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(risk_definition)=
# Definition of Risk
Almost all activities in life are characterized by some level of risk. Especially in the design and management of engineering systems, risk and safety are key concepts and need to be taken into account explicitly. In social discussions no unambiguous meaning is assigned to the concept of risk. Two definitions given in the Oxford dictionary are: 1) a situation involving exposure to danger; 2) the possibility that something unpleasant or unwelcome will happen.
The first definition focuses on the consequences, the second on the possibility or probability. Quantifying and evaluating risks based on either probabilities or consequences (not both) is less realistic. For example, the risk of losing € 100 with a probability of 50% is different than the risk of losing € 1000 with the same probability. Also, the risk associated with losing a given sum of money will depend on the probability of the event.
An often-used definition considers risk as an expected value:
*Risk is the probability of an undesired event multiplied by the consequences.*
The unit of risk now depends on the units of probability and consequences, where probability of an event is generally expressed as the probability per unit time, for example per year. The consequences of an undesired event often measure a diverse array of damages, such as material, ecological, injuries and fatalities. In many applications engineering consequences are expressed by means of a monetary value, in which case the unit of the risk (or expected value, $E(d)$) then becomes € per year. For a case with one event scenario $i$ with probability $p_{i}$ it yields:
$$
E(d_i) = p_{i} \cdot d_{i}
$$
A more general definition of risk has been given by Kaplan and Garrick (1981):
>Risk is a set of scenarios, $s_{i}$, each of which has a probability, $p_{i}$, and a consequence, $d_{i}$.
This definition allows the use of various so-called risk metrics (or risk measures) to quantify or depict risk. The expected value of the damage for a set of multiple discrete scenarios $i=1,....,n$ , can be expressed as:
<!-- $$
E(d) = \sum_{S_{i=1}}^{n}p_{i}d_{i}
$$ (eq_risk_definition) -->
```{math}
:label: eq_risk_definition
E(d) = \sum_{S_{i=1}}^{n}p_{i} \cdot d_{i}
```
For the set of scenarios considered, the expected value quantifies risk precisely; however, it does not give insight in the magnitude of probability and consequences and the contribution of individual scenarios. For example, a single value does not indicate whether the risk is governed by a large number of scenarios with small consequences, or a few scenarios with large consequences and low probabilities. Therefore, an often-used companion risk-based tool is the risk curve, which shows the probability of exceedance and consequence for all scenarios. A well-known example of such a risk curve is the *FN curve*, which displays the probability of exceedance associated with $N$ (human) fatalities, schematized in {numref}`fn_simple01`. It is easy to compare the probability of exceeding a relatively low or high number of fatalities.
```{figure} ../figures/fn_simple01.svg
---
height: 600
name: fn_simple01
---
FN curve, showing the probability of exceedance of a certain number of fatalities N on Log-Log scale.
```
The FN curve was originally introduced in the 1960's for the assessment of risks in the nuclear industry {cite}`farmer1967,kendall1977` and is now used to display and limit risks in a wide variety of industries around the world. It is an extremely useful way to quantitatively compare risk associated with a broad range of scenarios, and to make decisions. A famous example of this is shown in {numref}`risk-curve-baecher`, which compares the risk estimated for a wide variety of engineering infrastructure. This figure is described further in the {ref}`risk_curve` Section, and also illustrates the concept of acceptable risk, which is discussed in the Section on {ref}`safety_standards`. In short, it allows one to begin answering the question 'how safe is safe enough?'
```{figure} ../figures/risk-curve-baecher.PNG
---
height: 400px
name: risk-curve-baecher
---
Risk curve for various engineering projects, showing probability of exceeding a specific number of fatalities, $N$, and damages, $, on a Log-Log scale, making it both a FN and FD curve; see {ref}`risk_curve` Section. Source: {cite:t}`baecher2003`, based on {cite:t}`baecher1982` and described in {cite:t}`baecher2021`.
```
## Hazard, Vulnerability and Exposure
Engineering disciplines that are focused on decision making where natural hazards play a significant role often define risk as the combination of hazard, exposure and vulnerability, as shown in {numref}`risk-hazard`.
* Hazard: A dangerous phenomenon, substance, human activity or condition that may cause loss of life, injury or other health impacts, property damage, loss of livelihoods and services, social and economic disruption, or environmental damage
* Vulnerability: The characteristics and circumstances of a community, system or asset that make it susceptible to the damaging effects of a hazard
* Exposure: People, property, systems, or other elements present in hazard zones that are thereby subject to potential losses
```{figure} ../figures/risk-hazard.svg
---
height: 200px
name: risk-hazard
---
Alternative definition of risk as intersection of hazard, exposure and vulnerability. Risk is the small triangular-shaped area where all three circles overlap.
```
Often these terms are useful for framing qualitative discussions around complex topics where a quantitative analysis may be out of reach, for example, as done in IPCC reports on climate change {cite:p}`cardona2012`. However, they can also be used to simplify the risk analysis by subdividing it into parts that are (computationally) independent. An analytic expression can easily be formulated by expanding the probability term of Equation {eq}`eq_risk_definition`:
```{math}
:label: eq_risk_definition_HEV
p_{i} = P(N_i,E_j,H_k) = P(N_i|E_j,H_k) \cdot P(E_j|H_k) \cdot P(H_k)
```
where $N_i$, $E_j$ and $H_k$ are the vulnerability (fatalities), exposure and hazard of scenario $i$, respectively. Combining with Equation {eq}`eq_risk_definition` and taking the sum over all combinations of variables ($i$, $j$, $k$) would give the total risk. In the case of our flood management case, an example of each term could be:
- Hazard, $H_k$: a high water level.
- Exposure, $E_j|H_k$: breach (failure) of a dike due to the high water level, $H_k$.
- Vulnerability, $N_i|E_j,H_k$: fatalities given the broken dike and high water level. $N$ is usually modeled as a random variable as it is dependent on factors such as evacuation effectiveness and flood warning systems.
% Point out that this formulation simplifies the calculation, promots the use of tools like fragility curves, and give earthquake hazard assessment as an example (DRMS?).
## Other Risk Definitions
This book uses Equation {eq}`eq_risk_definition` as the primary definition of risk. However, it is useful to highlight some risk concepts used in other domains.
Within economics, risk is generally associated with a deviation from the expected return or the probability of loss. In social sciences risk is often considered as a contextual notion or social construct. {cite:t}`vlek1996` has summarized 11 formal definitions used in social sciences, see Table {numref}`risk_definitions`. In some of these definitions (e.g. numbers 2 and 4) the perceived seriousness of the undesired consequences plays an important role. Examples of other, more informal risk definitions used in psychology are “the lack of perceived controllability”, “set of possible negative consequences” and “fear of loss” {cite:p}`vlek1996`.
Substantial research has also focused on factors that determine the perception of risk (e.g. {cite:t}`slovic1987`, {cite:t}`vlek1996`) including: degree of damage, controllability of and familiarity with hazards, extent of benefits from an activity, and voluntariness of exposure.
:::{list-table} Formal definitions of risk used in social sciences {cite:p}`vlek1996`
:header-rows: 1
:name: risk_definitions
* - No.
- Definition
* - 1
- Probability of undesired consequence
* - 2
- Seriousness of (maximum) possible undesired consequence
* - 3
- Multi-attribute weighted sum of components of possible undesired consequences
* - 4
- Probability $\cdot$ seriousness of undesired consequence (‘expected loss’)
* - 5
- Probability-weighted sum of all possible undesired consequences (‘average expected loss’)
* - 6
- Fitted function through graph of points relating probability to extent of undesired consequences
* - 7
- Semi variance of possible undesired consequences about their average
* - 8
- Variance of all possible consequences about mean expected consequence
* - 9
- Weighted sum of expected value and variance of all possible consequences
* - 10
- Weighted combination of various parameters of the probability distribution of all possible consequences
* - 11
- Weight of possible undesired consequences (‘loss’) relative to comparable possible desired consequences
:::

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# Risk Analysis
This book splits risk into two parts: risk *analysis* and risk *evaluation.* This chapter considers the analysis of risks from a broad perspective, beginning with definitions and key steps of a risk analysis, illustrated quantitatively through an FN curve. The next two chapters explore quantitative analyses (component and system reliability) before returning to risk evaluation concepts of decision theory, cost benefit analysis and safety standards, where the principal focus is on making decisions and answering the question 'how safe is safe enough?'
---
_The primary authors for this chapter are Bas Jonkman and Robert Lanzafame._
% The relationship between safety standards and engineering design is indicated in section 3.6...may be useful to add
% It is noted that a (probabilistic) risk analysis is different from a (deterministic) scenario analysis. A risk analysis is based on quantitative analysis of all (known) undesired events and their probabilities and consequences. A scenario analysis considers one (or a limited number) of design scenarios, often without considering its failure probability. Both approaches are complementary, as a scenario analysis considers one of the scenarios from the risk analysis.
% In the Netherlands, scenario analysis is often used for evaluating disaster preparedness and designing the capabilities of emergency management services. For example, in the design of the Green Heart Tunnel which is a part of the high speed rail line in the Netherlands, a single accident scenario was chosen to design the emergency exits out of the tunnel. Analysing a high impact, low probability scenario in detail may provide helpful clues about the effectiveness of particular safety measures or crisis management actions. While such analyses may be helpful, it would be wrong to assume that such high-impact, low probability scenarios should always be properly manageable by emergency responders. This is because the probability of harm should be an integral part of any cost-benefit or cost-effectiveness analysis (Jongejan et al., 2011).

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(risk_curve)=
# Risk Curve
As introduced in the {ref}`risk_definition` Section, the risk curve is useful for evaluating the expected damages for a wide range of scenarios. In contrast to the total expected damages, it allows the comparison of relative magnitude of probability of occurrence and associated consequences. An FN curve tends to be the most widely used, where the 'N' refers to number of fatalities and the 'F' in comes from the CDF of the probability distribution of $N$, $F_N(n)$, although the curve itself actually represents the complementary CDF, $1-F_N(n)$. It is often plotted on a log-log scale to allow comparison over a wide range of scenarios. While in theory a risk curve can be derived from the probability density function (or probability mass function for discrete cases), the distribution of damages is rarely known and must be approximated by considering a finite and suitable set of discrete events. Thus, in practice, the curve is constructed using scenarios as defined previously in our risk definition (Equation {eq}`eq_risk_definition`).
Although the FN curve is widely used, the risk curve is not exclusively applied to evaluate expected number of fatalities, $N$. In principle any measurable quantity can be used to evaluate risk, referred to as a *metric,* although it can be difficult to apply this to some situations. For example, assessing the value of environmental characteristics can be challenging because the benefit is not easily related to a measurable quantity. The area of impacted wetland habitat during a flood (e.g., km$^2$) can be used as a risk metric, but this may not properly represent actual damage, which may recover quickly or slowly depending on the frequency of inundation. Financial damage (e.g., €) is often considered as a risk metric, in which case the FN curve is called an *FD curve.*
```{admonition} Fatalities and Damages
:class: tip, dropdown
In some cases it is useful for planning and design purposes to equate $N$ and $D$, but this places a specific financial value on human life, which raises difficult moral and ethical concerns. This topic is discussed further in the {ref}`value_human_life` Section.
```
%(risk_curve_examples)=
<!-- ## Risk Curve Examples
Why is an FN Curve useful?
```{warning}
Add Dike Ring 14 and nuclear (with EQ?) case.
```
### Flood Risk in South Holland
### Nuclear Power Plants -->
(risk_curve_famous)=
## A Famous Risk Curve
A risk curve is often useful because it can convey a large amount of complex information in an easily recognizable format. University lectures and conference presentations around the world which focus on risk, especially for civil engineering applications, will often begin with a casual reminder that 'risk is probability times consequences' and show some version of a risk curve. {numref}`risk-curve-baecher-2` is the most famous risk curve, having been shared widely over the last decades, perhaps because it so nicely compares a wide range of engineering projects.
```{figure} ../figures/risk-curve-baecher.PNG
---
height: 400px
name: risk-curve-baecher-2
---
Risk curve for various engineering projects, showing probability of exceeding a specific number of fatalities, $N$, and damages, $, on a Log-Log scale, making it both a FN and FD curve; this figure is also included in the {ref}`risk_definition` Section. Source: {cite:t}`baecher2003`, based on {cite:t}`baecher1982` and described in {cite:t}`baecher2021`.
```
{numref}`risk-curve-baecher-2` was created as part of a risk analysis for a petrochemical storage facility in Tokyo Bay, Japan, where large tanks, constructed on soft soils (hydraulic fill), are vulnerable to damage during an earthquake. Of primary concern was to prevent the petrochemicals entering bay waters, thus not only failure of the storage tanks, but also the containments systems in place to collect spilled material (a complicated system reliability problem). The client needed to convince the regulator that the final design was sufficient for keeping the risk of loss of containment to a sufficiently small level, which the project team was able to do by comparing the risk calculated for the Tokyo site to other engineering works around the world. The figure clearly illustrates what the estimated risk is for each project, and by making the assumption that these already-constructed projects provide a level of safety that is acceptable to society (primarily via regulatory agencies), the 'acceptable' line can be used to approve other projects. This concept of acceptable standards based on risk curves is covered in depth in the Section on {ref}`safety_standards`.
The origin of this figure was described a Terzaghi Lecture by Professor Gregory Baecher, which can be viewed on YouTube: [*IFCEE 2021: Karl Terzaghi Lecture: Greg Baecher: Geotechnical Systems, Uncertainty, and Risk*](https://www.youtube.com/watch?v=Y5w1p3uAe0I&ab_channel=Geo-InstituteofASCE) {cite:p}`baecher2021`}. The introduction to the Tokyo Project begins at [16:16](https://www.youtube.com/watch?v=Y5w1p3uAe0I&t=976s), and the explanation of the risk curve is given at [27:00](https://www.youtube.com/watch?v=Y5w1p3uAe0I&t=1620s); however, the entire lecture is well worth listening to and closely related the subject of this book.
## Construction of an FN Curve
The following three figures illustrate the computation of an FN curve for a system with two mutually exclusive event scenarios:
- Accident 1 with $N_{1}=10$ fatalities and a probability of $P_{1} = 10^{-2}$ per year
- Accident 2 with $N_{2}=100$ fatalities and a probability of $P_{2} = 10^{-3}$ per year
First, the probability mass function is created $f_N(n)=P(N=n)$, taking care that the probability of no fatalities is included:
$$ P(0) = 1 - P_{1} - P_{2} = 1 - 0.01 - 0.001 = 0.989 $$
The cumulative distribution function, $F_N(n)=P(N\leq n)$, can thus be easily computed, as well as the probability of exceedance, $1-F_N(n)$, also known as the *FN curve.* Note that if a risk metric such as damage in Euros was being used, we would call this the *FD curve* and use $D$ to denote the random variable, as in $f_D(d)$ and $F_D(d)$
While the curves below illustrate probability and consequences associated with specific scenarios, the expected value of the distribution can be computed, which is equivalent to the area under the FN curve. This is useful can be used to assess the entire system. For this example, the expected value of fatalities per year for all scenarios can be computed as follows:
$$E(N) = P_{1}N_{1} + P_{2}N_{2} = 0.2\quad \textsf{(fatalities per year)}$$
```{figure} ../figures/ex_FN_curve_step_01_py.svg
---
width: 400
name: ex_FN_curve_step_01
---
Probability mass function (PMF), $f_N(N)$.
```
```{figure} ../figures/ex_FN_curve_step_02_py.svg
---
width: 400
name: ex_FN_curve_step_02
---
Cumulative distribution function (CDF), $F_N(n)$.
```
```{figure} ../figures/ex_FN_curve_step_03_py.svg
---
width: 400
name: ex_FN_curve_step_03
---
Exceedance probability (FN curve), $1-F_N(n)$.
```
```{admonition} Practice Exercise
:class: tip
You can practice constructing a simple FN curve in {ref}`ex_fn_curve`, although some of the questions require knowledge of limit lines, introduced in the {ref}`safety_standards` Section.
```

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(risk_steps)=
# Steps in a Risk Analysis
The previous section made it clear that risk is a function of probabilities and consequences. Risk analysis, therefore, consists of an analysis of probabilities and consequences associated with undesired events in a given system. Alternative terms used in literature are risk assessment and quantitative risk analysis (QRA).
A risk analysis is carried out because involved parties (e.g. designers, managers, decision makers) want to identify and evaluate the risks and decide on their acceptability. Outcomes of risk analysis can be used in the design process to decide on the required safety levels of new systems (e.g. a new tunnel) or to support decisions on the acceptability of safety levels and the need for measures in existing systems (e.g. a flood defence system). A quantitative measure of some form is needed to transfer decisions on acceptable safety into a technical domain {cite:p}`voortman2004`, such as the height of a flood defence or the strength of a building. In addition, risk analysis can be used to analyse the effectiveness of risk reduction measures, including management and maintenance strategies. Whereas the primary goal of a risk analysis is often to support rational decision-making regarding risk-bearing activities {cite:p}`apostolakis2004`, it also provides insight in the way a system may fail. As such, it can also serve as a tool of communication and management. Insights drawn from a risk analysis can be used to optimize system design and management and often there is a direct link to quality assurance.
In general, the following elements can be identified within a risk analysis, (based on: {cite:t}`CUR1997`, {cite:t}`CIB2001`, {cite:t}`faber2003` and {cite:t}`jongejan2008`):
1. System definition and setting the scope and objectives of the analysis;
2. Qualitative analysis of undesired events;
3. Quantitative analysis of the risk;
4. Risk evaluation (of the acceptability of the risk)
Each of these steps is illustrated in {numref}`risk-steps` and described in more detail below, providing context for the following chapters in this book. In addition to the risk analysis of a given system, *risk management* also includes the element ‘risk reduction and control’. Depending on the outcome of the first four steps of the analysis, measures can be taken to reduce the risk, which will lead to changes in the system configuration and risk level. Steps 1--4 are often repeated several times with adjusted system specifications to obtain an optimal design.
```{figure} ../figures/risk-steps.PNG
---
width: 250px
name: risk-steps
---
Schematic view of steps in risk analysis (risk assessment) and risk management.
```
## 1. System definition
This step entails the definition and description of the system as well as the scope and objectives of the analysis. Usually a system is divided into components and subsystems, which can be schematised graphically. Further information on the decomposition and analysis of system is provided in the {ref}`rel_sys` Chapter of this book, and illustrated through the example exercises.
A system can be represented in terms of physical components, organizations and users, and an external environment. In order to analyse failure and risks, not only physical components, but also organizations and operators and users need to be considered (see e.g. {cite:t}`bea1998`), as different groups of organizations and persons will be involved in different roles. Consider the aviation industry, where professionals may be responsible for operation and management of the system (e.g. the pilots and crew in an aircraft), but important roles are also played by the users of a system (passengers) and external parties (people living near the airport exposed to risk and noise). Each group has a different relationship and attitude towards the risk, which could affect its acceptability. For example, a higher risk might be acceptable for pilots and crews in an aircraft (who derive a direct benefit from flying) than for regular passengers. Finally, the external environment (e.g. wind or runway conditions) will determine the loads on the system and affect potential failure mechanisms.
## 2. Qualitative analysis
In this step, potential hazards, undesired events, failure mechanisms and scenarios are identified and described. An important goal of this phase is to gain insight, as complete as possible, into all possible undesired events and their consequences. For most systems, multiple undesired events can be distinguished. For example, two events with different impacts that can both lead to flooding of a polder are 1) the inflow of large amounts of water due to a dike failure; 2) the inflow of smaller amounts of water when a sluice gate is not closed.
Failure occurs when a system no longer fulfils one or more desired functions, where the failed state can be reached through different failure mechanisms (or failure modes). For example, a dike can fail due to overtopping, but also due to geotechnical failure mechanisms such as instability or piping. A **limit state** is a condition of a structure beyond which it no longer fulfils the relevant design criteria {cite:p}`eurocode2001`, of which two types are distinguished in practice: serviceability and ultimate limit states (SLS and ULS, respectively).
- **Ultimate limit state (ULS)**: failure or collapse of a system or structure occurs, for example, the breakwaters of a harbour entrance are destroyed as a result of extreme conditions. An example from structural engineering is the collapse of a roof of a building.
- **Serviceability limit state (SLS)**: exceedance leads to temporary and/or partial failure. For example, inability to use the harbour due to waves that are (temporarily) too high. Another example could be too much vibration of a structure, such that users experience discomfort.
Further information on the use of the SLS and ULS concepts in civil engineering is included in the {ref}`rel_comp`.
In a risk analysis, it is very important to get an overview of the various undesired events and failures before proceeding with a quantitative analysis, as many accidents in practice are caused by failing to identify failure modes. Unfortunately, creating a complete list of threats and failure modes is not an easy task. Support for this process can be found through are failure databases, literature studies, interviews, experiences with comparable systems, brainstorm sessions, et cetera. In addition, formal techniques for systematically identifying undesired events exist, such as Failure Modes and Effects Analysis.
## 3. Quantitative Analysis
The probabilities and consequences of the undesired events identified in step 2 are quantified in this step, and risk is quantified in as expected damage(s) or with a risk graph as described in the next Section.
### Probability
In simple cases the probability can be computed directly, but generally the undesired event must broken down into multiple steps or processes to facilitate the computation of probability, which are then recombined using integration or the total probability theorem. Such an approach results in systems of systems, each of which is composed of individual *components* (or elements), each of which is typically evaluated using the limit state concepts defined above. Often this the probability of a limit state being exceeded is referred to as the probability of failure, which of course reflects the component, not necessarily the system. Techniques for computing the probability of failure are covered in the {ref}`rel_comp` and {ref}`rel_sys` Chapters.
Whereas the most critical aspect of the previous step is identifying the most important failure modes, in the quantitative analysis it is critical to precisely define the quantity of interest, as ambiguity can lead to misunderstandings and incorrect assessment of risk. Often such ambiguities have a direct relationship with the conditional terms used inthe probability computations, which imply specific statements about dependence (or independence). Consider the previous example of river level exceeding the dike height $P(h_w>h_{dike})$: the probability can be significantly different if a design lifetime of one year or fifty years is considered. In addition, the time of year may also play a role as different types of floods may occur. If the seasonal probability of flooding is binary and mutually exclusive with wet and dry season each lasting 6 months of the year, the following expression holds (all values computed an a *per year* basis):
$$
P(h_w>h_{dike}) = 0.5 \cdot P(h_w>h_{dike}|\mathrm{wet}) + 0.5 \cdot P(h_w>h_{dike}|\mathrm{dry})
$$
Thus, if the risk analysis were only focusing on floods during the wet season, the probability of flooding would be overestimated by a factor on the order of 2 (assuming the probability is roughly equal in dry and wet seasons).
### Consequences
After failure has been defined, consequences of the events are quantified. First, physical effects associated with an undesired event are considered, sucha s heat and/or smoke from a fire, or inflow of water due to dike breach. Depending on the exposure of people or objects to the physical effects, damages, life loss or other impacts may occur. As an example, a specific consequence, $D$, of dike failure during a flood is considered as a sequence of three discrete events:
- The probability that a dike fails, $P(E_{1})$
- The conditional probability that water flows into the polder given a dike breach $P(E_{2}|E_{1})$
- The probability of damage given dike breach and inflow into the polder $P(D|E_{1}\cap E_{2})$
The probability of damage can now be computed by combining these terms:
$$P(D)=P(E_{1})P(E_{2}|E_{1})P(D|E_{1}\cap E_{2})$$
which is analagous to the hazard, vulnerability and exposure formulation of risk introduced in the previous Section.
Multiple types of consequences can be caused by one disaster, which are illustrated in {numref}`risk_damage_types` for the failure of a flood protection system. The damage is divided into tangible and intangible damage, depending on whether or not the losses can be assessed in monetary values. Another distinction is made between the direct damage, caused by physical effects of the event, and damages occurring outside the directly exposed area. The latter occurs when companies outside an impacted area experience damages, due to loss of demand from customers in the flooded area. In a risk analysis it is desirable to take into account a complete set of impacts; unfortunately many of the items from the table cannot be easily quantified, if at all, thus the quantitative analysis and risk evaluation are often focused on economic damages and life loss.
:::{list-table} General classification of damages, based on {cite:p}`vrouwenvelder1996`
:header-rows: 1
:name: risk_damage_types
* -
- Tangible
- Intangible
* - Direct
-
-
* -
- Residences
- Fatalities
* -
- Structure inventory
- Injuries
* -
- Vehicles
- Animals
* -
- Agriculture
- Utilities and communication
* -
- Infrastructure and other public facilities
- Historical and cultural losses
* -
- Business interruption
- Environmental losses
* -
- Evacuation and rescue operations
-
* -
- Reconstruction of flood defences
-
* -
- Clean up costs
-
* - Indirect
-
-
* -
- Damage for companies outside the exposed area
- Societal disruption
* -
- Substitution of production outside the area
- Damage to government
* -
- Temporary housing of evacuees
-
:::
## 4. Risk Evaluation
In the risk evaluation phase a decision is made whether the risk is acceptable or not and whether risk reduction measures should be implemented. In other words, a direct answer is sought to the question “how safe is safe enough?” {cite:p}`starr1967`. Results of the quantitative analysis provide input for risk evaluation and decision making, and different quantitative approaches can be used to support risk evaluation, which are outlined here and introduced formally in the {ref}`risk_eval` Chapter.
**Decision Analysis**: Also called 'decision making under uncertainty,' this approach aids the decision-making process by recording different possible outcomes along with the associated risks, costs and benefits, leading to the optimal selection from a number of alternatives. Often visual tools are used such as a risk matrix or decision tree.
**Cost Benefit Analysis**: costs and benefits of risk reduction measures are considered. When a large number of design choices are possible, an **economic optimization** can be applied to select an optimal system design, based on the costs and benefits of risk reduction.
**Safety Standards**: risk is compared with predetermined safety standards to directly determin acceptability. Such standards are typically imposed by government organizations or standards of practice and often focus on loss of life as the primary risk metric.
Given the nature of the key question "how safe is safe enough?", several political, psychological and social processes play a role in the evaluation of risk---in other words: risk evaluation is not purely a technical process, but involves many subjective elements and decisions. One difficulty facing regulators is that preferences and risk attitudes within society may diverge and that costs and benefits may not be distributed evenly, and that a single, collective decision has to be based on strongly divergent individual preferences. In practice, this implies that the establishment of collective decision making procedures is inevitably a political process. This ambiguity can also be found in the numerous interpretations of “the” precautionary principle, which is interpreted by some as a decision making criterion that requires proof of harmlessness (a scientific impossibility), whereas it is seen by others as a decision making procedure that puts emphasis on dialogue and stakeholder involvement, (e.g. {cite:t}`jongejan2008`).
## 5. Risk Reduction and Risk Control
If the risks evaluated in first four steps of a risk analysis are considered unacceptable, several forms of risk reduction can be implemented, such as changes to the engineered system, or changes to its organization and management. It can be helpful to determine how risk can be controlled, for example by monitoring, inspection or maintenance.
Post-accident analysis indicates that human and organizational errors are still a major cause of failure in civil engineering systems, and it seems that the only suitable way to reduce human errors is by the incorporation of sufficient control in different phases of the construction process {cite:p}`taerwe1986`, and by a thorough education of all personnel involved. Therefore, an extensive interaction between the safety methodology and quality management is a necessity in order to guarantee the safety of infrastructure.