November/December 2000
Using
Monte Carlo Simulation for Pavement Cost Analysis
by:
Keith D. Herbold
Life-cycle
cost-analysis (LCCA) models that are currently used by state highway
agencies treat input variables as discrete, fixed values. In actuality,
many input variables are not fixed; they are uncertain.
In
conducting an LCCA, it is important to recognize the uncertainty of
input variables and the uncertainty that this variability creates in
the results. Uncertainty comes from the assumptions, estimates, and projections that are made for the input parameters, such as the initial
and future agency and user costs and the timing of rehabilitation activities.
The
Federal Highway Administration's Demonstration Project No. 115 (DP-115),
Life-Cycle Cost Analysis in Pavement Design, is a technology-transfer
effort to demonstrate the application of LCCA to pavement design. DP-115
was initiated in fall 1996 and included an introduction to the use of
a probabilistic approach -- sometimes called risk analysis -- in the
treatment of uncertain LCCA data inputs.
To
better understand the practical aspects of the use of probability and
Monte Carlo simulation in LCCA, the Federal Highway Administration (FHWA)
developed a model and made arrangements with 10 state highway agencies,
the American Concrete Pavement Association, and the National Asphalt
Pavement Association to prepare case studies illustrating the application
of risk analysis to LCCA in pavement design. Case study participants
were requested to perform an LCCA on an existing project using their
current deterministic procedures and also using the procedures outlined
in FHWA's Life-Cycle Cost Analysis in Pavement Design, a technical bulletin.1
This
article summarizes the results of the case studies, which show that
with limited training in probabilistic principles and in the application
of risk-analysis software, state highway agency personnel can apply
the probabilistic approach to their current life-cycle cost-analysis
procedures. Then they can answer three basic questions: What can happen?
How likely is it to happen? What are the consequences if it happens?
By exposing these areas of uncertainty typically hidden in the traditional
deterministic approach to LCCA, the analyst can advise the decision-maker
about the risks associated with various courses of action and the probability
of an outcome actually occurring. By applying this information, the
decision-maker can select the best, most cost-effective solutions to
provide the greatest long-term benefits.
Current
Procedures
The traditional approach to addressing uncertainty in LCCA has been
to ignore it. The analyst makes a series of "best guesses" for the values
of each of the input variables and then computes a single deterministic
result. The problem with this approach is that information that could
improve the decision is often excluded.
In
some cases, a limited sensitivity analysis may be conducted in which
various combinations of inputs are selected to qualitatively assess
their effect on analysis results. However, even with a sensitivity analysis,
this deterministic approach to LCCA can conceal areas of uncertainty
that may be a critical part of the decision-making process. Often, stakeholders
seize upon the uncertainty associated with LCCA inputs and vigorously
debate the validity of the results. Traditional, deterministic LCCA
results can generate endless debate over which alternative really has
the lowest life-cycle cost. This process encourages division and unproductive
debate.
The
need to make strategic long-term investment decisions under short-term
budget constraints forces state highway agencies to incorporate risk
(either implicitly or explicitly) as a criterion in judging courses
of action. Also, decision-makers need an analysis tool that exposes
areas of uncertainty of which they may not be aware. Based on this new
information, the decision-maker then has the opportunity to take mitigating
action to decrease exposure to risk.
Risk
analysis is a technique that exposes areas of uncertainty that are typically
hidden in the traditional deterministic approach to LCCA, and it allows
the decision-maker to weigh the probability of any particular outcome.
Risk analysis combines probability descriptions of uncertain variables
and computer simulation to characterize risk associated with the outcome.
By
including all possible inputs into the analysis and weighing the probability
of occurrence of each, risk analysis elevates the debate from the validity
of LCCA results to deciding best public policy. With the emergence of
user-friendly computer software, quantitative risk-analysis concepts
and techniques can be easily integrated into the decision-making process
of a state highway agency.
In
October 1997, prior to developing a case study, each participant received
risk-analysis training using a software package called "@Risk." @Risk,
an add-on to Microsoft Excel, provides an efficient means to incorporate
simulation capability into a spreadsheet.
Without
the aid of simulation, a spreadsheet model will only reveal a single
outcome, generally the most likely or average scenario. Spreadsheet
risk analysis uses both a spreadsheet model and simulation to automatically
analyze the effect of varying input on the output of the modeled system.2
Table
1 summarizes the economic inputs used by each study participant. As
shown, all state participants currently use net present value (NPV)
as the economic indicator of choice, long periods of analysis, and a
discount rate of around 4 percent. All participants currently use a
deterministic procedure. It should be pointed out that Ohio incorporates
a sensitivity analysis of discount rates into the analysis.
Table
1 - LCCA Input Variables Used by Participants
|
Study
Participant |
Analysis
Method |
Analysis
Period (years) |
Discount
Rate (%) |
Kentucky |
NPV |
35
- 40 |
2
- 10 |
Nevada |
NPV |
4 |
|
North
Carolina |
NPV |
30 |
4 |
Ohio |
NPV |
35 |
Sensitivity
analysis 0 to 6 |
Pennsylvania |
NPV |
20
- 40 |
6 |
Texas |
NPV |
Not
applicable |
|
Washington |
NPV |
40 |
4 |
Wisconsin |
NPV |
50 |
5 |
Table
2 shows that all participating states address costs by using uninflated
(real) costs.
Table
2 - Treatment of Agency Costs
|
State
|
Agency
Cost Treatment
|
Kentucky |
Real
costs for both initial construction and future rehabilitation activities
are based on average unit bid prices. Traffic control costs are
estimated based on projects of similar scope. Salvage value is determined
by calculating the total quantity of materials (both original construction
and rehabilitation) in place on the roadway and giving them the
value of in-place dense graded aggregate. |
Nevada |
Includes
real costs for functional and structural overlays and traffic control
based on the tabulation of bids received from contractors for similar
projects. Maintenance costs are estimated at $1000 per year per
directional mile for flexible pavements. Salvage value is defined
as the last rehabilitation cost (excluding surface seals and annual
maintenance) multiplied by the ratio of the last rehabilitation's
remaining service to total life. |
North
Carolina |
Real
costs for initial and rehabilitation construction. Annual maintenance
costs, traffic control costs, and salvage value are typically not
available and, therefore, not included in the analysis. These costs
are included if they are available. |
Ohio |
Real
costs for both initial construction, future rehabilitation, and
contract maintenance. Routine maintenance by Ohio DOT forces is
ignored. Salvage value is not used. Designer tries to balance the
maintenance strategies such that the alternatives have approximately
the same condition in year 35. |
Pennsylvania |
Real
costs for construction expenditures over the life of the project,
including traffic control and maintenance. Annual maintenance costs
are estimated at $825 per lane-mile for rigid pavement and $1,825
per lane-mile for flexible pavement. Salvage value is not included
since it is assumed that the pavement will need reconstruction at
the end of the analysis period. |
Texas |
Real
costs for initial construction. |
Washington |
Real
costs based on the most recent bid item costs and quantity for initial
and future rehabilitation. Maintenance costs are not included because
these costs are low and their differences are negligible between
alternatives. Salvage value determined by multiplying the last rehab
cost by the ratio of its remaining life to its expected life. |
Wisconsin |
Real
costs for construction, future rehabilitation, and maintenance.
|
User
costs are defined as costs incurred by users of a highway facility,
including excess costs to those who cannot use the facility because
of agency or self-imposed detour requirements. User costs are a combination
of vehicle-operating, delay, and crash costs. Most of the participants
do not directly calculate user costs; however, these costs can be considered
indirectly. For example, Ohio uses the number of lane-closure days to
indirectly measure the costs to the user. A summary of the states' current
procedures for determining user costs is shown in table 3.
Table
3 - Current Procedures for User Costs
|
State |
User Cost Procedure
|
Kentucky
|
A
fixed user cost of $5,000/day is used. This is multiplied by the
number of days required to complete the work to get the total cost.
Typically, 120 days is assumed for initial construction and 30 days
for each rehabilitation. |
Nevada |
Not
calculated. |
North
Carolina |
Not
calculated. |
Ohio |
User
costs are not calculated. An alternative is to determine the number
of lane closure days. |
Pennsylvania |
Reduced
speed delay traversing work zone. Analysis does not include user
delay costs due to queuing. Use stopping, idling, and added time
according to NCHRP 133 (adjusted for inflation). Value of time $3
per hour for passenger cars and $5 per hour for all trucks. |
Texas
|
Not
calculated. While user delay is not generally considered on a dollar
basis, negative user impact is often a major factor for phasing
and material choices. |
Washington
|
Reduced
speed delay traversing work zone. Average value of time is $6.25
for all vehicle types. |
Wisconsin |
Not
calculated. |
Finally,
the values reported by each state for the performance lives of alternatives
are summarized in table 4.
Table
4 - Treatment Pavement Performance Lives by State
|
State |
Treatment
Performance Lives |
|
State |
Treatment
Performance Lives |
Kentucky |
Flexible:
Year 10 - mill 37.5 mm, overlay 37.5 mm Year 20 - mill 37.5 mm,
overlay 100 mm Year 30 - mill 37.5 mm, overlay 37.5 mm Rigid: Year
15 - clean and reseal joints Year 30 - clean and reseal joints |
|
Ohio
(cont'd) |
Composite:
Full-depth rigid repairs, milling, and an overlay every 8-12
years Unbonded concrete overlay: Maintenance similar to that
of a rigid pavement Fractured slab techniques: Year 8-12 -
thin overlay, 32 mm to 80 mm, with or without milling Year
16-22 - thick overlay, 100 mm to 200 mm, with milling, pavement
repair Year 24-32 - thin overlay, 32 mm to 80 mm, with or
without milling, microsurfacing, crack sealing Whitetopping:
Maintenance similar to that of a rigid pavement |
Pennsylvania |
Flexible:
8 years to first resurfacing 8-year interval between resurfacing
Rigid: 30 years to first resurfacing 8-year interval between
resurfacing |
Texas |
Expert
opinion |
Washington |
Flexible:
8-15 years to first resurfacing 8- to 15-year interval between
rehabilitation Rigid: 20 years to first rehabilitation |
Wisconsin |
Combination
of pavement management data and expert opinion |
|
Nevada |
Expert
opinion, pavement performance analysis program |
|
North
Carolina |
Flexible:
Year 10 - mill and replace Year 20 - mill, replace, and overlay
Rigid: Year 10 - saw and reseal joints Year 20 - saw and reseal
joints |
|
Ohio |
Flexible:
Year 10-15 - thin overlay, 32 mm to 75 mm, with or without milling
Year 18-25 - thick overlay, 75 mm to 180 mm, with milling Year 28-32
- thin overlay or microsurfacing or crack sealing Rigid: Year 18-25
- 2 percent to 10 percent full-depth rigid repairs, 1 percent to
5 percent partial-depth bonded repairs, diamond grinding, 75-mm
to 150-mm overlay, sawing, and sealing Year 28-32 - 1 percent to
3 percent full- and/or partial-depth repair, thin overlay, 32 mm
to 50 mm, with or without milling, sawing and sealing, microsurfacing,
crack sealing, diamond grinding |
|
Probabilistic
Approach
Each participant was requested to perform the analysis using the procedures
explained in FHWA's technical bulletin. As noted earlier, NPV is the
economic indicator of choice, and the basic NPV formula for discounting
discrete future amounts at various points in time back to some base
year is shown as follows:
where: i = discount rate and
n = number of years into future
N= number of rehabilitations
The
component of the above formula is referred to as the present value (PV)
factor for a single future amount. PV factors for various combinations
of discount rates and future years are available in discount factor
tables, which are more commonly referred to as interest rate tables.
The present value for a particular future amount is obtained by multiplying
the future amount by the appropriate PV factor.
In
the probabilistic approach, uncertain variables (for example, initial
cost, future rehabilitation cost, discount rate, and year of rehabilitation)
are modeled using probability distributions. Random sampling is then
used to compute NPV, and through many iterations, a probability distribution
of results is obtained. The results generated from each iteration are
captured for later statistical analysis.
Computers
can readily perform these calculations thousands of times in a few seconds,
and the most common approach is called Monte Carlo simulation. Monte
Carlo simulation is named after the casino in Monte Carlo, Monaco, where
games of chance exhibiting random behavior are played. The way that
Monte Carlo simulation selects variable values at random to simulate
a model is similar to the casino's games of chance that have a known
range of values but an uncertain value for any particular time or event.2
Monte
Carlo simulation is easy to use. Determining the uncertainty of the
results when combining several uncertain values can be very complex.
However, using Monte Carlo simulation, this and similar effects are
handled automatically so you do not have to know much about statistics
to get accurate results.3
The
features of each state's risk-analysis model are summarized in table
5. Many areas of commonality are evident in how the participating states
constructed their probabilistic models. For example:
- All of them used multiple sheets within their spreadsheet to construct
the model (rather than placing everything in one very large table).
- Only one used drop-down menus and navigation buttons in construction
of the model on the spreadsheet.
- All but one included a probability distribution for the performance
life of the initial construction.
- The performance life of future rehabilitation activities was modeled
as a probability distribution by all.
- All included different types of rehabilitation activities.
- All derived agency costs from unit quantities.
Many
other features were commonly used by all because they had been directed
to follow the recommendations of the technical bulletin.
On
the other hand, differences were found in some key areas:
- Some used the software's capability to provide explanatory cell comments
on their spreadsheet to document their model while others did not.
- Some set aside an area of the spreadsheet to contain all user inputs,
but others had input cells included in various parts of the spreadsheet.
- Different approaches were used to input the hours of operation for
work zones.
- The modelers were equally divided on whether the discount rate should
be fixed or have a probability distribution.
- The values of time and agency costs were similarly modeled by some
as a fixed value and by others as a probability distribution.
- Finally, some considered salvage value for user costs, but others
did not. This can have a considerable influence, especially for alternatives
that incur a rehabilitation near the end of the analysis period.
Conclusions
This study has shown that the state highway agencies can readily use the
latest in analytical tools to assess the risk of the decisions they
make. They can expose areas of uncertainty typically hidden in the traditional
deterministic approach to LCCA, advise the decision-maker of the risks
associated with various courses of action and the probability of an
outcome occurring, and thereby aid the selection of cost-effective solutions
that provide the greatest long-term benefits.
References
-
Life-Cycle
Cost Analysis in Pavement Design, Publication No. FHWA-SA-98-079,
Federal Highway Administration, Washington, D.C., 1998.
- "What
is Monte Carlo Simulation?" Decisioneering Web site, www.decisioneering.com/monte-carlo-simulation.html.
- "Simulation
Basics," Vanguard Software Corp. Web site, www.vanguardsw.com/Dphelp/dph00118.htm.
Keith
D. Herbold is a pavement engineer in FHWA's Midwestern Resource
Center in Olympia Fields, Ill. He has managed the Midwestern pavement
program since 1985. In addition to his service to state highway agencies,
he works on a variety of pavement-related expert task groups and panels,
serves as secretary of the Maintenance Management Task Force of the
American Association of State Highway and Transportation Officials (AASHTO)
Subcommittee on Maintenance, is a member of AASHTO's Lead State Team
on Pavement Preservation, and interacts with several Transportation
Research Board committees. He is very active as a trainer and has presented
courses throughout the United States and overseas. Herbold joined FHWA
in 1968, and his career has included geotechnical, materials, and pavement
engineering assignments in both the Washington, D.C., headquarters and
the former Region 5. He has a bachelor's degree in civil engineering
from North Dakota State University and a master's degree in civil engineering
from the University of Kentucky. Herbold is a member of American Society
of Civil Engineers, and he is a registered professional engineer in
Illinois.
Other Articles in this Issue:
Using Monte Carlo Simulation for Pavement Cost Analysis
ITS Peer-to-Peer Program
Design Evaluation and Model of Attention Demand (DEMAnD): A Tool for In-Vehicle Information System Designers
Studying the Reliability of Bridge Inspection
Ultrasonic Inspection of Bridge Hanger Pins
The Northwest Transportation Technology Exposition
Faster, Easier, Cheaper - Pyrotechnical Anchoring
Practical Research Answers Real-Life Questions
A Nondestructive Impulse Radar Tomography Imaging System for Timber Structures
Strategic Work-Zone Analysis Tools