The Simulation and Outputs section of the Switchboard is where deterministic life-cycle costs and simulations of probabilistic life-cycle costs are performed. Deterministic analysis is done using either deterministic inputs or most likely values for probabilistic inputs; in either case, point values are used in a deterministic analysis.
Opening the Deterministic Results form, shown in Figure 21, calculates deterministic present values for both agency and user costs and displays those values. The lowest cost alternatives for both agency and user are labeled. The form also provides a direct link to the Deterministic Results Excel worksheet, which contains all of the information required to investigate deterministic results.
Figure 21. Deterministic Results form.
Running a simulation is a necessary step toward performing a probabilistic analysis. To conduct probabilistic analysis, RealCost uses Monte Carlo simulation, which allows modeling of uncertain quantities in the model with probabilistic inputs. The simulation procedure randomly samples these inputs and produces outputs that are described by both a range of potential values and a likelihood of occurrence of specific outputs. The simulation produces the probabilistic outputs; without running a simulation, probabilistic outputs are not available. The Simulation form is shown in Figure 22.
The Sampling Scheme section of the form determines from where the software will draw its simulation numbers. Choosing Random Results causes the simulation seed value (where the simulation starts) to come from the computer's internal clock. While not truly random, this seed value cannot be influenced by the software user, and it produces different values with each simulation.
Figure 22. Simulation form.
The value of choosing Reproducible Results is that the analyst may perform separate simulation runs to compare more than two alternatives. "Reproducible Results" causes the same simulation values for each given seed value. Using reproducible results causes the same random numbers to be generated. This removes random variability associated with the seed value from the comparison, allowing the analyst to focus on actual input changes.
The Sampling Scheme section of the form determines how the software will draw its simulation numbers from the pseudo-random number generator (these numbers are not truly random but are satisfactory for simulation purposes). Choosing Random Results causes the simulation to start the random number sequence from a "seed" value taken from the computer's internal clock. This seed value cannot be influenced by the software user and produces different values with each simulation.
The Reproducible Results option allows the analyst to specify the specific seed value to be used in all simulations. This causes the same set of random numbers to be generated from the pseudo-random number generator. Choosing Reproducible Results allows the analyst to perform separate simulation runs to compare multiple alternatives, knowing that variations from run to run will be caused by actual input changes and not variability associated with different seed values.
Tail Analysis Percentiles are used to conduct analysis on the total cost probability distribution graphics provided by RealCost, discussed on page 44. Percentile values should be entered in ascending order.
The Iteration section is used to determine the number of iterations to be performed and whether the simulation will be monitored for convergence. Output convergence can be used by the analyst to determine that a simulation has run a sufficient number of iterations to properly define its outputs. Convergence is monitored by comparing the change in the mean and standard deviation of the cumulative outputs each time a specified number of iterations is completed (specified in the Monitoring Frequency box). Once the level of change falls below the specified Convergence Tolerance, RealCost will end the simulation run without completing any remaining iterations-yielding probabilistic results while significantly shortening the time it takes to complete the analysis. The number of iterations should be 2,000 at a minimum. Monitoring Frequency is adequate at 100 iterations, and, when used, a Convergence Tolerance of 2.5 (percent) should provide appropriate probabilistic outputs.
Figure 23 shows a simulation that ended due to simulation convergence of less than 2.5 percent. Note that the convergence error is listed at the bottom of the form. This convergence error is monitored and reported during the simulation.
Figure 23. Simulation form at the conclusion of a simulation run.
After a simulation run, probabilistic results are available for analysis. A simulation must be run prior to viewing probabilistic results. Figure 24 shows the results of a probabilistic simulation.
Figure 24. Probabilistic Results form.
Four worksheets are accessible from the Probabilistic Results form. The Probabilistic Results worksheet and the Output Distributions worksheet both provide probability distribution and cumulative density functions that describe outputs. Examples of these graphs are shown in Figure 25.
Figure 25. Probabilistic distribution density and cumulative density functions describing outputs.
The Tornado Graphs worksheet provides tornado graphs that describe how inputs affect outputs (example shown in Figure 26). For example, the input Initial Construction Cost has a significant effect on the output Alternative 1: Agency Costs. A correlation coefficient value of 1 would indicate a complete positive correlation between two variables. A value of -1 would indicate a complete inverse correlation between two variables. The value of 0 would indicate that there is no correlation between variables: they are independent. Other correlation values indicate a partial correlation; the output is affected by changes in the selected input, but may be affected by other variables as well.
Figure 26. Correlation coefficient graph (aka "tornado graph").
While the correlation coefficient graphics describe the sensitivity of outputs to individual inputs, the total cost probability distribution graphics provided by RealCost describe the sensitivity of outputs to combinations of inputs. Particular emphasis is given to the tails of the distribution, which encompass the most extreme outcomes encountered in the analysis. The analyst may enter four Tail Analysis Percentiles (see Figure 22, Simulation form) to define the areas of the tails of most interest. RealCost demonstrates how various inputs act together to produce these four defined tail areas.
For example, Figure 27 shows an agency project alternative total cost distribution. The shaded area on the left side of the distribution curve represents a 10 percent tail of the area under the curve (the 10th percentile, or the most favorable 10 percent of all agency cost outputs). Table 8 describes the combination of input values that would lead to that outcome. The values under the "10%" heading represent the number of standard deviations from the mean value for each of the inputs needed to fall within the 10 percent tail.
Figure 27. Tail analysis outputs.
|Input Variable Name||Alternative 1: Agency Cost|
|Initial Construction Cost||-1.76||-1.44||1.36||1.41|
|Initial Construction Life||0.65||0.57||-0.54||-0.81|
|Rehab 1 Life||0.86||0.54||-0.37||-0.46|
The value of tail analysis is that it identifies those inputs that contribute to the success, or failure, of a project alternative. Pavement design decisionmakers are able to quantitatively identify alternatives that they believe they are able to positively influence and also those alternatives that they are not able to influence. More discussion on interpreting probabilistic outputs is given in FHWA's LCCA Technical Bulletin, Life-Cycle Cost Analysis in Pavement Design (FHWA-SA-98-079).