Religious-pilgrimage system

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Lecture 4.2a_Analyzing Risk Management of Natural Hazards HRV KFSC, Fall 2020

I. Introduction

a. Review of risk management options i. What happens when you increase spending on mitigation investment?

ii. Risk(M) = Hazard Probability x Asset Vulnerability Probability(M) x Asset Loss

iii. So, decreasing the probability of vulnerability with higher mitigation investments will decrease the risk.

iv. Examples? 1. More investment in public education about safe

driving when facing dust storms, heavy rain, and floods to lower or mitigate crash and drowning probabilities of vulnerability.

2. More investment in inspecting hotels for violations of fire safety regulations to lower or mitigate fire probabilities of vulnerability.

3. More investment in heat-tolerant materials in Saudi Aramco oil refineries to lower or mitigate vulnerability probabilities to global warming.

b. So R(M) should be a downward sloping curve.

II. Optimizing Mitigation Investment a. Basic strategic problem: what is the best level of mitigation

investment from a Net-Utility Gain (NUG) perspective? i. If the mitigation investment, M, is too small, then R(M) will scarcely be greater than R(0) so NUG will be negative – around the value of –M, and the risk will therefore remain very high.

ii. This is like the situation where the fire fighters use a low powered, and therefore low cost water pump that scarcely does anything to put out a fire.

iii. If the mitigation investment, M, is too large, the risk R(M) will be much lower, which is good, but M will be much higher than R(M), so NUG will very negative since M is so large.

1. This is like the situation when corrupt city officials receive bribes from fire-truck manufacturers to by many more fire trucks than needed.

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2. As a result, the local house fires are quickly put out, but most of the fire trucks are never used.

3. So, the NUG of the investment in fire trucks is very negative.

b. So, what is the best or “optimum” level of mitigation

investment, M? i. This is when a particular level of investment, M*, will generate the highest possible net utility gain, NUG(M*).

ii. This problem is easy to solve with calculus theory. iii. One has to find at what value of M will make the slope of

the total benefit function, TB(M), equal to the slope of the total cost function TC(M).

iv. The calculus expression, d[TB(M)]/d[M] is a fancy way of saying what the slope of the function TB(M) is for a given level of M: the change or “delta” in TB(M)/change or “delta” of M.

v. So, calculus theory teaches that the greatest difference between total benefit function, TB(M) and total cost function, TC(M) occurs when d[TB(M)/d[m] or dTB(M)/dM = dTC(M)/dM.

vi. You can see what this means in the following example in Fig. 1

TB(M)

$

M

TC(M)

M*

NUG(M*) = TB(M*) – TC(M*)

dTB(M*)/dM or slope of TB(M) when M = M*

dTC(M*)/dM or slope of TC(M) when M = M*

Note how when dTC(M)/dM is equal to dTB(M)/dM at M*, NUG(M) is at its highest value

Fig. 1

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c. In order to determine where NUG is maximum, we first need to determine what R(0), R(M), R(0) – R(M), and M look like from a graphical perspective.

d. Interpretation of Fig 2

i. Risk with no mitigation investment, R(0) is a constant. ii. But as mitigation investment increases, it drives down

risk, R(M) by decreasing vulnerabilities or lowering the hazard magnitude’s impact, but only to a point.

iii. Why? 1. R(M) = Probability of Hazard x Probability of

Vulnerability (M) x Loss. 2. By increasing mitigation investment, M, you

decrease the probability of vulnerability, thus decreasing risk, R(M).

iv. R(M) reaches a level of “diminishing returns” where ever greater increases of mitigation investment leads to every smaller decreases of risk.

1. For example, you can build higher and higher walls to block flood water from a storm surge during a hurricane, but it will not stop the risk from wind and electrical-blackout damage.

R(0)

M

R(M)

M

R(0)

Fig. 2

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2. Or, you can build a house with more and more fire-proof material, but that will not stop the furniture from catching fire.

v. What does the function of M look like? It is a straight line at a 45 degree angle, of course because the function is M=M.

e. Graphing the total benefit function, R(0) – R(M).

i. So, now that we have functions for R(0) and R(M), we can get a total benefit function, TB(M), by calculating R(0) – R(M) as shown in Fig. 3.

ii. Once we have the upward sloping total benefit function, TB(M)=R(0) – R(M), we can determine what level of mitigation investment, M, will deliver the maximum level of NUG.

iii. As shown in Fig. 4, NUG is maximum at M* because that is where the slope of TC(M) and the slope of M are equal, or where the distance between these two curves in the greatest.

R(0)

R(M)

M

R(0)

Fig. 3

R(0) – R(M1)

R(0) – R(M1)

R(0)- R(M2)

M1 M2

R(0)- R(M3)

R(0)- R(M4)

R(0)- R(M5)

M3 M4 M5

R(0)- R(M2)

R(0)- R(M3)

R(0)- R(M4)

R(0)- R(M5)

$

TB(M) = R(0) – R(M)

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TB(M) = R(0) – R(M)

M

$

TC(M) = M

M*

NUG(M*) = TB(M*) – TC(M*)

Fig. 4

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Sample Problem 1: Determine best level of M

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R(M)

M

Solution I

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R(M)

M

R(0)

TC(M) = M

x x x

x x

x x

TB(M) = R(0) – R(M)

M*

$

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Sample Problem 2: Effect of Vulnerability Increase

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R1(M): Risk with existing vulnerability

M

R2(M): Risk with increased vulnerability

R2(M) has the same slope as R1(M)

Sample Solution 2: Effect of Vulnerability Increase

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R1(M): Risk with existing vulnerability

M

R2(M): Risk with increased vulnerability

R1(0)

R2(0)

R1(0) – R1(M) = R2(0) – R2(M) = TB(M)

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R1(M)

M

$

R2(M)

Sample Problem 3: Effect of Vulnerability Increase

R2(M) has a steeper slope than R1(M)

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R1(M)

M

R1(0)

TC(M) = M

TB1(M) = R1(0) – R1(M)

M1*

$

R2(0)

R2(M)

TB2(M) = R2(0) – R2(M)

M2*

Sample Solution 3: Effect of Vulnerability Increase