Estimating Life Expectancies of Highway Assets, Volume 1: Guidebook (2012)

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124 Analytical models such as those used for life expectancy analysis can be characterized as “garbage-in/garbage-out,” in that the credibility of the results can be highly dependent on the quality of the inputs. When predicting asset life expectancy, various uncertainties exist (Lin 1995): • Inherent randomness of structural characteristics (e.g., material properties, section dimen- sions, loads); • Inherent randomness of external effects (e.g., environmental conditions, extreme events); • Maintenance uncertainties (e.g., effectiveness, frequency); • Statistical uncertainty (e.g., incomplete or errant data from inspections, or errors in estimat- ing parameters of probability models); and • Model imperfection (e.g., error created through idealized mathematical modeling attempting to describe complex physical phenomena). Therefore, the prediction of life expectancy is uncertain. The credibility of the results is very important if the investment in the models is to pay off. So it is important to test the models systematically for weaknesses, in a way that sets priorities for improvement. Sensitivity analysis is a good tool to do this. Through sensitivity analysis, agencies can identify the inputs with the most influence on the life expectancy estimate, quantify the range in potential asset life caused by the uncertain input, and assess the life extension or contraction caused by a unit change in the input. If the effect of an input is considered unreasonable, then the model may require improvements. Alternatively, if the effect of an input is considered reasonable, then data collection efforts may be focused on trying to reduce that uncertainty or contingency funds may be set aside. Furthermore, this discussion of uncertainty can be taken a step further with the recognition that some planning decisions may be inherently linked to asset life. As a result, there is a risk that less-than-optimal planning decisions may be made as a result of uncertain life expectancy factors and life estimates. Therefore, risk analysis techniques may be appropriate. Agencies applying risk analysis can make more informed decisions through the probabilistic description of potential asset life and other planning factors such as lifecycle costs and project utility. Unlike sensitivity analysis, risk analysis allows for quantification of the likelihood of vari- ous outcomes, upon which agencies can apply risk management techniques to protect against uncertainty. A further description of sensitivity and risk analysis techniques, as well as examples, is provided in the following sections. C h a p t e r 6 Accounting for Uncertainty: How to Improve Life Expectancy Models

accounting for Uncertainty: how to Improve Life expectancy Models 125 6.1 Sensitivity Analysis of Life Expectancy Models Sensitivity analysis is a simple method of assessing uncertainty that quantifies how outputs may change when input values are systematically varied on a unit-by-unit basis. In doing so, it is possible to • Identify the most critical factor driving the output (i.e., the factor that leads to the most wide- spread range in output values or the largest change in outputs on a unit basis); • Assess weaknesses in the model (i.e., if the range of outputs produced by a particular input is unreasonable, then the model may require revision); • Focus data collection (i.e., in order to reduce the uncertainty of an input within control of the agency, additional data collection may be needed); • Justify contingency plans (i.e., to reduce uncertainty of an input outside the control of the agency (e.g., climate conditions), contingency plans to deal with potential outputs may be needed); and • Set priorities for improvements (i.e., if an input produces more (or less) favorable outputs, then attempts can be made to maximize (or minimize) the input in future cases). The most common presentations of sensitivity analysis results are through the use of tornado (Figure 6-1), spider (Figure 6-2), and elasticity diagrams which describe how the output changes when each input is varied from its minimum to maximum values while holding all others at their average values. A tornado diagram presents the range of outputs produced by each input in a descending order of influence. A spider diagram graphically portrays the influence of each input, where the largest magnitude slope is the most influential and the sign of the slope indicates a positive or negative effect on the output. An elasticity diagram is similar to a spider diagram, except that the percent change in output is assessed against a percent change in input for different points in time. Additionally, the influence of a unit change from the current input value is often assessed as a function of the parameter estimate or coefficient. Linear regression models have the simplest sensitivity interpretation. In these models, the coefficient directly indicates how much the output changes for every unit change in an input. For example, in the following life expectancy model, an asset life extension of 2 years is predicted for every unit change in y. Asset life x y= − +35 3 2 Input 1 Input 2 Input n Range of Outputs . . . . . . . . Increase in Input leads to a Decrease in Output Increase in Input leads to an Increase in Output Increasing Influence on Output Figure 6-1. Example of a tornado diagram (FHWA 2006).

126 estimating Life expectancies of highway assets When dealing with transformed variables, the coefficient will have to be transformed back. For instance, in the following model, an asset life extension of just over 7 years (=exp(2)) is obtained for every unit change in y. Natural Log of Asset life x y= − +35 3 2 For non-parametric models, sensitivity analysis can still be performed by comparing different groupings of data. For instance, if Markov chains are used to analyze bridge life, the life estimate of bridges with one level of maintenance can be compared against the life estimate with a higher level of maintenance. Although conceptually the same, various terms are applied to the description of a factor’s sensitivity. For instance, in survival models, this unit change is often termed an acceleration parameter. These parameters represent the stretching or contracting of the survival curve for every unit change in one of the inputs. In ordered probit models, unit changes are often termed marginal effects. These effects refer to the change in probability of being in one state given a unit change in an input. A direct comparison of coefficients does not always indicate which input has the great- est influence on the output. For a fair comparison of the influence of each input, the relative parameter strength can be used. The coefficients can be normalized by dividing each input by its average unit value, which results in a unitless comparison of the influence of each factor. To demonstrate how to interpret the results, the researchers conducted a sensitivity analysis of the pipe culvert life expectancy model in Section 4.1.1.5. If one input at a time is varied from its minimum to maximum values (Table 6-1) while holding all others at their average values, the asset predictions in Table 6-2 are obtained. The resulting tornado diagram visualizing the ranges in estimates in Figure 6-3 is then produced. As is apparent in the tornado diagram and in tabular form, the most influential factors for this life expectancy model are the climate conditions. For this analysis, the range of factors was set based on the minimum, average, and maximum values for the entire collected pipe culvert database. However, when assessing the sensitivity of life at a single location, far more certainty may be incorporated into the assessment. Percent Change in Input Value Output Value Increasing influence an input has on Increasing the output Increasing influence an input has on Decreasing the output Figure 6-2. Example of a spider diagram (van Dorp 2009).

accounting for Uncertainty: how to Improve Life expectancy Models 127 Additionally, for factors within the asset manager’s control, this particular model sug- gests that using metal culverts can add 13 years to the asset’s life, replacing corrosive soils may extend its life 9 years, coating an asset may extend its life 6 years, and using ditch inlets/ outlets to filter contaminants may extend its life 4 years. For every additional unit of precipi- tation from the average, asset life is predicted to decline by 6.2 years {67 * [exp(-.097)-1]}. Similarly, asset life is predicted to increase by 6.8 years for every unit change in temperature from the average and decrease by -0.6 years for every change in freeze/thaw cycles from the average. Life expectancy factor Minimum value Average value Maximum value Metal material type indicator (1 if metal, 0 otherwise) 0 1 1 Average annual freeze/thaw cycles 95 130 150 Soil corrosiveness potential (1 if high, 0 otherwise) 0 0 1 Ditch inlet/outlet indicator (1 if ditch inlet/outlet, 0 otherwise) 0 1 1 Coating application indicator (1 if coated, 0 otherwise) 0 1 1 Average annual temperature in °F 45 49 53 Average annual precipitation in inches 38 43 47 Table 6-1. Range of values for example sensitivity analysis. Life expectancy factor Asset life at minimum values Asset life at maximum value Range Metal material type indicator (1 if metal, 0 otherwise) 54 67 +13 Average annual freeze/thaw cycles 93 56 -37 Soil corrosiveness potential (1 if high, 0 otherwise) 67 58 -9 Ditch inlet/outlet indicator (1 if ditch inlet/outlet, 0 otherwise) 63 67 +4 Coating application indicator (1 if coated, 0 otherwise) 61 67 +6 Average annual temperature in °F 46 99 +53 Average annual precipitation in inches 109 46 -63 Asset life at Average Values 67 Table 6-2. Range of asset life estimates for example sensitivity analysis. -60 -20-40 0 20 40 60 Change in Service Life (years) Li fe E xp ec ta nc y Fa ct or Precipitation Temperature F-T Cycles Metal Material Soil Corrosiveness Coating Application Ditch Inlet/Outlet Figure 6-3. Tornado diagram of example sensitivity analysis.

128 estimating Life expectancies of highway assets 6.2 Risk Analysis of Life Expectancy Models A more in-depth assessment of uncertainty in life expectancy estimates can be done by way of risk analysis. Risk is defined as an uncertain outcome with an inherent likelihood and conse- quence (typically, an undesirable consequence). Due to the uncertainties associated with asset life expectancy and life expectancy factors, agencies stand at a risk of making less-than-optimal planning decisions. Examples, provided in this guidebook and the accompanying report, include an assessment of the uncertainty of future asset life due to uncertain climate and the uncertainty of over/underestimating of long-term planning needs due to uncertain asset life. Risk analysis can be incorporated into asset management through four steps (Ford 2009): • Risk Identification—describe the consequences and the conditions that may influence the likelihood of the risk (e.g., risk of scheduling asset replacement project before the full asset life is reached, leading to increased lifecycle costs caused by uncertain life expectancy estimates or factors); • Risk Assessment—quantify the consequences and likelihood of the risk (e.g., consequence = increase in lifecycle cost; and likelihood = probability of lifecycle cost increase given the survival probabilities of the asset); • Risk Management—decide on a mitigation strategy based on the consequences and likelihood of the risk (e.g., conduct additional asset inspections/mechanistic testing); and • Risk Monitoring—measure the effectiveness of the mitigation strategy (e.g., were lifecycle costs reduced by applying the management strategy?). Of these steps, the most relevant to the asset manager’s task of life expectancy determination is the risk assessment step. This assessment differs from sensitivity analysis in that the likelihood of a range of outputs can be quantified. A typical risk assessment involves two statistical techniques: distribution fitting and Monte Carlo Simulation (Ashley et al. 2006). Distributions can be fit using software or by conducting various goodness-of-fit tests (e.g., Kolmogorov-Smirnov, Anderson Darling, Chi-squared). Life expectancy factors such as climate variables have relatively well-known distributions. For instance, long-term NOAA data are generally assumed to be normally distributed (Whitehurst 2008). To assess the likelihood of outputs, it is then a matter of conducting a Monte Carlo Simulation. Monte Carlo Simulation is the process of randomly sampling values from each input distribu- tion, inputting these values into the model, and finally assessing the likelihood of the outputs (Figure 6-4). In the context of life expectancy, risk analysis can be conducted in two stages: 1. Assess the likelihood of asset life estimates due to uncertain life expectancy factors; and 2. Assess the likelihood of lifecycle costs and other planning factors due to uncertain asset life estimates. Vulnerability relates to hazardous or threatening events and vulnerability analysis often simu- lates attacks on a system and evaluates the system responses. Significant amounts of literature exist on the vulnerability of major assets (e.g., pavements and bridges) to hazardous events (e.g., earthquake, flood, landslide, and fire). Approaches discussed in the literature could also be used to analyze vulnerability of less-studied assets such as pavement marking, traffic signs, and signal and lighting structures. Historical data on flooding, landslides, fire, and earthquakes are typically available in the public domain; thus, assessing the vulnerability of less-studied assets to these events can be analyzed using the data available. However, data on other events, such as collisions between boats and bridge piers, or between vehicles and guardrails or with other road appurtenances, hazmat spills, and terror events are quite rare. As such, any analysis of vulner- ability to such events will be expected to rely heavily on expert opinion.

accounting for Uncertainty: how to Improve Life expectancy Models 129 6.2.1 Example Risk Assessment of Uncertain Life Expectancy Factors Continuing with the sensitivity analysis example in Section 6.1, let us suppose an agency now wishes to know the likelihood of asset life at one location with uncertain temperature and precipitation values. Via risk analysis, this can be done by fitting distributions and applying Monte Carlo Simulation techniques. For this example, let us assume the distributions in Table 6-3. By randomly sampling these distributions, a planner recognizes that expected climate conditions over the life of an asset are not certain, and the life expectancy predicted therefore is not certain. By randomly sampling these distributions, a range of survival curves is obtained (Figure 6-5). Wider confidence intervals represent more uncertainty in the estimate. For instance, from Figure 6-6, it can be seen that the uncertainty surrounding asset survival probability is relatively low within the first 20 years but then increases until around year 80 before decreasing again. The uncertainty surrounding the asset life prediction can be assessed by analyzing how the 50th percentile asset life changes for each random sample of the inputs. As a result of this analysis, the distribution (Figure 6-7) representing how the average life changes, given random tempera- ture and precipitation values, is obtained. Although the median life of the distribution remains at 67 years (see Section 6.1), the most likely life estimate now is actually calculated to be 48 years. Given the uncertainty in temperature and Precipitation values, this analysis suggests a 90% confidence interval of [26 years, 173 years] and a 68% confidence interval of [38 years, 119 years]. O Z Y X Figure 6-4. Monte Carlo simulation process (van Dorp 2009). Life Expectancy Factor Mean Standard Deviation Normal annual temperature in °F 49 1 Normal annual precipitation in inches 43 6 Table 6-3. Distributions for example risk analysis.

130 estimating Life expectancies of highway assets 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 120 140 160 180 200 Su rv iv al P ro ba bi lit y Age in Years Average 68% Confidence Interval 90% Confidence Interval Figure 6-5. Example uncertainty surrounding asset survival curve. 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 Si ze o f 9 0% C on fid en ce In te rv al o f A ss et Su rv iv al P ro ba bi lit y 0 20 40 60 80 100 120 140 160 180 200 Age in Years Figure 6-6. Example uncertainty by assessment of confidence interval size. 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 Pr ob ab ili ty 0 20 40 60 80 100 120 140 160 180 200 Service Life Estimate Figure 6-7. Example uncertainty surrounding life expectancy estimate.

accounting for Uncertainty: how to Improve Life expectancy Models 131 The wide variation in asset life estimates demonstrates that care must be taken when basing planning decisions on remaining life. Actual climate conditions are likely to be more certain, resulting in a narrower range of predictions. To further illustrate the risk associated with uncer- tain asset life, the following section demonstrates how a risk analysis can be repeated with asset life as the uncertain input and various planning decisions as the outputs. 6.2.2 Example Risk Assessment of Uncertain Estimates of Asset Life Asset life estimates can be incorporated into various business processes such as assessing bud- get needs, calculating lifecycle costs, and ranking projects. If setting budget needs, the expected amount of money that should be set aside for replace- ment can be taken as the product of the probability of needing to replace an asset within a certain planning horizon and the cost of replacement for that asset. The expected network needs are then the total for all assets. If the time of replacement is considered the same as the predicted asset life, then the expected budget needs can be readily calculated. For example, consider a pipe culvert that is estimated to cost $1,000 to replace and the planned time for replacement taken as the distribution in Figure 6-7. The expected needs for this one asset in a 25-year planning horizon are then E $ Replacement Cost P SL years[ ] = ≤( ) 25 The probability of an asset life estimate being less than 25 years is equivalent to the area under the curve shown in Figure 6-8, assuming new construction. In this case, there is only a 4% chance of a planner predicting the asset to need replacement within the planning horizon. Therefore, only $44 ($1,000 * 0.044) may need to be added to the total budget on account of this asset. Similarly, the risk of planning for inaccurate lifecycle costs can be calculated. For example, if a manager is interested in an asset’s present value, assuming no maintenance or rehabilitation, and the time of replacement is considered to be the estimated asset life, then E PV i FV P SL SL [ ] = +   ×     × ( )∑ 1 1 P(SL 25) = 0.044 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 Pr ob ab ili ty 0 20 40 60 80 100 120 140 160 180 200 Service Life Estimate Figure 6-8. Example probability of replacement in 25-year planning horizon.

132 estimating Life expectancies of highway assets If a discount rate of 4% is assumed, with the same replacement cost and asset life distribu- tion, then the distribution of present value in Figure 6-9 is obtained, with an expected present value of $113. Additionally, for agencies that use remaining asset life as a factor in ranking projects, the utility associated with a project may be considered uncertain due to the risk of inaccurate life estimates. For example, let us consider a utility curve developed through surveying INDOT officials (Figure 6-10). Assume now that a culvert with the estimated life distribution in Figure 6-7 is 45 years old and we would like to predict the change in utility associated with a replacement project in 5 years. If we assume a life of 67 years (the median life predicted for this example—calculated in Section 6.1), then the remaining asset life at the time of potential replacement for this asset is 17 years. From our utility curve, we could then conclude that planning for replacement at this time would not improve our utility. $200 $400 $600 $800 $1,000 Present Value 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 Pr ob ab ili ty $0 Figure 6-9. Example probability of estimated present value. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 Ut ili ty Remaining Service Life in Years Figure 6-10. Example remaining life utility curve (Li and Sinha, 2004).

accounting for Uncertainty: how to Improve Life expectancy Models 133 However, given that asset life is uncertain, there is some probability associated with this proj- ect being worthwhile. For instance, the probability of this project actually having the highest possible change in utility is P U 1 P RSL and P SL∆ =( ) = ≤( ) ≥( )0 10 For the distribution in this example, this probability turns out to be 30.6%. Similarly, the prob- ability of the asset having no change in utility is 58.7% and the expected utility for this potential project is 36. This finding shows that the confidence that this project will have the predicted utility is lower than some planners may assume, showing the risk of planning and potentially program- ming less-than-optimal projects. Uncertainty surrounding life expectancy factors and estimates can highlight deficiencies in the model, identify the most influential factors, and quantify the effect on basic planning deci- sions. Therefore, it is up to the agency to sift through the quality of its life estimates and to man- age any potential risk in planning for an errant forecast.