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Uncertainty in Future Events

The ability to forecast with high accuracy on future events is an important aspect of engineering. However, even with the best estimates, it is impossible to guarantee a 100% accuracy. Disruptive changes in technology, economy, and culture are usually bound to occur and alter the manner that future events occur. Specifically, disruptive changes can lead to a loss in value due to obsolescence, reduction in demand, or increase in operation cost (Sullivan, Wicks, & Koelling, 2014). In light of this, engineering economic analysis assesses the long-term consequences of projects while considering the time value of money.

Due to the inability to accurately foretell the future, engineers assess the cost and benefits of assets by comparing various scenarios such as most optimistic, most pessimistic, and most likely events that affect a product. They then give a weight to each scenario, which they later use to estimate the future cost or benefit of an event. A break-even analysis is conducted to examine the impact of the variability of estimates of the outcome (Newnan, Lavelle, & Eschenbach, 2013). It enables an engineer to estimate the amount of variability that a parameter may have before an initial decision is affected. However, the breakeven analysis does not show how an individual may take the inherent variability of parameters into account in an economic analysis (Blank & Tarquin, 2013). It is important to note that the salvage value is the component that mostly affects the differences between the economic alternatives. When different projects or assets have an almost similar salvage value, their economic differences may be small. On the contrary, a large difference in salvage values may make different projects to have huge economic differences.

The three range of scenarios; optimistic, pessimistic, and most likely are useful in determining the cost or benefits of an event. In a real-world scenario, an engineer is normally faced with a range of variables that fall within these categories. Ordinarily, most of these variables lie within the most likely range, while few of them are in the optimistic and pessimistic range (Newnan, Eschenbach, & Lavelle, 2004). The following equation, which considers the difference in weight of the variables is used to estimate the mean value of an asset.

Mean value= (optimistic+ (most likely)4+ pessimistic value))/6

**Probability**

Probabilities can also be used to determine the cost or benefits of an event. In this case, the weight of each class of costs or benefits is calculated using the probabilities. In probability, the sum of the probable ratios lies between zero and one. Although most research and analytical works use a lot of samples to estimate the most accurate probabilities, engineering normally uses between 2 and 5. The reason for the use of a small sample is because it is estimated by experts in their respective fields. In addition, each data requires detailed analysis to find if it is accurate (Newnan, Eschenbach, & Lavelle, 2004). Therefore, a small data compensates with the detailed analysis done on each sample.

When using probability to estimate the benefits or cost, the weight of each category, most likely, optimistic, and pessimistic are multiplied with the weights and then added up to estimate the overall cost or benefit of an event.

Joint Probability

In some cases, an asset or a project may have more than one probability due to various conditions that it must fulfill. For instance, a project may have the 0.2, 0.5, and 0.3 as the ratios for optimistic, most likely, and pessimistic respectively. Further, the asset may have a ratio of remaining in operation for 6 years as 0.7 and for 12 years as 0.3. This example represents a case of joint probability since each score (optimistic, most likely, and pessimistic) is joined with various probable scores of the time that the project may be in operation (6 years or 7 years). Moreover, both of these classes are independent of each other (Sullivan, Wicks, & Koelling, 2014).

The probability of A and B independent variable is presented as, *P*(A and B)= *P*(A) * *P*(B)

**Expected Value**

The expected value refers to the arithmetic mean that is calculated in probability distributions. The expected value is calculated by getting the sum of the weight of each scenario. The weights are calculated by multiplying each outcome with its probability of occurring (Blank & Tarquin, 2013).

Expected Value (A to N) = Outcome _{A}* P(A)+ Outcome _{B}*P(B)+… + Outcome _{N} * P(N)

**Economic Decision Trees**

An economic decision tree refers to a tree like structure that is used to determine the probable outcomes of various decisions that engineers make. It usually has the following shape

D_{1}

Decision node D_{2 }Decision maker chooses either of the available paths

D_{x}

p_{1} C_{1}

Chance node p_{2} C_{2 }Represents probabilistic chance for each event_{. }p_{3}

C_{3}

Outcome (C_{1}, C_{2},…,C_{Y}) Has probability (p_{1}, p_{2}, p_{3, }…p_{y}) respectively.

Outcome node shows the results of a particular path in a decision tree.

Pruned branch indicates that the branch has been pruned and another path has been chosen.

The chance nodes, decision node, and outcome nodes illustrate the problems structure.

Engineers make decisions of the best asset using the final nodes in a tree. The criteria used in the decision-making entails maximizing the present worth (PW) or minimizing the equivalent uniform annual cost (EUAC). The expected value for PW and EUAC is calculated at the chance nodes. A branch is chosen if it originates from the decision node that has the best PW or EUAC (Newnan, Lavelle, & Eschenbach, 2013). If it originates from a chance node, an individual calculates the expected value and inputs the value in the node. This process is important since it rolls back the values from the terminal nodes to the initial decision.

**Risk**

Risk refers to the probability that the company will get results that are not similar to the expected values, especially ones that lead to a loss or make the decision worse off. Risk is mainly measured using the probability of a loss. Alternatively, the standard deviation is used to measure how much the outcomes are dispersed from the expected values (Newnan, Eschenbach, & Lavelle, 2004). The standard deviation (α) is derived from the square root of the variance. The variance, on the other hand, refers to the difference between the mean and the actual values.

Standard deviation (α) = √ [EV(X – mean)^{2}]

Standard deviation (α) = √ {EV(X2) – [EV(X)]^{2}}

Standard deviation (α) = √ {Outcome^{2}_{A} x P (A) + Outcome^{2}_{B} x P(B)+… – expected value^{2}}

**Risk versus Return**

The risk versus return refers to the probability of gaining in comparison to the probability of losing. A measure of the risk, which is calculated using standard deviation is placed on the x-axis where it is compared with the return, which is measured using the expected value that is placed on the y-axis. When assessing the projects to be considered, an individual checks for those that are not dominated by other projects (Newnan, Eschenbach, & Lavelle, 2004). Dominated projects are the ones that lie below the efficient frontier and have other projects above them in the chart. Projects that are in the efficient frontiers are the ones that are selected.

**Simulation**

Simulation refers to the process of assessing how an asset will be in its future time. Economic simulation uses random sampling of one of the variables to analyze an economic model for many iterations. The sampled values are used to calculate the PW, IRR, or EUAC for each iteration (Blank & Tarquin, 2013). The results from the iteration are then combined to create a probability distribution for the PW, IRR, or EUAC. Various analytical software such as Excel, @Risk, and Crystal Ball generate random numbers, which are used in simulation to assess possible future events.

**Questions**

**10.1**

Telephone poles exemplify items that have varying useful lives. Telephone poles, once installed in a location, remain in useful service until one of a variety of events occurs.

- Name three reasons why a telephone pole might be removed from useful service at a particular location.

Solution

Damage by mites or lightning

Changes in technology resulting in the use of mobile phones instead of landline connections

Migration of individuals to various areas making the infrastructure unimportant

(b) You are to estimate the total useful life of telephone poles. If the pole is removed from an original location while it is still serviceable, it will be installed elsewhere. Estimate the optimistic life, most likely life, and pessimistic life for telephone poles. What percentage of all telephone poles would you expect to have a total useful life greater than your estimated optimistic life?

Solution

Optimistic Life is 25 years

Most likely life is 20 years

Pessimistic life is 10 years.

Total useful life= (optimistic+ (most likely)4+ pessimistic value))/6

Total useful life= (25+ (20* 4)+ 10)/6= 19.17 Years

**10-6**

Annual savings due to an energy efficiency project have a most likely value of $30,000. The high estimate of $40,000 has a probability of 0.2, and the low estimate of $20,000 has a probability of 0.30. What is the expected value for the annual savings?

Solution

Expected Value (A to N) = Outcome _{A}* P(A)+ Outcome _{B}*P(B)+… + Outcome _{N} * P(N)

Sum of all probabilities = 1

Therefore, Most likely probability = 1- (0.2+ 0.3)= 0.5

Expected value = ((40,000* 0.2) + (30,000* 0.5) + (20,000* 0.3)

Expected value= 8,000+15,000+ 6,000= 29,000

**10-15 **

A decision has been made to perform certain repairs on the outlet works of a small dam. For a particular 36-inch gate valve, there are three available alternatives:

(a) Leave the valve as it is.

(b) Repair the valve.

(c) Replace the valve.

Leave | Repair | Replace | Present Worth | |

Valve Seat | 0.6 | 0.4 | 0.3 | 10,000 |

Valve Stem | 0.5 | 0.3 | 0.2 | 20,000 |

Valve Body | 0.4 | 0.2 | 0.1 | 30,000 |

Cost | 0 | 10,000 | 20,000 |

Loss due to leaving

0.6*10,000+ 0.5* 20,000+ 0.4* 30,000= 28,000

Cost of leaving is zero.

Net gain/ loss 28000+ 0= 28,000

Loss due to Repair

0.4*10,000+ 0.3* 20,000+ 0.2* 30,000= 16,000

Cost of Repair is 10,000

Net gain/ loss 16,000-+10,000= 26,000

Loss due to Replacement

0.3* 10,000+ 0.2* 20,000+ 0.1* 30,000= 10,000

Cost of Replacement= 20,000

Net gain/ loss 10,000+ 20,000= 30,000

Repairing is the best option. It results in the least expense for the dam owners, which is $26,000. Leaving the dam unrepaired costs $28,000 and replacing the valves costs $30,000.

**10-24**

Find the expected PW and probability distribution

First Cost 80000

Salvage value 0

Interest rate 9%

Shift/ day Saving per year Probability Useful life Probability

1 15000 0.3 3 0.6

2 30000 0.5 5 0.4

3 45000 0.2

Solution

PW (EV)= -80000 + (15,000 or 30000 or 45000){P/A 10%, 3 or 5}

P/A 9%, 3 year P/A 9%, 5 year

2.5313 3.88962

2.5313 | 3.88962 | ||||

Annual benefit | Probability | Life | Joint Probability | PW | PW* Joint Probability |

15000 | 0.3 | 3 | 0.18 | -42030.5 | -7565.49 |

15000 | 0.3 | 5 | 0.12 | -24966.7 | -2996 |

30000 | 0.5 | 3 | 0.3 | 30088.6 | 9026.58 |

30000 | 0.5 | 5 | 0.2 | 33908.5 | 6781.7 |

**10.25**

Extend project life by 3 years at a cost of $50,000 at end of initial useful life.

Find the worth to the firm.

Solution

Present value of extension capital of 50000

PV= Capital* (1/(1+i)^{n}

PV After 3 years After 5 years

38609.17 32496.57

PW (EV)= -118609.17 + (15,000 or 30000 or 45000){P/A, 9%, 6 years}

PW (EV)= -112496.57 + (15,000 or 30000 or 45000){P/A, 9%, 10 years}

P/A 9%, 6 year P/A 9%, 8 year

4.4859 5.5348 ** **

Annual benefit | Probability | Life | Joint Probability | PW | PW* Joint Probability |

15000 | 0.3 | 6 | 0.18 | -51320.7 | -9237.72 |

15000 | 0.3 | 8 | 0.12 | -29474.6 | -3536.95 |

30000 | 0.5 | 6 | 0.3 | 15967.83 | 4790.349 |

30000 | 0.5 | 8 | 0.2 | 53547.43 | 10709.49 |

45000 | 0.2 | 6 | 0.12 | 83256.33 | 9990.76 |

References

Blank, L. & Tarquin, A. (2013). *Basic engineering economy* (2 Ed.). New York, NY: McGraw-Hill Education.

Newnan, D., Eschenbach, T., & Lavelle, J. (2004). *Engineering economic analysis* (9^{th} Ed.). New York, NY: Oxford University Press.

Newnan, D., Lavelle, J., & Eschenbach, T. (2013). *Engineering economic analysis* (12^{th} Ed.). New York, NY: Oxford University Press.

Sullivan, W., Wicks, E., & Koelling, P. (2014). *Engineering economy* (16th Ed.). Upper Saddle River, NJ: Person Publishers.