Decision Making Analysis Domalaon, Romeuel M.Marbella, Janecca BMariñas, Patricia Grazielle

This Topic further expands our ability to analyze variation in estimates, to consider probability, and to make decisions under risk. Fundamentals discussed include variables; probability distributions, especially their graphs and properties of expected value and dispersion; random sampling; and the use of simulation to account for estimate variation in engineering economy studies. Through coverage of variation and probability, this chapter complements topics in the first sections of Chapter 1: the role of engineering economy in decision making and economic analysis in the problem-solving process. These techniques are more time-consuming than using estimates made with certainty, so they should be used primarily for critical parameters.

CERTAINTY, RISK, UNCERTAINTYDECISION MAKING UNDER

This is what we have done in most analyses thus far.Deterministic estimates are made and entered into measure of worth relations—PW, AW, FW,ROR, BC—and decision making is based on the results. The values estimated can be consideredthe most likely to occur with all chance placed on the single-value estimate

CERTAINTY

When there may be two or more observable values for a parameter and it is possible to estimatethe chance that each value may occur, risk is present. Virtually all decision making is performed under risk

RISK

Decision making under uncertainty means there are two or more values observable, but thechances of their occurring cannot be estimated or no one is willing to assign the chances. Theobservable values in uncertainty analysis are often referred to as states of nature.

UNCERTAINTY

Decision Making under Certainty

This is what we have done in most analyses thus far.Deterministic estimates are made and entered into measure of worth relations—PW, AW, FW,ROR, BC—and decision making is based on the results. The values estimated can be consideredthe most likely to occur with all chance placed on the single-value estimate.

CERTAINTY

A typical example is an asset’s first cost estimate made with certainty, say, P $50,000. A plot of P versus chance hasthe general form of Figure 19–1 a with one vertical bar at $50,000 and 100% chance placed on it.The term deterministic, in lieu of certainty , is often used when single-value or single-pointestimates are used exclusively.

CERTAINTY

Decision Making under Risk

RISK

When there may be two or more observable values for a parameter and it is possible to estimate the chance that each value may occur, risk is present. Virtually all decision making is performed under risk .

RISK

Some basics of probability and statistics are essential to correctly perform decision making under risk via expected value or simulation analysis. They are the random variable, probability, probability distribution , and cumulative distribution , as defined here.

A random variable or variable is a characteristic or parameter that can take on any one of several values. Variables are classified as discrete or continuous. Discrete variables have several specific, isolated values, while continuous variables can assume any value between two stated limits, called the range of the variable.

ELEMENTS IMPORTANT TO DECISION MAKING UNDER RISK

The estimated life of an asset is a discrete variable. For example, n may be expected to have values of n 3, 5, 10, or 15 years, and no others. The rate of return is an example of a continuous variable; i can vary from −100% to , that is, −100% i . The ranges of possible values for n (discrete) and i (continuous) are shown as the x axes in Figure 19–2 a . (In probability texts, capital letters symbolize a variable, say X , and small letters x identify a specific value of the variable. Though correct, this level of rigor in terminology is not applied in this chapter.)

ELEMENTS IMPORTANT TO DECISION MAKING UNDER RISK

Random Variable

A probability distribution describes how probability is distributed over the different values of a variable. Discrete variable distributions look significantly different from continuous variable distributions, as indicated by the inset at the right.

ELEMENTS IMPORTANT TO DECISION MAKING UNDER RISK

The individual probability values are stated asP ( X i ) probability that X equals X i

The distribution may be developed in one of two ways: by listing each probability value for each possible variable value (see Example 1.2) or by a mathematical description or expression that states probability in terms of the possible variable values

The estimated life of an asset is a discrete variable. For example, n may be expected to have values of n 3, 5, 10, or 15 years, and no others. The rate of return is an example of a continuous variable; i can vary from −100% to , that is, −100% i . The ranges of possible values for n (discrete) and i (continuous) are shown as the x axes in Figure 19–2 a . (In probability texts, capital letters symbolize a variable, say X , and small letters x identify a specific value of the variable. Though correct, this level of rigor in terminology is not applied in this chapter.)

ELEMENTS IMPORTANT TO DECISION MAKING UNDER RISK

Probability Distribution

Cumulative distribution, also called the cumulative probability distribution , is the accumulation of probability over all values of a variable up to and including a specified value.

ELEMENTS IMPORTANT TO DECISION MAKING UNDER RISK

Identified by F ( X i ), each cumulative value is calculated asF ( Xi ) sum of all probabilities through the value Xi

P ( X X i ) As with a probability distribution, cumulative distributions appear differently for discrete (stair stepped) and continuous variables (smooth curve). Examples 1.2 and 1.3 illustrate cumulative distributions that correspond to specific probability distributions. These fundamentals about F ( X i ) are applied in the next section to develop a random sample.

EXAMPLES

Simulation analysis. Use the chance and parameter estimates to generate repeated computations of the measure of worth relation by randomly sampling from a plot for each varying parameter similar to those in Figure 19–1 . When a representative and random sample is complete, an alternative is selected utilizing a table or plot of the results. Usually, graphics are an important part of decision making via simulation analysis. Basically, this is the approach discussed in the rest of this chapter.

THERE ARE TWO WAYS TO CONSIDER RISK IN AN ANALYSIS:

Expected value analysis. Use the chance and parameter estimates to calculate expected values E (parameter) via formulas such as Equation [18.2]. Analysis results in E (cash flow), E (AOC), and the like; and the final result is the expected value for a measure of worth, such as E (PW), E (AW), E (ROR), E (BC). To select the alternative, choose the most favorable expected value of the measure of worth. In an elementary form, this is what we learned about expected values in Chapter 18. The computations may become more elaborate, but the principle is fundamentally the same.

RISK