Risk Analysis and Decision-Making

Abstract

As stated in Chap. 1, inverse modeling is not an end by itself but a precursor to model building needed for either better understanding of the process or for decision-making that results in some type of action. Decision theory is the study of methods for arriving at rational decisions under uncertainty. This chapter provides an overview of quantitative decision-making methods under uncertainty along with a description of the various phases involved. The decisions themselves may or may not prove to be correct in the long term, but the scientific community has reached a general consensus on the methodology to address such problems. Different sources of uncertainty are described, and an overall framework along with various elements of decision-making under uncertainty is presented. How Bayes’ approach can play an important role in decision-making is also briefly presented. Decision problems are clouded by uncertainty, and/or by having to meet multiple objectives; this chapter also presents basic concepts of multi-attribute modeling. The role of risk analysis and its various aspects are covered along with applications from various fields. General principles are presented in somewhat idealized situations for better understanding, while several illustrative case studies are meant to provide exposure to real-world situations. The topics which relate to decision analysis are numerous, and at varying levels of maturity; only a basic overview of this vast body of knowledge has been provided in this chapter.

Problems

Pr.12.1

Identify and briefly discuss at least two situations (one from everyday life and one from an engineering or environmental field) which would qualify as problems falling under the following categories:

• (a) Low epistemic and low uncertainty

• (b) Low epistemic and high uncertainty

• (c) High epistemic and low aleatory

• (d) High epistemic and high aleatory

Pr. 12.2

The owner of a commercial building is considering improving the energy efficiency of his building systems using both energy management options (involving little or no cost) and retrofits to energy equipment prepare:

• (a) An influence diagram for this situation

• (b) A decision tree diagram (similar to Fig. 12.3)

Pr. 12.3

An electric utility is being pressured by the public utility commission to increase its renewable energy portfolio (in this particular case, energy conservation does not qualify). The decision-maker in the utility charges his staff to prepare the following (note that there are different possibilities):

• (a) an influence diagram for this situation,

• (b) a decision tree diagram.

Pr. 12.4

Compute the representative discretized values and the associated probabilities for the Gaussian distribution (given in Appendix A3) for:

• (a) a 3-point approximation,

• (b) a 5-point approximation.

Pr. 12.5

Consider Example 12.2.4.

• (a) Rework the problem including the effect of increasing unit electricity cost (linear increase over 10 years of 3% per year). What is the corresponding probability level of moving at which both AC options break even, i.e., the indifference point?

• (b) Rework the problem but with an added complexity. It was stated in the text (Sect. 12.2.6a) that discounted cash flow analysis where all future cash flows are converted to present costs is an example of an additive utility function. Risk of future payments is to be modeled as increasing linearly over time. The 1st year, the risk is low, say 2%, the 2nd year is 4% and so on. How would you modify the traditional cash flow analysis to include such a risk attitude?

Pr. 12.6

Traditional versus green homes

An environmentally friendly “green” house costs about 25% more to construct than a conventional home. Most green homes can save 50% per year on energy expenses to heat and cool the dwelling.

• (a) Assume the following:

  • It is an all-electric home needing 8 MWh/year for heating and cooling, and 6 MWh/year for equipment in the house

  • The conventional home costs $ 250,000 and its life span is 30 years

  • The cost of electricity is $ 15 cents with a 3% annual increase

  • The “green” home has the same life and no additional value at the end or 30 years can be expected

  • The discount rate (adjusted for inflation) is 6% per year.

• (b) Is the “green” home a worthwhile investment purely from an economic point of view?

• (c) If the government imposes a carbon tax of $ 25/ton, how would you go about reevaluating the added benefit of the “green” home? Assume 170 lbs of carbon are released per MWh of electricity generation (corresponds to California electric mix).

• (d) If the government wishes to incentivize such “green” construction, what amount of upfront rebate should be provided, if at all?

Pr. 12.7

Risk analysis for building owner being sued for indoor air quality

A building owner is being sued by his tenants for claimed health ailments due to improper biological filtration of the HVAC system. The suit is for $ 500,000. The owner can either settle for $ 100,000 or go to court. If he goes to court and loses, he has to pay the lawsuit amount of $ 450,000 plus the court fees of $ 50,000. If he wins, the plaintiffs will have to pay the court fees of $ 50,000.

• (a) Draw a decision tree diagram for this problem

• (b) Construct the payoff table

• (c) Calculate the cutoff probability of the owner winning the lawsuit at which both options are equal to him.

Pr. 12.8

Plot the following utility functions and classify the decision-maker’s attitude towards risk:

• (a) $U(x) = \displaystyle\frac{{100 + 0.5x}}{{250}}{\rm{ }}\quad- 200 \le x \le 300$

• (b) $U(x) = \displaystyle\frac{{{{(x + 100)}^{1/2}}}}{{20}}{\rm{ }}\quad- 100 \le x \le 300$

• (c) $U(x) = \displaystyle\frac{{{x^3} - 2{x^2} - x + 10}}{{800}}{\rm{ }}\quad0 \le x \le 10$

• (d) $U(x) = \displaystyle\frac{{0.1{x^2} + 10x}}{{110}}{\rm{ }}\quad0 \le x \le 10$

Pr. 12.9

Consider payoff Table 12.20 with four alternatives and four chance events (or states of nature).

• (a) Calculate the alternative with the highest expected value (EV).

• (b) Resolve the problem assuming the utility cost function is exponential with zero minimum monetary value and risk tolerance R = $ 2 k.

Table 12.20 Payoff table for Problem 12.9

Pr. 12.10

Monte Carlo analysis for evaluating risk for property retrofits

The owner of a large office building is considering making major retrofits to his property with system upgrades. You will perform a Monte Carlo analysis with 5,000 trials to investigate the risk involved in this decision. Assume the following:

• The capital investment is $ 2 M taken to be normally distributed with 10% CV.

• The annual additional revenue depends on three chance outcomes:

  • $ 0.5 M under good conditions, probability p(Good) = 0.4

  • $ 0.3 M under average conditions, p(Average) = 0.3

  • $ 0.2 M under poor conditions, p(Poor) = 0.3

• The annual additional expense of operation and maintenance is normally distributed with $ 0.3 M and a standard deviation of $ 0.05.

• The useful life of the upgrade is 10 years with a normal distribution of 2 years standard deviation.

• The effective discount rate is 6%.

Pr. 12.11

Risk analysis for pursuing a new design of commercial air-conditioner

A certain company which manufactures compressors for commercial air-conditioning equipment has developed a new design which is likely to be more efficient than the existing one. The new design has a higher initial expense of $ 8,000 but the advantage of reduced operating costs. The life of both compressors is 6 years. The design team has not fully tested the new design but has a preliminary indication of the efficiency improvement based on certain limited tests have been determined. The preliminary indication has some uncertainty regarding the advantage of the new design, and this is quantified by a discrete probability distribution with four discrete levels of the efficiency improvement goal as shown in the first column of Table 12.21. Note that the annual savings are incremental values, i.e., the savings over those of the old compressor design.

Table 12.21 Data table for Problem 12.11

• (a) Draw the influence diagram and the decision tree diagram for this situation

• (b) Calculate the present value and determine whether the new design is more economical

• (c) Resolve the problem assuming the utility cost function is exponential with zero minimum monetary value and risk tolerance R = $ 2,500

• (d) Compute the EVPI (expected value of perfect information)

• (e) Perform a Monte Carlo analysis (with 1,000 trials) assuming random distributions with 10% CV for p(L = 90) and p(L = 30), 5% for p(L = 70). Note that p(L = 50) can be to be determined from these, and that negative probability values are inadmissible, and should be set to zero. Generate the PDF and the CDF and determine the 5% and the 95% percentiles values of EV

• (f) Repeat step (e) with 5,000 trials and see if there is a difference in your results

• (g) Repeat step (f) but now consider the fact that the life of both compressors is normally distributed with mean of 6 years and standard deviation of 0.5 years.

Pr. 12.12

Monte Carlo analysis for deaths due to radiation exposure

Human exposure to radiation is often measured in rems (roentgen-equivalent man) or millirem (mrem). The cancer risk caused by exposure to radiation is thought to be approximately one fatal cancer per 8,000 person-rems of exposure (e.g., one cancer death if 8,000 people are exposed to one rem each, or 10,000 exposed to 0.8 rems each,…). Natural radioactivity in the environment is thought to result in an exposure of 130 mrem/year.

• (a) How many cancer deaths can be expected in the United States per year as a result (assume a population of 300 million).

• (b) Reanalyze the situation while considering variability. The exposure risk of 8,000 person-rems is normally distributed with 10% CV, while the natural environment radioactivity has a value of 20% CV. Use 10,000 trials for the Monte Carlo simulation. Calculate the mean number of deaths and the two standard deviation range.

• (c) The result of (b) could have been found without using Monte Carlo analysis. Using formulae presented in Sect. 2.3.3 related to functions of random variables, compute the number of deaths and the two standard deviation range and compare these with the results from (b).

Pr. 12.13

Radon is a radioactive gas resulting from radioactive decay of uranium to be found in the ground. In certain parts of the country, its levels are elevated enough that when it seeps into the basements of homes, it puts the homeowners at an elevated risk of cancer. EPA has set a limit of 4 pCi/L (equivalent to 400 mrem/year) above which mitigation measures have to be implemented. Assume the criterion of one fatal cancer death per 8,000 person-rems of exposure,

• (a) How many cancer deaths per 100,000 people exposed to 4 pCi/L can be expected?

• (b) Reanalyze the situation while considering variability. The exposure risk of 4pCi/L is a log-normal distribution with 10% CV, while the natural environment radioactivity is normally distributed with 20% CV. Use 10,000 trials for the Monte Carlo simulation. Calculate the mean number of deaths and the 95% confidence intervals.

• (c) Can the result of (b) be found using formulae presented in Sect. 2.3.3 related to functions of random variables? Explain.

Pr. 12.14

Monte Carlo analysis for buying an all-electric car

A person is considering buying an all-electric car which he will use daily to commute to work. He has a system at home of recharging the batteries at night. The all-electric car has a range of 100 miles while the minimum round trip of a daily commute is 40 miles. However, due to traffic congestion, he sometimes takes a longer route which can be represented by a log-normal distribution of standard deviation 5 miles. Moreover, his work requires him to visit clients occasionally for which purpose he needs to use his personal car. Such extra trips have a lognormal distribution of mean of 20 miles and a standard deviation of 5 miles.

• (a) Perform a Monte Carlo simulation (using 10,000 trials) and estimate the 99% probability of his car battery running dry before he gets home on days when he has to make extra trips and take the longer route.

• (b) Can you verify your result using an analytical approach? If yes, do so and compare results.

• (c) If the potential buyer approaches you for advice, what types of additional analyses will you perform prior to making your suggestion?

Pr. 12.15

Evaluating feed-in tariffs versus upfront rebates for PV systems

One of the promising ways to promote solar energy is to provide incentives to homeowners to install solar PV panels on their roofs. The PV panels generate electricity depending on the collector area, the efficiency of the PV panels and the solar radiation falling on the collector (which depends both on the location and on the tilt and azimuth angles of the collector). This production would displace the need to build more power plants and reduce adverse effects of both climate change and health ailments of inhaling polluted air. Unfortunately, the cost of such electricity generated in yet to reach grid-parity, i.e., the PV electricity costs more than traditional options, and hence the need for the government to provide incentives.

Two different types of financing mechanisms are common. One is the feed-in tariff where the electricity generated at-site is sold back to the electric grid at a rate higher than what the customer is charged by the utility (this is common in Europe). The other is the upfront rebate financial mechanism where the homeowner (or the financier/installer) gets a refund check (or a tax break) per watt-peak (W_p) installed (this is the model adopted in the U.S.). You are asked to evaluate these two options using the techniques of decision-making, and present your findings and recommendations in the form of a report. Assume the following:

• The cost of PV installation is $ 7 per Watt-peak (W_p). This is the convention of rating and specifying PV module performance (under standard test conditions of 1 kW/m² and 25°C cell temperature)

• However, the PV module operates at much lower radiation conditions throughout the year, and the conventional manner of considering this effect is to state a capacity factor which depends on the type of system, the type of mounting and also on location. Assume a capacity factor of 20%, i.e., over the whole year (averaged over all 24 h/day and 365 days/year), the PV panel will generate 20% of the rated value

• Life of the PV system is 25 years with zero operation and maintenance costs

• The average cost of conventional electricity is $ 0.015/kWh with a 2% annual increase

• The discount rate for money borrowed by the homeowner towards the PV installation is 4%.