Ben-Akiva and Gopinath, 1995) have
shown that latent performance models are appropriate to generate condition forecasts of trans-
portation infrastructure, i.e., the goodness-of-fit measures are better than those reported using
other other statistical methods. This lead Madanat and Ben-Akiva (1994) to include latent per-
formance models into a framework to support M&R investments by formulating the underlying
optimization problem as a latent MDP. The measurement-error model in latent MDP formulations
is represented by a (discrete) probability mass function that relates the condition variable to the
distress measurements. Mathematically,
Prob (Zt = k|Xt = i) , i, k S, t = 1, · · · , T + 1
(1)
where i and k are elements in a finite set of possible conditions S, and Expression (1) represents
14

Exogenous Factors
Affecting Condition
Age, traffic loading,
cumulative precipitation,
structural characteristics
Latent/Unobserved
Infrastructure
Condition
Distress
Measurements
Roughness, cracking,
rut depth, surface
patching, etc.
Figure 2: Latent Performance Modeling Approach
the conditional probability of collecting a measurement Zt = k at the start of period t given that
the true condition is Xt = i.
Unfortunately, discrete measurement-error models are virtually useless when multiple technolo-
gies are used simultaneously to measure different distresses (i.e.: when an array of measurements
Zt is collected) because it is necessary to specify a probability for every possible combination of
measurements (a number that grows exponentially with the number of technologies and the num-
ber of distresses being measured). The computational complexity to find optimal M&R policies
also increases exponentially with the size of Zt. An example and further discussion of limitation is
addressed further in the following section. To a large extent, this difficulty explains why previous
studies in the literature have only considered the case of inspections that yield a single distress
measurement (c.f. Madanat and Ben-Akiva (1994), Smilowitz and Madanat (2000), Guillaumot,
Durango-Cohen, and Madanat (2003)). In any case, it is clear that the emergence of advanced
monitoring technologies poses serious methodological and computational challenges because of the
potentially large quantities of data being generated. This limitation serves as motivation for the
framework proposed in this paper which constitutes an alternative to the latent MDP that is both
statistically rigorous and computationally efficient.
15

2.3.1
Computational Limitations of the Latent-MDP approach: An Example
Here we present an example to illustrate the computational problems that are associated with
discrete measurement-error models. Consider a situation where five technologies, are used to collect
five distress measurements at the start of each period. Lets assume that each technology collects
continuous measurements in a range [0, R] and that each of the ranges is discretized into 11 points,
{0, 1, 2, · · · , 10}. This means that measurements are rounded up or down when they are collected. A
measurement-error model for this situation must specify a probability for every possible combination
of measurements. That is, 115 = 161, 051 probabilities need to be specified. This, in turn, poses
two fundamental problems:
1. Statistically, we note that the schemes to specify these probabilities are based on approxima-
tion schemes that induce errors; and
2. Computationally, we see that the number of probabilities that need to be specified increases
exponentially with the number of technologies/distress measurements that are collected. Un-
fortunately, the computational effort required to obtain optimal M&R policies using the
latent-MDP approach also increases exponentially with the number of technologies/distress
measurements. This is because every possible outcome of the measurement process in every
period needs to be considered to obtain optimal M&R policies. In dynamic programming,
these problems are referred to as the "curse of dimensionality" and are described in detail in
references such as Dreyfus (1977) and Bertsekas (1995).
The above leads to the observation that state-of-the-art optimization models to support M&R
investments are not suited to address the challenges posed by developments that allow agencies to
simultaneously collect multiple distress measurements using multiple technologies.
16

3
Model Formulation and Solution
This section addresses Task 2 of the project (Model Formulation and Solution). First, we describe
in very specific terms the problem that we address. We then present a mathematical formulation
for the problem and the approach we propose to solve it.
3.1
Problem Description
We consider an agency that manages a facility under a periodic review policy over T periods.7 At
the start of every period, t = 1, · · · , T , the agency collects sets of distress measurements. The data
are related to a facility's state represented with the random variable Xt. The measurements taken
at the start of t are represented by the vector Zt. We use It to represent the set of information
that an agency has at its disposal at the start of period t. Using the above notation,
It Z1, A1, Z2, A2, · · · , Zt-1, At-1, Zt = It-1, At-1, Zt , t = 1, · · · , T + 1, and, I0
(2)
Based on the available information, an agency decides to apply an action to the system, At, and
incurs a cost, g(Xt, At) , that depends both on the action and on the current state of the system.
Ê
This cost structure can be used to capture both agency and operating costs. In the management
of transportation infrastructure, agency costs correspond to the costs of applying M&R actions
and operating costs to (a fraction of) the users' vehicle operating costs. Vehicle operating costs
depend on condition and are associated with travel time, fuel consumption, vehicle maintenance,
etc. At the end of the planning horizon facilities have a salvage/residual value of s(XT+1)
that
Ê
depends on the terminal state of the system.
The costs to operate a facility increase as it deteriorates. By making M&R investments, agencies
can mitigate and even reverse the effects of deterioration and, consequently decrease current and
future operating costs. As a result, an agency's choice of M&R investments trades off investments
with operating costs. These trade-offs are illustrated in Figure 3.
The figure depicts a situation where at the start of the third period an agency is choosing
between either a small investment, S, or a large investment, L. The figure on the left is for facility
condition vs. time. The one on the right is for cumulative discounted costs over time. As is
illustrated, the large investment results in greater improvement in condition and in additional costs
incurred at the start of the third period. However, as a result of the improvement in condition, the
rate at which costs are accrued after the investment is greater for the small investment. Ultimately,
an agency's choice of actions is intended to minimize the sum of expected discounted (social) costs
7In the management of transportation infrastructure, planning horizons tend to be long and uncertain. Therefore,
it is often acceptable to consider the case when T .
17

h
th
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e

process. The structure of the deterioration model, D·(·), is determined by factors such as material
and construction quality, environmental conditions, etc. The arguments of the system equation
may include deterministic or stochastic exogenous inputs that are captured in the vector t. These
inputs may include environmental factors, traffic loadings, etc. Equation set (5) is a measurement
error model. It establishes the relationship between the underlying, true state of the system and
the distress measurements. The measurement error model M·(·) includes a set of exogenous (deter-
ministic or stochastic) inputs captured in the matrices t (one vector associated with each distress
measurement). Equation (6) specifies the initial condition of the system.
3.2.1
Assumptions
We proceed to state and discuss the assumptions that we use to solve the mathematical model used
to compute M&R policies. The assumptions are:
1. We assume that X
n
t, At
, t = 1, · · · , T + 1 and that Z
, t = 1, · · · , T + 1.
Ê
t Ê
2. The period cost function can be represented (or approximated) by a second order polynomial,
i.e., g(Xt, At) = aX2t + bXtAt + cA2t + dXt + eAt + f. We define (at, bt, ct, dt, et, ft) =
t-1(a, b, c, d, e, f ). The cost structure may also be used to account for non-stationary costs.
3. The salvage value function can be represented (or approximated) by a second order polyno-
mial, i.e., s(XT+1) = -pT+1X2T+1 - qT+1XT+1 - rT+1.
4. The system equation can be represented (or approximated) by a linear polynomial, i.e.,
Dt(Xt, At) = gtXt + htAt + t. The polynomial can be obtained by estimating an AutoRe-
gressive Moving Average with eXogenous input (ARMAX) model.
5. We assume that t follow a Normal Distribution with mean ¯t and finite variance 2 .
t
6. We assume that the measurement error model can be represented (or approximated) by a
linear polynomial, i.e., Mt(Xt) = HtXt + t. We discuss the nature of t in Section 3.3.2.
Prior to discussing a solution procedure for the problem, we note that the above assumptions
are not overly restrictive. Specifically,
· The linear structure assumed for D(·) and M(·) is actually quite general as a number of trans-
formations can be employed to capture complex patterns/structures in the data. ARMAX
models represent a broad class of time series models.
· The assumptions about costs are not restrictive because, for example, it is possible to obtain
optimal M&R investment policies for general cost structures by solving a finite sequence of
problems. In each problem the cost structure is approximated by a second-order Taylor Series
expanded about a different point.
19

3.2.2
Optimization Problem
With the assumptions discussed in the preceding section, the optimization problem presented above
can be formulated as a dynamic program with state-space It, t = 1, · · · , T +1. The optimal objective
function, vt(It) is defined as the minimum expected discounted cost from the start of t until the
end of the horizon given the information available at the start of t, It. The recurrence relation is
as follows:
vt(It) = min EX
a
t|It
tX2
t + btXtAt + ctA2
t + dtXt + etAt + ft
At
+E t [vt+1 (gtXt + htAt + t)]]}
(7)
The boundary condition for the problem is as follows:
vT+1(IT+1) = EX
p
T +1|IT +1
T +1X2
T +1 + qT +1XT +1 + rT +1
(8)
3.3
Solution Procedure
The solution procedure we propose involves solving two subproblems: an optimization problem and
a state-estimation problem. Both are described below.
3.3.1
Optimization Problem
The dynamic programming formulation allows for a solution that can be obtained inductively.
For the above problem, the solution can be expressed in closed-form with parameters computed
recursively as follows:
2p
A
t+1ht¯t + qt+1ht + et
t
=
- bt + 2pt+1gtht E[X
, t = T, · · · , 1
(9)
2c
t|It] +
t + 2pt+1h2
t
2ct + 2pt+1h2t
pt = at + pt+1g2 - (bt + 2pt+1gtht)2
t
, t = T, · · · , 2
(10)
4[ct + pt+1h2t]
qt = dt + 2pt+1¯tgt + qt+1gt - [bt + 2pt+1gtht][et + 2pt+1ht¯t + qt+1ht], t = T, · · · , 2 (11)
2[ct + pt+1h2t]
rt = ft + pt+1(¯2t + 2 ) + q
, t = T, · · · , 2
(12)
t
t+1¯t + rt+1 - [et + 2pt+1ht¯t + qt+1ht]2
4[ct + pt+1h2t]
These equations are derived from the first-order/necessary conditions for the problem. The
second-order/sufficiency conditions are satisfied because the objective function is convex.
The
equations are evaluated recursively noting that pT+1, qT+1, rT+1 are the parameters that define the
salvage value function. Using the solution to the above system of equations allows us to write the
20

optimal objective value function as:
vt(It) = ptE [Xt|It]2 + qtE [Xt|It] + rt, t = 1, · · · , T
(13)
We note that in order to compute the optimal policy, At, t = 1, · · · , T and to evaluate the optimal
objective value function it is necessary to compute the expected state given the set of information
in each period, i.e., E[Xt|It], t = 1, · · · , T . This step is referred to as the state estimation problem
and it is discussed further in the following section. We note that the key to processing distress
measurements generated simultaneously by multiple technologies is to compute these expectations
efficiently.
3.3.2
State Estimation Problem
The state estimation problem consists of finding the expected state for a given information set.
This is in general a hard problem, however, under the assumption that the error terms t and t
follow a Gaussian Distribution with zero mean and finite second moments, then the expectation
can be computed with a recursive algorithm known as the Kalman Filter. These assumptions
are mild because they are consistent with obtaining adequate estimations of the models (unbiased
parameters). The algorithm is presented below:
Kalman Filter Algorithm
Repeat at the start of each period:
Given E[Xt-1|It-1], Var(Xt-1|It-1), At-1, and Zt = zt
Define: ^
Xt-1 E [Xt-1|It-1],
Pt-1 Var (Xt-1|It-1), and
It = {It-1, At-1, zt}
Time Update:
^
X-
t = gt-1 ^
Xt-1 + ht-1At-1
P -
t
= g2t-1Pt-1 + 2t-1
Measurement Update:
Kt = P -
t H (P -
t HH + R)-1
E[Xt|It] ^
X-
t + Kt(zt - H ^
X-
t )
Var(Xt|It) (1 - KtH)P -
t
The time update step uses the system equation to project the estimates of the conditional
expectation and variance. The measurement update step updates (with Bayes' Law) the expectation
and variance taking into account the new set of measurements obtained at the start of period t,
zt. The computational complexity of the Kalman Filter increases polynomially with the size of the
vectors Zt which means that the framework does not suffer from the shortcomings of the latent
21

MDP approach. This is because with it is only necessary to update the first two moments of the
state-distribution as opposed to the probability mass function.
22

4
Application Case Studies
This section addresses Task 3. We present a computational study where we:
1. Provide numerical examples to illustrate the methodology presented above to address the
state-estimation problem in the above framework. In particular, we show how the Kalman
Filter processes distress measurements to update the state distribution. In Section 4.1 we use
a set of simulated data and in Section 4.2 we use the framework to address the state-estimation
problem using data collected with a strain sensor.
2. Use the framework to study the effect of uncertainties in the deterioration process and in
the process of collecting distress measurements on the optimal life-cycle cost of managing
infrastructure facilities.
3. Use the framework to quantify the value of combining different technologies for condition
assessment.
4.1
Numerical Example: State-Estimation Problem
To illustrate how the Kalman Filter addresses the state-estimation problem in the above framework,
we consider the management of a pavement over a 40-year planning horizon. The initial condition of
the pavement is 10 given in a PCI-like scale with range [0, 100]. The deterioration and measurement-
error models in the example are given by:
Xt+1 = Xt + 8 - At
(14)
Zt = Xt + t; where t is Normally distributed with µt = 0 and 2 .
(15)
t
That is, we assume that the pavement deteriorates deterministically at a rate of 8 PCI units per
year, and that the distress measurements correspond to the actual condition plus a random error
term/white noise. As stated earlier, the parameter 2 represents the precision of the technology
t
used to collect the distress measurements.
We also assume that the pavement is restored to its initial condition every ten years, i.e.,

80; t = 11, 21, 31, 41
At =
(16)
0;
otherwise
Finally, we assume that the initial, estimated state-distribution is Normal with E[X1|I1] = 25
and Var(X1|I1) = 20.
To illustrate how the Kalman Filter uses the sequence of distress measurements to update the
state-distribution we simulated an instance of the above process. The solid line in Figure 4 rep-
23

resents the true condition of the pavement over time. The triangles represent a set of randomly
generated distress measurements that are consistent with the measurement-error model in Equation
(15). In this part of the study we use 2 = 10. The dashed line corresponds to the first moment
t
of the estimated state-distribution. The figure shows how the condition estimate converges to the
true condition of the pavement (over time, the dashed line traces the solid line).
90
80
70
60
50
40
Facility State
30
20
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
Start of Year
Estimated
True Condition
Measurements
Figure 4: Updated state-distribution: First moments
Figure 5 shows how the Kalman Filter updates the second moment of the estimated state-
distribution. In this part of the study we considered the effect of technologies of different precisions
to collect distress measurements. Specifically, we considered cases where 2 = 0, 2, 10, i.e., "per-
t
fect", "fine", and "coarse" technologies used to collect measurements. We also considered the case
where two "coarse" technologies with 2 = 10 where used to collect distress measurements si-
t
multaneously (the technologies were assumed to be independent of each other and therefore this
case does not correspond to using the same technology to collect two sets of measurements). We
observe that the variance in the estimated state-distribution becomes very small very quickly. The
asymptote and the convergence rate are properties of the technologies. The key observation is that
the variance in the estimated state distribution is well within the precision of each technology, i.e.,
the procedure filters out the random error/noise in the measurements. For example, the variance
in the state distribution when measurements are collected with the "coarse" technology (2 = 10)
t
converges to approximately 1 after 10 years.
24

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In this part of the study we considered the same technology choices that we used in the pre-
vious section. However, instead of considering a deterministic deterioration process we let 2 be
t
0.1, 1, 2, 4, and 8. For each combination of technology and deterioration process variance we calcu-
lated the average optimal expected cost of managing 100 pavement sections. The optimal policies
were obtained by solving the optimization model presented earlier. The results appear in Figure 7.
350
348
) 346
2 m
344
st ($/o
342
340
iscounted C 338
336
Expected D 334
332
330
Var() =0.1
Var() =1
Var() =2
Var() =4
Var() =8
Deterioration Process Variance
Var() = 0
Var()= 2
Combination of 2 Var()=10
Var() = 10
Figure 7: Life-Cycle Costs vs. Deterioration Process Variance
From the figure we observe that, as expected, the costs to manage the pavements increase as
the uncertainty in the deterioration process grows. Also as expected, "coarser" data collection
technologies result in higher costs incurred. As mentioned earlier, this is the first study in the
area of transportation infrastructure management to quantify the costs incurred when inspection
technologies are combined for condition assessment. We notice that the cost of using the combina-
tion of "coarse" technologies falls roughly in between the costs of using the "coarse" or the "fine"
technologies independently. This type of analysis can be used together with the costs of adopting
technologies to obtain an effective data collection strategy. For example, it is conceivable that the
cost of adopting the "fine" technology is not justified by the benefits that will accrue from using
it in the management process. In the next section, we further explore the issue of quantifying the
value of combining different technologies for condition assessment.
4.4
Combining Multiple Technologies for Condition Assessment
As discussed earlier, agencies often use multiple technologies to collect distress measurements si-
multaneously. A question that arises is: what is the use of collecting these data? From the previous
27

section, technology choice would seem to be dictated by precision. If this were true, then what
would be the value of collecting additional data (with "coarse" technologies? Here, we show that
combining various technologies can result in benefits even when compared to a single technology
that with high precision. The analysis also provides insights that can be used when adopting in-
spection technologies.
To obtain a measurement-error model for the analysis, we build on the statistical work in
Ben-Akiva and Gopinath (1995). They consider the case of collecting five distress measurements:
roughness (RQI), cracking (CRX), rutting (RDMN), surface patching (SPAT), and raveling (RAV).
The model is presented in Appendix B.
4.4.1
Numerical Results
In our study we considered the same setup as in Section 4.3 (deterioration model, cost functions,
planning horizon, and discount rate). We assumed that the deterioration model described the pro-
gression of roughness. In order to highlight the effect of technology precision we set the variance
in the deterioration process to be 2 to be 0.1.
t
Table 2 presents the average (over 100 instances) of the minimum expected costs for all possible
combinations of the different technologies.
The third column in the table corresponds to the
percentage of the difference between the costs of using a particular combination of technologies
and a case of perfect inspections yielding the true condition of the pavement (average minimum
expected costs $337.4802). This difference is taken relative to the difference in costs that results
when the system is managed while collecting only raveling measurements.
The main observations from the simulation are as follows:
· We can see from Table 2 that the least minimum expected cost occurs when all the five
technologies are combined together. This shows that we do obtain a better performance by
combining different technologies.
· We also notice that the costs are not only dependent on 2 but, also on the 's in the
t
measurement error model. This shows that we not only need precise measurements but also
relevant measurements of the latent condition. In this case, measurements that are highly
related to roughness. For example, the technology used to collect measurements of surface
patching (SPAT) is highly precise when compared to other technologies. However, the value
of 4 is 0.167. Cracking (CRX) is the least accurate of all the technologies but 2 = 1.503
which means that the measurements are closely related to the latent variable that we are
trying to measure. We notice that collecting measurements of cracking is more cost-effective
than collecting measurements of raveling.
28

Technology
Minimum Expected Cost
Percentage of Largest Difference
RQI CRX RDMN SPAT RAV
338.5218
4.89
RQI CRX RDMN RAV
338.5797
5.16
RQI CRX RDMN SPAT
338.5975
5.25
RQI CRX RDMN
338.663
5.56
RQI CRX SPAT RAV
338.9166
6.75
RQI CRX RAV
339.0244
7.26
RQI CRX SPAT
339.045
7.36
RQI RDMN SPAT RAV
339.1679
7.93
RQI CRX
339.1701
7.94
RQI RDMN RAV
339.3124
8.61
RQI RDMN SPAT
339.3278
8.69
CRX RDMN SPAT RAV
339.3692
8.88
RQI RDMN
339.4972
9.48
CRX RDMN RAV
339.5526
9.74
CRX RDMN SPAT
339.5679
9.81
CRX RDMN
339.7869
10.84
RQI SPAT RAV
340.1278
12.45
RQI RAV
340.463
14.02
RQI SPAT
340.4689
14.05
CRX SPAT RAV
340.6569
14.93
RQI
340.8862
16.01
CRX SPAT
341.1373
17.19
CRX RAV
341.1445
17.23
RDMN SPAT RAV
341.6213
19.47
CRX
341.7704
20.17
RDMN SPAT
342.365
22.96
RDMN RAV
342.4338
23.29
RDMN
343.4865
28.23
SPAT RAV
349.6943
57.42
SPAT
356.5109
89.46
RAV
358.7532
100.00
Table 2: Expected Costs for all Technology Combinations
29

5
Summary and Conclusions
We have developed an optimization framework to provide support for investments in preservation
and improvement of transportation infrastructure facilities that are inspected periodically with
sensors or other advanced technologies. This work is motivated by recent developments in remote
sensing and communications technologies that have increased the availability and cost-effectiveness
of using advanced technologies; and by statistical and computational limitations associated with
existing optimization models to support investment decisions. These limitations are related to their
inability to process condition data collected by simultaneously by multiple technologies.
The framework we present involves formulating the underlying decision problem as a discrete-
time, stochastic optimal control problem and consists of two components: a state-estimation prob-
lem that involves processing vast arrays of condition data and using them to develop condition
forecasts; and an optimization problem whose solution yields M&R investment policies. Our ap-
proach differs from the literature in that both elements are fully integrated. This, in turn, leads
to a framework that is both statistically rigourous and computationally efficient, i.e., capable of
providing effective decision-support.
Through four application case studies, we provide numerical examples to illustrate the methodol-
ogy presented above to address the state-estimation problem in the above framework. In particular,
we show how the Kalman Filter processes distress measurements to update the state distribution.
For this part of the study we use both a set of simulated data as well as sensor data. The sensor
data was provided by the MnROAD project. We then show how the the proposed framework can be
used to study the effect of uncertainties in the deterioration process and in the process of collecting
distress measurements on the optimal life-cycle cost of managing infrastructure facilities. We also
illustrate how the framework can be used to quantify the value of combining different technologies
for condition assessment. This is the first study in the area of transportation infrastructure man-
agement to quantify the costs incurred when inspection technologies are combined for condition
assessment.
Finally, we gratefully acknowledge the support for this project. This support enabled Naveen
Tadepalli, a graduate student in the Department of Civil and Environmental Engineering at North-
western University, to complete the requirements for a M.S. degree.
30

A
Cost Parameter Generation
The objective in the numerical study presented in Section 4.3 is to illustrate how the methodology
can be used to obtain optimal M&R policies and to quantify the benefits of using different com-
binations of inspection technologies. To interpret the results of the study as being representative,
we chose to use data "inspired by" studies in the pavement management literature. To estimate
the parameters needed to specify the period cost function g(Xt, At), we used data from the empir-
ical study presented in Madanat and Ben-Akiva (1994) with minor modifications. In that study,
the states used to represent pavement condition are obtained by discretizing the PCI scale into 8
states. From this discretization we constructed a modified roughness scale to be consistent with the
assumption that as the variable used to represent condition, Xt, increases, the condition worsens.
The two scales are shown in Table 3.
State:
PCI Range
Modified Roughness Scale
0
0­20
80­100
1
20­40
60­80
2
40­50
50­60
3
50­60
40­50
4
60­70
30­40
5
70­80
20­30
6
80­90
10­20
7
90­100
0­10
Table 3: Discretization and Transformation of PCI Scale
The agency and user costs used in the aforementioned study are presented in Table 4 and are
a function of the discrete states shown above and four M&R actions. Each entry in the table is
labeled with a corresponding state-action pair in the domain of the period cost function. The "do
nothing" action was assumed to have no effect on facility condition, i.e., At = 0. "Routine main-
tenance" was assumed to prevent the facility from deteriorating, i.e., At = 8. The effects of the
other two actions on improvements (measured as reductions on the modified roughness scale) was
obtained by calculating the expected improvement using the transition probabilities in Madanat
and Ben-Akiva (1994).
To obtain the parameters presented in Table 1 we assumed that g(Xt, At) could be represented
as a second-order polynomial and estimated the parameters using linear regression. The data come
from Table 4.
31

State:
Do Nothing
Routine
2" overlay
Reconstruction
User
Maintenance
Costs
0
(90, 0) 0
(90, 8) 6.9
(90, 51.5) 21.81
(90, 91.5) 25.97
100
1
(70, 0) 0
(70, 8) 2
(70, 41.5) 12.31
(70, 71.5) 25.97
26
2
(55, 0) 0
(55, 8) 1.4
(55, 36.5) 10.69
(55, 56.5) 25.97
22
3
(45, 0) 0
(45, 8) 0.83
(45, 36.5) 9.06
(45, 46.5) 25.97
14
4
(35, 0) 0
(35, 8) 0.65
(35, 36.5) 6.64
(35, 36.5) 25.97
8
5
(25, 0) 0
(25, 8) 0.31
(25, 26.5) 4.11
(25, 26.5) 25.97
4
6
(15, 0) 0
(15, 8) 0.15
(15, 16.5) 3.91
(15, 16.5) 25.97
2
7
(5, 0) 0
(5, 8) 0.04
(5, 6.5) 3.81
(5, 6.5) 25.97
0
Table 4: Agency and User Costs from Madanat and Ben-Akiva (1994) ($/m2)
B
Latent Performance and Measurement-Error Models
The performance and measurement-error models from Ben-Akiva and Gopinath (1995) are pre-
sented below.
AGER
ESAX
CP
X
=
1
+ 2
+ 3
+
(19)
SN C
SN C
SN C





RQI
1X + 1




CRX




2X + 2




RDMN =

(20)


3X + 3




SP AT


4X + 4

RAV
5X + 5
where X is the latent variable representing condition. The condition is specified to be influenced
by the following factors:
AGER: The lapsed time since the last major rehabilitation.
SN C: The pavement structural number.
ESAX: The cumulative equivalent standard axle loads since the last rehabilitation.
CP : The cumulative precipitation since the last rehabilitation.
In the measurement error model, the distress measurements correspond to:
RQI: Roughness (in quarter-car index with units in count/km)
CRX: Cracking (in percentage of surface area)
RDM N : Rut Depth (in mm)
32

SP AT : Patching (in percentage of surface area)
RAV : Raveling (percentage of surface area).
The models were estimated using pavement deterioration data gathered in Brazil between 1975
and 1982. Statistics that describe the data used to estimate the model are presented in Table 5.
Condition Indicator
Mean
Std. Deviation
RQI
40.53
15.36
CRX
15.16
25.70
RDMN
3.71
2.19
SPAT
2.47
8.65
RAV
7.61
19.70
Table 5: Descriptive statistics
The estimated parameters are presented in Table 6.8
Parameter
Value
SMC
1
3.562
2
3.897
3
1.654
1
1.000
0.37
2
1.503
0.35
3
0.167
0.45
4
0.256
0.09
5
0.531
0.04
Table 6: Parameter Estimation Results
The 's describe how the different technologies complement each other. In order to compute
the expected costs associated with different combinations of technologies it is also necessary to
obtain the precisions associated with each of the technologies. The next subsection describes how
we estimated the precisions associated with the technologies.
B.1
Estimation of Precisions Associated with Technologies: A Note on Linear
Regression
A regression equation is of the form:
Y = mX +
(21)
8SMC for structural model = 0.62
33

where Y is the observed variable, X is a vector of explanatory variables, and
is a random error
term (assumed to have zero mean and finite variance). m is the set of parameters that are estimated.
Linear regression models try to explain the variance of Y using the variance of X and the
variance of the error term. The variance of Y is termed total variance (SST), and the variance of
mX is called the explained variance (SSR) and the error variance is called the unexplained variance
(SSE). The relationship between these is as follows:
SST = SSR + SSE
(22)
The goodness-of-fit for a regression equation is measured using a statistic called R2, defined as
the proportion of the variance of Y explained by X. Mathematically, this can be represented as:
SSR
R2 =
= 1 - SSE
(23)
SST
SST
Ben-Akiva and Gopinath (1995) present the Squared Multiple Correlation (SMC) for each of
the the measurement error model. SMC is analogous to R2. From the definition of R2, we have
that SSE = SST (1 - SM C). SSE is nothing but the variance of error term in the regression
model. We use SST and SM C to estimate the precisions of each of the technologies. The results
are presented in Table 7.
Technology
Variance
Roughness
148.6356
Cracking
429.3185
Rut Depth
7.570225
Surface Patching
68.088
Raveling
372.5664
Table 7: Precision Estimates
34

References
Bellman, R. E. (1955). "Equipment replacement policy." Journal of the Society of Industrial and
Applied Mathematics, 8(3), 133­146.
Ben-Akiva, M. and Gopinath, D. (1995). "Modeling infrastructure performance and user costs."
ASCE Journal of Infrastructure Systems, 1(1), 33­43.
Ben-Akiva, M., Humplick, F., Madanat, S., and Ramaswamy, R. (1991). Infrastructure Manage-
ment under Uncertainty: The Latent Performance Approach. Department of Civil Engineering,
Massachussetts Institute of Technology.
Ben-Akiva, M. and Ramaswamy, R. (1993). "An approach for predicting latent infrastructure
facility deterioration." Transportation Science, 27(2), 134­153.
Bertsekas, D. (1995). Dynamic Programming and Optimal Control. Athena Scientific.
Carnahan, J. (1988). "Analytical framework for optimizing pavement maintenance." Journal of
Transportation Engineering, 114(3), 307­318.
Carnahan, J., Davis, W., Shahin, M., Keane, P., and Wu, M. (1987). "Optimal maintenance
decisions for pavement management." Journal of Transportation Engineering, 113(5), 554­573.
Dreyfus, S. (1960). "A generalized equipment replacement study." Journal of the Society of Indus-
trial and Applied Mathematics, 8(3), 425­435.
Dreyfus, S. (1977). The Art and Theory of Dynamic Programming. Academic Press.
Durango, P. (2002). "Adaptive optimization models for infrastructure management," PhD thesis,
University of California, Berkeley.
FHWA (1979). Federal Highway Administration ­ Recording and Coding Guide for Structure In-
ventory and Appraisal of the Nations Bridges. US Department of Transportation. Washington,
DC.
FHWA (1999). Asset Management Primer. FHWA's Office of Asset Management.
Friesz, T. and Fernandez, E. J. (1979). "A model of optimal transport maintenance with demand
responsiveness." Transportation Research Part B, 13(4), 317­339.
Gendreau, M. and Soriano, P. (1998). "Airport pavement management systems: An appraisal of
existing methodologies." Transportation Research Part A, 32(3), 197­214.
Golabi, K., Kulkarni, R., and Way, G. (1982).
"A statewide pavement management system."
Interfaces, 12(6), 5­21.
Golabi, K. and Shepard, R. (1997). "Pontis: A system for maintenance optimization and improve-
ment of u.s. bridge networks." Interfaces, 27(1), 71­78.
35

Guillaumot, V., Durango-Cohen, P., and Madanat, S. (2003). "Adaptive optimization of infras-
tructure maintenance and inspection decisions under performance model uncertainty." ASCE
Journal of Infrastructure Systems, 9(4), 133­139.
HRB, H. R. B. (1962). The AASHO Road Test, Report 5: Pavement Research. Special Report No.
61E.
Hulten, C. (1996). "Infrastructure capital and economic growth: How well you use it may be more
important than how much you have. Mimeo, University of Maryland.
Humplick, F. (1992). "Highway pavement distress evaluation: Modeling measurement error." Trans-
portation Research Part B, 26, 135­154.
Jaselskis,
J.
E.,
Thomas,
C.
E.,
Stephen,
A.
J.,
and
Zhili,
G.
(2003).
"Pi-
lot study on improving the efficiency of transportation projects using laser scanning.
<http://www.ncgia.uscb.edu/ncrst/research.html >.
Kane, A. (2000). "Why asset management is more critically important than ever before?." Public
Roads, 63(5), 22­24.
Madanat, S. and Ben-Akiva, M. (1994). "Optimal inspection and repair policies for infrastructure
facilities." Transportation Science, 28(1), 55­61.
Murakami, K. and Turnquist, M. (1985). "A dynamic model for scheduling maintenance of trans-
portation facilities." Transportation Research Record, (1030), 8­14.
Prine,
D.
and
Fish,
P.
(Dec.
5,
2003).
"Remote
global
monitoring
of
the
michigan
street
bridge,
sturgeon
bay,
wisconsin.
ITI
Technical
Report,
<http://iti.northwestern.edu/publications/technical reports/tr19.html >.
Shahin, M. and Kohn, S. (1981). "Pavement maintenance management for roads and parking lots.
Technical Report M-29, US Army Corps of Engineers.
Small, K., Winston, C., and Evans, C. (1989). Road Work: A new highway pricing and investment
policy. The Brookings Institution.
Smilowitz, K. and Madanat, S. (2000). "Optimal inspection, maintenance and rehabilitation policies
for networks of infrastructure facilities under measurement and forecasting uncertainty." Journal
of Computer-Aided Civil and Infrastructure Engineering, 15(1), 5­13.
Terborgh, G. (1949). Dynamic Equipment Replacement Policy. McGraw-Hill.
Washer, G. (2000). "Developing non-destructive evaluation technologies for infrastructure assess-
ment." Public Roads, 63(4), 44­50.
36