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(cost_benefit)=
# Cost Benefit Analysis
This section deals with simplified cost benefit analysis for risk reduction interventions in the engineering domain. An important question in evaluating (engineering) projects is whether the benefits outweigh the costs. Cost benefit analysis (CBA) is generally used for appraisal of a wide range of effects of projects or interventions in order to support decision making. The cost benefit analysis starts with defining the system and existing situation. Then, a broad range of effects of the proposed project and intervention can be identified.
{numref}`costs_and_benefits` below shows an overview of effects of the Delta works that were built after the 1953 flood disaster in the Netherlands. The main aim of the delta works was to provide flood protection to the Southwest of the Netherlands. However, other effects included the agricultural benefits to the region (benefits) and the effects on environmental quality in the estuaries in which dams were built (costs or negative effects).
```{list-table} Costs and benefits of the delta works
:header-rows: 1
:name: costs_and_benefits
* - Costs
- Benefits
* - Construction and maintenance costs
- Reduction of flood risk (damage, loss of life)
* - Loss of environmental quality
- Improvement of infrastructure
* -
- Recreation
* -
- Agricultural benefits
* -
- Economic stimulus for the region
* -
- Economic stimulus for the water engineering sector
* -
- National Pride
```
A choice has to be made for which effects in the CBA are evaluated in monetary terms. For some items, such as construction costs or economic risk reduction (see below) this is straightforward. For other items such as environmental effects or reduction of risk to life, approaches for monetary evaluation exist, but they are not standardized or undisputed.
## Risk Reduction via Probability
When considering investments that primarily focus on risk reduction (e.g. dike reinforcement or reinforcing of buildings for earthquakes), the main benefits will consist of the reduction of expected economic damages. For a measure to be cost effective, the investments should be smaller than the risk reduction.
$$
I < \Delta E(D)
$$ (investment)
where:
- $I$ is investments
- $\Delta E(D)$ present value of risk reduction
This formula can also be used to calculated the benefit/cost ratio, i.e. $\Delta E(D)/I$, which should be greater than 1 for cost effective projects. Note that all costs and the risk reduction are given in terms of present value with a unit of €.
For investments that focus on prevention (i.e. reducing the failure probability of the system) {eq}`investment` can be formulated as follows.
$$
I < (P_{f,0} - P_{f, new}) D
$$
where:
- $P_{f,0}$ the initial failure probability
- $P_{f,new}$ the failure probability after risk reduction investment
```{admonition} Example: Drainage of a construction site - costs and benefits
The example from the previous section could also be formulated as a simplified cost-benefit analysis. The same values for the variables are used. The costs of the drainage are €150,000. The benefits are equal to the risk reduction, i.e.
$$
\Delta E(D) = (P_{f,0} - P_{f,d}) = (0.12 - 0.04)\text{€} 5,000,000 = \text{€} 400,000
$$
This shows that costs are smaller than the risk reduction benefits, i.e. $I<\Delta E(D)$. The benefit/cost ratio equals €400,000/€ 150,000 = 2.66
One can easily show that the investment would just be acceptable if the benefits and costs are equal, i.e. when the benefit / cost ratio equals 1. This would still be the case if the probability of failure with drainage equals $P_{f,d} = 0.09$.
```
## Risk Reduction via Damages
Other types of interventions do not affect the probability of an accident, but focus on reducing the damages. They are often indicated as mitigation. An example in the field of flood management concerns raising buildings instead of reinforcing the dikes. In this case the criterion becomes:
$$
I < P_{f,0}(D_0 - D_{new})
$$
where:
- $D_0$ the initial damage [€]
- $D_{new}$ the damage after investments in reducing the consequences [€]
## Net Present Value
The foregoing assumes that both the costs and benefits are expressed in the same unit, generally in the form of a present value [€]. The net present value (NPV) represents the sum of the present values of the benefit and cost cash flows over a period of time. In engineering we often deal with situations with larger initial investments, whereas the risk reduction benefits are spread out over a longer time period. For example, when a government invests this year in flood protection, the costs are made in year 0, but the benefits will be spread over the coming decades.
For such situations the failure probability is generally expressed per unit of time, mostly per year. That means that the risk (reduction) is expressed in terms of € per year, whereas the initial investments have the unit of €. The net present value of cost or benefit values over a future range of years can be calculated with {eq}`npv`. To calculate the net present value NPV [€] a discount rate should be used. The discount rate represents a required return on an investment.
$$
NPV = \sum_{t=1}^{T} \frac{C_t}{(1+r)^t}
$$ (npv)
where:
- $C_t$ the costs in year $t$ [€]
- $T$ the reference period [years]
```{admonition} Interest Rates
:class: tip, dropdown
The discount rate, $r$, is analagous to opportunity costs and interest rates that you may be familiar with from the field of economics. It provides a way to evaluate the true cost of investing in a particular project over a long time period, for example: if the value of a stock market increases with an annual interest rate of 4%, how does that compare to the expected earnings on the alternative investment under consideration?
Although civil engineering projects are typically not associated with financial investments like stocks and bonds, the need to consider the time value of money is identical. In fact, many governments use the financial instruments like bonds to finance projects, which is easy to see the relationship to interest rates.
```
The contribution of costs in a certain year to the net present value depend on the discount rate and the reference year. Costs or benefits closest to the present will have the greatest contribution. {numref}`present-value` below shows the value of $1/(1+r)^t$ for a given year $t$ for different discount rates. The higher the discount rate, the smaller the contribution to the net present value of costs or benefits that are far away from the present. For an infinite time horizon it can be shown that $\sum 1/(1+r)^t \approx 1/r$. This approximation can be used in evaluating engineering projects with a long life time.
```{figure} ../figures/present_value.svg
---
width: 600
name: present-value
---
Normalized present value of a cost or benefit in year $t$ for different discount rates.
```
(value_human_life)=
## Valuation of Human Life
The previous paragraphs have focussed on the evaluation of economic costs and benefits of risk reduction interventions. Many of these interventions also directly focus on reducing injuries and fatalities. Examples are regulations and investments in traffic safety which have introduced measures such as airbags and seat belts.
In literature on risk management the economic valuation of human life is often depicted as a difficult problem as it raises numerous moral questions. Some claim it is unethical to put a price on human life because life is priceless. The actual expenditures on risk reducing prospects show however that the investment in the reduction of risks to humans is always finite. Different approaches are available for evaluating the costs of interventions in relation to the reduction of risk to human life, see {cite:t}`vrijling2000`; {cite:t}`jongejan2005` for a further discussion of the various approaches. One of the options is to add the economic value of human fatalities to the economic damages, i.e. $D+N \cdot d$. The value of the number of lives lost can be determined with different approaches.
Several approaches are based on so-called stated preferences. A survey can reveal how much people are willing to pay, e.g. for safety measures. In these cases the value of statistical life (VoSL) is obtained from the willingness to pay expressed by respondents in surveys. For example, in the cost benefit analysis for the flood defences in the Netherlands a value of a statistical life of €6.7 million per fatality is used {cite:p}`jeuken2013`. The Value of a Statistical life lost in traffic accidents is estimated at €2.6 million {cite:p}`SWOV2012`.
One alternative approach is based on so-called revealed preferences. The costs of saving and extra (statistical) life ($CSX$) for actual life-saving interventions that have been taken in the past can be determined.
$$
CSX = \frac{I}{\Delta E(N)}
$$
where:
- $CSX$ the costs of saving an extra life [€/life year]
- $\Delta E(N)$ the reduction of the expected number of fatalities per year
An extensive study on values in various sectors, see {cite:t}`tengs1995` showed that these vary widely across sectors and even within sectors. The highest $CSX$ values are found for risks for small probability – large consequence events, for example in the nuclear domain. For such cases the expected number of fatalities is already small and investments in incremental safety are large.
One other approach is to base the value of a human life on macro-economic indicators. Several metrics have been proposed that relate this value to a person’s economic production.
Given the difficulties associated with economic valuation of human life and the associated risk reduction, it is decided in some domains to develop separate criteria for considering the risk to life. This topic is further elaborated in ⚠️section 3.5.

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(decision)=
# Decision Analysis
Decision analysis, or decision-making under uncertain conditions is part of everyday life: when choosing to buy a lottery ticket or choosing to take an umbrella during cloudy weather. In contrast to the rather intuitive decision making in everyday matters, a structured analysis of different alternatives with associated risks, costs and benefits is very useful for decisions in (civil) engineering. This chapter offers a very basic introduction into the decision theory with applications to decision problems in the civil engineering domain. Further reference is made to the work by other scholars for more rigorous and detailed treatment of this topic. See, for example {cite:t}`raiffa1961`; {cite:t}`benjamin1970`.
## Introduction
Making a decision is in fact choosing from alternatives. The decision theory is based on the classic “Homo Economicus” model assumes that the decision-maker:
- has complete information about the decision situation;
- knows all the alternatives;
- knows the existing situation;
- knows which advantages and disadvantages each alternative provides, be it in the form of random variables;
- strives to maximise that advantage (formally called utility).
The decision-making concept discussed in these lecture notes assumes this model, although decision-making in practice is often different since the above conditions are often not fulfilled. For example, there can be multiple decision makers and multiple objectives, and in reality the decision maker does not know all the alternatives or their outcomes. For many practical cases this has led to an extension of the decision model, but not to a fundamental adjustment of the classical model described here.
Within a decision problem the following characteristics can be distinguished:
- the set of all possible actions or decisions $a$, from which the decision maker can choose
- the set of all (natural) circumstances, $\theta$ that influence the outcomes*
- the set of the set of all possible results, $\omega$, which are functions of the actions and circumstances: $\omega = f(a,\theta)$.
The actions, natural circumstances and the outcomes can be shown in a so-called decision tree (Figure {numref}`decision-tree-blank`).
```{figure} ../figures/decision-tree-blank.png
---
height: 300px
name: decision-tree-blank
---
Decision tree.
```
Based on the possible results a choice is made for an action. To be able to assess the different results, a numerical value is assigned to each outcome, $\omega$, which can be used to establish the benefit of each outcome. This number can be a monetary value, a number on an arbitrary scale or utility--as long as the decision maker(s) can establish a consistent ranking of the outcomes with it. In the last two cases the benefit has no absolute value, but only gives the relative value of the different outcomes.
Utility, $u$, is a concept used to rank the possible outcomes according to the preferences of the decision maker, with possible values $0\leq u \leq 1$ . A utility function can be used to characterize the relative utility of various outcomes. The elaborations below are based on the monetary values as a measure for the outcomes and assume a risk neutral decision maker. This is a decision maker who is indifferent between choices with equal expected outcomes, even if one choice is riskier than the other. For example, a risk neutral decision maker would have the same preference for a € 400 pay out, or a 50/50 bet with a coin toss with outcomes of € 0 (head) or € 800 (tail). Utility and risk aversion are further discussed in ater sections.
## Decision rules
Once a set of actions, circumstances and outcomes is known, various approaches can be used to come to a preferred decision. Various deterministic decisions rules are available which do not take into account the probabilities of the possible circumstances and outcomes. One example of such a decision rule is the minimax criterion: a decision maker wants to minimize maximum losses. This is in fact a risk-averse criterion. Another example is the maximax criterion: a decision maker chooses the option with the maximum income and is risk seeking.
Although these decision rules are helpful in some cases, the probability of occurrence of certain circumstances is a key feature of the decision problem. Information on the probability of outcomes is needed for an optimal choice of action(s). For example, when making a decision to start a business in soup or ice cream, the decision maker would want to know what the probabilities of rainy or sunny weather are. Selling ice cream in Dutch winter will probably not make a good (expected) profit, but it would be a profitable business in a Mediterranean summer.
Therefore it is necessary to include information on the probabilities of circumstances and outcomes, in order to determine a rational action with the highest expected value of the benefit. This theory is known as the Bayesian decision theory. In a probabilistic or Bayesian decision framework the optimal action $a*$ is defined as the one maximizing the expected utility. The following formula is found for the case with discrete outcomes.
$$
a^*: \max_a E(u(a, \theta))=\max _a \sum_\theta u(a, \theta_i) P(\theta_i)
$$
In which $u(a, \theta)$ is the utility of action a under circumstance $\theta$ and $P(\theta)$ is the probability that circumstance $\theta_{i}$ occurs.
A rational decision is choosing the action with the highest expected (utility) value or highest benefit if outcomes are expressed in monetary values. This is illustrated in the example below. Note that other examples in these lecture notes will also be based on monetary values.
````{admonition} Example: buying shares or bonds?
Suppose a person has €1000 at his or her disposal and is given the choice to invest this money in bonds or in shares of a given company. It is known that, on a yearly basis, 3 % of the current market value is distributed as interest on the bonds. The dividend of the shares depends on the company’s profit. Suppose that the board of directors have made the following agreements:
- for a profit smaller than 5% of the shareholders capital, no dividend is paid;
for a profit larger than 5% of the shareholders capital, dividend is paid, the percentage of which corresponds to 3% of the current market value of the shares;
- for a profit larger than 10% of the shareholders capital, the dividend corresponds to 6% of the current market value of the shares.
The set of actions A has two elements: $a_{1}$= investing in shares AND $a_{2}$= investing in bonds
The set or market circumstances N has three elements, namely:
$\theta_{1}$ = company profit ≤5 %
$\theta_{2}$= 5 % < company profit ≤ 10 %
$\theta_{3}$= company profit > 10 %
Assume the inflation amounts to 2 %. The set of outcomes ω contains three possible outcomes for the shares:
$\omega_{1}$= return (0 % -2 %) = -2 % per annum
$\omega_{2}$= return (3 % -2 %) = 1 % per annum
$\omega_{3}$= return (6 % -2 %) = 4 % per annum
Note that for the bonds the net outcome always yields $\omega_{2}$ =1% (i.e. 3% interest – 2% inflation). The outcomes can be illustrated using a decision tree (see {numref}`decision-tree-example`) or a table (see {numref}`ex_market_outcomes`).
```{list-table} Outcomes given the decisions ($a_1$, $a_2$) and market conditions ($\theta_{1}$, $\theta_{2}$, $\theta_{3}$).
:header-rows: 1
:name: ex_market_outcomes
* -
- Market circumstances
-
-
* - Actions
- $\theta_{1}$
- $\theta_{2}$
- $\theta_{3}$
* - $a_{1}$: buy shares
- $-$2 %
- 1 %
- 4 %
* - $a_{2}$: buy bonds
- 1 %
- 1 %
- 1 %
```
```{figure} ../figures/decision-tree-example.png
---
height: 300px
name: decision-tree-example
---
Decision tree for the example of buying shares ($a_{1}$) or bonds ($a_{2}$).
```
The deterministic decision rules can be applied to this example. Minimax would result in investing in bonds ($a_{2}$), maximax would result in buying shares.
The optimal decision can be found by taking into account the probabilities of the market circumstances. These three circumstances are assumed to be exhaustive and mutually exclusive (i.e. outcomes cannot overlap and the sum of probabilities equals 1). The probabilities are estimated at $P(\theta_{1})$ = 0.2; $P(\theta_{2})$ = 0.3; $P(\theta_{3})$ = 0.5. These probabilities can now be included in the decision tree. The expected value of the return of the actions is as follows:
Buying shares: 0.2(-2 %) + 0.3 (1 %) + 0.5 (4 %) = 1.9 %.
Buying bonds: 1%
In this case the expected outcome is larger for buying shares than for buying bonds. So for a risk neutral decision maker buying shares (a1) would be the preferred action. Note that this action also includes a probability of 0.2 of a loss. This is also expressed by a higher standard deviation of the expected outcomes for buying shares. The above example can also be extended by applying different utility functions for various outcomes.
````
In the previous example, the number of circumstances is limited and the probability distribution of the circumstances is discrete. For many decision problems this is not the case. The state of nature, for instance, can assume many values that cannot be made discrete. This, for example, would have been the case if the dividend in example 0 had been a percentage of the profit. In such cases a probability density function can be used to characterize the spectrum of outcomes. Using a continuous form of formula (3.5), the expected value of various actions, and the optimal action / decision can be identified.
In taking decisions with uncertainties, it appears that probabilistic calculation techniques are a valuable aid to reach a rational choice. This is particularly the case if risks are dependent on the possible decisions. In such cases, Bayesian decision theory minimizes the total costs (i.e. investment costs plus risk in terms of potential losses). This can best be illustrated by means of an example from the civil engineering domain.
````{admonition} Example: drainage of a construction site – decision tree
In a river polder a basement has to be built in an excavated construction site. The construction site is made of sheet piling and the bottom is sealed off with a clay layer with a thickness of approximately 2.5 m. The thickness is not known exactly but measurements indicate that a normal distribution can be used with mean $\mu_d$= 2.5 m and a standard deviation of $\sigma_{d}$ = 0.2 m.
The river cuts through the clay layer and the underlying sand layer is fed by the river, which causes high uplift pressures on the clay which is exposed in the excavation (see Figure {numref}`example-river-excavation`). If the pressure becomes too high the excavation base will push upward and water will enter the excavation in an uncontrolled way, damaging property and perhaps risking human life.
```{figure} ../figures/example-river-excavation.png
---
height: 300px
name: example-river-excavation
---
Cross-section of excavation and river.
```
Measurements of the groundwater levels indicate the maximum upward pressure, $h$, under the sealing layer in the construction period has an a normal distribution with an expected value of $\mu_{h}$ = 4 m water column and a standard deviation of $\sigma_h=0.75 \mathrm{m}$ (see Figures {numref}`example-river-excavation-2`). Since $h$ is a hydraulic head (units of $[\mathrm{L}]$), pressure is calculated by multiplying the unit weight of water ($10\:\mathrm{kN/m^3}$).
Failure occurs when the uplift pressure exceeds the weight of the clay layer, which can be described by a limit state function $Z = R - S$ with resistance $R$ and load $S$:
$$
Z=\rho_c\cdot d-\rho_w \cdot h
$$
Where $\rho_c$ is the density of clay ($20\:\mathrm{kN/m^3}$), and is assumed here to be a deterministic value. The probability of failure is thus $P(Z<0)$, which for this situation can be found by calculating the mean and standard deviation of $Z$:
$$
\mu_Z=\rho_c \mu_d-\rho_w \mu_h=20(2.5)-10(4)= 10\:\mathrm{kN/m^2}
$$
$$
\sigma_z=\sqrt{\rho_c^2 \sigma_c^2+\rho_w^2 \sigma_{\mathrm{h}}^2}=8.5\:\mathrm{kN/m}^2
$$
Since $Z$ also has the normal distribution, we find the failure probability:
$$
p_f=P(Z<0)=\Phi\left(-\mu_z / \sigma_z\right)=\Phi(-1.17)=0.12
$$
```{figure} ../figures/example-river-excavation-2.png
---
height: 300px
name: example-river-excavation-2
---
Cross-section of excavation near a river. Hashed area indicates the clay layer, which is underlain by sand. The water pressure is given by $h$ and is directly related to the river level (Figure {numref}`example-river-excavation`).
```
The effect of a drainage system in the construction site (see {numref}`example-river-excavation`) on the groundwater levels has been reviewed using groundwater flow calculations. It appears that it reduces the mean value of the maximum water levels to $\mu _{h}$ = 3.52m and the standard deviation remains the same. In this case the failure probability is reduced to 0.04. Such a drainage system costs €150,000.
The flooding of the construction site will result in estimated damages of €5,000,000, and the designer of the construction site is faced with the choice whether drainage facility is a worthwhile investment. To provide insight, the decision problem is illustrated with a decision tree, but first the set of actions, circumstances and outcomes are defined:
The set of actions A consists of:
- $\textit a_{1}$= excavation without drainage
- $\textit a_{2}$= excavation with drainage
The set of circumstances N is formed by:
- $\theta_{1}$= the sealing layer remains intact
- $\theta_{2}$= the water pressure exceeds the weight of the sealing layer
The set of outcomes $\Omega$ consists of:
- $\omega_{1}$ = nothing happens; loss = € 0
- $\omega_{2}$ = the construction excavation is flooded: loss = €5,000,000
The previous analysis has shown that the probability of flooding of the excavation equals $p_{f}$ = 0.12 for a situation without drainage and $p_{fd}$ = 0.04 with drainage.
Without drainage, the risk, defined as the expected value of the loss, is
$$
\textrm{risk, without drainage = }0.12 \cdot \euro{} 5,000,000 = \euro{} 600,000
$$
With drainage the risk is:
$$
\textrm{risk, with drainage = }0.04 · \euro{} 5,000,000 = \euro{} 200,000
$$
Costs and probabilities can also be shown in the decision tree (see Figure {numref}`example-river-excavation-3`). The expected values of the costs can be calculated for the different actions by adding the present values of the cost of actions and risk:
- $\textit a_{1}$ : expected value (additional) costs = risk =€ 600,000
- $\textit a_{2}$ : expected value (additional) costs
= extra costs + risk = € 150,000 + € 200,000 = € 350,000
This implies that the construction of the drainage system is rationally speaking the best decision for a risk neutral decision maker.
```{figure} ../figures/example-river-excavation-3.PNG
---
height: 300px
name: example-river-excavation-3
---
Decision tree with probabilities and costs.
```
````

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(econ_optimization)=
# Economic Optimization
The previous sections have focussed on decisions for which the number of actions was limited, e.g. excavation with or without drainage and the associated costs and benefits. However, there are several situations in which the number of actions is unlimited. This occurs when the failure probability level has to be decided for a system that is yet to be designed, with an infinite number of design options. An example of this type of decision problem is the heightening of dikes, as in theory an unlimited amount of values can be chosen for the elevation, e.g. 2m, 5m, 5.1m, 5.11m, 5.1111m 6m etc.
For this situation an economic optimization that takes into account the costs of increasing the safety level and reducing the risks can be applied to derive an optimal level of safety (or the optimal “failure probability”). The economic optimization was developed and applied by {cite:t}`vandantzig1956`, to derive the optimal dike height for South Holland after the 1953 storm surge disaster, as will be further elaborated in the next section.
In the economic optimization the total costs ($C_{tot}$[€]) are determined, consisting of the investments $I$[€] in a safer system and the present value of the risk $R$[€].
$$
C_{tot} = I + R
$$
The annual risk, or expected economic dagmage is found by:
$$
E(D) = P_f D
$$
where:
- $E(D)$ the expected value of the risk [€/year]
- $P_f$ the failure probablity of the system per year [1/year]
- $D$ the damage in case of failure [€]
In this approach it is thus assumed that all damages are expressed in monetary terms. Additional criteria for separately considering the loss of human life are included in the next section.
The present value of the risk for an infinite time horizon can be found as follows:
$$
R = \frac{P_f D}{r}
$$
The risk can be reduced by constructing a safer system (a lower $P_f$), or limiting the damage (smaller $D$). In this case we assume prevention measures that focus on reducing the failure probability. The investments will become a function of the failure probability of the system, since increasing the safety will lead to an increase of costs.
$$
I = I(P_f)
$$
{numref}`economic_optimum` shows the costs and risks as a function of the accepted failure probability of a system. The economic optimum is found when the total costs are minimal.
```{figure} ../figures/economic_opt.svg
---
width: 600
name: economic_optimum
---
Economic optimization: costs, risks and total costs as a function of the
failure probability of the system.
```
For this situation the following is valid:
$$
\frac{d C_{tot}}{d P_f} = 0
$$
This approach can be applied to various decisions problems, such as the optimal dike height (see next section) but also the dimensioning of other interventions such as sprinklers or ventilation in tunnels to reduce fire risks. In cases where no continuous functions are available to create a curve as in {numref}`economic_optimum`, the analyst can consider a limited number of design options (e.g. no, small, medium or large sprinklers in a tunnel) and determine investments, risks and total costs for these options.
In addition to the determination of the optimal failure probability, the cost benefit criterion should still be verified. (It is possible that we find an economic optimum with higher total costs than in the current situation without interventions). It is checked whether the benefits of risk reduction are larger than the costs of the dike heightening, i.e. $(P_{f,0} - P_{f,opt}) D > I$.

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(ex-dike-height)=
# Optimization Example
Economic optimization is illustrated through the determination of an optimial dike height to protect against flooding.
Before the major floods of 1953, dikes in the Netherlands were not designed for a specified safety level but mainly strengthened based on practical experience. One of the main questions after the disaster was the optimal dike height and the “acceptable” probability of flooding. Van Dantzig was a professor in mathematics and a member of the first Delta committee. He developed an econometric approach to determine the optimal probability of flooding (or protection level) and the corresponding dike height {cite:p}`vandantzig1956`.
## Probability and Risk
The approach only considers failure of the dikes due to overtopping. The probability distribution of water levels along the Dutch coast can be approximated by an exponential distribution.
$$
F(h) = 1 - e^{(-\frac{h-A}{B})}
$$ (prob_dist_water_levels)
In which:
- $h$ the water level [m]
- $A,B$ constants of the exponential ditribution [m]
Neglecting wave run-up, the probability of failure of the dikes - leading to flooding - can be approximated by the probability of exceedance of the dike height $h_d$, i.e.
$$
P_f = P(h>h_d) = e^{(-\frac{h_d - A}{B})}
$$ (exceedence_dike_height)
It is assumed that there is total damage to all objects and infrastructures in the flooded area if the dikes fail. For the discount rate Van Dantzig applied a reduced interest rate $r^\prime$ = economic growth – inflation. The net present value of the expected damages (i.e., the risk) is found as follows, assuming an infinite time horizon:
$$
R = \frac{P_fD}{r^\prime} = \frac{e^{-\frac{h_d-A}{B}}D}{r^\prime}
$$ (dike_height_risk)
## Costs and Benefits
As the risk is dependent on height, the dikes can be heightened further to limit the failure probability and thus risks. Investments in dike heightening are determined by initial costs for mobilization and variable costs that are dependent on the level of dike heightening (see also {numref}`dike_heightening_schematic`):
$$
I = I_0 + I(h_d - h0)
$$ (dike_investments)
```{figure} ../figures/dike_heightening_schematic.png
---
height: 200px
name: dike_heightening_schematic
---
Schematic view of dike heightening
```
Now, the total costs $C_{tot}$ are the sum of the investment costs and the risk.
$$
C_{tot} = I_0 + I(h_d - h_0) + \frac{e^{-\frac{h_d-A}{B}}D}{r^\prime}
$$ (total_costs)
The optimal dike height is found when the total costs are minimal. This is the case when $dC_{tot}/dh_d=0$.
$$
\frac{dC_{tot}}{dh_d} = I - \frac{e^{-\frac{h_d-A}{B}}D}{Br^\prime} = 0
$$ (optimal_dike_costs)
Since the $P_f = e^{-(h_d-A)/B}$, we can find the optimal flooding probability $P_{f,opt}$:
$$
P_{f,opt} = \frac{IBr^\prime}{D}
$$ (optimal_flooding_prob)
The larger the damage $D$, the smaller the optimal failure probability and thus the higher the level of protection. If incremental protection is expensive – i.e. for high values of $I$ – a higher optimal failure probability and thus a lower level of optimal safety will be found. The optimal failure probability is also dependent on the discount rate. Combination with {eq}`exceedence_dike_height` gives the optimal dike height $h_{d,opt}$:
$$
h_{d,opt} = A - B \cdot \ln(P_{f,opt})
$$ (optimal_dike_height)
## Verification
In addition to the determination of the optimal dike height the cost benefit criterion should still be verified by checking whether the benefits of risk reduction are larger than the costs of dike heightening.
$$
I(h_{d,opt})<\frac{(P_{f,0} - P_{f,opt})D}{r^\prime}
$$ (risk_reduction_check)
Where $P_{f,0}$ the flooding probability in the initial situation. It can be easily shown that this is equivalent to a check of whether the total costs in the optimal situation are smaller than those in the original situation. The first Delta committee used the following values for South Holland:
- $D$ = fl 24.2$\cdot 10^9$ [unit is Dutch guilders]
- $r^\prime$ = 0.015
- $I$ = 40.1$\cdot 10^6$ fl/m
- $A$ = 1.96 m
- $B$ = 0.33 m
- $h_0$ = 3.25 m
- $I_0$ = fl 110$\cdot 10^6$
Using these values, the following optimal dike height and optimal failure probability were derived:
- $h_{d,opt}$ = 5.83 m
- $P_{f,opt}$ = $8 \cdot 10^{-6}$ per year
## Discussion
Although the optimal safety level was determined at a failure probability of $P_{f,opt}$ = 1/125,000 per year, in later political discussions a value of 1/10,000 per year was selected for the probability of exceedance of design water levels. This implied that the dikes of South Holland would need to be designed for hydraulic conditions (water levels and waves) with a probability of exceedance of---on average---1/10,000 per year. In later decision-making, safety standards were derived for other dike rings (see {numref}`dike_rings`), for example, flood defenses in the river system are designed for a safety standard of 1/1250 per year.
```{figure} ../figures/dike_rings.jpg
---
width: 400px
name: dike_rings
---
Dike Rings in the Netherlands, showing probability of exceedance of the design water levels per dike ring. Note that dike rings along the River Meuse are not shown on this map. Most of these dike rings have a safety standard of 1/250 per year. This map defined design levels until the safety standards changed in 2017.
```
It was expected that the actual failure probability for dikes designed for this design load, would be smaller than 1/10,000 per year. Recent risk analysis in the project VNK have shown that this is not the case due to the geotechnical failure mechanisms. For most dike rings the estimated failure probabilities are larger than the probability of exceeding the design loads. For example, for riverine dike rings that have been designed for design levels with a probability of exceedance of 1/1250 per year, failure probabilities are often in the order of magnitude of 1/100 per year {cite:p}`vnk2014`. This is part of the reason for revising the safety standards and flood defence assessment criteria in the Netherlands, which since 2017 is based on an allowable probability of flooding rather than a load-based exceedance probability approach.
Historic discussions and developments in the Netherlands illustrate how the risk analaysis and risk management for a given system can change over time. Several extensions and additions to the van Dantzig model have been proposed over the years. For example, the inclusion of sea level rise {cite:p}`vrijling1990`, modelling of the damage as dependent on the water depth in the polder {cite:p}`vandantzig1956`, the inclusion of the economic value of loss of life as part of the damage, and inclusion of risk aversion by giving quadratic weight to the damages {cite:p}`vangelder1997`.
The model by van Dantzig was primarily focused on finding what the optimal safety at that time (when van Dantzig published his model in the 1960’s) should have been. However, a single optimal level is not always the best solution. While considering longer timescales and changing conditions, such as economic growth and sea level rise, the model needs to address the possibility of multiple interventions during the period considered. This leads to questions regarding the timing of interventions (when? At which intervals?), as well as the optimal strengthening or raising of the dikes (how much?). {cite:t}`Eijgenraam2006` developed an economic decision model that takes into account both questions.
{numref}`periodic_investments` below depicts the “saw-tooth” curve that shows the periodic interventions: both the extent of the intervention (vertical) as well as the timing between interventions (horizontal). In between interventions, the safety will gradually decrease due to sea level rise and / or subsidence of the dike. Over a longer period of time, the dike heightening (or strengthening) should follow sea level rise. Additional dike heightening (lower the failure probability even further) could be considered in case of economic growth, which will lead to an increase of damages and increase of protection standards. The length of the optimal interval between interventions is largely dependent on the initial or mobilization costs. If these are high, for example in the case of structural interventions such as storm surge barriers, a long interval or life time of almost 100 years can be chosen. For regular dike reinforcements this interval will be more in the range of several decades (e.g. 50 years). For interventions with no or very small initial costs, such as nourishments along the coast, it is optimal to intervene more frequently.
```{figure} ../figures/periodic_investments.png
---
width: 400px
name: periodic_investments
---
Periodic investments in dike reinforcement for a situation with sea level rise.
```

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# Risk Evaluation
This chapter focuses on the evaluation of risk (Step 4) by applying several different techniques. While the Sections may seem long, the methods are relatively simple and a lot of examples and explanations are provided to illustrate how they can be applied in practice. After this chapter, you will be ready to try all of the practice problems in the Exercise Chapter.
---
_The primary author for this chapter is Bas Jonkman. Robert Lanzafame contributed through rearrangement of the topics and formatting._

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# Safety Standards
When answering the question “how safe is safe enough” a merely economic treatment with cost benefit analysis or economic optimization is often not sufficient for activities with risks to people. Therefore, criteria have been developed that focus on risks to human life. This section focuses on safety standards and criteria for evaluating the risk to life.
Two aspects are typically considered when evaluating and regulating risks to the public: the total or population-wide effects, and the distribution of effects within the affected population. {numref}`risk_evaluation` summarizes these perspectives. The societal perspective is concerned with ‘total effect’ and the effects of large-scale accidents on the society, in terms of economic damages and life loss. The individual perspective is concerned with distributive justice (‘equity’), i.e. the distribution of harm over the population.
As risk is often the by-product of an otherwise legitimate and advantageous activity, such as production or transportation, regulating risks is essentially a balancing act between economic and social activities on the hand and a sufficiently safe society on the other hand (e.g. {cite:t}`jongejan2008`): just as too lenient regulations are suboptimal, too stringent ones are too.
```{list-table} Overview of perspectives on risk evaluation
:header-rows: 1
:name: risk_evaluation
* - Aspect
- Relevant risk metric
- Rationale
* - Equity ('distributive justice')
- Individual risk
- Regulation is to prevent that individuals are exposed to disproportionally large risks
* - Total effect
- Societal and economic risk
- Regulation is to limit large-scale accidents and optimize the costs and benefits of risk reduction
```
## Three Types of Risk
Based on the general concepts described above, it has been proposed to evaluate risks based on three criteria {cite:p}`TAW1985,vrijling1995,vrijling1998`:
- limit the **invidual risk** to prevent that certain people are exposed to disproportionally large risks;
- limit the **societal risk** to limit the risks of large scale accidents with many fatalities;
- **Economic optimization** to balance investments in risk reduction from an economic point of view.
### Societal and Economic Risk
Two of these criteria have already been discussed elsewhere in this book. **Societal risk** refers to the probability of an accident with multiple fatalities and is often graphically represented by an FN-curve ({ref}`risk_curve` Section). **Economic optimization** has already been discussed in the {ref}`econ_optimization` Section.
### Individual Risk
**Individual risk** concerns the probability of death of a person due to an accident. Various related definitions exist. The “average individual risk” for a certain activity can be calculated, e.g. the individual risk due to smoking or the risk of an airplane crash for a frequent flyer. {numref}`death_probability` compares the probability of various types of fatal accidents.
```{list-table} Average probability of death due to several activities. (statistics based on various sources for the Netherlands and Europe)
:header-rows: 1
:name: death_probability
* - Activity
- Probability (per year)
* - Mountain climbing
- $10^{-2}$
* - Traffic (young men)
- $10^{-4}$
* - Accidents at home
- $10^{-4}$
* - Structural failures
- $10^{-7}$
```
The individual risk due to an accident can be calculated with:
$$
IR = P_f P_{d|f}
$$
where:
- $IR$ the individual risk [1/year]
- $P_{f}$ the probability of an accident [1/year]
- $P_{d|f}$ the conditional probability of death given an accident
Both the individual risk and $P_{d|f}$ can be formulated as a function of a location near a risk source. If it is assumed that a person is permanently present at that location, the individual risk becomes a property of that location and it can be used for zoning and land use planning. This is applied in the industrial safety policy in the Netherlands (see next section).
```{admonition} Example: Difference between individual and societal risk.
To illustrate the difference between the individual and societal perspective, consider a fairly safe car that has a probability of 10$^{-5}$ per year of causing a fatality due to technical failure. This may well be acceptable to an individual. Such a probability is in the same range as the average death rate in traffic (650 fatalities / 17 million people which is about 4.10$^{-5}$ per year). If only 1000 cars are sold in the Netherlands, the number of fatal accidents with this car is rare, i.e. 0.01 fatalities per year. However, when the car becomes popular and 10 million cars are sold, the average number of fatalities due to technical failure becomes 100 per year. This may well lead to public concerns and indignation. From a societal point of view this may no longer be acceptable, whereas the individual risk has not changed.
```
## Limits for Individual and Societal Risk
Safety standards can be used to set limits to the individual and societal risk. A **limit** value can be set for **individual risks**. Such a limit value is to avoid disproportionate exposures by laying down a minimum safety level for all individuals. In various fields of applications limit values in the range of 10$^{-4}$ to 10$^{-6}$ per year are used (see below for more information). To put the stringency of the individual risk criteria into perspective, one could consider the effect of the probability of an accident on life expectancy. When an average person would be constantly exposed to a maximum allowable level of risk of 10-6 per year, the decrease of his or her life expectancy would be only 1 day as shown in {numref}`individual-risk-life-expectancy`.
```{list-table} Relationship between the additional risk and life expectancy
:header-rows: 1
:name: individual-risk-life-expectancy
* - Additional probability of death in a year
- Life expectancy in years
- Decrease of life expectancy in years
- Decrease of life expectancy in days
* - $10^{-3}$
- 74.97
- 3.16
- 1153
* - $10^{-4}$
- 77.81
- 0.32
- 117
* - $10^{-5}$
- 78.10
- 0.03
- 11
* - $10^{-6}$
- 78.13
- 0.00
- 1
* - $10^{-7}$
- 78.13
- 0.00
- 0
```
Societal risk can be evaluated by means of an FN limit line. The calculated FN curve of the system should, in principle, not exceed the limit line. An FN-criterion is defined by three variables:
1. A base point: the exceedance probability of 1 fatality (i.e. $C$),
2. A slope, $\alpha$, and
3. Probability and/or consequence cut-offs ($A$ and $B$).
{numref}`limit_line` schematically illustrates the different criterion that could make up an FN limit line.
```{figure} ../figures/fn_limit00.svg
---
height: 600px
name: limit_line
---
Schematic FN curve illustrating limit line formulation with horizontal and vertical cut-offs.
```
The general formulation for such a limit line without horizontal of vertical cut-off equals:
$$
1 - F_N(n) \leq \frac{C}{n^\alpha}
$$ (limit_line)
where:
- $C$ a constant that determines the vertical position of the limit line
- $\alpha$ a coefficient that determines the steepness of the limit line
The limit line is called *risk neutral*[^neutral] if $\alpha=1$, since it places equal weight on exceedance probabilities and numbers of fatalities. If $\alpha = 2$, the limit is *risk averse.* This means that given an arbitrary point, the exceedance probability of 10 times as many fatalities should be 100 times lower. This has been motivated by public aversion to large numbers of fatalities. For example, the loss of 1000 people in one accident (e.g. a major explosion) could be valued differently than 1000 losses of 1 person in separate accidents (e.g. in traffic).
For different applications limit lines have been developed with varying constants and steepness. Examples of application areas include industrial risks in the Netherlands (next section), dams in the United States and Canada, and chemical risks in Hongkong and the UK {cite:p}`jonkman2003`.
````{admonition} Exaple: Risk Matrix
Risk matrices are often used in various industries for risk evaluation and decision support, for example, to quickly prioritize actions, especially in time-sensitive situations. For a given undesired event the extent of probability and consequences are estimated on a qualitative or semi-quantitative scale, see {numref}`risk_matrix` for an example. Ranges of failure probabilities or consequences can be assigned to the qualitative terms on the axes in the example. The combination of probability and consequence determines the level of risk, and depending on the application, whether it is acceptable or whether it requires more attention and risk reduction efforts. However, unlike the FN curve, the cumulative effects of multiple possible events are generally not considered in a risk matrix.
```{figure} ../figures/risk_matrix.png
---
width: 500px
name: risk_matrix
---
Example of a risk matrix used by various engineering and military branches of the United States Army Corps of Engineers. Source: [this USACE document](https://home.army.mil/lee/application/files/7815/3809/1878/Tables.pdf) (opens a PDF); general explanation can be found via Engineering Manual EM-385-1-1 {cite:p}`usace14`, or online at USACE [Risk Management Center](https://www.rmc.usace.army.mil/) or [Risk Analysis Gateway](https://www.iwr.usace.army.mil/Missions/Risk-Analysis-Gateway/Risk-Assessment/).
% Note: real source is a PDF [here](https://home.army.mil/lee/application/files/7815/3809/1878/Tables.pdf)
% Formal definition is in this: Manual EM 385-1-1 https://www.publications.usace.army.mil/portals/76/publications/engineermanuals/em_385-1-1.pdf
```
````
## Case Study: Industrial Hazards
The Dutch major hazards policy deals with the risks to those living in the vicinity of major industrial hazards such as chemical plants and LPG-fuelling stations. The development of the Dutch major hazards policy was strongly incident driven, as were European efforts aimed at the prevention of major industrial accidents. After a number of severe industrial accidents, including the Bhopal accident in 1984 which killed an estimated 3000 people and severely injured over 200.000, a European directive was drafted concerning the prevention of major accidents: the 1982 Seveso Directive. This was later replaced by the Seveso II Directive.
The cornerstones of the Dutch major hazards policy are a) the use of quantitative risk analysis (QRA); b) comparison of QRA outcomes with limits to individual and societal risks {cite:p}`bottelberghs2000`.
Within the Dutch major hazards policy, individual risk is defined as the probability of death of an average, unprotected person that is constantly present at a certain location. It is thereby a property of location and iso-risk contours can be plotted on a map (see {numref}`risk_contour`). Individual risk is therefore also named local risk (“plaatsgebonden risico”) in the Netherlands. The shape of the risk contours for other applications will look different. For airports the contours will follow the shape of the runway and flight paths, for polders and flooding the risk contours will be highest in the deepest part of the polder.
```{figure} ../figures/risk_contour.png
---
height: 400px
name: risk_contour
---
Example of a schematic individual risk contour (circles) for a chemical installation.
```
```{list-table} Individual risk criteria used in the Netherlands.
:header-rows: 1
:name: individual-risk-criteria
* -
-
- Individual (or local) risk criterion
* - **Existing situations**
- Vulnerable
- $10^{-6}$ per year
* -
- Limitedly vulnerable
- $10^{-5}$ per year, strive for $10^{-6}$ per year
* - **New situations**
- Vulnerable
- $10^{-6}$ per year
* -
- Limitedly Vulnerable
- $10^{-6}$ per year
```
A distinction is made between vulnerable objects such as schools and houses and limitedly vulnerable objects such as small offices. The following criteria apply. For new situations a limit of $10^{-6}$ per year applies. A comparison with {numref}`individual-risk-criteria` shows that this individual risk value has a negligible effect on life expectancy.
The criterion for societal risk that is used in the Netherlands for evaluating the third party risks posed by major industrial hazards is shown in {numref}`fn_limit` below. It serves as a reference in the broader assessment of third party risks by competent authorities. Exceedances of the criterion line also have to be motivated by competent authorities. When the criterion line is not exceeded, the acceptability of the third party risk still has to be motivated. The limit line is characterized by $C=10^{-3}$ and a steepness of $\alpha = 2$, making it a risk averse criterion. The criterion is used to assess the acceptability of the risks of individual facilities.
```{figure} ../figures/fn_limit01.svg
---
width: 600
name: fn_limit
---
FN limit line used for installations in the Netherlands, with hypothetical risk curve.
```
## Case Study: Flood Protection
For systems for which no regulations are available the question 'how safe is safe enough?' can be difficult to resolve. A general model has been developed by the Technical Advice Committee for Water defences (TAW)[^taw] in the process of deriving safety standards for flood protection in the Netherlands, which, in a formal sense, has been an ongoing process since the 1950's. This approach is derived on the basis of accident statistics {cite:p}`TAW1985,vrijling1995,vrijling1998`, where underlying assumption of the model is that accident statistics are the result of a process of societal optimisation and that they thereby reflect what is apparently considered acceptable by society at large. Such an approach is commonly referred to as a 'revealed preference' approach, and was already introduced in a a general sense in the Chapter on Risk Analysis, Section {ref}`risk_curve_famous` ({numref}`risk-curve-baecher`).
### Individual risk
Accident statistics reveal that the extent to which participation in the activity is voluntary strongly influences the level of risk that is accepted by individuals. Relatively high individual risks are accepted for activities that are voluntary and have a (personal) benefit, such as mountain climbing. Much smaller individual risk values are accepted for involuntary activities for which the risks are imposed by others, e.g. for chemical and nuclear industry. A policy factor ($\beta$) is therefore introduced to account for voluntariness of exposure. This factor is set at $\beta=1$ for an individual risk value of $10^{-4}$ per year. This represents the “baseline” individual risk for the group young men[^men] who are most at risk of dying in a traffic accident.
:::{list-table} Accident statistics and proposed policy factor and characteristics of the activity {cite:p}`CUR1997,vrijling1998,vrijling2001`.
:header-rows: 1
:name: accident_statistics
* - Prob. of death(per year)
- Exmaple/application
- $\beta$
- Voluntariness
- Benefit
* - $10^{-2}$
- Mountain climbing
- 100
- Voluntary
- Direct benefit
* - $10^{-3}$
- Driving a motor cycle
- 10
- Voluntary
- Direct benefit
* - $10^{-4}$
- Driving a car
- 1
- Neutral
- Direct benefit
* - $10^{-5}$
- Flooding
- 0.1
- Involuntary
- Some benefit
* - $10^{-6}$
- Factory/LPG station
- 0.01
- Involuntary
- No benefit
:::
The proposed individual risk limit becomes:
$$
IR < \beta \cdot 10^{-4}
$$ (individual_risk)
An appropriate value for the policy factor $\beta$ can be chosen depending on the characteristics of the activity. If the conditional probability of death due to an accident $P(d|f)$ is known, the acceptable failure probability can be computed:
$$
P_f \leq \frac{\beta \cdot 10^{-4}}{P_{d|f}}
$$ (acceptable_failure_prob)
Note that the Dutch individual risk criterion for hazardous installations would be obtained for $\beta = 0.01$ and a conditional probability of death of 1 ($P_{d|f} = 1$).
### Societal risk
The societal risk criterion proposed by the TAW is based on the thought that societal risk should be evaluated primarily at a national level as local developments may lead to a situation that is considered unacceptable by society as a whole {cite:p}`vrijling1995`. The societal risk criterion at a national scale proposed by the TAW is:
$$
E(N) + k \sigma(N) \leq \beta \cdot 100
$$ (societal_risk_criterion)
where:
- $E(N)$ the expected number of fatalities per year
- $k$ the risk aversion index (proposed value, $k=3$)
- $\sigma(N)$ the standard deviation of the number of fatalities per year
- $\beta$ the policy factor
A risk aversion index $k$ has been introduced to account for risk aversion. For accidents with small probabilities and large consequences the standard deviation $\sigma(N)$ is large relative to $E(N)$, see example below. The total risk takes a risk aversion index $k$ [-] into account. For $k>1$, the “cost of risk bearing” exceed expected loss, implying a risk averse attitude.
````{admonition} Expected value and standard deviation for two systems
We consider two sytems
1. This sytem has a high failure probability of 0.01 per year and 1 fatality
2. The second system has a smaller failure probability of 0.0001 per year but a higher number of 100 fatalities.
For both systems a binomical distribution of the number of fatalities is applied meaning that the number of fatalities in case of failure is exactly known. The expected value and standard deviation of the number of fatalities are found as follows:
$$
E(N) = P_fN \;\;\;\; \sigma^2(N) = P_f(1-P_f)N
$$ (binomial_moments)
The resulting expected value and standard deviation are shown in {numref}`exp_val_std_dev_two_activities` below. Although both systems have the same expected value, the standard deviation for the “small probability – large consequence” event for system 2 is higher. Taking into account the standard deviation in the TAW criterion thus accounts for risk aversion against accidents with large numbers of fatalities.
```{list-table} Calculation of expected value and standard deviation for two activities.
:header-rows: 1
:name: exp_val_std_dev_two_activities
* -
- $P_f$
- $N$
- $E(N)$
- $\sigma(N)$
* - 1
- $10^{-2}$
- 1
- $10^{-2}$
- 0.0099
* - 2
- $10^{-4}$
- 100
- $10^{-2}$
- 0.995
```
The next step would be to distribute this maximum allowable level of societal risk over individual installations. After all, locally imposed societal risk criteria are necessary for achieving the desired national level of societal risk. The translation of the nationally acceptable level of risk to a criterion for a single local installation depends on the type of probability distribution of the number of fatalities. In {cite:t}`vrijling1998` a formulation of the risk acceptance at a local level is presented conform {eq}`limit_line`:️
$$
1 - F_n(n) \leq \frac{C}{n^\alpha}
$$
For a binomial distribution this yields:
$$
C = \left( \frac{\beta \cdot 100}{k \sqrt{N_a}} \right)^2
$$
in which $N_a$ is the number of independent locations where the activity takes place. This requirement corresponds to the requirement set for the chemical installations if $\beta=0.03$, $N_a=1000$ and $k=3$.
````
### Combination of Risk Types
According to the approach by TAW the three approaches (individual, societal and economic risk) lead to three acceptable failure probabilities. The most stringent of the three criteria can be chosen to determine the acceptable probability of failure of the system and to make sure that all three conditions are fulfilled. This can best be illustrated with an example (see below). The principles of this approach have been applied to derive the proposed new safety standards for flood defences in the Netherland by the {cite:t}`delta2014`.
````{admonition} Example: Combination of individual, societal and economic risk for a dike ring area
We consider application of the three criteria to the case of dike rings in the Netherlands. There are about 100 dike rings in the Netherlands of different sizes. A dike ring is a flood prone area protected from flooding by flood defences and high grounds. The conditional probability of death given flooding depends on the depth of the polder and flood characteristics. Research on loss of life due to floods shows that a conservative estimate would lead to $P_{d|f} = 0.1$.
First, we consider the individual risk. A value of $\beta = 0.1$ is proposed as being exposed is considered as an involuntary activity with some benefit (i.e. living in a prosperous delta). This leads to an acceptable individual risk value of $10^{-5}$ per year. This limit has also been proposed by the Dutch government in the year 2014 (“basisveiligheid”). The acceptable flooding probability according to the individual risk becomes:
$$
P_f \leq \frac{\beta \cdot 10^{-4}}{P_{d|f}} = \frac{0.1 \cdot 10^{-4}}{0.1} = 10^{-4}\text{ per year}
$$ (acceptable_flooding_probability)
The societal risk criterion can be determined according to {eq}`limit_line`. Assuming a risk averse criterion $\alpha=2$. We can determine the constant $C$ of the limit line for $N_a = 100$ installations and $\beta=0.1$.
$$
C = \left( \frac{\beta \cdot 100}{k \sqrt{N_a}} \right)^2 = \left( \frac{0.1 \cdot 1000}{3 \sqrt{100}} \right)^2 = 0.11
$$
The limit line for societal risk becomes $1 - F_N(n) \leq 0.11/n^2$. Both the individual and societal risk criteria are plotted in below. As a third criterion the economic optimization can be added. The optimal or acceptable probability of failure depends on the damage and investment costs. A relationship with the graph below can be established by assuming that the number of fatalities is related to the economic damages. A dike ring with many inhabitants and potential fatalities will generally also represent a large economic value. For the sake of the example we assume that every fatality corresponds to an economic damage of €$5 \cdot 10^7$ (note: this is not equal to the value of a human life). To calculate the economic optimum for the example we assume arbitrary values of $r=0.025$ and €$5 \cdot 10^6$ $B=0.33$. {numref}`combined_risk_criteria` below shows the combination for the three criteria.
For a given number of fatalities in a dike ring the acceptable failure probability according to the three criteria can be derived. The individual risk criterion is independent on the number of fatalities. The economic criterion shows a linear relation between the failure probability and damage or number of fatalities. The societal criterion is risk averse so shows a decreasing quadratic relationship between acceptable failure probability and consequences.
For a given number of inhabitants and potential fatalities for a dike ring, the acceptable failure probability can be determined. For dike rings with between 1 and 10 fatalities – generally small areas-the individual risk criterion is the most stringent. For dike rings with large numbers of fatalities, the risk averse societal risk criterion becomes dominant.
Several extensions of this model are possible. One can consider to add the economic value of life loss or consider a different distribution of the nationally acceptable societal risk over dike rings with different sizes.
```{figure} ../figures/risk_types.svg
---
width: 600
name: combined_risk_criteria
---
Combination of individual, societal and economic risk criteria for a
hypothetical example.
```
````
One field of application where these concepts have been applied concerns the flood protection standards in the Netherlands. Since 2017 new safety standards have been in place for primary flood defences in the Netherlands that have been derived based on a risk assessment, with standards been formulated as a (tolerable) probability of failure of the flood defences.
[^neutral]: It should be noted that the usage of the term “risk neutral” to describe the FN-curve limit line for $\alpha = 1$ is widespread but, strictly speaking, incorrect in the context of decision theory (see {ref}`decision` Section). This is because the cost of risk bearing to a risk neutral decision maker equals expected loss (i.e. the product of probabilities and damages). The FN-curve shows cumulative probabilities, however. Also, an individual crossing of the limit line would not necessarily disturb a risk neutral decision maker, provided the other accident scenarios have relatively small probabilities.
[^taw]: TAW is nowadays called ENW: Extertise Network on Flood Protection.
[^men]: The group 'young men' is used as a baseline because this was the only information available at the time the study was completed.