Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
9Â Â AppendixÂ B of this report is a primer that focuses on the mathematical and statistical concepts underlying quantitative risk assessment. The primer provides a quick refresher of risk terminol- ogy and concepts for the practitioner who may not have studied these concepts for many years. It also provides for the nonpractitioner a basic understanding of the concepts deployed later in this report. Key concepts contained in the primer are summarized in this section. The primer explains the fundamental principles behind quantified risk management that are commonly used by risk and asset managers in the private sector and applicable to transportation agencies. Most of the concepts in the primer will be familiar to anyone who has taken courses in microeconomics, modeling, optimization, asset management, and some aspects of engineering. The primer is intended to refresh readers with these concepts and relate those commonly under- stood concepts to risk assessment. 4.1 Glossary of Terms and Refresher of Concepts The primer includes a glossary with definitions for common risk terms such as probability, expected value, consequence, impact, utility, risk appetite, risk aversion, and risk premium. These definitions bring clarity and common understanding to the risk analysis process. The primer restates several fundamental aspects of probability analysis. One fundamental concept is probability. The primer defines probability (expressed as a ratio or percentage) as the likelihood of possible outcomes for an event. Probability can be generated mathematically or from expert opinion. The primer also explains the concept often used in quantitative risk management of con- ditional probability. Conditional probability addresses the instance when two events occur together; either they can be completely independent of each other, or the probability of the second event occurring may be conditional on whether the first event occurs. Mathematically, the notation â|â can be read as âgiven.â For instance, the probability of A occurring given that B occurred is written P(A|B). The primer uses the example of the probability of a culvert failing during a storm to illustrate the concept of conditional probabilities. Another basic concept that the primer restates is the requirement that probabilities have a value between 0 and 1, and the total probability of an event occurring and that of it not occurring must equal 1. If there are multiple possible outcomes, then the probabilities of those outcomes may be expressed in the form of a probability distribution, and the aggregate probability of all those outcomes must equal 1. Probability is one factor in determining impact or expected value. The probability of an event is multiplied by its consequence or impact to determine the expected value or total risk. Expected S E C T I O N 4 Summary of Risk Management Primer
10 Risk Assessment Techniques for Transportation Asset Management: Conduct of Research value is the positive or negative value of the probability of the occurrence of an event multiplied by its impact. The primer illustrates these concepts with an asset management example: the computation of the probability of a culvert failing if a severe storm were to occur. If the probability (P) of a severe storm (S) occurring in a particular year, P(S), is 1:100, or 0.01, and the probability of the culvert failing (F) during the storm, P(F|S), is 0.5, then the probability of failure of the culvert in a severe storm can be calculated as . . . , . %0 5 0 01 0 005 0 5P P SP F SF and S or# #= = =` a `j k j Then, assuming the consequence of a culvert washout was $50,000, the expected cost of the culvert washing out in any one year would be $50,000 Ã 0.005 = $250. (Over a longer 10-year analysis period, this can simply be summed so that the expected cost would be $250 Ã 10 = $2,500.) These types of calculations are then applied in quantified risk mitigation analyses. If the expected value of potential loss is $250 per year, any mitigation effort that reduces the probability of failure and costs less than $250 per year could be considered as a potential mitigation strategy. 4.2 Probabilities and Utilities Other foundational concepts the primer includes are independent events and probabilities. If two events are independent, then it is assumed that the probability of either event happening has nothing to do with whether the other event has occurred. Independent events and prob- abilities allow the calculation of multiple events not happening together. An example could be the probability of two roads remaining open during a hurricane compared with the probability of both failing. Another important concept in quantitative risk analysis is utility. The primer describes utility as a measure of a decision makerâs preference for a state or outcome. Typically, the higher the value, the more the outcome is preferred. However, the utility is often tempered by the concept of declining marginal utility, or declining marginal benefit. As the measure of a condition rises, the incremental benefit of one additional unit of utility often decreases. The primer uses an example of pavement conditions to illustrate both utility and declin- ing marginal utility (FigureÂ 4-1). A state DOT may place great value, or utility, on improving Condition Index 8060 U(100) 100 U(80) U(60) U til ity o f C on di tio n In de x [U (C on di tio n In de x) ] Figure 4-1. An illustration of declining marginal utility, also called declining marginal benefit.
Summary of Risk Management Primer 11Â Â pavement conditions from an average condition of 60 to an average condition of 80. However, the value of raising average conditions above 80 rapidly declines. The utility of raising an average condition to 90 from 80 is less than raising the condition from 60 to 70. The concepts of utility and declining marginal utility feature in decision making about invest- ment trade-offs, such as investments to reduce threats. The primer explains that asset managers deal with thousands of assets, each with numerous components. Therefore, equally numerous treatments can be applied to the assets and their components. For example, each bridge compo- nent can be treated in multiple ways. Computing the utility of each component is made easier through software programming. Equations can be applied to compute the utility of actions among multiple options in an asset management software. To help decision makers address large numbers of asset components, such formulae can be incorporated into the softwareâs program- ming logic, and utilities for actions can be shown as outputs. For example, when decisions are made about bridge treatments, the utility of specific treat- ments on an asset component is computed formulaically by a BMS. What the primer and this report demonstrate is how these concepts of utility can be applied to reduce threats to assets. By calculating the value of an assetâs loss or deterioration, the benefits of actions to reduce the prob- ability of loss or deterioration can be calculated. 4.3 Risk Tolerance and Translating Policies into Values The primer also describes risk tolerance, a means to translate policy preferences into quanti- fied risk assessments. Risk tolerance is described as a decision makerâs ability to tolerate a loss, or the amount of uncertainty or variability they are willing to accept. Risk tolerance is sometimes called the risk appetite or risk threshold. Risk tolerance or appetite could be qualitative or quantitative. A qualitative risk tolerance could be as simple as the agency affirming that it tolerates less risk regarding highway safety targets than it is willing to tolerate regarding pavement targets. A numeric risk tolerance could be a simple approach to limit how many âriskyâ assets to tolerate. An agency could adopt as policy that it does not want any increase in the lane miles subject to tidal flooding. Its tolerance for additional flooded miles is zero, which could trigger investments to mitigate additional flooding. Risk tolerance could be statistical or mathematical. Statistical or mathematical risk tolerance relates to variability surrounding an expected outcome. In simpler terms, an agency could state that it wants no more than a 5Â percent probability that it will not achieve its Interstate pavement and bridge targets. An example analysis could be conducted for a set of pavements. By comparing the deviation over sections between the actual performance and the modeled performance, the agency can capture the amount of risk within a pavement forecast. In transportation asset man- agement, a high degree of variance between predicted and actual performance could trigger decisions to improve data for modeling, classification of the sections with the highest deviation, further analysis, or other actions. A risk-based performance measure for modeling would be that the modeled results correlate to actual results by some risk factor. For example, a 95Â percent correlation requirement can be instituted. 4.4 Attitudes of Tolerance Decision makers may take one of three attitudes to risk tolerance: risk neutral, risk averse, or risk seeking. Each attitude represents the value of a factor that influences risk-based decisions. Risk neutral is when the risk of the outcome or expected value is below or meets the risk toler- ance. Risk averse is when decisions made are safe and do not involve taking any risks. Risk-seeking decisions reflect that an agencyâs risk tolerance is high. These attitudes can be applied to various
12 Risk Assessment Techniques for Transportation Asset Management: Conduct of Research practical aspects of an agencyâs functions. For example, risk averseness or low risk tolerance often applies to decisions related to safety. Risk tolerance can also be applied to help decision makers show their stakeholders that a certain level of risk may sometimes be necessary or that trade-offs may have to be made when revenues are scarce. 4.5 Proxy Indicators Proxy indicators can be surrogates for risk when a particular risk is hard to quantify or there is not a direct indicator of risk. An example of a proxy indicator for risk to bridges from flood damage could be their geomorphic compatibility and vulnerability to channel erosion. Presumably, reducing those characteristics could reduce the risk of bridge damage during floods. 4.6 Decision Tree and Monte Carlo Simulation Because the NCHRP 08-118 research demonstrates inexpensive off-the-shelf tools, the primer introduces two such Excel-based tools. One is a probabilistic decision tree. It differs from a stan- dard decision tree by including probabilities and expected values. At each decision point, the tree includes the likelihood of success of a decision path as well as the likely value of the outcome. This allows for the expected value of each potential decision to be compared. The primer uses a typical culvert replacement analysis to illustrate a decision tree. It compares the cost of mitigating a culvert to reduce the risk of its failure with a âdo nothingâ decision. The decision tree can be easily modified to reflect different levels of risk of culvert failure and different costs to mitigation. Depending on the assumed cost and the assumed risk of failure, different treatment options can be compared. The second tool described is Monte Carlo simulation. Monte Carlo simulation is an iterative method of estimating the value of an unknown parameter using sets of random input variables with uncertain distributions in a mathematical model to arrive at a large sampling of the out- comes. This sampling of outcomes can then be analyzed using statistical or probabilistic tools to arrive at the expected value of the unknown parameter. The primer describes how Monte Carlo simulation of future asphalt pavement costs could be a key input variable for a 10-year pavement program. The probability distribution that best represents this historical data can be used to randomly select inflation rates to compute the fore- casted cost. The Monte Carlo simulation can then run as many forecasts as instructed about future asphalt pavement costs. Innumerable forecasts are possible. Monte Carlo simulations do not only run a forecast with 1Â percent inflation, another with 2Â percent, a third with 3Â percent, and so forth. Rather, the simulation may randomly select yearly inflation rates of 1.01Â percent one year, 2.05Â percent in another year, 3.81Â percent in a third year, or any other number of calculations based on the probability distribution assumed for inflation rates. Most off-the-shelf software tools allow the selection of probability distributions for input variables from a variety of choices and accommodate hundreds of thousands of output scenarios.