Blank L., Tarquin A. Engineering Economy (McGraw-Hill Series in Industrial Engineering and Management)
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SECTION
18.5
Staged Evalua
ti
on of Alternatives Using a Decision Tree
Only
in
an
expanding economy wi
ll
the cash flows return the
15%
and justify the in-
vestment.
The
expected present worth is
E(PW) = I
PW)P(j)]
j =
R.5
,E
= - 656(0.2) - 434(0.6) + 1309(0.2)
=
$-
130
At
15
%, E(PW) <
0;
the
equipment
is not justified using an expected value analysis.
Comment
It
is also correct to calculate the E(cash flow) for each year and then determine
PW
of
the E(cash flow)
ser
ies, because the
PW
co
mputation
is
a linear function
of
cash flows.
Computing E(cash flow)
fir
st may be easi
er
in
that it reduces the number
of
PW com-
putation
s.
In
this example, calculate E(CF,) for each year, then
detennine
E(PW).
E(CF
o
)
= $- 5000
E(CF
1
)
= 2500(0.
2)
+ 2000(0.6) + 2000(0.2) =
$2
100
E(CF
2
)
= $2200
E(CF) = $2100
E(PW)
= - 5000 + 2100(P/ F,
15
%,
J) + 2200(P/
F,l5
%,2) + 2J
OO(P
/ F,
J5
%,3)
= $- 130
18.5
STAGED EVALUATION OF ALTERNATIVES
USING A DECISION
TREE
Alternative evaluation may require a ser
ie
s
of
decisions where the outcome from
one stage is important to the next stage
of
decision making. When each alterna-
tive
is
clearly d
efi
ned and probability estimates can be made to account for risk,
it
is helpful to perform the evaluation us
in
g a decision tree. A decision tree includes
• More than
one
stage
of
alternative
se
lection.
• Selection
of
an alternative at one stage that leads to another stage.
• Expected results from a decision at each stage.
• Probability estimates for each outcome.
• Estimates
of
economic value (cost
or
revenue) for each outcome.
• Measure
of
worth as the selection criterion, such as E(PW).
The
decision tree is constructed left to right and includes each possible decision
and outcome. A square represents a
decision node with the possible alternatives
indicated on the
branches from the decision node (Figure 18-7a). A circle repre-
sents a
probability node with the possible outcomes and estimated probabilities
on the branches (Figure
18
- 7b). Since outcomes always foll
ow
decisions, the
treelike structure
in
Figure
18
- 7c results.
Usually each branch
of
a decision tree has some estimated economic value
(often referred to as
payo
}f)
in
cost, revenue,
or
benefit. These cash flows are
635
636
Figure
18-7
Decision a
nd
probability
nodes used to construct a
decision tree.
CHAPTER 18 Fo
rm
alized Sensitiv
it
y Analysis and Expected Va
lu
e Decisions
Decision Alte
rn
atives
ffK
~----------------
(a) Decision node
Outcomes
pw
f§
-c-/
_______
:_:
j
~
0
.3
Probabilities
(b)
Pr
obabi]jty node
D
Final outcomes
(c)
Tr
ee structure
expressed
in
terms of PW,
AW
, or
FW
values and are shown
to
the right of each
final outcome branch. The cash flow a
nd
probability estimates on each outcome
branch a
re
used
in
calculating the expected economic value
of
each decision
branch. This proces
s,
called solving the tree or joldback, is explained after Ex-
ample 18.7, which illustrates the construction
of
a decision tree.
EXAMPLE 18.7
:,
Jerry
Hill
is president and CEO
of
an
American-based food processing company, Hill
Products and Service
s.
He was recently approached by an international supermarket
chain that wants to market in-country
its
own brand
of
frozen microwaveable dinners.
The offer made to Jerry
by
the supermarket corporation requires that a series
of
two de-
cisions be made, now and 2 years hence. The current decision involves two alternatives:
(1) Lease a facility
in
the United Arab Emirates (DAE) from the supermarket chain,
which has agreed to convert a current processing facility for immediate use by Jerry's
compa
ny
; or (2) build and own a processing and packaging facility in the UAB. Possi-
ble outcomes
of
this first decision stage are good market or poor market depending
upon the public's response.
Tbe decision choices 2 years hence are dependent upon the lease-or-own decision
made
now.
If
Hill decides to lease, good market response means that the future decision
alternatives are to produce at twice, equal to, or one-half
of
the original volume. This
SECTION 18.5
D1
Figure
18-8
Lease
(L)
Own
(0)
Staged Evaluation
of
Al
ternatives Using a Decision Tree
Good
2X
Poor
Good
Poor
Good
Good
Poor
Good
Poor
Out
of
bu
s
in
ess
Good
Poor
Good
Good
Poor
Poor
Good
Poor
OUlof
business
A two-stage decision tree
id
entifying alternatives a
nd
possible outcome
s.
will be a mutual decision between the supermarket chain and Jerry's company. A poor
market response will indicate a one-half level
of
production, or complete removal from
the
UAE market. Outcomes for the future decisions are, again, good and poor market
responses.
As agreed
by
the supermarket company, the current decision for Jerry to own the
facility will allow him to set the production level 2 years hence.
If
market response is
637
638
CHAPTER 18 Formalized Sensitivity Analysis and Expected Value Decis
ion
s
good, the decisi
on
alternatives are four or two times original level
s.
The reaction to poor
market response will be production at the same level or no production at all.
Construct the tree
of
decisions and outcomes for Hill Products and Services.
Solution
This
is
a two-stage decision tree that has alternatives now and 2 years hence. Identify
the decision nodes and branches, and then develop the tree using the branches and the
outcomes
of
good and poor market for each decision. Figure
18
- 8 details the decision
stages and outcome branches.
Decisi
on
now:
Label
itDl.
Alternatives: lease (L) and own
(0).
Outcomes: good and poor markets.
Decisions 2 years hence:
Label them D2 through
DS
Outcomes: good market, poor market, and out-of-business.
Choice
of
production levels for D2 through
05:
Quadruple production (4X); double production (2X); level production
(I X); one-half production
(O.SX); stop production (OX)
The alternatives for future production levels (D2 through DS) are added to the tree and
followed by the market responses
of
good and poor.
If
the stop-production
(O
X) deci-
si
on
is
made at D3 or
OS,
the only outcome
is
out
of
business.
To utilize the decision tree for alternative evaluation and selection, the fol-
lowing additional information
is
necessary for each branch:
•
The
estimated probability that each outcome may occur. These probabilities
must sum to
l.0
for each set
of
outcomes (branches) that result from a
decision.
• Economic information for each decision alternative and possible outcome,
such as initial investment and estimated cash flows.
Decisions are made using the probability estimate and economic value estimate
for each outcome branch. Commonly the present worth at the
MARR
is
used
in
an
expected value computation
of
the type
in
Equation [18.2]. This is the general
procedure to solve the tree using
PW
analysis:
I.
Start at the top right
of
the tree. Determine the
PW
value for each outcome
branch considering the time value
of
money.
2.
Calculate the expected value for each decision alternative.
E(decision) = I (outcome estimate)P(outcome) [18.3]
where the summation is taken over all possible outcomes for each decision
alternative.
SECTION 18.5 Staged Evaluation
of
Alternatives Using a Decision Tree
3.
At
each decision node, select the best E(decision)
value-minimum
cost or
maximum va
lu
e (if both costs and revenues are estimated).
4.
Continue moving
to
the l
ef
t
of
the tree
to
the root decision in order
to
select
the best alternative.
5.
Trace the best decision path back through the tree.
EXAM
PLE
18.8 .
A decision is needed to either market or se
ll
a new invention.
If
the product is marketed,
th
e next decision is
to
take it international or national. Assume the details
of
the out-
come branches result
in
the decision tree
of
Figure 18-9.
The
probabilities for each out-
come and
PW
of
CFBT (cash
flow
befo
re
taxes) are indicated. These payoffs are
in
mil-
li
ons
of
dollars. Determine
th
e best decision at the decision node OJ.
Hi
gh
@
0.2
0.8
Market
Low
4.2
9
ru
Se
ll
Q)
o Expected
va
lu
es for each
decision alte
rn
ative
Figure
18-9
®
@
PWof
CPBT
($ million)
0.5
12
0.5
16
0.4
4
0.4
-3
0.2
- 1
0.8
6
0.2
- 3
0.4
6
0.4
-2
0.2
2
1.0
9
Solution
of
a decision tree with pres
ent
worth
of
estimated
CPBT
values,
Example 18.8.
Solution
Use the procedure above to determine that the
01
decision altern
at
ive to
selJ
the inven-
tion should m
ax
imize
E(
PW
of
CFBT).
1.
Present worth
of
CFBT is supplied.
639
640
CHAPTER
18
Formalized Sens
iti
v
ity
Analysis and Expected Va
lu
e Decis
ion
s
2.
Calc
ul
ate
th
e expected PW for alternatives from nodes D2 and D3, using Equa-
tion
[1
8.3].
In
Figure 18-9, to the right
of
decision node D2, the expect
ed
values
of
14
and 0.2 in ovals are determined as
E(international decision)
= 12(0.5) + 16(0.5) =
14
E(national decision) = 4(0.4) - 3(0.4) - 1(0.2) = 0.2
The expected PW values
of
4.2 and 2 for D3 are calculated
in
a similar fashion.
3. Select the larger expected value at each decision node. These are 14 (international)
at D2 and 4.2 (international) at D3.
4.
Calculate the expected PW for the two DJ branches.
E(marketdecision)
= 14(0.2) + 4.2(0.8) = 6.16
E(sell decision)
= 9(1.0) = 9
The expected value for the sell decision
is
simple since the one outcome has a pay-
off
of
9.
The se
ll
decision yields the larger expected PW of
9.
5.
The largest expected PW
of
CFBT path is
to
select the sell branch at D J for a guar-
anteed
$9,000,000.
CHAPTER SUMMARY
In this chapter the emphasis
is
on sensitivity to variation
in
one
or
more pat'ame-
tefs using a specific measure
of
worth, When two alternatives are compared,
compute and graph the measure
of
worth for different values
of
the parameter to
determine when each alternative is better.
When several parameters are expected to vary over a predictable range, the
measure
of
worth is plotted and calculated using three estimates for a
parameter-
most likely, pessimistic, and optimistic. This formalized approach can help deter-
mine which alternative
is
best among severaL Independence between parameter
is assumed
in
all these analyses.
The combination
of
parameter and probability estimates results in the ex-
pected value relation
E(X) =
I.XP(X)
This expression is also used to calculate E(revenue), E(cost), E(cash flow),
E(PW), and
E(i)
for the entire cash flow sequence
of
an alternative,
Decision trees are used to make a series
of
alternative selections. This
is
a way
to explicitly take risk into account. It is necessary to make several types
of
esti-
mates for a decision tree: outcomes for each possible decision, cash flows, and
probabi
li
tie
s.
Expected value computations are coupled with those for the mea-
sure
of
worth to solve the tree and find the best alternatives stage by stage.
PROBLEMS
Sensitivity to
Parameter
Variation
18.1 The Central Drug Distribution Center
wants to evaluate a new mat
er
ials han-
dling system for fragile products. The
complete device will cost
$62,000 and
have an 8-year life and a salvage value
of
$1500. Annual maintenance, fuel, and
overhead costs are estimated at
$0.50 per
metric ton moved. Labor cost will be $8
per h
our
for regular wages and $16 for
overti me. A total
of
20 tons can be moved
in an 8-hour period. The center handles
from
10
to
30 tons
of
frag
il
e products per
day. The center
lls
es a
MARR
of
10%.
Determine the sensiti vity
of
present worth
of
costs
to
the annual volume moved. As-
s
um
e the operator is paid regular wages
for
200 days
of
work per year. Use a
I O-metric-ton increment for the analysis.
18
.2
An
equipment alternative is being eco-
nomically evaluated separately
by
three
eng
in
eers at Raytheon. The first cost will
be
$77,000, and the
lif
e is estim ated at
6 years with a salvage value
of
$10,000.
The engineers disagree, however, on
th
e
estimated revenue the equipment
wi
ll
generate. Joe has made
an
estimate
of
$10,000 per year. Jane states
th
at this
is
too low a
nd
estimates $14,000, while
Carlos estimates
$18,0
00
per year.
If
the
before-tax
MARR
is 8% per year, use
PW to determine if these different esti-
mates will change the decision to pur-
chase the equipment.
18.3
Perform the analysis in Problem 18.2 on
a spreadsheet, and make it an after-tax
cons
id
era
ti
on using 5-year MACRS de-
preciation and a 35% effective tax rate.
Use estimated annual expenses
of
$20
00
.
Determine the effective after-tax
MARR
from the before-tax
MARR
of
8%.
PROBLEMS
641
18.4 A
manuf
acturing company needs 1000
square meters
of
storage space. Purchas-
ing land for $80,000 and erecting a tem-
porary metal building at
$
70
per square
meter are one option. The president ex-
pects to sell the land for
$100,000 a
nd
the
building for
$20,000 after 3 years. A
n-
other option is to lease space for $2.50
per square meter per month payable at
the beginning
of
each year. The
MARR
is
20
%.
Perform a present worth analysis
of
the building and leasing alternatives to
determine the sensitivity
of
the decision
if
construction costs go down 10% and
th
e lease cost goes up to $2.75 per square
meter per month.
18
.5 A new demonstration system h
as
been
designed by Custom Baths
& Showers.
For the data shown, determine the sensi-
tivity
of
the rate
of
return to the amount
of
the revenue gradient G for values from
$1500 to $2500.
If
the
MARR
is 18% p
er
year, would this variation in the revenue
gra
di
ent affect the decision to build the
demonstration system? Work this prob-
l
em
(a) by hand and (b) by comput
er.
P = $74,000 n = 10 years S = 0
Expense: $30,000 first year, increas
in
g
$3000 per year thereafter
Revenue:
$63,000 first year, decreasing
by G per year thereafter
18
.6 Consider the two air conditioning sys-
tems detailed below.
System 1
System 2
First cost, $
-
10
,000
- 17,000
Annual operating cost, $/year
- 600
- l50
Salvage
va
lu
e,
$
- 100
- 300
New compressor a
nd
motor
- 1,750
- 3,000
cost at
mid
lif
e,
$
Life, years
8
l2
642
CHAPTER
18
Formalized Sens
iti
v
it
y Analysis and Expected Value Decisions
Use A W analysis to determine the sensi-
tivity
of
the economic decision to MARR
values
of
4%, 6%, and 8
%.
Plot the sen-
sitivity curve. Work this problem
(a) by
hand and
(b) by computer.
18
.7 Clint and Anne plan to either purchase a
weekend home or buy a travel trailer and
a four-wheel-drive vehicle to pull the
trailer for vacations. They have found a
5-acre tract with a small house 25 miles
from their home. It will cost them
$130,000, and they estimate they can sell
the place for
$145,000 in 10 years when
their children are grown. The insurance
and upkeep costs are estimated at
$1500
per year, but this weekend site is expected
to
save the family $150 every day they
don't go on a traveling vacation. The cou-
pl
e estimates that even though the cabin
is only
25
miles from home, they will
travel 50 miles per day when at the cabin
while working on it and visiting neigh-
bors and local events. Their car averages
30 miles per gallon
of
gas.
The trailer and vehicle combination
would cost
$75,000 and could be sold for
$20,000
in
10
year
s.
Insurance and oper-
ating costs will average
$1750 per year,
but this alternative is expected to save
$125 per vacation day.
On a normal vaca-
tion, they travel
300 miles each day.
Mileage per gallon for the vehicle and
trailer is estimated at
60% that
of
the fam-
ily car. Assume gas costs
$1.20 per U.
S.
gallon. The money earmarked for this
purchase currently earns
10% per year.
(a) Compute the breakeven number
of
vacation days per year for the two
plans.
(b)
Determine the sensitivity
of
A W for
each plan if vacation time may vary
as much as
±4
0% from the break-
even number.
(c)
If
Anne just started a new job and will
have only
14
days for the next few
years, which alternati
ve
is less costly?
1
8.8
(a) Calculate by hand and plot the sensi-
tivity
of
rate
of
return versus the bond
interest rate
of
a $50,000 fifteen-year
bond that h
as
bond interest paid quarterly
and is discounted to
$42,000. Consider
bond rates
of
5%, 7%, a
nd
9%.
(b)
Use a
spreadsheet to solve this problem.
18
.9 Leona has been offered an investment
opportunity that will require a cash out-
lay
of
$30,000 now for a cash inflow
of
$3500 for each year
of
investme
nt.
How-
ever, she must state now the number
of
years she plans to retain the investmen
t.
Additionally,
if
the investment is retained
for 6 years,
$25,000 will be returned to
investors, but after
10 years the return
is
anticipated to be only
$1
5,000, and after
12 years it
is
estimated to be $8000.
If
money
is
currently worth 8% per year,
is
the decision sensitive to the retention
period?
18.10 An asset costs
$8000 and has a maximum
life
of
15
years. Its AOC
is
expected to be
$500 the first year and increase
by
an
arithmetic gradient G between
$60 and
$140 per year thereafter. Determine the
sensitivity
of
the economic service life to
the cost gradient in increments
of
$40,
and plot the results on the same graph.
Use an interest rate
of
5% per year.
18
.
11
For
plans A and E, graph the sensitivity
of
PW
values at 20% per year for
th
e
range
-50
% to + 100%
of
the following
single-point estimates for each
of
the pa-
rameters:
(a) first cost,
(b)
AOC, a
nd
(c) annual revenue.
First cost, $
AOC, $/year
Annual revenue, $/year
Salvage value, $
Expected life, years
Plan A
-500
,000
-
75
,000
150,000
50,000
5
Plan B
- 375,000
- 80,000
130,000
37,000
5
18
.12
18.
13
18.14
Use a spreadsheet to determine and
graph the sensitivity
of
the rate
of
return
to a
±25
%
change
in
(a) purchase price
and
(b)
selling price for the fo
ll
owing
in
-
vestment. An engineer purchased an an-
tique
car
for $25,000 with the plan to
"make
it original" and se
ll
it at a profit.
Improvemen
ts
cost
$5500
the first year,
$1500 the second year, and
$1300
the
third year. He sold the
car
after 3 years
for
$35,000.
Use
a spreadsheet to plot on one graph
(s
imilar to Figure
18-3)
the sensitivity
of
A W over the range - 30% to + 50%
of
the parameters (a) first cost,
(b)
AOC,
and (c) annual revenue. Use a
MARR
of
18% per year.
Process
First
co
st, $
Salvage value, $
Life, years
Aoe
, $/year
Annual revenue, $/year
Estimate
- 80,000
10,000
10
-
15
,000
39
,000
Graph the sensitivity
of
what a person
should be
wi
ll
ing to pay now for a
9%
$ 10,000 bond
due
in
10
years if there is a
±30
% change
in
(a) face value,
(b)
divi-
dend rate,
or
(c) required nominal rate
of
return, which is expected to be 8% per
year, compounded semiannually.
The
bond dividends are paid semiannually.
Three Estimates
18
.
15
An engineer must decide between two
ways to pump concrete up to the top
floors
of
a seven-story office building to
be constructed.
Plan I requires the pur-
chase
of
eq
ui
pment for $6000 which
costs between
$0.40 and $0.75 per metric
ton to operate, with a most likely cost
of
$0.50 per metric ton.
The
asset is
able to pump
100 metric tons per day.
If
18.16
PROBLEMS
643
purchased, the asset
wi
ll
last for 5 years,
have no salvage value, and be used from
50 to 100 days per year. Plan 2 is an
equipment-leasing option and
is
ex-
pected to cost the company
$2500 per
year for equipment with a low cost esti-
mate
of
$1800 and a high estimate
of
$3200 per year. In addition, an extra $5
per hour labor cost will be incurred for
operating the leased equipment per 8-
hour day.
Plot the
AW
of
each plan versus
total annual operating cost
or
lease cost
at
i = 12%. Which plan should the engi-
neer recommend
if
the most likely esti-
mate
of
use is (a) 50 days per year and
(b)
100 days per year?
A meat packing plant must decide be-
tween two ways to cool cooked hams.
Spraying cools to
30
°C using approxi-
mately
80 liters
of
water for each ham.
The immersion method uses
40 liters per
ham, but an extra initial cost for equip-
ment
of
$2000 and extra maintenance
costs
of
$100 per year for the lO-year life
are estimated. Ten million hams per year
are cooked, and water costs
$0.12 per
1000 liters. Another cost is $0.04 per
1000 liters for wastewater treatment,
which
is
required for either method.
The
MARR
is 15% per year.
If
the spray method is selected, the
amount
of
water used can vary from an
optimistic value
of
40
liters to a pes-
simistic value
of
100 liters with 80 liters
being the most likely amount.
The
im-
mersion technique always takes
40
liters
per ham. How will this varying use
of
water for the spray method affect the eco-
nomic decision?
18.
17
When the country's economy
is
expand-
in
g, AB Invest
ment
Company is opti-
mistic and expects a
MARR
of
15
% for
new investments. However,
in
a receding
economy the expected retu
rn
is 8%.
Normally a 10% return is required. An
644
CHAPTER
18
Formalized Sensitivity Analysis and
Expected
Value Decisions
expanding
economy
causes the estimates
of
asset life to go down about 20%, and a
receding
economy
makes the n values in-
crea
se about 10%. Plot the sensitivity
of
present worth versus (a) the
MARR
and
(b) the life va
lu
es for the
two
plans de-
tailed below, using the most likely esti-
mates for the
other
factors. (c) Con-
sidering all the analyses, und
er
which
scenario,
if
any, should plan M
or
Q
be
rejected?
Initial investment, $
Ca
sh flow,
$/year
Life, years
Expected Value
Plan M
-
100
,000
+ 15,000
20
Plan Q
-
IJO,OOO
+ 19,000
20
18.18 Calculate the expected flow rate for each
oil well using the estimated probabilities.
North well
East we
ll
Expected
Flow,
Barrels/Day
100 200 300 400
0.15
0.35
0.
75
0.15
0.10
0.45 0.05
18.19
There
are four estimates made for the an-
ticipated cycle time to produce a sub-
component.
The
estimates,
in
seconds,
are
10,20,30,
and 70. (a)
If
equal weight
is placed on each estimate what is the ex-
pected time to plan for?
(b)
If
the largest
time is disregarded, estimate the ex-
pected time. Does the large estimate
seem to significantly increase the ex-
pected value?
18.20
The
variable Y
is
defined as
3/1
for n = 1,
2, 3, 4 with probabilities
of
0.4, 0.3,
0.233, and 0.067, respectively. Deter-
mine the expected value
of
Y.
18.21
The
AOC
value for an alternative is ex-
pected to be
one
of
two values. Your
office partner told you that the low value
is $2800
per
year.
If
her computations
show a probability
of
0.75 for the high
value and an expected
AOC
of
$4575,
what is the high
AOC
value she used
in
the computation
of
the average?
18.22 A total
of
40
different proposals were
evaluated by the
IRAD
(Industrial Re-
search and Development)
committee
during the past year. Twenty were
funded.
Their
rate
of
return estimates are
summarized below with the
i*
values
rounded to the nearest integer.
For
the
accepted proposals, calculate the ex-
pected rate
of
return E(i).
Proposal
Number
Rate
of
Return, %
of
Proposals
- 8
- 5
I
0 5
5 5
8
2
10
3
15
3
20
18.23 Starbreak Foods has performed an eco-
nomic analysis
of
proposed service
in
a
new region
of
the country.
The
three-
estimate approach to sensitivity analysis
has been applied.
The
optimistic and pes-
simis
ti
c values each have an estimated
15% chance
of
occurring.
Use
the AW
values shown to
compute
the expected
AW.
AW value,
$/year
Optimistic
Most
Likely Pessimistic
+ 300,000 + 50,000
-
25
,000