The economic analysis involves assessing the future outcomes of an option. Generally, a single value is picked to represent the best estimate that can be made. It is known that estimates will not always end up being correct but in the economic analysis, we usually treat them as if they were right.
A Range of Estimates
When parameters are explained using a range of values they are more elaborate as compared to when a single value is used to explain them. The range includes a pessimistic, most likely and optimistic estimates. could be comprised of an optimistic estimate, most likely estimate, and a pessimistic estimate. Through an economic analysis, one can determine if the decision made takes into consideration the range of projected values. We can calculate the approximate mean value for a given parameter under Beta distribution using the following formula
(10-1)
Probability
Probability is a technique for arriving at the estimates of future estimates. Probability is built on data or expert judgment, or a combination of both. It can also be described as the chances of an event occurring in a single trial. When considering a number of trials, probability can then be defined as the number of time that a certain result is likely to be observed. Probability should follow a set of rules as shown below:
0 £ probability £ 1 (10.2)
(outcomej) = 1, K = outcomes (10.3)
In a probability course, there are many probability distributions including normal, uniform, and beta. Engineering economy analysis utilizes 2 -5 outcomes which represent a variety of possibilities since if more outcomes are estimated using expert judgment, they would be inaccurate.
Joint Probability Distribution
In the joint probability distribution, random variables are taken to be statistically independent. It estimates the outcomes when over two variables are involved in the event. All random variables have their probability distribution. For example, if the events, for the independent variables Y and Z, the joint probability for both events to occur is as follows:
P (Z and Y) = P (Z) x P (Y) (10.4)
Expected Value
The expected value refers to the weighted average of every possible outcome by their probabilities. To calculate the expected value, one is required to calculate every outcome by its probability and adding up the results. This is as shown below:
Expected value = Outcome A x P (Y) +Outcome B
x P (Z) +…. (10.5)
Economic Decision Trees
A decision tree is a graphical representation of all the decision in a challenging project and all the possible outcomes and their probabilities. These are used to model complex engineering projects whose evaluations are correspondingly more complex. Forming a decision using decision trees is represented as follows:
Decision No Here the person making the decision chooses one path fro the available ones.
Chance Node Each likely outcome has a an associated path.
Risk
Risk refers to the chance of arriving at an outcome other than the expected value, with an emphasis on something negative. Risk can be measured using the probability of a loss and through standard deviation. Standard deviation measured the distribution of the outcomes about the expected value. Standard deviation is calculated as follows:
Standard deviation (s) = (10-6)
= (10-6′)
Simulation
Simulation refers to a complex methodology of considering risk in engineering economic analysis. This approach utilizes random sampling from the probability distribution of one or more variables to evaluate an economic paradigm for various duplications. For every duplication, each variable with a probability distribution is randomly sampled. Obtained values are then used in calculating the IRR, NPW, or EUAW. Finally, the results of all duplications are merged to generate a probability distribution for the IRR, NPW, or EUAW. Replication is then done manually with a table of random number through excel function, or through stand-alone simulation programs. Examples are as shown in the excel spreadsheet below:
Exercise
Problem 10-60
A project’s first cost is $25,000, and it has no salvage value. The interest rate for evaluation is 7%. The project’s life is from a discrete uniform distribution that takes on the values 7, 8, 9, and 10. The annual benefit is normally distributed with a mean of $4400 and a standard deviation of $1000. Using Excel’s RAND function, simulate 25 iterations. What are the expected value and standard deviation of the present worth?
Solution
Problem 10-61
A factory’s power bill is $55,000 a year. The first cost of a small geothermal power plant is normally distributed with a mean of $150,000 and a standard deviation of $50,000. The power plant has no salvage value. The interest rate for evaluation is 8%. The project’s life is from a discrete uniform distribution that takes on the values 3, 4, 5, 6, and 7. (The life is relatively short due to corrosion.) The annual operating cost is expected to be about $10,000 per year. Using Excel’s RAND function, simulate 25 iterations. What are the expected value and standard deviation of the present worth?
Solution
