diff --git a/README.md b/README.md index 293c30b..b77fd83 100644 --- a/README.md +++ b/README.md @@ -1,5 +1,7 @@ # Template +_Update: Feb 23 transferred files from [Risk and Reliability Book](https://gitlab.tudelft.nl/interactivetextbooks-citg/risk-and-reliability/) to set things up._ + This will be the future home of the template book. Design philosophy: - people will fork this to create a book in as easy-as-possible a way - minimal changes should be made to the book contents itself, to prevent downstream merge conflicts diff --git a/book/exercises/example-quiz.json b/book/exercises/example-quiz.json new file mode 100644 index 0000000..53de808 --- /dev/null +++ b/book/exercises/example-quiz.json @@ -0,0 +1,204 @@ +[ + { + "question": "Choose all of the following that can be included in Jupyter notebooks?", + "type": "many_choice", + "answers": [ + { + "answer": "Text and graphics output from Python", + "correct": true, + "feedback": "Correct." + }, + { + "answer": "Typeset mathematics", + "correct": true, + "feedback": "Correct." + }, + { + "answer": "Python executable code", + "correct": true, + "feedback": "Correct." + }, + { + "answer": "Formatted text", + "correct": true, + "feedback": "Correct." + }, + { + "answer": "Live snakes via Python", + "correct": false, + "feedback": "I hope not." + } + ] + }, + { + "question": "Which of these are used to create formatted text in Jupyter notebooks?", + "type": "multiple_choice", + "answers": [ + { + "answer": "Wiki markup", + "correct": false, + "feedback": "False." + }, + { + "answer": "SVG", + "correct": false, + "feedback": "False." + }, + { + "answer": "Markdown", + "correct": true, + "feedback": "Correct." + }, + { + "answer": "Rich Text", + "correct": false, + "feedback": "False." + } + ] + }, + { + "question": "Enter the value of $\\pi$ to 2 decimal places.", + "type": "numeric", + "answers": [ + { + "type": "value", + "value": 3.14, + "correct": true, + "feedback": "Correct." + }, + { + "type": "range", + "range": [ + 3.142857, + 3.142858 + ], + "correct": true, + "feedback": "True to 2 decimal places, but you know $\\pi$ is not really 22/7, right?" + }, + { + "type": "range", + "range": [ + -100000000, + 0 + ], + "correct": false, + "feedback": "$\\pi$ is the AREA of a circle of radius 1. Try again." + }, + { + "type": "default", + "feedback": "$\\pi$ is the area of a circle of radius 1. Try again." + } + ] + }, + { + "question": "Enter the value of $\\pi$ to 2 decimal places.", + "type": "numeric", + "precision": 2, + "answers": [ + { + "type": "value", + "value": 3.14, + "correct": true, + "feedback": "Correct." + }, + { + "type": "range", + "range": [ + 3.142857, + 3.142858 + ], + "correct": true, + "feedback": "True to 2 decimal places, but you know $\\pi$ is not really 22/7, right?" + }, + { + "type": "range", + "range": [ + -100000000, + 0 + ], + "correct": false, + "feedback": "$\\pi$ is the AREA of a circle of radius 1. Try again." + }, + { + "type": "default", + "feedback": "$\\pi$ is the area of a circle of radius 1. Try again." + } + ] + }, + { + "question": "Determine the output of the following Python code:", + "code": "a=\"1\"\nb=\"2\"\nprint(a+b)", + "type": "multiple_choice", + "answers": [ + { + "answer": "1", + "correct": false, + "feedback": "No. When strings are operated on by +, they are concatenated." + }, + { + "answer": "2", + "correct": false, + "feedback": "No. When strings are operated on by +, they are concatenated." + }, + { + "answer": "3", + "correct": false, + "feedback": "No. When strings are operated on by +, they are concatenated." + }, + { + "answer": "12", + "correct": true, + "feedback": "Yes. The + operator will concatenate the strings \"1\" and \"2\"." + }, + { + "answer": "error", + "correct": false, + "feedback": "No. The + operator for strings performs string concatenation." + } + ] + }, + { + "question": "The variable mylist is a Python list. Choose which code snippet will append the item 3 to mylist.", + "type": "multiple_choice", + "answers": [ + { + "code": "mylist+=3", + "correct": false + }, + { + "code": "mylist+=[3]", + "correct": true + }, + { + "code": "mylist+={3}", + "correct": false + } + ] + }, + { + "question": "Which of these is the ratio of a circle's circumference to its diameter?", + "type": "multiple_choice", + "answers": [ + { + "answer": "$\\pi$", + "correct": true, + "feedback": "Correct." + }, + { + "answer": "$\\frac{22}{7}$", + "correct": false, + "feedback": "$\\frac{22}{7}$ is only an approximation to the true value." + }, + { + "answer": "3", + "correct": false, + "feedback": "This is a crude approximation to the true value." + }, + { + "answer": "$\\tau$", + "correct": false, + "feedback": "True for the ratio of the circle's circumference to its radius, not diameter." + } + ] + } +] \ No newline at end of file diff --git a/book/exercises/exercise-dam.md b/book/exercises/exercise-dam.md new file mode 100644 index 0000000..8adb8f6 --- /dev/null +++ b/book/exercises/exercise-dam.md @@ -0,0 +1,154 @@ +# Dam and River + +You are tasked to analyse the safety of a dam and the downstream river system. Downstream of the dam there is a dike ring protected by two main dike sections which must be able to contain water released from the reservoir. The dike sections are connected to each other and form a continuous boundary along the same side of the river. Let $P(F_1)=0.01$ and $P(F_2)=0.01$ denote failure probability of dike section 1 and section 2 respectively. There appears to be a correlation between failures of both sections, $\rho_{1,2}=0.9$, and the figure below shows the effect of the correlation coefficient on the joint failure of the two sections. + +%commented paragraphs were used for an exam question with a fault tree + +%You are tasked to analyse the safety of a dam. The dam itself can fail due to instability, and another failure mechanism is failure of the spillway due to uncontrolled erosion that could reach the body of the dam. If water levels in the reservoir become high, water is released from the reservoir into the spillway due to overflow, which happens approximately once every 10 years. + +%When water is released through spillway, there are two situations that could lead to a weak spillway prone to erosion and failure: 1) hidden damage to the spillway structure that cannot be detected in yearly inspections, and 2) a crack in the concrete of the spillway which has been missed during the yearly inspection. + +%Downstream of the dam there is an area protected by two main dike sections which must be able to contain water released from the reservoir. A dike section fails if the river discharge exceeds a critical discharge $Q_c$ with a probability of exceedance of 1/100 per year, which leads to flooding of the area. There appears to be a correlation between failures of both sections, $\rho_{1,2}=0.9$, and the figure below shows the effect of the correlation coefficient on the joint failure of the two sections. + +```{figure} ../figures/system-corr-coeff.PNG +--- +width: 400px +name: corr-coeff-or +--- +Effect of correlation coefficient joint failure, $P(A \cap B)$ (assume that $P(A \cap B)=0.003$ at $\rho=+0.9$, where the arrow is located). +``` + +**Question 1:** Estimate the probability of flooding of the area protected by the two dike sections. Assume that the failure probability of each mechanism is 0.01, and they are _not_ independent; however, you can use the figure above to quantify this. + +```{admonition} Answer +:class: tip, dropdown + +Flooding of the protected area occurs when at least one of the two dike section fails. This is a series system. Thus, probability of flooding of the protected area can be determined by: + +$$ + P(\textrm{flooding}) &= P(\textrm{dike sections 1 fails}\:\cup \textrm{dike section 2 fails})\\ + P(F_1 \cup F_2) &= P(F_1) + P(F_2) - P(F_1 \cap F_2) +$$ + +Based on the figure and the correlation $\rho_{1,2} = 0.9$, the joint probability of failure $P(F_1 \cap F_2)=0.003$ and the series failure probability is + +$$ + P(\textrm{flooding}) = 0.01 + 0.01 - 0.003 = 0.017 \: \mathrm{/year} +$$ +``` + +## Dam repair: probabilistic planning + +The dam has to be repaired. The responsible minister wants to make sure the whole project is finished within 12 months, and only accepts a probability of 10% that actual construction works exceed this duration. The dam reinforcement consists of two activities. The second activity can only start when the first one has been finished. The durations of both activities are uncertain and normally distributed. The first activity has a mean duration of $\mu_1=4$ months and a standard deviation of $\sigma_1 = 1$ month; for the second activity $\sigma_2 = 2$ months. + +**Question 2:** Compute the mean duration of the activity 2, so that the project duration would fall within the criterion given by the minister. + +```{admonition} Answer +:class: tip, dropdown + +Let $T_1$ and $T_2$ denote the duration of activities 1 and 2; $T$ denotes the total duration; $T_1$ and $T_2$ are normally distributed: + +$$ + T_1 \sim \mathcal{N}( + \mu_1=4,\sigma_1=1) \\ + T_2 \sim \mathcal{N}(\mu_2=?,\sigma_2=2) +$$ + +$T = T1 + T2$ is also normally distributed, where + +$$ +\mu_T &= \mu_1 + \mu_2 \\ +\sigma_T &= \sqrt{\sigma_1^2 + \sigma_2^2} = \sqrt{5} = 2.326 +$$ + +The probability of project duration greater than 12 months can be found with the complementary standard normal CDF, and should be a maximum of 0.1: + +$$ +1-\Phi\left[\frac{12\:\mathrm{months}-\mu_T}{\sigma_T}\right]=0.1 +$$ + +Applying the inverse CDF allows us to solve for $\mu_1$: + +$$ +\frac{12-\mu_T}{\sigma_T} &=\Phi^{-1}[0.9] \\ +\frac{12-(\mu_1 + \mu_2)}{\sigma_T} &=1.2816 +$$ + +$$ +\mu_1 &= 12 - 1.2816 \cdot \sigma_T - \mu_2 \\ +\mu_1 &= 12 - 1.282(2.326) - 4 = 5.018\:\mathrm{months} +$$ + + +``` + +## Acceptable risk + +The government wants to study the acceptable risk and the acceptable probability of dam failure. Two criteria are used for acceptable risk: + +- Societal risk (SR): an FN limit line: $P_f \leq 10^{-2}/n^2$. +- Individual risk (IR): $IR \leq 10^{-4}$ per year. + +We know that mortality in case of dam failure is $F_{d|f} = 0.1$ and that the population living downstream of the dam is 1000 people. + +**Question 3:** Plot the FN limit line and indicate the acceptable probability of failure according to both criteria (individual risk, societal risk). + +````{admonition} Answer +:class: tip, dropdown + +```{figure} ../figures/exercise-dam-fn-limit-line.png +--- +width: 500px +--- +``` +```` + +**Question 4:** Provide an advice on the acceptable probability of failure for the dam. Give the proposed numerical value and a short motivation. + +```{admonition} Answer +:class: tip, dropdown + +Based on individual risk criterion, we have: + +$$ + IR = P_f P_{d|f} \leq IR_{acc} +$$ + +where $P_{d|f}=0.1$. Hence, the acceptable probability of failure is: + +$$ + P_f \leq \frac{IR_{acc}}{P_{d|f}} = \frac{10^{-4}}{0.1} = 10^{-3} \: \mathrm{/year} +$$ + +Based on societal risk criterion (FN-Curve): + +Acceptable failure probability can be determined by the FN-Curve: + +$$ + P_f = 1 - F_N(n) \leq \frac{C}{n^\alpha} = \frac{10^{-2}}{n^2} +$$ + +Based on given information above, expected number of casualties per dam failure event is $n = F_{d|f}\cdot 1000 = 100$. + +$$ + P_f &\leq \frac{10^{-2}}{100^2} \\ + P_f &= 10^{-6} \: \mathrm{/year} +$$ + +The most stringent of the two criteria applies, so the proposed safety standard would be $10^{-6}$ per year for the given inputs. +``` diff --git a/book/exercises/exercise-fn-curve.md b/book/exercises/exercise-fn-curve.md new file mode 100644 index 0000000..62f9102 --- /dev/null +++ b/book/exercises/exercise-fn-curve.md @@ -0,0 +1,68 @@ +(ex_fn_curve)= +# FN Curve + +A small town is located in a seismic region threatened by two faults. Fault A has a 10% probability of occurring each year, which would cause 10 fatalities, and Fault B has a 1% chance of occurring, which would cause 100 fatalities. Construct an FN curve for the town. Assume that the town is risk averse and that the tolerable risk limit is governed using a constant of C=1. + +**Question 1:** Compute the expected annual fatalities for the town. + +```{admonition} Answer +:class: tip, dropdown + +$$ +E(N) = P_1 \cdot N_1 + P_2 \cdot N_2 = 0.1 \cdot 10 + 0.01 \cdot 100 = 2 +$$ +``` + +**Question 2:** If the town improved the safety of its buildings, what would be the maximum allowable number of fatalities that would be considered acceptable due to an earthquake on fault A? + +```{admonition} Answer +:class: tip, dropdown + +Using the limit curve + +$$ +1 - F_n(n) = C / n^\alpha +$$ + +$$ +0.1 = 1 / n^2 +$$ + +$$ +n = \sqrt{10} \approx 3 +$$ +``` + +**Question 3:** What is the maximum value for the probability of an earthquake on Fault B that would be considered acceptable by the town? + +```{admonition} Answer +:class: tip, dropdown + +Using the limit curve + +$$ +1 - F_n(n) = C / n^\alpha +$$ + +$$ +1 - F_n(100) = C / 100^2 = 0.0001 = 10^{-4} +$$ +``` + +**Question 4:** Create an FN curve for this situation, including the limit line. Be sure to clearly label the axes and the values of key points on your diagram. + +```{note} +If you are not sure what the limit line is, wait for this question until you read the Safety Standards Section (Risk Evaluation chapter). +``` + +````{admonition} Answer +:class: tip, dropdown + +```{figure} ../figures/exercise-FN-curve.png +--- +width: 600px +name: FN-curve +--- +FN-curve for exercise 4 +``` +```` \ No newline at end of file diff --git a/book/exercises/exercise-paint.md b/book/exercises/exercise-paint.md new file mode 100644 index 0000000..c7197db --- /dev/null +++ b/book/exercises/exercise-paint.md @@ -0,0 +1,173 @@ +# Paint System + +In this assignment we will consider a paint system that prevents rust and corrosion on a steel structure. A high quality paint system costs €40 per square meter and has a failure probability of 0.002, which assumes that the old paint system is completely stripped off and the steel surface is cleaned prior to installation, which has an additional cost of €20 per square meter. + +Common cost cutting measures are to use a cheaper paint that costs half as much, but is five times more likely to fail, or to simply sand the old paint rather than stripping it off prior to applying a new paint, which increases the failure probability by 10, but only costs €5 per square meter. + +During operation of the structure, if a rust spot is found it must be repaired immediately by removing the damaged paint and replacing it. The repair has an additional cost of €500 per square meter. + +Construct a decision tree to evaluate the different options for steel preparation and paint, then use it to answer the following questions. Assume that the owner wants to minimize total costs, that the paint system only fails once in any reference period and that the probability of failure in each year is independent. All costs and probabilities are calculated on a square meter basis. + +%**Question 6:** Upload a photo/scan of your decision tree. Make sure the values and parameters for each branch are clear. + +````{admonition} Decision Tree +:class: tip, dropdown + +This problem can be visualized using a decision tree as shown below. However, it results in a different interpretation of the problem than that shown in the solution. For example, consider the top branch at the far right, where 'No rust spot' is assigned a probability of 0.998. This is consistent with the formulation of a decision tree, but not consistent with the problem statement, which implies that the €60 cost associated with cleaning and painting is always done when replacing the paint system (thus a probability of 1.0). This is why the values in the decision tree are slightly different than those in Question 3, for example. + +Although the decision tree is not part of the exam, this illustrates an important lesson: it is very easy to accidentally make probability calculations that are inconsistent with the situation you are trying to evaluate! + +```{figure} ../figures/exercise-decision-tree.png +--- +width: 600px +name: decision-tree +--- +Decision tree. +``` +```` + +**Question 1:** What is the probability of each paint system failure per square meter over one year? + +```{admonition} Answer +:class: tip, dropdown + +| Paint | Cleaning | $p_f$ [1/yr] | +|------- |--- | --- | +| High-Quality | Stripping | 0.002 | +| High-Quality | Sanding | 0.02 | +| Low-Quality | Stripping | 0.01 | +| Low-Quality | Sanding | 0.10 | +``` + +**Question 2:** What is the probability of each paint system failure per square meter over a 10 year period? + +```{admonition} Answer +:class: tip, dropdown + +$$P_{f,10y} = 1 - (1 - p_{f,1y})^{10}$$ + +| Paint | Cleaning | $p_f$ [1/10-yr] | +|------- |--- | --- | +| High-Quality | Stripping | 0.02 | +| High-Quality | Sanding | 0.18 | +| Low-Quality | Stripping | 0.10 | +| Low-Quality | Sanding | 0.65 | +``` + +**Question 3:** What is the expected cost per square meter over a one year period if the owner uses the high quality cleaning and high quality paint? + +```{admonition} Answer +:class: tip, dropdown + +$$ +40 + 20 + 0.002 \cdot (500 + 40 + 20) = \euro{} 61.12 /\mathrm{m^2} +$$ +``` + +**Question 4:** What is the optimal paint system for a one year period? + +```{admonition} Answer +:class: tip, dropdown + +$$C_{paint} + C_{cleaning} + P_{f,1y} \cdot (C_{paint} + C_{cleaning} + C_{repair})$$ + +| Paint | Cleaning | Expected cost [€/yr] | +|------- |--- | --- | +| High-Quality | Stripping | 61.12 | +| High-Quality | Sanding | 55.90 | +| Low-Quality | Stripping | **45.40** | +| Low-Quality | Sanding | 77.50 | + +Low Quality Paint + High Quality Cleaning +``` + +**Question 5:** What is the optimal paint system for a 10 year period? + +```{admonition} Answer +:class: tip, dropdown + +$$C_{paint} + C_{cleaning} + P_{f,10y} \cdot (C_{paint} + C_{cleaning} + C_{repair})$$ + +| Paint | Cleaning | Expected cost [€/10 yr] | +|------- |--- | --- | +| High-Quality | Stripping | **71.10** | +| High-Quality | Sanding | 144.70 | +| Low-Quality | Stripping | 91.63 | +| Low-Quality | Sanding | 366.94 | + +High Quality Paint + High Quality Cleaning + +**NOTE:** If you're getting slightly different results, don't panic! Most likely you have approximated the probabilities computed above in question 2. +``` + +Assume the owner has decided to use the cheap paint system, and is simply going to sand it off (i.e. the cheap cleaning method) and re-apply the same system every year (i.e., for these problems consider a 1-year reference period). You have been asked to assess whether it is worthwhile to use a quick sonic test system to prevent failures during the year, and if it is, the number of tests that should be performed per year. Each test costs about €3 per m2, and if a weak spot is found, a new layer of paint can be easily applied, lowering the failure probability by a factor of 0.50. + +**Question 6:** What is the annual expected cost due to paint failure if no sonic inspections are done? + +```{admonition} Answer +:class: tip, dropdown + +$$ +E(D) = P_f \cdot D = 0.1 \cdot 525 = 52.50 +$$ +``` + +**Question 7:** What is the annual expected total cost for the paint system if no sonic inspections are done? + +```{admonition} Answer +:class: tip, dropdown + +Using the decision tree from above: €77.50/m2 +``` + +**Question 8:** What is the change in risk for 3 sonic inspections? + +```{admonition} Answer +:class: tip, dropdown + +$$ +\Delta E(D) = (P_{f,0} - P_{f,new}) \cdot D = 45.94 +$$ +``` + +**Question 9:** What is the annual expected cost due to paint failure if 3 sonic inspections are done? + +```{admonition} Answer +:class: tip, dropdown +The expected damage is: +$$ +E(D)=P_f \cdot D={0.5}^3 \cdot 0.1 \cdot 525=6.56 +$$ +But the 3 inspections cost €9, thus the total expected cost is €15.56. It is incorrect to multiply the inspection cost by the failure probability. NB: this calculation implies the inspections occur regardles of when the failure happens. + +``` + +**Question 10:** What is the maximum number of inspections per year that can be performed that are cost-effective? Note: Round your answer towards the smallest integer. + +```{admonition} Answer +:class: tip, dropdown + +For the investments to be cost effective: + +$$ +\frac{\Delta E(D)}{I}>1 +$$ + +Which results in a maximum of 17 inspections per year. +``` + +**Question 11:** What is the optimum number of sonic inspections to perform each year? Plot a graph, with the Risk, Investment and Total cost curves and clearly indicate the optimum. + +````{admonition} Answer +:class: tip, dropdown + +As shown in the figure, the optimum is 3 per year. + +```{figure} ../figures/exercise-optimization.png +--- +width: 600px +name: optimization-curve +--- +Optimization curve for exercise 12 +``` +```` \ No newline at end of file diff --git a/book/exercises/exercise-sample-exam.md b/book/exercises/exercise-sample-exam.md new file mode 100644 index 0000000..8c8ac10 --- /dev/null +++ b/book/exercises/exercise-sample-exam.md @@ -0,0 +1,248 @@ +# Sample Exam Questions + + +## General Questions + +**Question 1:** Indicate which of the following statements is true (multiple answers possible): + 1. A parallel system gets stronger if there is a strong positive correlation between failure of the elements. + 2. Dependence between elements is only determined by dependence in loads. + 3. In case of a full dependence between two elements $A$ and $B$, it holds that $P(B|A)=1$. + 4. A flood defence system that is a single line of defence composed of multiple dike sections would best be characterized as a series sytem, not as a parallel system. + 5. A prison with multiple walls and fences would best be characterized as a series system, not as a parallel system. + +```{admonition} Answer +:class: tip, dropdown + +Options 3 and 4 are true. +``` + +## Flood Protection for a Bay + +A flood protection system for an area at a bay consists of a first line of defence (a dam on the coast) and a second line (a dike around the bay) to protect the city against a storm surge that can cause a high water level in the bay. The design team considers two strategies: 1) a single dike around the bay, or 2) a dike around the bay plus a dam (see figure). + +```{figure} ../figures/exercise-sample-exam-bay.png +--- +width: 400px +name: flood-protection +--- +``` + +**Question 1:** from the perspective of flooding in the city, the function of the dam and dike together is best described as a (choose one): +a. series system +b. parallel system +c. a single component +d. none of the above + +```{admonition} Answer +:class: tip, dropdown + +Option b is true. +``` + +**Question 2:** how will dependence between the dam and dike change the system failure probability? (choose one) +a. increase +b. decrease +c. no change + +```{admonition} Answer +:class: tip, dropdown + +Option a is true. +``` + +**Question 3:** describe whether or not there is dependence between the dam and dike, and what the source could be. + +```{admonition} Answer +:class: tip, dropdown + +It is likely that both structures are constructed of the same material (e.g., local soil), which would respond in a similar way to the storm surge and waves. The structures are also likely to be operated and maintained by the same government agency, which would result in dependence due to human error and maintenance problems. +``` +--- +The damage in case of flood protection system failure is 1 billion $10^9$ Euros. We compare two different investments in the system. In strategy 1 the system failure probability becomes $10^{-3}$ per year at a cost of 10 million ($10 \cdot 10^{6}$). In strategy 2 the system failure probability becomes $3 \cdot 10^{-4}$ per year at a cost of 50 million. + +**Question 4:** for which value of the interest rate $r$ is strategy 2 the most interesting? (You may consider an infinite lifetime of investments). + +```{admonition} Answer +:class: tip, dropdown + +Find $r$ such that the difference in expected damage between strategy 1 and 2, $\Delta p_{f,sys}D/r$, is greater than the difference between the investments, $I_2-I_1$, where the net present value over an infinite lifetime is $1/r$. + +$$ +r < \frac{\Delta p_{f,sys} D}{I_2-I_1} = \frac{(0.0010-0.0003)\cdot 1000 \text{m€}}{50\: \text{m€}-10\: \text{m€}}=0.0175 +$$ + +Can also set up as solving for $r$ when comparing total costs: + +$$ +I_1+\frac{p_1 D}{r}>I_2+\frac{p_2 D}{r} +$$ +``` +--- + A full scale risk assessment is made for the system with probabilities $P(N>10)=10^{-3}$ per year and $P(N>100)=5 \cdot 10^{-5}$ per year. A so-called limit line is given with values with $C = 10^{-1}$; and $\alpha = 2$. + +**Question 5:** determine whether this situation is acceptable. You may answer using a few sentences, calculations or by submitting a sketch, but either way, you must support your answer quantitatively. If you use a sketch, indicate the scale and relevant points on the plot. + +````{admonition} Answer +:class: tip, dropdown + +First create an FN curve for the risk assessment + +**NOTE:** _the figure below is slightly incorrect; the horizontal lines should be to the right of the "dots;" in other words, the 1E-05 value should be above N=100. This makes it more obvious why the system does not meet the safety criteria._ + +```{figure} ../figures/exercise-sample-exam-limit-line.png +--- +width: 400px +name: limit-line +--- +``` + +Next, check that each point on the line is below the limit line: + +$$ +1-F_N(n)<\frac{C}{N^α} +$$ + +The probability of exceeding 100 fatalities is too high since $5E-5>0.1/100^2=1E-5$. The same is true for 10 fatalities. +```` +--- +The flooding probability is 1/50 per year and damage is 150 million € ($10^6$). It is questioned whether additional protection is needed and what would be the best solution between the following: + +- Strategy A: raise the dike +- Strategy B: raise the dike and build a mangrove forest in front which reduces waves + +The table below indicates the costs of the strategies and how much they reduce the initial failure probability. Assume the interest rate is 4%. + +```{list-table} Mitigation Strategy A and B. +:header-rows: 1 + +* - Strategy + - Failure probability factor + - Costs, million € ($10^6$) +* - Do Nothing + - 1.0 + - 0 +* - A + - 0.5 + - 10 +* - B + - 0.2 + - 18 +``` + +**Question 6:** what is the most cost-effective or optimal strategy? Explain your answer. + +1. Strategy A +2. Strategy B +3. neither A nor B + +````{admonition} Answer +:class: tip, dropdown + +Strategy B is best. + +Various costs (NPV value) are calculated in the table below, and Strategy B is the lowest total cost we well as highest risk reduction. + +```{figure} ../figures/exercise_sample_opt.png +--- +width: 500px +--- +``` +```` +--- +Consider the individual risk in two areas of the city by the bay, above. The acceptable individual risk is $10^{-5}$ per year. The conditional probability of death due to flooding is dependent on the water depth and indicated next to the graph. The individual risk level in area A is $10^{-5}$ per year and the water depth is 3 m. The individual risk level in area B is $10^{-5}$ per year and the water depth is 2 m. + +**Question 7:** what should be the maximum allowable failure probability of the dike on the bay, $P_{\text{flooding}}$, to meet the standard for individual risk, $IR$? + +```{figure} ../figures/exercise-sample-exam-mortality.png +--- +width: 400px +--- +``` + +````{admonition} Answer +:class: tip, dropdown + +Individual risk is: + +$$ +IR = P_{\text{flooding}} \cdot P_{\text{death}|\text{flooding}} +$$ + +where the values for the conditional term can be found using the mortality figure, and solve for $P_{\text{flooding}}$. + +```{figure} ../figures/exercise_sample_IR.png +--- +width: 500px +--- +``` +The norm is governed by the most strict safety requirement, thus, $P_{\text{flooding}}$ should be 0.0002 per year. +```` + +## Component Reliability + + + +The shaded region, $\Omega$, in the figure below is the failure domain of an unknown object (the shaded zone extends further towards positive infinity for $x$ and $y$). $p_f$ is the probability of failure of that object, such that: + +$$ + p_f = \iint_\Omega f(x,y)dxdy +$$ + +Where $x$ and $y$ are random variables and $f(x,y)$ is the joint PDF. + +```{figure} ../figures/exercise-sample-exam-failure-domain.png +--- +width: 400px +name: failure-domain +--- +``` + +**Question 1:** identify a specific real-life object of your choosing (it can be anything!) that can be described by this diagram and failure probability. Describe the object and provide a definition of failure using words only, no equations. Mention whether $X$ and $Y$ each acts as a load or a resistance. + +```{admonition} Answer +:class: tip, dropdown + +It is clear that $X$ and $Y$ are both loads, since high values of both variables lead to failure. There are many possible ‘objects’ that are governed by two loads: + +- Dike: x and y are water level and wave load, failure is excessive overtopping +- Steel rod: 2 loads exceed strength +- Beam: self-weight and length are loads (but this is non-linear) +- Island: x and y are sea level rise and settlement, capacity is elevation; failure is flooding +- 2 things on object with limiting capacity. E.g.: people on a couch or bench, birds on a tree branch +- Literally any two loads would be acceptable, as long as the capacity of the object was a sum of both of them (if not a sum, it’s non-linear) +%Honorable mentions: exam grading, stoof pot, student knowledge on exam material, ear damage at a music festival, dog and cat in box, exam desk stability, fingers squeezing a pen, 2 heaters in a room + + + diff --git a/book/exercises/overview.md b/book/exercises/overview.md new file mode 100644 index 0000000..e0df0bc --- /dev/null +++ b/book/exercises/overview.md @@ -0,0 +1,7 @@ +# Exercises + +A number of exercises are included in this chapter that illustrate the previously discussed concepts of risk and reliability in a probabilistic design context. At the end a set of **sample exam** questions are included. + +--- + +_Exercises are largely drawn from decades of old course material from many contributors; this chapter can be referred to using standard Editor attribution._ \ No newline at end of file diff --git a/book/other-topics/test_codeblocks.md b/book/other-topics/test_codeblocks.md new file mode 100644 index 0000000..826213d --- /dev/null +++ b/book/other-topics/test_codeblocks.md @@ -0,0 +1,49 @@ +# Including a code block and a figure + +Here is a code block that is parsed from a python file: + +```{eval-rst} +.. literalinclude:: ../code/sinewave.py + :language: python +``` + +Which is the result of: + + ```{eval-rst} + .. literalinclude:: sinewave.py + :language: python + ``` + +We can toggle the code block by nesting it inside a `toggle` directive: + + ```{toggle} + ```{eval-rst} + .. literalinclude:: sinewave.py + :language: python + ``` + +Note the single `-triplet at the end of the toggle. + +````{toggle} +```{eval-rst} +.. literalinclude:: ../code/sinewave.py + :language: python +``` +```` + +The Python file saves the figure, so we can include it using the regular markdown recipe: + + +```{figure} ../figures/sinewave.svg +--- +name: sine_wave +--- +A sine wave +``` + +```{figure} ../figures/cosinewave.svg +--- +name: cosine_wave +--- +A cosine wave +``` \ No newline at end of file diff --git a/book/prob-design/01-one-random-variable.md b/book/prob-design/01-one-random-variable.md new file mode 100644 index 0000000..2555a3d --- /dev/null +++ b/book/prob-design/01-one-random-variable.md @@ -0,0 +1,78 @@ +(prob_design_1_rv)= +# One Random Variable + +Consider a river that is protected by dikes, earthen embankments with the purpose of keeping water in the river channel and preventing flooding of the hinterland. The dikes should be designed and build such that the height allows the river to safely pass the maximum discharge every year, measured in m$^3$/s. A rating curve ({numref}`rating_curve`) gives us the relationship between discharge and water depth, $h_w$, so if we know what the maximum discharge is, we can build our dikes to the critical height, $h_{dike}\geq h_w$. But there is a problem---what is the maximum discharge in the river each year? How would we even go about determining this value? One way is to make observations and find the highest recorded data point. For example, in 1995 the Rhine River reached a discharge of around 12,000 m$^3$/s, the highest recorded level to date. But even with over 100 years of measurements, the record is not complete: there could always be a bigger discharge. Another approach could be to come up with a number that is, beyond a reasonable doubt, impossible to exceed. Many experts believe that 16,000 m$^3$/s is getting close to an upper limit, and 20,000 m$^3$/s may be physically impossible. Which of these two values should we choose for the design of our dike, and why?[^expensive-option] + +```{figure} ../figures/rating_curve.svg +--- +height: 400px +name: rating_curve +--- +Rating curve for hypothetical river. Relationship is $q=C \cdot h_w^\beta$ where $\beta$=2.0 and $C$=4.0. +``` + +This is a difficult question because the actual discharge observed in a given year is random. In addition to other variables, it is largely dependent on rainfall in the drainage basin, which is impossible to predict far in advance---certainly not for the entire year. Furthermore, we don't have a way to accurately extrapolate our past observations to identify a logical maximum discharge. For this reason, we will use a continuous probability distribution to quantify the probability associated with a specific maximum discharge, $q$, that might be observed in a given year, defined by the PDF $f_{Q}(q)$ and CDF $F_{Q}(q)$. The only other piece of information needed is a design standard---a probability that defines the discharge to use for our design. Other chapters in this book cover the derivation of a design standard, so for this discussion we will assume that a flooding probability of 1/100 has already been determined as the acceptable level. Because there is only one random variable, we can use the PDF to determine the design discharge ({numref}`design_1_rv`). + +```{figure} ../figures/design_1_rv.svg +--- +height: 400px +name: design_1_rv +--- +PDF and design value for river discharge with 0.01 probability of exceedance: $q_{design}=155 \: \text{m}^3\text{/s}$. +``` + +In particular, we note that failure occurs for high discharge, so we must use the complementary CDF to find the design value, in this case probability of exceeding $q_{\mathrm{design}}$: + +$$ +1-F_{Q}(q_{\mathrm{design}})=0.01 +$$ + +$$ +q_{\mathrm{design}}=F_{Q}^{-1}(1-0.01)=155\:\text{m}^3\text{/s} +$$ + +after which the water depth and minimum dike height can easily be determined from the rating curve: + +$$ +h_{dike}=h_{w,design}=\left(\frac{q_{design}}{C}\right)^{1/\beta}=6.23 \text{m} +$$ + +Recalling the *safety margin* limit state from the previous section, $M=R-S$, we can see that randomness plays a role through the load variable, $S$, such that + +$$ +M=R-S=h_{dike}-h_{w,design} +$$ + +If the condition $M<0$ defines flooding (i.e., failure) and if $q$ is the only random variable, we have successfully ensured a situation where the flooding probability is 0.01. + +**Reflection on the Simple Example** + +Is this an oversimplification of reality? Perhaps, but it gets pretty close. Many flood protection systems in the world use an exceedance proabability approach to probabilistic design: choosing a specific and making sure the component or system can handle the load. This works well when the primary source of uncertainty is the load: the river discharge in this case. + +We also took for granted that a simple relationship between river discharge and water depth is available. However, this is never the case, as such a relationship depends on the cross-sectional shape and roughness of the floodplain, which not only varies along the river trajectory, but also changes due to natural phenomenon and human interventions. This introduces additional uncertainty into our assessment of whether the dikes are high enough. + +Fortunately there is also extra conservatism built into this approach. For example, duration of high water plays a role: if discharge exceeds the capacity of the dike system, but only lasts for a short time (minutes or a couple hours), perhaps the dikes can withstand the overflow without eroding and causing flooding. This can be conceptualized by considering the joint probability of a high discharge *and* degradation of the dike leading to flooding: + +$$ +P(\text{flooding},h_w>h_{dike})=P(\text{flooding}|h_w>h_{dike})\cdot P(h_w>h_{dike}) +$$ + +where the conditional term represents the probability that the dike erodes (fails) given that the water depth exceeds the height. Assuming failure when the critical water depth is exceeded is conservative because it implies $P(\text{flooding}|h_w>h_{dike})=1.0$, whereas in reality this is not the case, and the 'true' flooding probability is less than that computed in the equation above. + +Later chapters of this book will introduce methods for taking these realities into account in the design and decision-making process. For now, we will see how our probabilistic design changes when more than one random variable must be considered. + + + + +[^expensive-option]: Although the 20,000 m$^3$/s value is tempting because it is the most safe option, it is probably prohibitively expensive to build! \ No newline at end of file diff --git a/book/prob-design/02-two-random-variables.md b/book/prob-design/02-two-random-variables.md new file mode 100644 index 0000000..2843d63 --- /dev/null +++ b/book/prob-design/02-two-random-variables.md @@ -0,0 +1,191 @@ +(prob_design_2_rv)= +# Two Random Variables + +The previous section illustrated how a decision for dike height, $h_{dike}$, can take into account uncertainty in river discharge. We used a maximum allowable probability of flooding to derive the specific value of discharge, $q_{design}$, that should be used to choose $h_{dike}$. This was necessary because there was otherwise no way of establishing what the maximum discharge should be. The only variable considered to be random was the maximum river discharge that occurs each year. This section considers how the design process becomes more complicated when *two* random variables are considered (two load variables). + +````{admonition} Distribution for $q_1$ and $q_2$ +:class: dropdown +On this page all calculations assume the same lognormal distribution for random variables $q_1$ and $q_1$, where the first and second moments of the distribution are: + +$$ +\begin{align*} +E[X] &= \exp\left(\mu + \frac{\sigma^2}{2}\right) = 100\\ +VAR[X] &= \left( \exp(\sigma^2)-1 \right)\exp(2\mu + \sigma^2)= 400 +\end{align*} +$$ + + +For the `lognorm` method in `scipy.stats` we need to provide the parameters shape `s` = $\sigma$, `loc` = 0 and `scale` = $\exp(\mu)$ (see explanation [here](https://mude.citg.tudelft.nl/book/probability/Loc-scale.html)), in this case: +```python +import scipy.stats as stats +s = 0.198 +loc = 0.000 +scale = 98.058 +q_1 = stats.lognorm(s=s, loc=loc, scale=scale) +q_2 = stats.lognorm(s=s, loc=loc, scale=scale) +``` +```` + +## Discharge from Two Rivers + +In this scenario our objective for choosing $h_{dike}$ is still the same, except now we recognize that our location on the river is downstream of a confluence of two smaller rivers. The discharge at the location of our dike is thus the sum of the dicharge from Rivers 1 and 2: + +$$ +q=q_1+q_2 +$$ + +where the same rating curve applies, giving $h_w$ as a function of $q$. As before, acknowledging that the maximum discharge per year in each river is a random variable, we can approach the design problem in the same way: choose $h_{dike}$ such that it retains water when $q$ takes a value with probability of exceedance 0.01. For now, we can assume the distribution for River 1 and River 2 are the same as in the previous Section, and that they are independent. + +*Compared to the previous example, now there are two distributions---how do we find the design discharge and dike height?* + +Before presenting the correct approach, three common mistakes are illustrated. + +### Incorrect Approach 1 + +Find the discharge in each river, $q_1$ and $q_2$, such that the probability of exceeding each discharge is 0.01. As in the previous section, + +$$ +q_{1,\mathrm{design}}=q_{2,\mathrm{design}}=F_{Q}^{-1}(1-0.01)=155\:\text{m}^3\text{/s} \;\rightarrow\; q_{design}=310\:\text{m}^3\text{/s} +$$ + +*Why is this approach incorrect?* + +It considers the simultaneous occurrence of *both* rivers exceeding a design discharge with probability 0.01. This is a joint probability of occurrence, $P(Q_1>q_{1,design},Q_2>q_{2,design})$, which can be evaluated with a multivariate probability distribution (or the marginal distributions in the independent case). The scenario is equivalent to observing two sixes after tossing two dice simultaneously: if the dice are fair, the probability is $1/36$. Thus, for this incorrect case the probability $P(Q_1>q_{1,design},Q_2>q_{2,design})$ is not 0.01, it's actualy 0.01$^2$=0.0001, illustrated in the figure below. + +```{figure} ../figures/design_2_rv_error_1.svg +--- +height: 400px +name: design_2_rv_error_1 +--- +'And' probability for Incorrect Approach 1. +``` + +### Incorrect Approach 2 + +Find the discharge in each river, $q_1$ and $q_2$, such that the joint probability of exceeding these values is 0.01. Since the two rivers are independent, we can solve for our design probability, $p_{design}$: + +$$ +\left(p_{design}\right)^2=0.01\: \rightarrow \: p_{design}=\sqrt{0.01}=0.1 +$$ + +$$ +q_{1,\mathrm{design}}=q_{2,\mathrm{design}}=F_{Q}^{-1}(1-0.1)=126\:\text{m}^3\text{/s} \;\rightarrow\; q_{design}=252\:\text{m}^3\text{/s} +$$ + +```{figure} ../figures/design_2_rv_error_2a.svg +--- +height: 400px +name: design_2_rv_error_2a +--- +'And' probability for Incorrect Approach 2. +``` + +*Why is this approach incorrect?* + +It only considers one scenario! + +There are infinite alternative scenarios that also result in a joint probability of exceedance of 0.01, for example, $0.2\cdot 0.05 = 0.01$: + +$$ +q_1=F_Q^{-1}(1-0.20)=116\:\text{m}^3\text{/s}\;\; \text{and}\;\; q_2=F_Q^{-1}(1-0.05)=136\:\text{m}^3\text{/s} +$$ + +which results in a totally different dike height. + +```{figure} ../figures/design_2_rv_error_2b.svg +--- +height: 400px +name: design_2_rv_error_2b +--- +Comparison of two 'and' probabilities for Incorrect Approach 2. +``` + +### Incorrect Approach 3 + +Since the 'and' case didn't work we can use the 'or' probability, or union! In other words: find the discharge in each river, $q_1$ and $q_2$, such that the probability of either of them being exceeded is 0.01. + +The union probability can be found as follows: + +$$ +P(Q_1>q_1 \cup Q_2>q_2)=1-P(Q_1q_1 \cup Q_2>q_2)=1-P(Q_1q_1 \cup Q_2>q_2)=0.01$ implies the exceedance probability for each river discharge is $\sqrt{1-0.01}=\sqrt{0.99}=0.995$, thus: + +$$ +q_{1,\mathrm{design}}=q_{2,\mathrm{design}}=F_{Q}^{-1}(0.995)=163\:\text{m}^3\text{/s} \;\rightarrow\; q_{design}=326\:\text{m}^3\text{/s} +$$ + +```{figure} ../figures/design_2_rv_error_3.svg +--- +height: 400px +name: design_2_rv_error_3 +--- +'Or' probability for Incorrect Approach 3. +``` + +*Why is this approach incorrect?* + +As with the first two incorrect approaches, it also only considers one specific scenario! There are infinite combinations of $Q_1$ and $Q_2$ such that the union is 0.01. + +### Correct Approach + +The discharge in the main river is the sum of the discharge of its tributaries, thus it is a function of random variables which has a distribution $f_Q(q)$. Thus we can use the distribution of $Q$ to find the design value $q_{design}$ directly. Unfortunately, although $q$ is a linear combination of $q_1$ and $q_2$, because they are lognormally distributed we cannot find a simple analytic expression for the distribution of $q$. However, using Monte Carlo sampling from the distributions of $q_1$ and $q_2$ gives: + +$$ +q_{design}=F_Q^{-1}(1-0.01)=275\:\text{m}^3\text{/s} +$$ + +This is equivalent to integrating the joint probability distribution of $q_1$ and $q_2$ over the region $\Omega$ where $q>275\:\text{m}^3\text{/s}$, illustrated in the figure below, along with the contours of joint probability density. + +```{figure} ../figures/design_2_rv_correct.svg +--- +height: 400px +name: design_2_rv_correct +--- +Correct approach: $P(Q>275\:\text{m}^3\text{/s})=0.01$. Also illustrated is the region $\Omega$ where $q>275\:\text{m}^3\text{/s}$ and contours of joint probability density. +``` +### Comparison of Approaches + +All four design values of $Q$ are collected in the table below, along with the required dike height. Clearly the method of incorporating uncertainty makes a difference, as the height varies over 1 m across all options, which is a significant increase in cost for a structure that is many kilometers long. + +```{list-table} Comparison of design values for all four approaches. +:header-rows: 1 +:name: design_2_rv_comparison + +* - Approach + - Design Value, $q_{design}$ [m$^3$/s] + - Design Value, $h_{dike}$ [m] +* - Incorrect 1 + - 310 + - 8.80 +* - Incorrect 2 + - 252 + - 7.94 +* - Incorrect 3 + - 326 + - 9.03 +* - **Correct** + - **275** + - 8.29 +``` + +## Reflections on the Simple Example + +% Dependence considered in componenet reliability part (and system!) +% Design point consideration NOT included here, and probably not in this book!!! + +We need to be explicit in what we are trying to evaluate. In this case, it is the distribution of a function of random variables, $Q$. + +For non-Gaussian distributions and for non-linear functions of random variables it is often not possible to get the distribution of the function of random variables analytically. Fortunately it is easy to find it numerically through sampling techniques. This will be discussed elsewhere. + +The 'and' and 'or' approaches (intersection and union) are simple, but really only limited to discrete cases. As illustrated above, they cannot be used in the case of continuous random variables, or with functions of random variables because one must integrate across all possible scenarios leading to failure. Nonetheless, these approaches are extremely useful for many situations, and are the basis of *system reliability* problems. + +The situation illustrated here is often referred to as a *component reliability* problem, where the 'component' is defined by a function of random variables. + Although nothing more than a function of random variables,is nothing more than an integration over a specific region of a multivariate probability density function. Often this region describes failure of a component, which we will try to keep below an acceptable level. \ No newline at end of file diff --git a/book/prob-design/example-river-system.md b/book/prob-design/example-river-system.md new file mode 100644 index 0000000..384bb3d --- /dev/null +++ b/book/prob-design/example-river-system.md @@ -0,0 +1,24 @@ +# Flood Management Case Study + +```{note} +The flood management example is covered in the plenary lecture on Jan 9, 2023 (see Brightspace), and has not yet been added to this book. +``` + +As mentioned on Introduction page: +> Throughout this book we will repeatedly draw on the field of flood management to illustrate risk and reliability concepts as this field requires expertise from all perspectives in civil and environmental engineering and geosciences, from climate and hydrologic processes to evacuation and recovery. + + + +```{figure} ../figures/simple-city.png +--- +width: 600px +name: simple-city +--- +A simple city that we would like to protect against flooding by the river. +``` \ No newline at end of file diff --git a/book/prob-design/exercise.md b/book/prob-design/exercise.md new file mode 100644 index 0000000..d834ed6 --- /dev/null +++ b/book/prob-design/exercise.md @@ -0,0 +1,13 @@ +# Exercises + +Elsewhere in this book we have covered methods that will allow you to solve the discharge problem on the previous page, for example, propagation of uncertainty, functions of random variables, etc. Try the following exercises to evaluate the distribution of $q$. In each case, you should be able to define both the distribution of $q$ as well as the value of $q_{design}$: + +1. Define the distribution of $q_1$ and $q_2$ as LogNormal using Scipy, then evaluate the distribution of $q$ using Monte Carlo Simulation +2. Approximate the distributions of $q_1$ and $q_2$ using only their moments and use error propagation to approximate the distribution of $q$ +3. Use the moments to define Normal distributions for $q_1$ and $q_2$, then find the distribution of $q$ (hint: it should also be Normal, and you can solve for it analytically) +4. Use the moments to define Normal distributions for $q_1$ and $q_2$, then use the multivariate normal distribution in Scipy to define a correlation coefficient and explore what the effect of dependence between $q_1$ and $q_2$ has on the result. + +```{note} +Solutions to these exercises are on the next page. +``` + diff --git a/book/prob-design/exercise_solutions.ipynb b/book/prob-design/exercise_solutions.ipynb new file mode 100644 index 0000000..2b6f30c --- /dev/null +++ b/book/prob-design/exercise_solutions.ipynb @@ -0,0 +1,308 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "12d4c736-839c-41c2-974e-e472550b1ec4", + "metadata": {}, + "source": [ + "# Solutions\n", + "\n", + "All code required to solve the exercises is provided on this page, with the output included by default. To use the code interactively, click {fa}`rocket` --> {guilabel}`Live Code` on the top right corner of this screen and then wait until all cells are executed. You can also download this page as a notebook to work with on your personal computer.\n", + "\n", + "```{note}\n", + "On this page we adopt the notation of lower-case $q$ used in other sections of this chapter, and do not use the upper-case convention to denote a random variable.\n", + "```\n", + "\n", + "First we import a few packages and define the distributions of $q_1$ and $q_2$ as described on the previous pages." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "baaf5a2b-e837-46f1-bf9d-56bf9cec708c", + "metadata": { + "tags": [] + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "import scipy.stats as stats\n", + "\n", + "s = 0.198\n", + "loc = 0.000\n", + "scale = 98.058\n", + "q_1 = stats.lognorm(s=s, loc=loc, scale=scale)\n", + "q_2 = stats.lognorm(s=s, loc=loc, scale=scale)" + ] + }, + { + "cell_type": "markdown", + "id": "a2724e09-9815-4f02-af62-2d595019c1b3", + "metadata": { + "tags": [] + }, + "source": [ + "## Part 1\n", + "\n", + "We can use Monte Carlo simulation to evaluate the problem easily. Remember, there are four essential steps to Monte Carlo Simulation:\n", + "\n", + "1. Define distributions for random variables (probability density over a domain) \n", + "2. Generate random samples \n", + "3. Do something with the samples (deterministic calculation) \n", + "4. Evaluate the results: e.g., “empirical” PDF, CDF of samples, etc.\n", + "\n", + "The solution below also uses the method of moments on the MCS sample to compare the Normal and Lognormal parametric distributions, as well as includes the \"exact\" analytic solution for Parts 2 and 3.\n" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "0b46ebdf-1a24-4de9-a426-286f3f2720a7", + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The 99th percentile is index: 9900\n", + "The 99th percentile from the sample is: 275.9 m^3/s.\n", + "The inverse CDF of the Normal dist gives: 266.3 m^3/s.\n", + "The inverse CDF of the LogNormal dist gives: 276.4 m^3/s.\n" + ] + }, + { + "data": { + "image/png": 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+ "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "N = 10000\n", + "sample_1 = q_1.rvs(size=N)\n", + "sample_2 = q_2.rvs(size=N)\n", + "q = sample_1 + sample_2\n", + "\n", + "q_as_N = stats.norm(*stats.norm.fit(q))\n", + "q_as_LN = stats.lognorm(*stats.lognorm.fit(q))\n", + "\n", + "plot_values = np.linspace(100,400,50)\n", + "plt.hist(q, bins=plot_values,\n", + " density=True,\n", + " edgecolor='black')\n", + "plt.plot(plot_values, q_as_N.pdf(plot_values),\n", + " linewidth=2.5, label='Fit with N')\n", + "plt.plot(plot_values, q_as_LN.pdf(plot_values),\n", + " linewidth=2.5, label='Fit with LN')\n", + "plt.plot(plot_values,\n", + " stats.norm.pdf(plot_values, loc=200, scale=np.sqrt(800)),\n", + " ':', linewidth=2.5, label='Part 2 and 3')\n", + "plt.xlabel('Discharge, $q$ [m$^3$/s]')\n", + "plt.ylabel('Density [--]')\n", + "plt.legend()\n", + "\n", + "get_index = int(np.floor(0.99*(N + 1)))\n", + "print(f'The 99th percentile is index: {get_index}')\n", + "q_percentile = np.sort(q)[get_index - 1]\n", + "print(f'The 99th percentile from the sample is: {q_percentile:.1f} m^3/s.')\n", + "print(f'The inverse CDF of the Normal dist gives: {q_as_N.ppf(0.99):.1f} m^3/s.')\n", + "print(f'The inverse CDF of the LogNormal dist gives: {q_as_LN.ppf(0.99):.1f} m^3/s.')\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "1fae3f87-afc0-4c3d-a6f3-f634a5422304", + "metadata": { + "tags": [] + }, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "id": "13773486-ddd6-4bc7-a012-b44fc1f67ed8", + "metadata": {}, + "source": [ + "## Part 2 and 3\n", + "\n", + "Error propagation, as introduced in previous chapters, is a method that relies on a linearized approximation of the function of random variables and assumes that the random variables themselves are normally distributed. This results in the relatively simple equations for computing the moments of the function based on the moments of the random variables, applied below. In this case, since we are only using the moments of the lognormal variables in Part 2, the distribution is identical to that described by Part 3, so the solution is the same.\n", + "\n", + "The expectation is simply:\n", + "\n", + "$$\n", + "\\mathbb{E}(q(q_1,q_2))= q(\\mu_{q_1},\\mu_{q_2}) = \\mu_{q_1} + \\mu_{q_2}\n", + "$$\n", + "\n", + "And, given independent random variables, the variance is:\n", + "\n", + "$$\n", + "\\sigma_q^2 = \\sigma_{q_1}^2 + \\sigma_{q_2}^2\n", + "$$\n", + "\n", + "The partial derivatives have been left out, as they are all +1, and the covariance term is zero. Plugging in values from the exercise:\n", + "\n", + "$$\n", + "\\mathbb{E}(q)= 100 + 100 = 200\n", + "$$\n", + "\n", + "And the variance:\n", + "\n", + "$$\n", + "\\sigma_q^2 = 400 + 400 = 800\n", + "$$\n", + "\n", + "Finally, we can compare the distibution to those found above (see figure). The two normal distributions are very similar to each other, which tells us that the asymettry of the lognormal distribution does not have a big influence on the distribution of $q$." + ] + }, + { + "cell_type": "markdown", + "id": "6da9930f-5409-4bee-a8ea-e4c505355425", + "metadata": {}, + "source": [ + "## Part 4\n", + "\n", + "Here we will use the analytically computed value of the distribution of $q$ and compare the design value $q_{design}$ to the value found with a Monte Carlo Simulation using `scipy.stats.multivariate_norm` and a covariance matrix to generate a plot and compute values. The solution is more complicated that you were expected to provide, in order to illustrate various approaches and insights.\n", + "\n", + "First we find the variance of $q$, which adds an additional term now that the random variables are dependent:\n", + "\n", + "$$\n", + "\\sigma_q^2 = \\sigma_{q_1}^2 + \\sigma_{q_2}^2 + 2\\sigma_{q_1}\\sigma_{q_2}\\rho_{q_1,q_2}\n", + "$$\n", + "\n", + "Resulting in:\n", + "\n", + "$$\n", + "\\sigma_q^2 = 400 + 400 + 2 \\cdot 20 \\cdot 20 \\rho_{q_1,q_2} = 800(1+\\rho_{q_1,q_2})\n", + "$$\n", + "\n", + "Design values are computed below, and as the plot illustrates, increasing correlation coefficient results in a higher likelihood of observing high values of $q$, and therefore the design value increases. The opposite is true with no dependence, as well as negative dependence.\n", + "\n", + "The code cell below does the following:\n", + "- define the analytically derived moments of $q$\n", + "- set up the expectation and covariance matrix for $q_1$ and $q_2$\n", + "- sample from the joint distribution of $q_1$ and $q_2$ and compute the corresponding sample of $q$\n", + "- evalue the empirical design value, $q_{design}$\n", + "- print and plot relevant results\n", + "\n", + "_Change the values in the cell below to explore the problem, especially $\\rho$ and $N$, which can have profound influences on the result: $q_{design}$ and the accuracy of the MCS, respectively._" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "5122d9c9-f239-4914-aa54-3e0cfef79f2f", + "metadata": { + "tags": [] + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The correlation coefficient is: 0.50\n", + "The mean value of q is: 200\n", + "The variance of q is: 1200.0\n", + "The design value for q~N(mu,sigma) is: 280.6 m^3/s\n", + "\n", + "For N = 10000 samples, the 99th percentile is index: 9900\n", + "The 99th percentile from the sample is: 279.3 m^3/s\n", + " ^ this is the 1% design value, q_design, from MCS\n", + "\n" + ] + }, + { + "data": { + "image/png": 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", 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "rho = +.50\n", + "N = 10000\n", + "\n", + "mu_q = 200\n", + "var_q = 800*(1 + rho)\n", + "\n", + "mu_q1q2 = np.array([100, 100])\n", + "cov_q1q2 = 400*np.array([[1, rho],\n", + " [rho, 1]])\n", + "\n", + "q1q2 = stats.multivariate_normal(mu_q1q2, cov_q1q2)\n", + "sample = q1q2.rvs(size=N)\n", + "\n", + "q_1_dependent = sample[:,0]\n", + "q_2_dependent = sample[:,1]\n", + "q_dependent = q_1_dependent + q_2_dependent\n", + "\n", + "get_index = int(np.floor(0.99*(N + 1)))\n", + "q_percentile = np.sort(q_dependent)[get_index - 1]\n", + "\n", + "fig, ax = plt.subplots()\n", + "ax.scatter(q_1_dependent, q_2_dependent,\n", + " color='black', label='All samples of q')\n", + "ax.scatter(q_1_dependent[q_dependent>q_percentile],\n", + " q_2_dependent[q_dependent>q_percentile],\n", + " color='red', label='Top 1% of q')\n", + "\n", + "print(f'The correlation coefficient is: {rho:.2f}')\n", + "print(f'The mean value of q is: {mu_q}')\n", + "print(f'The variance of q is: {var_q:.1f}')\n", + "print(f'The design value for q~N(mu,sigma) is: ',\n", + " f'{stats.norm.ppf(0.99, loc=mu_q, scale=np.sqrt(var_q)):.1f} m^3/s\\n')\n", + "print(f'For N = {N} samples, the 99th percentile is index: {get_index}')\n", + "print(f'The 99th percentile from the sample is: {q_percentile:.1f} m^3/s')\n", + "print(' ^ this is the 1% design value, q_design, from MCS\\n')\n", + "\n", + "plt.xlabel('Discharge of river 1, $q_1$ [m$^3$/s]')\n", + "plt.ylabel('Discharge of river 2, $q_2$ [m$^3$/s]')\n", + "plt.title('Samples of river discharge')\n", + "ax.axis('equal');\n", + "ax.legend()\n", + "plt.show()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.4" + }, + "widgets": { + "application/vnd.jupyter.widget-state+json": { + "state": {}, + "version_major": 2, + "version_minor": 0 + } + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/book/prob-design/overview.md b/book/prob-design/overview.md new file mode 100644 index 0000000..2f447ef --- /dev/null +++ b/book/prob-design/overview.md @@ -0,0 +1,41 @@ +(prob_design)= +# Probabilistic Design + +Stated as simply as possible, the term *probabilistic design* implies the use of probability to design something. Although this may sound complex, it is actually not much different than a deterministic approach, where values used to design something are not assumed to be random. For example, the load on a beam, or the maximum discharge expected in a river. In a typical design process, the problem and functional requirements are first defined, the system or component is evaluated and then refined {cite:p}`voorendt2017`. It is ideally an iterative process, where tradeoffs between competing requirements must be considered in order to arrive at an optimal configuration. During much of your university education you have probably learned a variety of techniques to evaluate interesting problems in your field of study; however, only towards the end of a bachelor or master program do you begin to *design* things and take into count information beyond what is written in a simple exam problem to make decisions. The design process is often challenging because there is no single right answer, and it turns out that incorporating probability in the design process can help. + +Consider some arbitrary object--let's call it a thingamajig--which we must design and eventually build. We would like to apply the *factor of safety* approach, which compares load[^solicitation], $S$, to resistance, $R$, such that $FS=R/S$. As long as we make sure that $S$ is less than $R$, or $FS>1$, our thingamajig will never fail. We could also use the *safety margin* approach, where $M=R-S$, which also quantifies the point between safe and unsafe, although here we make sure $M>0$. These so-called *limit states* make the design process quite simple and is a great engineering approach for many situations, for example, if: +1. the values of $R$ and $S$ are precisely known, or the range of values is negligibly small +2. $R$ and $S$ never change (at least not over short time or length scales) +3. $R$ and $S$ are easy to measure in all locations of interest +4. the model used to design the thingamajig is nearly perfect +5. the consequence of a failure is negligible + +In situations like this you can make your design decision and rest assured that there won't be any lawyers knocking on your door in the near future because the thingamajig broke. + +For many situations, however, a deterministic analysis is not wise because we cannot guarantee that our thingamajig will always survive. When $R$ and/or $S$ can take on a range of possible values, all of a sudden there is a chance that our thingamajig is *not* reliable: there is a distinct probability it may *fail.* Part of this book is concerned with the quantification of what is often called the *failure probability,* $p_f$, for which there are many different techniques. These techniques all rely on probability theory because it allows us to quantitatively account for the following in our engineering process: + +1. imprecise measurements +2. model error +3. randomness in some of our important design variables +4. risk-based design criteria and safety standards + +These are just a few examples, where items 3 and 4 are the focus of the following chapters, and items 1 and 2 have already been considered elsewhere in MUDE. + +%Sometimes we have criteria, sometimes we do not. + +## Chapter Overview + +%Terms and definitions first?? +%Aleatoric and epistemic. +%Sometimes we don't have a structure. Then we can consider just the *hazard*. + +A probabilistic design is illustrated in the following sections, first with the simple cases of one random variable. Then an example with two random variables is used, which can be referred to as a *bivariate* case. This is more insightful than the univariate case because it allows us to consider the quantitative and qualitative influence of dependence, as well as the functional relationship between two variables and their joint probability density (e.g., union, intersection or limit-states). + +The evaluation and design of a river flood protection system is used to introduce key aspects of risk and reliability, as well as the design process. A distinction is made between using probability to assess engineering components and systems (reliability analysis) and to derive design criteria (risk evaluation), which are introduced formally in later chapters after the general risk analysis framework is discussed. + +--- + +_The primary author for this chapter is Robert Lanzafame._ + +[^solicitation]: $S$ stands for solicitation. While this letter and word are much more pedantic-sounding than simply using load, or $L$, it is widely used in the structural engineering field, where component reliability methods were pioneered. Here we take a broader approach on the subject. Classical texts are {cite:t}`adk2022`, {cite:t}`moss2020` and {cite:t}`ditlevsen1996`. +%[^pattern]: The word *paradigm* is not used because it suggests completeness. The examples here, are meant to be used as illustrations for key concepts, and extended to other situations and higher dimensions, so *pattern* seems more appropriate. \ No newline at end of file diff --git a/book/reliability-system/exercise-simple-city.md b/book/reliability-system/exercise-simple-city.md new file mode 100644 index 0000000..d8e852a --- /dev/null +++ b/book/reliability-system/exercise-simple-city.md @@ -0,0 +1,136 @@ +# Example System + +A pre-fabricated bridge design is being considered for river crossings in a remote region of the world, as shown in Figure 1. Cities 1, 2 and 3 are labelled C1, C2 and C3, with bridges labelled B1-B4. + +```{figure} ../figures/simple-city.png +--- +width: 600px +name: exercise-simple-city +--- +``` + +**Question 1:** If the probability of failure for an individual bridge is 0.1 per year, compute the probability that you cannot drive from City 1 to City 2 + + +```{admonition} Answer +:class: tip, dropdown + +$$ + P_{f_{C1-C2}} = P_{f_{B1}} * P_{f_{B4-B3-B2}} = P_{f_{B1}} * (1 - (1 - P_{f_{B4}})(1 - P_{f_{B3}}*P_{f_{B2}})) +$$ +$$ + P_{f_{C1 - C2}} = 0.1 * (1 - (1 - 0.1)(1 - 0.1 * 0.1)) = 0.0109 +$$ +``` + +**Question 2:** If the probability of failure for an individual bridge is 0.1 per year, compute the probability that you cannot drive from City 2 to City 3. + +```{admonition} Answer +:class: tip, dropdown + +$$ + P_{f_{C2 - C3}} = P_{f_{B3}} * P_{f_{B2}} * P_{f_{B1 - B4}} = P_{f_{B3}} * P_{f_{B2}} * (1 - (1 - P_{f_{B1}})(1 - P_{f_{B4}})) +$$ +$$ + P_{f_{C2 - C3}} = 0.1 * 0.1 * (1 - (1 - 0.1)(1 - 0.1)) = 0.0019 $$ +``` + +**Question 3:** If the probability of failure for an individual bridge is 0.1 per year, compute the probability that you cannot drive from City 1 to City 3 + +```{admonition} Answer +:class: tip, dropdown + +$$ + P_{f_{C1 - C3}} = P_{f_{B4}} * P_{f_{B1-B3-B2}} = P_{f_{B4}} * (1 - (1 - P_{f_{B1}})(1 - P_{f_{B3}}*P_{f_{B2}})) +$$ +$$ + P_{f_{C1 - C3}} = 0.1 * (1 - (1 - 0.1)(1 - 0.1 * 0*1)) = 0.0109 +$$ +``` + +**Question 4:** Suppose there is a critical facility (e.g., a hospital) in City 3. How many bridges would you need to install between City 2 and 3 to ensure that the probability of someone not reaching City 3 from City 2 is less than $10^{-4}$? + +```{admonition} Answer +:class: tip, dropdown + +$$ + P_{f_{C2 - C3}} = P_{f_B}^n * (1 - (1 - P_{f_{B1}})(1 - P4)) +$$ +$$ + 0.0001 = 0.1^n * (1 - (1 - 0.1)(1 - 0.1)) +$$ +$$ + n = 3.3 +$$ + +It is necessary to install 2 more bridges between City 2 and 3. +``` + +**Question 5:** If you had 1 extra bridge to place, which bridge (i.e., B1, 2, 3 or 4) would you place it next to in order to have the biggest decrease of the probability of not reaching City 3 from City 1? + +```{admonition} Answer +:class: tip, dropdown + +I would locate the extra bridge next to Bridge 4. In this case, adding an extra element in the parallel system, making it more redundant. + +- $P_f$ next to B1 = 0.00199 +- $P_f$ next to B2 or B3 = 0.0101 +- $P_f$ next to B4 = 0.00109 +``` + +**Question 6:** If you only had 3 available bridges, which of the 4 bridges in the figure would you remove first, to cause the smallest reduction in failure probability? + +```{admonition} Answer +:class: tip, dropdown + +I would remove Bridge 2 or 3. If bridge 1 or 4 is removed, a whole path is removed from the system, while by removing only Bridge 2 or 3, there is still an available bridge to cross that river. +``` + +**Question 7:** What should the probability of each bridge be in order to make the probability of not crossing from C1 to C3 less than 0.01? + +```{admonition} Answer +:class: tip, dropdown + +$$ + P_{f_{C1-C3}} = P_{f_{B4}} * P_{f_{B1-B3-B2}} = P_{f_{B4}} * (1 - (1 - P_{f_{B1}})(1 - P_{f_{B3}} * P_{f_B2})) +$$ +$$ + 0.01 = X * (1 - (1 - X)(1 - X * X)) +$$ +$$ + X = 0.0960 +$$ + +Since the bridges have identical designs and have all been manufactured by the same contractor, positive dependence is expected between performance of bridges B1-B4. +``` + +**Question 8:** State whether or not dependence would increase or decrease the probability calculated in question 3, and explain why with 1 or 2 sentences. + +```{admonition} Answer +:class: tip, dropdown + +Increase. + +The probability of failure would increase because the system mainly works like a parallel system. In this case, when there is full dependence between the elements, the probability of failure equals the probability of failure of the strongest element. For instance, $P_{f_{\text{bridges 2, 3}}}=0.01$ for independent case and 0.1 for full dependence case. +``` + +Assume each prefabricated bridge is made up of a single road deck and 5 separate frames that are joined together. Each bridge will fail if either the road deck fails, or 1 of the frames fails. + +**Question 9:** If the probability of a frame failure is 0.01, what is the required failure probability of the road deck to make sure the bridge failure probability is no more than 0.1? + +```{admonition} Answer +:class: tip, dropdown + +$$ + P_f = 1 - (1 - P_{f_{text{deck}}})(1 - P_{f_{\text{one of the frames}}}) +$$ +$$ + P_f = 1 - (1 - P_{f_{text{deck}}})(1 - (1 - (1 - P_{f_{\text{frame}}})^5)) +$$ +$$ + 0.1 = 1 - (1 - P_{f_{text{deck}}})(1 - (1 - (1 - 0.01)^5)) +$$ +$$ + P_{f_{\text{deck}}} = 0.0536 +$$ +``` \ No newline at end of file diff --git a/book/reliability-system/overview.md b/book/reliability-system/overview.md new file mode 100644 index 0000000..9ecd09b --- /dev/null +++ b/book/reliability-system/overview.md @@ -0,0 +1,137 @@ +(rel_sys)= +# System Reliability + + + +A structure or system in civil engineering generally consists of a set of *components* (also called *elements*). Whereas the previous chapter focused on the reliability of a single component, this chapter addresses reliability analysis for systems that consist of multiple components. The consequences of the failure of a component will depend on the type of system that is considered. In some cases, the failure of a component leads to failure of the whole system (progressive collapse). In other cases (and system configurations) there could be no failure if a single component fails and other components take over the function of the failed one. + +The techniques introduced in this chapter essential for carrying out step 3 in a risk analysis: quantitative analysis. + +## Series and Parallel Systems + +In reliability analysis two types of basic systems can be distinguished: series and parallel (see {numref}`sys-series-parallel`). + +```{figure} ../figures/system_series_parallel.png +--- +height: 400px +name: sys-series-parallel +--- +Series (top row) and parallel (bottom row) systems: the building blocks of system reliability analysis. +``` +Within a series system, failure of a single component will always lead to the failure of the entire system. An example is a chain, which breaks if any individual link fails. In our flood management example, a dike ring protecting a single area that is made up of multiple segments is also a series system. Within a parallel system, failure of one component can be compensated by the performance of another component; it represents the concept of redundancy. Drawing on the chain example, it would imply using more than one chain capable of holding the maximum load. An example from flood management would be to build a second dike ring inside the first dike ring. In practice, a complex system can generally be represented by means of a combination of parallel and series subsystems, just as the parallel system of multiple chains (or dikes) is a parallel system of series systems of links (or dike segments). + +```{figure} ../figures/system_series_parallel_ex.png +--- +height: 400px +name: sys-series-parallel-ex +--- +Example of series (top row) and parallel (bottom row) systems, illustrated with a chain and dike ring. Each component in the parallel case is itself a series system: links in the chain and dike segments in the dike ring. +``` + +Systems are often schematized using a system diagram, as in {numref}`sys-series-parallel`. This also illustrates the function of a system as an input-output diagram, where one can considers the ability to travel from left to right on the figure. It is analagous to many scenarios involving flow: electricity ({numref}`sys-lightbulbs`), water in pipes or transportation routes. + +```{figure} ../figures/system_lightbulbs.png +--- +width: 400px +name: sys-lightbulbs +--- +Series (left) and parallel (right) systems illustrated as a string of lightbulbs. +``` + +### Definition of Failure + +Whereas in component reliability problems we defined failure using a function of random variables, in system reliability problems, we define failure based on the continuity of the system diagram. Failure occurs when we are not able to travel from left to right on the diagram due to one or more broken links. + +## Reliability Calculations + +Equations for computing parallel and series system reliability are described here with the notation $F_i$ and $\bar{F_i}$ to denote failure and non-failure of component $i$, respectively. A system is made up of $i=1,...,n$ components where the probability of failure of component $i$ is $P(F_i)$. Note that as a concept, "system reliability" is almost always going to be quantified using the failure probability of the system, $p_{f,sys}$. This is sometimes misleading or confusing because the mathematical definition of reliability is the complement of reliability, $1-p_{f,sys}$. +### Parallel System + +A parallel system is often described as redundant since *all* components must fail to cause system failure. This is also simple to calculate as it can be described mathematically as the joint failure probability, or intersection, of all components: + +$$ +p_{f,sys} = P(F_1 \cap F_2 \cap ... \cap F_n) +$$ + +If the components are independent, this becomes + +$$ +p_{f,sys} = \prod_i^n P(F_i) +$$ + +Although we will not quantify the effect of dependence in depth here, it is easy to see semi-quantitatively that positive dependence increases the intersection probability, and therefore also increases $p_{f,sys}$. Negative dependence has the opposite effect. In fact, if parallel components are mutually exclusive the system is guaranteed to survive. The case of negative dependence is trivial in engineering reliability computation, but something that is used to motivate the incorporation of resilience or robustness in our risk reduction decisions. + +### Series System + +A series system fails if *any* component fails, which is + +$$ +p_{f,sys} = P(F_1 \cup F_2 \cup ... \cup F_n) +$$ + +For two components this can easily be re-written as + +$$ +p_{f,sys} = P(F_1) + P(F_2) - P(F_1 \cap F_2) +$$ (sys_p_series) + +Although it is straightforward to write out terms for 3 or more components, it is easier to re-write the union formulation as the complement of joint *survival* of the system: + +$$ +p_{f,sys} = 1 - P(\bar{F}_1 \cap \bar{F}_2 \cap ... \cap \bar{F}_n) +$$ + +As $\bar{F}_i=1-F_i$, we arrive at the following simple relationship: + +$$ +p_{f,sys} = 1 - \prod_i^n \left(1-P(F_i)\right) +$$ + +The influence of dependence on a series system is most easily understood through consideration of {eq}`sys_p_series`, which allows a direct analogy to the parallel case through the intersection term. With positive dependence, the system failure probability decreases + +% add exercise to show equivalence for 3 or more components. + +### Quantification of Dependence + +{numref}`sys_dependence` quantifies the effect of dependence on the join probability of failure of two components. The joint failure probability is calculated using the multivariate normal distribution which has a linear dependence measure, the correlation coefficient, $\rho$. The blue line shows the series system failure probability, equivalent to {eq}`sys_p_series`. The figure allows one to evaluate the quantitative effect of dependence on series and parallel systems, and even illustrates the bound of $p_{f,sys}=P(F_1)$ for the case of $\rho=+1$. + + +```{figure} ../figures/system-corr-coeff.PNG +--- +width: 400px +name: sys_dependence +--- +Effect of correlation coefficient on failure probability for a series system via joint failure, $P(F1 \cap F2)$ (computed with a Gaussian copula). The arrow is located at $\rho=+0.9$, where $P(F1 \cap F2)=0.003$. +``` + +````{admonition} Tips for Interpreting the Figure +:class: tip, dropdown + +{numref}`sys_dependence` is interesting because it shows the effect of dependence on *both* series and parallel systems, since they are related with {eq}`sys_p_series`, the value that is shown in the blue line. Note that the distance between the blue line and the top reference line is the value of the intersection, $P(F1\cap F2)$ (also called the "joint" or "and" probability). Note that without a joint probability distribution (in this case a bivariate Gaussian), you can't compute the joint probability of two events, except for three special cases, the vertical black lines in the figure---do you know what those values are? This is related to the concepts of dependent, mutually exclusive and statistically independent events, which are easy to confuse with each other. + +It is also common to think about dependence in the sense of how "one event happens given that the other event has happened," but this is not quite right. Dependence is describged simply by the definition of conditional probability + +$$ +P(F1 \cap F2) = P(F1|F2)\cdot P(F2) +$$ + +which means the correct statement is that dependence quantifies how the "*probability* of one event *changes* (or not) given that the other event has happened." +```` + +## Chapter Overview + +This page contains a brief introduction to the computation of failure probability for series and parallel systems, and describes the role of dependence. The following section illustrates these concepts with a simple exercise. + +### Additional Information + +The word 'system' in the title of this chapter refers to a reliability analyses made up of more than one distinct components. It may or may not be the same as the system for which the risk analysis is being done. For example, in the flood risk case we look at a dike ring, which is a series system of dike segments. Each dike segment can fail in a variety of ways (e.g., sliding or eroding), where if *any* of these types of failure occurs, the dike fails; a series system. Thus, the dike ring is a series system, where each component also a series system. + +% add a note about systems of systems, and being able to simplify problems. add figure too (there was one in Chapter 3, originally) + +%Sometimes the distinction between system and component reliability is made because components are very different. Sometimes it is a matter of convenience, as two components may share a number of random variables through their respective limit-state functions that influence failure. + +System reliability analysis can look very similar to component reliability problems, especially when various combinations of discrete and continuous random variables are incorporated. In reality, all of these problems are generalized by the concept of a multivariate probability distribution. The concepts are separated following conventions in engineering practice, as well as for the convenience of illustrating analytic and numerical solutions. + +--- + +_The primary author for this chapter is Robert Lanzafame._ \ No newline at end of file diff --git a/book/risk-analysis/definition.md b/book/risk-analysis/definition.md new file mode 100644 index 0000000..046f09b --- /dev/null +++ b/book/risk-analysis/definition.md @@ -0,0 +1,119 @@ +(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: + + + +```{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 +::: + diff --git a/book/risk-analysis/overview.md b/book/risk-analysis/overview.md new file mode 100644 index 0000000..2fd0574 --- /dev/null +++ b/book/risk-analysis/overview.md @@ -0,0 +1,13 @@ +# 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). \ No newline at end of file diff --git a/book/risk-analysis/risk-curves.md b/book/risk-analysis/risk-curves.md new file mode 100644 index 0000000..b76dc49 --- /dev/null +++ b/book/risk-analysis/risk-curves.md @@ -0,0 +1,86 @@ +(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_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. +``` \ No newline at end of file diff --git a/book/risk-analysis/steps.md b/book/risk-analysis/steps.md new file mode 100644 index 0000000..154fe6c --- /dev/null +++ b/book/risk-analysis/steps.md @@ -0,0 +1,144 @@ +(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. \ No newline at end of file diff --git a/book/risk-evaluation/cost-benefit.md b/book/risk-evaluation/cost-benefit.md new file mode 100644 index 0000000..cee0619 --- /dev/null +++ b/book/risk-evaluation/cost-benefit.md @@ -0,0 +1,138 @@ +(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. \ No newline at end of file diff --git a/book/risk-evaluation/decision.md b/book/risk-evaluation/decision.md new file mode 100644 index 0000000..9dfda7e --- /dev/null +++ b/book/risk-evaluation/decision.md @@ -0,0 +1,202 @@ +(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. +``` +```` \ No newline at end of file diff --git a/book/risk-evaluation/econ-optimization.md b/book/risk-evaluation/econ-optimization.md new file mode 100644 index 0000000..9ffdfc8 --- /dev/null +++ b/book/risk-evaluation/econ-optimization.md @@ -0,0 +1,58 @@ +(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$. \ No newline at end of file diff --git a/book/risk-evaluation/example-dike-height.md b/book/risk-evaluation/example-dike-height.md new file mode 100644 index 0000000..8aea567 --- /dev/null +++ b/book/risk-evaluation/example-dike-height.md @@ -0,0 +1,121 @@ +(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. +``` \ No newline at end of file diff --git a/book/risk-evaluation/overview.md b/book/risk-evaluation/overview.md new file mode 100644 index 0000000..372b495 --- /dev/null +++ b/book/risk-evaluation/overview.md @@ -0,0 +1,8 @@ +(risk_eval)= +# 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._ \ No newline at end of file diff --git a/book/risk-evaluation/safety-standards.md b/book/risk-evaluation/safety-standards.md new file mode 100644 index 0000000..312d6d0 --- /dev/null +++ b/book/risk-evaluation/safety-standards.md @@ -0,0 +1,371 @@ +(safety_standards)= +# 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. \ No newline at end of file