Engineering Economics in Canada Chapter 12 Dealing with Risk: Probability Analysis

Copyright © 2006 Pearson Education Canada Inc Introduction to Uncertainty and Risk We all encounter situations where we don’t know for sure what events will happen in the future. Only until the event occurs, we know the exact outcome. We always talk about the “chance” that the certain event may take place, which is mathematically described by probability theory.

Copyright © 2006 Pearson Education Canada Inc What will be covered in this chapter? Decision making schemes using the knowledge of the probability (chances) of uncertainties. decision trees, for decomposing a problem into its decision alternatives and uncertain events. decision criteria for evaluating alternatives in the decision tree obtained. a brief introduction of Monte Carlo simulation, for analyzing complex problems.

Copyright © 2006 Pearson Education Canada Inc Short introduction of probability theory Uncertainty is characterized by unknown outcomes. Unknown outcomes can be uniquely represented by a random variable. A random variable can take on a number of possible values. Only one of these values will eventually occur. A function describes the likelihood of each value of a random variable: probability distribution function

Copyright © 2006 Pearson Education Canada Inc Probability Distribution Function (PDF) Consider a random variable X (capitalized) that can take on m discrete outcomes x 1, x 2, …, x m (lowercase): discrete random variable If these outcomes are mutually exclusive and collectively exhaustive, a probability distribution function p(x) is a set of numerical measures p(x i ) such that: Pr(X= x i ) = p(x i ). Intuitively, the higher p(x i ), the more likely it is that x i will occur.

Copyright © 2006 Pearson Education Canada Inc Example 12.1 You are testing three solder joints on a printed circuit board. Every solder can either be open (O) or closed (C). determine the PDF for the random variable X, the number of open joints in three tested joints. The probability that a single tested joint will be open is 20%. Solution: X can take on four possible values: x 1 = 0, x 2 = 1, x 3 = 2, and x 4 = 3.

Copyright © 2006 Pearson Education Canada Inc Table 12.1 TestNumber of Sequence“Opens”Probability (O,O,O) = 0.2 × 0.2 × 0.2 (O,O,C) = 0.2 × 0.2 × 0.8 (O,C,O) = 0.2 × 0.8 × 0.2 (C,O,O) = 0.8 × 0.2 × 0.2 (O,C,C) = 0.2 × 0.8 × 0.8 (C,C,O) = 0.8 × 0.8 × 0.2 (C,O,C) = 0.8 × 0.2 × 0.8 (C,C,C) = 0.8 × 0.8 × 0.8

Copyright © 2006 Pearson Education Canada Inc Example 12.1… the PDF for X, the number of “open” joints in the three tests is: Pr(X = 0) = p(x 1 ) = Pr(X = 1) = p(x 2 ) = Pr(X = 2) = p(x 3 ) = Pr(X = 3) = p(x 4 ) = 0.008

Copyright © 2006 Pearson Education Canada Inc Cumulative Distribution Functions (CDF) The cumulative distribution function for a discrete random variable X is defined as follows: For Example 12.1, the CDF for the number of open joints is: Pr(X ≤ 0) = P(x 1 ) = Pr(X ≤ 1 ) = P(x 2 ) = Pr(X ≤ 2 ) = P(x 3 ) = Pr(X ≤ 3 ) = P(x 4 ) = 1.000

Copyright © 2006 Pearson Education Canada Inc Mean and Variance Two summary statistics useful in describing a random variable X is its expected value, or mean, E(X), and variance, Var(X). If X can take values x 1, x 2, …, x m, then its mean is: And its variance is:

Copyright © 2006 Pearson Education Canada Inc Mean and Variance The expected value of a random variable is much like the centre of mass for an object. –The expected value is simply the centre of the probability “mass.” The variance measures the degree of spread or dispersion of a random variable about the mean.

Copyright © 2006 Pearson Education Canada Inc Decision Trees Formal methods can help by: –providing a means of decomposing a problem and structuring it clearly. –suggesting a variety of decision criteria to help with the process of selecting a preferred course of action. decision trees, a graphical means of structuring a decision-making situation where uncertainties can be characterized by probability distributions.

Copyright © 2006 Pearson Education Canada Inc Decision Trees A decision tree is a graphical representation of the logical structure of a decision problem in terms of –the sequence of decisions to be made and –outcomes of chance events. It provides a mechanism to decompose a large and complex problem into a sequence of small and essential components. It clarifies the options a decision maker has and provides a framework with which to deal with the risk involved.

Copyright © 2006 Pearson Education Canada Inc Example 12.2 Edwin Electronics (EE) has a factory for assembling TVs. EE outsources the TV screen to a supplier, but are considering bringing screen production in-house. Uncertainty in demand for the company’s TVs has an important bearing on the decision. –If the future demand is low, outsourcing seems to be the reasonable option in order to save production costs. –On the other hand, if the demand is high, it may be worthwhile to produce the screens on-site due to economies of scale. EE’s engineers represented their decision in a graphical manner with a decision tree.

Copyright © 2006 Pearson Education Canada Inc Example 12.2…

Copyright © 2006 Pearson Education Canada Inc Components of a Decision Tree There are four main components in a decision tree: 1.A decision node represents a decision to be made by the decision maker. It is denoted by a square. 2.A chance node represents an event whose outcome is uncertain. It is denoted by a circle. 3.The branches of a tree are the lines connecting nodes from left to right, depicting the sequence of possible decisions and chance events. 4.Finally, the leaves indicate the values, or payoffs, associated with each terminal (rightmost) branch of the decision tree. A decision tree grows from left to right and usually begins with a decision node.

Copyright © 2006 Pearson Education Canada Inc Decision Criteria Once a complete decision tree is structured, an analyst is in a better position to select a preferred alternative from a set of possible choices. This section deals with several commonly used decision criteria for situations that involve uncertainty. –Expected Value –Dominance

Copyright © 2006 Pearson Education Canada Inc Expected Value One criterion for selecting among risky alternatives is expected value, EV. With a decision tree, this is carried out as follows: 1.Structure the problem: Develop a decision tree. 2.Rollback: moving from right to left on the tree: –At each chance node, compute the expected value of the possible outcomes. –At each decision node, select the option with the best expected value. This becomes the value associated with the decision node. Mark option(s) not selected with a double-slash (//) on the corresponding branch. –Continue until the leftmost node is reached. 3.Conclusion: The expected value associated with the final node is the expected value of the overall decision.

Copyright © 2006 Pearson Education Canada Inc Example 12.3 Carry out a decision tree analysis given the figure below. What decision should they make based on expected value?

Copyright © 2006 Pearson Education Canada Inc Example 12.7…

Copyright © 2006 Pearson Education Canada Inc Example 12.7…

Copyright © 2006 Pearson Education Canada Inc Dominance The expected value criterion is straightforward, but is only a summary measure. –It does not consider the dispersion of the outcomes associated with a decision. information from the probability distributions allows a decision maker to use dominance concepts to screen out less preferred alternatives, or to pick the best of several alternatives.

Copyright © 2006 Pearson Education Canada Inc Mean-Variance Dominance Suppose that an engineer is attempting to select between two projects X and Y where the outcomes are monetary (i.e., more is better). Alternative X is said to have mean-variance dominance over alternative Y: (a) If EV(X) ≥ EV(Y) and Var(X) < Var(Y), or (b) If EV(X) > EV(Y) and Var(X) ≤ Var(Y). an alternative is said to be mean-variance efficient if no other alternative has both a higher mean and a lower variance.

Copyright © 2006 Pearson Education Canada Inc Example 12.9 Prod’nDemand Pattern Strategy123Mean Variance Strategies 1 and 3 remain. A choice between the two will require management to assess its willingness to trade-off mean profits with variability in profits.

Copyright © 2006 Pearson Education Canada Inc Outcome Dominance Outcome dominance of alternative X over alternative Y can occur in one of two ways. –the worst outcome for alternative X is at least as good as the best outcome for alternative Y. –when one alternative is at least as preferred to another for each outcome, and is better for at least one outcome. Outcome dominance can be useful in screening out alternatives that are clearly worse than others Though it straightforward to apply, it may not remove many alternatives.

Copyright © 2006 Pearson Education Canada Inc Stochastic Dominance Example 12.10: Suppose that the probability distribution functions (risk profiles) of the outcomes for the two decision alternatives that EE is considering are as shown in Figure 12.10

Copyright © 2006 Pearson Education Canada Inc Stochastic Dominance… A look at the cumulative distribution functions, (cumulative risk profiles) for the two alternatives provides further insight.

Copyright © 2006 Pearson Education Canada Inc Stochastic Dominance… The cumulative risk profile for the outsource decision either overlaps with or lies to the left and above of the cumulative risk profile of the produce decision –for all outcomes, the probability that the outsource decision gives a lower profit per unit is equal to or greater than the corresponding probability for the produce decision. The produce strategy is said to dominate the outsource strategy according to (first-order) stochastic dominance.

Copyright © 2006 Pearson Education Canada Inc Dominance Stochastic dominance and outcome dominance can be used to screen alternatives, but they are often not able to provide a definitive best alternative. Despite this limitation, cumulative risk profiles can be very useful in making statements such as: –“Alternative A is more likely to produce a profit in excess of $ than alternative B, or –“Project C is more likely to suffer a loss than project D”.

Copyright © 2006 Pearson Education Canada Inc Monte Carlo Simulation Monte Carlo simulation can be very useful when analyzing complex problems characterized by multiple sources of risk. Each decision strategy is evaluated by repeatedly randomly sampling branches of the decision tree and then constructing risk profiles for the relevant performance measures.

Copyright © 2006 Pearson Education Canada Inc Probability Distribution Estimation Monte Carlo simulation attempts to construct the probability distribution of an outcome performance measure of a project (e.g., present worth) by repeatedly sampling from the input random variable probability distributions. Once the probability distribution is estimated, summary statistics associated with a decision strategy can be used to gain insight into the possible performance level for the project.

Copyright © 2006 Pearson Education Canada Inc The Monte Carlo Simulation Approach A Monte Carlo simulation model is constructed via a five step process: 1.Analytical model: Identify input random variables that affect the outcome performance measure of the project. Develop the equation(s) necessary to compute the outcome performance measure from a particular realization of the input random variables. 2.Probability distributions: Establish an appropriate probability distribution for each input random variable. 3.Random sampling: Sample a value for each input random variable from its associated probability distribution.

Copyright © 2006 Pearson Education Canada Inc The Monte Carlo Simulation Approach… 4.Repeat sampling: Continue sampling until a sufficient sample size is obtained for the sampled outcomes (the computed performance measure) 5.Summary: Summarize the frequency distribution of the sample outcomes using a histogram. Summary statistics, like the range of possible outcomes and expected value can also be calculated from the sample outcomes.

Copyright © 2006 Pearson Education Canada Inc Application Issues Decision trees and Monte Carlo simulation are powerful tools for analyzing decision making situations that involve risk. –Use of probability distributions permits the engineer to get an overall picture of risk. A drawback of these methods is that specifying the probability distribution of outcomes can be challenging, and at times, highly subjective. Despite this drawback, decision trees and Monte Carlo simulation are widely used.

Copyright © 2006 Pearson Education Canada Inc Summary Introduction to Uncertainty and Risk Basic Concepts of Probability Random Variables and Probability Distributions Structuring Decisions with Decision Trees Decision Criteria –Expected Value –Dominance Monte Carlo Simulation –Dealing with Complexity –Probability Distribution Estimation –The Monte Carlo Simulation Approach Application Issues