Review
This is a review of the solution alternatives section
of the course. This review cannot teach all the materials of
the section. A careful student will at least skim the contents
of each of the individual lessons focusing on concepts and techniques
not immediately clear. If extra problems are provided, the student
should try them and check against the answer keys.
It would also be useful for review to visit the
presentations list in the resources section. Here the lecture
presentations are listed for easy viewing. Use the calculators
in the Toolbox to solve the example problems.


Goals 

Each lesson in the section has a collection
of goals. Most of them describe an activity that you should understand
and be able to perform. The paragraphs below describe the purpose
of each lesson. Click the bush icon to see the goals of the individual
lessons.


Present
and Annual Worth
This lesson uses the present and annual worth criteria
for selection between mutually exclusive alternatives.
Both methods yield the same selection, but we will
see that there are important differences in application
and interpretation.

Rate
of Return
This lesson explores the rate of return method for
comparing alternatives. We learn the important lesson
that the rate of return method must be applied to increments
of investment rather than directly to the individual
alternatives.

Inflation
Inflation of costs and revenues is inevitable
in modern economies and it complicates economic decision
making. Although the present worth, annual worth, and
rate of return methods can be used when the effects
of inflation are included, this lesson shows how the
methods must be adjusted.

Comparisons
with Risk
This lesson extends the results of this
section to the explicit consideration of risk. No deterministic
analysis should be acceptable to the decision maker
when there is significant uncertainty in the parameters
of the analysis. The decision problem is made more
difficult by risk, but the lesson shows how risk can
be expressly included.




Text 

For most of this section, the problem
is to select a solution from a set of mutually exclusive alternatives. Most
of the materials in this section
are in chapter 3 of the text.


Chapter
3: Engineering Economic Analysis 




Present and Annual
Worth 

This lesson extends the present worth and annual
worth evaluation methods to the comparison of alternative solutions.
The extension is simple when the lives are equal. The best alternative
is the one with the greatest NPW value. This assumes revenues
are positive and costs are negative.

For
the Net Present Worth Method compute the net present
worth of the cash flows for the alternatives and choose
the best. Be careful when the lives of the alternatives
are not equal. 

When the alternatives have different lives, a study period must
be selected. A present worth comparison is meaningless unless
the time represented by the present worth is the same for all
alternatives. There are several ways to choose a study period,
but we usually use the leastcommonmultiple of the lives. Replacements
during the study period are likeforlike.

When
the alternatives have different lives, they must be
compared over a common study period. It is common to
choose the least common multiple of the lives. 

With the annual worth method, NAW values can be compared. Although
it is not necessary to select a study period, the implicit assumption
is the likeforlike replacements over the least common multiple
of the lives. This makes the NAW method easier for comparisons
with different lives. NPW and NAW comparisons always result in
the same selection.

For
the Net Annual Worth Method compute the equivalent
uniform annual worth of the cash flows for the alternatives
and choose the best. 

Try the problems to test your skills. Don't look at the answers
unless you solve the problem by yourself.
The homework for this section is in next lesson. 

Rate of Return 

This lesson considers the project selection problem
and the comparison of alternatives problem. They are quite different
in both statement and solution, but both use the IRR as a measure.
The
project selection problem chooses a subset of the projects from
a proposed set. For this problem compute the IRR for each alternative.

An
organization has a set proposed projects from which
some subset must be chosen. The IRR provides a measure
for project selection. 

When the MARR is given, all projects with IRR greater than the
MARR are acceptable. The solution assumes there is no budget
limit for capital expenditures.

With
a given MARR and no budget restriction, select the
projects that have IRR ≥ MARR 

When there is a budget limit, use the selection rule below.

With
restricted budget, rank the projects by IRR and in
order of the ranking choose the largest subset that
does not exceed the budget. 

The rate of return
method can also be used to select one solution from a set of
mutually exclusive alternatives. Be sure to learn the incremental
method. Generally, it is not necessary to calculate the IRR values
for the individual alternatives. You must calculate the IRR for
increments of investment. For examinations, you must follow the
procedure completely in order to receive full credit.

To choose among mutually exclusive
alternatives, you must use incremental analysis.
The IRR of each increment of investment must satisfy
the MARR requirement.


Try solving the problems with calculators in the toolbox before
looking at the answers.


Inflation 

Inflation complicates the evaluation
of projects because dollars at different times have different
buying power. This lesson introduces the language associated
with inflation and provides tools for evaluating solutions.
With inflation there are two kinds of dollars,
actual dollars and real dollars. There are
also two values for the MARR. The market MARR evaluates
cash flows expressed in actual dollars. The real MARR
evaluates cash flows expressed in real dollars. Cost or revenue
estimates are measured with today's prices. Escalation rates
convert the estimates to actual dollars. The linked formulas
calculate the economic measures for both real and actual dollars.


Economic
Analysis Formulas



Since NPW is the equivalent at time zero, the value
is the same whether the evaluation comes from real or actual
dollars.

When
the cash flow is in actual dollars use the market MARR
to find the NPW. When the cash flow is in real dollars
use the real MARR to find the NPW. The NPW values computed
with the two methods are the same.


When component escalation rates are the same as the rate of
general inflation, it is not necessary to change the estimates
to actual dollars. Use the estimated cash flow and the real MARR
to compute the NPW. This makes it easier to solve some problems.

When
the escalation rates of all cash flow components are
the same as general inflation, the estimated cash flow
is the same as the real cash flow. Do not adjust it
for inflation, but use the real MARR to find the NPW.


When inflation is present, comparison of alternatives may be
more complex. When using the NPW method evaluate each alternative
as indicated above and choose the best. For the NAW method, use real NAW
values to compare the alternatives.
For the ROR method, use incremental
analysis. When the cash flow is expressed in real dolloars, the
IRR is a real rate of return. When the cash flow is expressed
in actual dollars, the IRR is a market rate of return. When making
decisions to accept or reject incremental investments, be sure
to compare the IRR with the correct type of MARR. Real IRR values
compare to the real MARR. Market IRR values compare to the market
MARR.
Some the problems below include taxes. Neglect the parts that
mention taxes because they are not considered in this course.
Neglect the parts that mention taxes because they are not considered
in this course. Skip 1.b, 3.c, and 6.c.
Skip 5 and 6.


Risk 

We extend the present and annual worth methods
of comparison to include measures of risk. The primary measure
of an alternative is it's mean NPW or NAW. When uncertainty is
present, these measures are described by continuous probability
distributions.
From the distributions we compute the risk
measures that inform the decision maker about the risk associated
with the NPW or NAW.
To calculate the risk measures for an alternative,
we must know its probability distribution or use simulation.
The NPW or NAW might be specified as a sum of independent
random variables as below.
The moments are computed as the weighted sum of the
component moments.
For a normal distribution, all the risk measures
are computed from the given mean and standard deviation with the
help of the cumulative and inverse of standard normal distribution.
When the distribution is one of the named distributions, the risk
measures can be computed with formulas or by using the functions
of the Random
Variables addin.
When simulation is necessary, the risk measures can
be computed from the simulated moments and by using the histogram
of the NPW or NAW.
The risk measures provide additional information
that the decision maker might use for the selection of the best
alternative. With the normality assumption we may be able to eliminate
solutions because of dominance. We sometimes use dominance
even without the normality assumption.
The collection of undominated solutions is called
the efficient frontier. Considering only the mean and
standard deviation, the best solution is on the efficient frontier.
If there is more than one solution on the frontier, we might construct
a table as below showing the best and worst of each measure. The
decisionmaker must make the final choice.

A1 
A3 
A4 
A5 
Best 
Worst 
Mean 
60 
130 
25 
160 
A5 
A4 
Standard
Deviation 
75 
80 
55 
100 
A4 
A5 
IR 
0.80 
1.63 
0.45 
1.60 
A3 
A4 
PS(0) 
0.212 
0.052 
0.325 
0.055 
A3 
A4 
10%Percentile 
36.12 
27.48 
45.49 
31.84 
A5 
A4 
90%
VaR 
96.12 
102.52 
70.49 
128.16 
A4 
A5 


Summary 



Solution
Alternatives Summary 



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