Using Monte Carlo Simulation for Pavement Cost Analysis
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.
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|
|Ohio||NPV||35||Sensitivity analysis 0 to 6|
|Pennsylvania||NPV||20 - 40||6|
Table 2 shows that all participating states address costs by using uninflated (real) costs.
Table 2 - Treatment of Agency Costs
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|
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.|
|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.|
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||
|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|
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.
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.
- 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.