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Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Engineering Economy

Chapter 12: Probabilistic Risk

Analysis

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

The objective of Chapter 12 is to

discuss and illustrate several

probabilistic methods that are

useful in analyzing risk and

uncertainty associated with

engineering economy studies.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Decision making is fraught with risk and

uncertainty.

• Decisions under risk are those where the

decision maker can estimate probabilities of

occurrence of particular outcomes.

• Decisions under uncertainty are those

where estimates of probabilities of the

several unknown future states cannot be

estimated.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Four major sources of uncertainty are

present in engineering economy studies

• Possible inaccuracy of cash-flow estimates

• The type of business involved in relation to

the future health of the economy

• The type of physical plant and equipment

involved

• The length of the study period used in the

analysis

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Factors such as revenues, costs, salvage

values, etc., can often be considered

random variables.

For discrete random variables X, the

probability X takes on any particular value

xi is

where

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Some other properties of discrete random

variables.

Probability mass function

Cumulative distribution function

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

For continuous random variables…

The probability that X takes on any particular value is 0.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

The cumulative distribution function (CDF) is

which leads to

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

The expected value (mean, central

moment), E(X), and variance (measure of

dispersion), V(X), of a random variable

X, are

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Some properties of the mean and

variance.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Acme manufacturing has installed a much-needed

new CNC machine. The initial investment in this

machine is $180,000 and annual expenses are

$12,000. The life of the machine is expected to be 5

years, with a $20,000 market value at that time.

Acme’s MARR is 10%. Possible revenues follow the probabilities given below.

Revenue Probability

$35,000 0.1

$44,000 0.3

$50,000 0.4

$60,000 0.2

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

What are the expected value and

variance of Acme’s revenue?

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

What are the expected value and

variance of Acme’s revenue?

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Acme is considering purchasing a new vision system for

their production line. The vision system would allow them

to increase revenue due to improved quality and therefore

reduced warranty expense. The initial cost of the system is

$220,000, and the annual increase in net revenue is $45,000.

Acme’s MARR is 15%. However, the life of the system is uncertain, with the probability of different asset lives in the

following table. Given this information, should Acme

purchase the vision system?

Useful life, years (N) 8 9 10 11 12 13 14

p(N) 0.10 0.15 0.25 0.20 0.20 0.05 0.05

Pause and solve

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Probability tree diagrams can display

prospective cash flows, and their respective

probabilities that occur in each time period.

End-of-Year 0 1 2

0.5

0.6

0.4

$700

$825

$800

$1,000

$900

$1,100

0.3

0.4

0.3

0.7

$625

$700

$750

-$2,000

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Continuous random variables present special

challenges, and special opportunities.

• Two frequently used assumptions are

– cash-flow amounts are distributed according to a

normal distribution, and

– cash flows are statistically independent.

Thus, if

then

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Apply these concepts to cash flows over

time to fine the expected PW, and SD of

PW, for the expected values and standard

deviations in the table below. (Use

i=8%.)

End of

Year,k

Expected Value of Net

Cash Flow, Fk

SD of Net

Cash Flow, Fk

0 -$10,000 $0

1 $4,000 $400

2 $4,000 $600

3 $5,000 $650

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

For the expected PW.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

For V(PW) and SD(PW)

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

With our estimates of cash flow variables, and

using the normal distribution, we can find the

probability of events about the random

variable occurring. For instance, in the

previous example, what is the probability that

the PW of the cash flows is positive? Recall

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

The standard normal (mean=0 and

standard deviation=1) variable, Z, is

defined as

For our problem, since our random variable is

present worth,

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

The probability is found by looking up a

value in the standard normal table

(Appendix E).

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Another way to handle uncertainty is to

use Monte Carlo simulation.

• Based on the probability of different outcomes for each

random variable, a particular value is randomly generated.

• The numbers generated for all random variables constitute

an instance or realization reflecting a particular outcome.

• Hundreds or thousands of these instances are generated,

and these are examined to assist in decision making.

• A caution: these will yield long term, average results, and

you will be able to see the variation over time. However,

your decision may be a one-time decision, so don’t expect the “average” outcome to be your outcome.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Remember Acme Manufacturing and the new

CNC machine. The revenues and associated

probabilities are given below, and also now

the expenses have been given probabilities.

Revenue Probability

$35,000 0.1

$44,000 0.3

$50,000 0.4

$60,000 0.2

Expenses Probability

$6,000 0.1

$8,000 0.4

$11,000 0.3

$14,000 0.2

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Using the RAND() function in Excel, and

following the probabilities on the previous

slide, we generated 1000 revenue and expense

values (each using a separate random number

for independence), and found the resulting

PW for Acme. We found the following useful

information (we could discuss a lot more).

Not a good deal!

1. The average PW was -$20,670

2. The number of positive PW values, out of the

1000 simulated, was 216.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Monte Carlo simulation is very flexible.

• The example used a discrete distribution, but there is a way

to use any discrete or continuous distribution. If Excel is

used, it has several special functions that generate random

variates (e.g., NORMINV to generate normal random

variates and BETAINV for beta random variates.)

• There are many ways to look at the performance of an

alternative using Monte Carlo simulation (we examined

only two in the previous example). Graphs can be

especially valuable.

• Generate lots of data, through many trials. When average

values converge to a fairly constant amount, you probably

have enough data.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Decision trees can be helpful in

examining sequential decision problems

with outcomes that vary over time.

• Break down large problems into a series of smaller

problems.

• Provide objective analysis that explicitly considers

the risk and effect of the future.

• A decision tree is built from a series of nodes,

where decisions are made (square symbols) or

chance outcomes are noted (circle symbols), and

branches, which specify outcomes.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Revisiting Acme Manufacturing and

their CNC machine decision.

Acme already has a machine they can use that is adequate

for their needs. However, they wish to make a decision

about their purchase of the new machine. The time

horizon is 4 years, and they know that they won’t replace their existing machine if there is only one years of the

time horizon remaining. So, they will make a decision

today, and at the end of the first and second years

regarding the new machine. We assume no chance

elements, only a deterministic decision tree. The tree on

the following slide depicts the situation. Cash inflows

and durations are above the arrows, and capital

investments below the arrows.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Acme’s decision tree.

Old: 30k/yr

1 yr

Old: 25k/yr

1 yr

Old: 20k/yr

2 yr

-15k -20k -30k

New: 55k/yr

4 yr

New: 70k/yr

3 yr

New: 70k/yr

2 yr

0 1 2

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Analyze decision trees from the last

decision, backward to the first.

Decision Point Alternative Monetary Outcome Choice

2 Old $20k(2)-$30k = $10k

Old New $70k(2)-$180k = -$40k

1 Old $10k+$25k(1)-$20k = $15k

New New $70k(3)-$180k = $30k

0 Old $30k+$30k(1)-$15k = $45k

Old New $55k(4)-$180k = $40k

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Acme should keep their current machine

for one more year.

• The table reveals that at decision point 2, if Acme still has

the old machine, they should keep it.

• At decision point 1, purchasing the new machine provides

greater return than keeping the old one.

• At decision point 0 (today), it is more advantageous to

keep the old (current) machine for one more year, given

that the best decision at decision point 1 is to get the new

machine.

• It would be appropriate to include the time value of money,

so cash flows should be discounted to the present and the

analysis performed again.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Adding probabilities to decision trees.

• Most decisions also include chance outcomes, so we use

chance nodes.

• All alternatives emanating from either a decision or chance

node must be mutually exclusive (no more than one may be

selected) and exhaustive (contain all possible outcomes).

• The probabilities on the branches from a chance node must

sum to one (like probability tree diagrams).

• The value assigned to a chance node is the expected value

of the possible outcomes along each of the branches

leaving the node.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Mitselfik, Inc. believes new scheduling software (at

a cost of $150,000) will allow them to better manage

product flow and therefore increase sales. The

projection of increased annual sales (for the next 5

years), and the associated probabilities, are below.

The following slide shows the decision tree, and

resulting PW at a MARR of 12%.

Increased annual sales Probability

$75,000 0.35

$60,000 0.45

$40,000 0.15

$30,000 0.05

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

New

software

Sales increase

75,000 $120,360

$68,992

PW Probability

0.45

0.15

0.05

0.35

30,000

40,000

60,000

$0Current

software

$66,288

-$5,808

-$41,856

$68,992

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Mitselfik should purchase the software;

the expected PW of the investment is

$68,992.

• The PW of each annual sales increase amount is

given in the far right of the tree.

• The expected value of the annual sales increase is

$218,992 (the sum of the probabilities times the

respective PW).

• Subtracting the initial cost yields a net PW of

$68,992, which is superior to “do nothing” (which is eliminated, signified by the double lines on that

decision branch).

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

How much would we pay to have perfect

information about the future?

• Perhaps with additional information we might have a better estimate of sales, or exact knowledge of sales (“perfect” information).

• The cost of reducing the uncertainty must be balanced against the value.

• Perfect information is not obtainable, so the expected value of perfect information (EVPI) is an upper limit on what we would consider spending.

• EVPI = the value of the decision based on perfect information minus the value without the information.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Mitselfik could make the right decision

with perfect information.

Decision with perfect

information

Increased

sales Probability Decision Outcome

Prior

decision

$75,000 0.35 Purchase $120,360 $120,360

$60,000 0.45 Purchase $66,288 $66,288

$40,000 0.15 No purchase $0 -$5,808

$30,000 0.05 No purchase $0 -$41,856

Expected Value $71,956 $68,992

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

How much should Mitselfik pay for

perfect information?

• If Mitselfik had perfect information they would decide not to purchase the software if the increase in sales were $30,000 or $40,000.

• EVPI = $71,956 - $68,992 = $2,964.

• It is possible to find the expected value of any additional information that is not “perfect.” This is discussed in detail in the text.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Decision trees can be used to assist in

analyzing real options.

• Real options, similar to financial call

options, allow decision makers to invest

capital now or postpone all or part of the

investment until later.

• When a firm makes an irreversible capital

investment that could be postponed, it

exercises its call option, which has value by

virtue of the flexibility it gives the firm.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

A good example of postponable

investment is a plant addition.

Consider Mitselfik, Inc., which in addition to

purchasing software needs to expand the

facility. It can complete the entire expansion

now at a cost of $7 million, leading to

anticipated net cash flows (after tax) of $1.2

million for the next ten years. At an after-tax

MARR of 12%, is this an attractive

investment?

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

This does not look attractive for

Mitselfik.

What if demand should change, rising higher than

originally anticipated? Perhaps Mitselfik, Inc. could

be prepared and have an option available that would

allow them to respond to this increased demand.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Assume that demand could balloon to $3.5 million

(or, it could go to zero). The original expansion

could handle sales of $1.5 million, and Mitselfik

could acquire additional space (an option) to handle

the additional increased demand of $2 million at a

cost of $4 million. If this additional demand did not

materialize, the original expansion could be sold for

$1 million. What is the best decision for Mitselfik?

This is modeled as a decision tree on the next slide.

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Mitselfik’s decision tree

Buildinge

xpansion

$9.20mil

-$10.6mil

-$3.79mil

$1.48mil

-$6.55mil

-$0.22mil

-$6.11mil with

option

Sales

$2.1mil

Negligible

sales

Sales

$1.2mil

Add

Continue

Abandon

Add

Continue

Abandon

-$7.00mil

-$6.11mil

Expected value =

$0.96mil if all

outcomes are

equally likely.

-$6.11mil

$9.20mil

-$0.22mil

Upper Saddle River, New Jersey 07458

Engineering Economy, Sixteenth Edition

By William G. Sullivan, Elin M. Wicks, and C. Patrick Koelling

Mitselfik should strongly consider

investing since the option to add capacity

can provide a positive return.

• Using the decision tree model to assess the option

reveals that the expected return, given the best

decisions along the way, is $960,000.

• However, losses could be large, and there is a 2/3

chance of a loss (if all outcomes are equally

likely). Issues like this are covered in Chapter 14.