Blank L., Tarquin A. Engineering Economy (McGraw-Hill Series in Industrial Engineering and Management)
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Independe
nt
projects
A.
Initial
in
ves
tm
e
nt
SECTION
12.1
An Overview
of
Capital Rationing Among Projects
Estimated
net cash flows
)lJJlJ
T
Capital
investment
limit
In
vestment
""--
"
B
LBJ
-
T
c.
Hlffi3
--
Life
Select
0 to a
ll
3 projects
Objective: Maximize
PW
va
lu
e of selection
within capital limit
@
®
(can't
invest
more than
this much)
This guideline
is
not different from that used for selection in previous chap-
ters for independent projects. As before, each project
is
compared with the do-
nothing project; that is, incremental analysis between projects is not necessary.
The
primary difference now is that the amount
of
money available to invest is
limited. Therefore, a specific solution procedure that incorporates this constraint
is
needed.
Previously,
PW
analysis required the assumption
of
equal service between
alternatives. This assumption
is
not valid for capital rationing, because there is no
life cycle
of
a project beyond its estimated life. Yet, the selection guideline is based
on the
PW
over
the respective life
of
each independent project. This means there
is
an implied reinvestment assumption, as follows:
All positive net cash flows
of
a project are reinvested at the MARR from
the time they are realized until the end
of
the longest-lived project.
425
Figure 12-1
Ba
sic characteris
ti
cs of
a capital budgeting
problem.
PWanalysis
426
CHAPTER
12
Selection from Independent Projects Under Budget Limitation
This fundamental assumption is demonstrated to be correct at the end
of
Sec-
tion 12.3, which treats
PW-based capital rationing for unequal-life projects.
Another dilemma
of
capital rationing among independent projects concerns the
flexibility
of
the capital investment limit
b.
The limit may marginally disallow
an
acceptable project that
is
next in line for acceptance. For example, assume project
A has a positive
PW value at the MARR.
If
A will cause the capital limit
of
$5
,000,000
to
be exceeded
by
only $1000, should A be included in the PW analy-
sis? Commonl
y,
a capital investment limit is somewhat flexible, so project A should
be included. In the examples here, we will not exceed a stated investment limit.
It is possible
to
use a ROR analysis to select from independent projects.
As
we
have learned in previous chapters, the
ROR technique may not select the same
projects
as
a PW analysis unless incremental ROR analysis is performed over
the LCM
of
lives. The same is true
in
the case
of
capital rationing. Therefore, we
recommend the
PW method for capital rationing among independent projects.
12.2
CAPITAL RATIONING USING PW ANALYSIS
OF
EQUAL-LIFE PROJECTS
To
select from projects that have the same expected life while investing no more
than the limit
b, first formulate all mutually exclusive
bundles-one
project at a
time, two at a time, etc. Each feasible bundle must have a total investment that
does not exceed
b.
One
of
these bundles is the do-nothing (DN) project. The total
number
of
bundles for m projects is calculated using the relation
21n.
The number
increases rapidly with
m. For m = 4, there are
24
= 16 bundles, and for m =
6,
2
6
= 64 bundles. Then the PW
of
each bundle is determined at the MARR. The
bundle with the largest
PW value is selected.
To
illustrate the development
of
mutually exclusive bundles, consider these
four projects with equal lives.
Project
A
B
C
D
Initial
Investment
$-10
,000
-5,000
- 8,000
-
15
,000
If
the investment limit is b = $25,000,
of
the 16 bundles, there are
12
feasible
ones to evaluate. The bundles ABD, ACD, BCD, and ABCD have investment
totals that exceed
$25,000. The viable bundles are shown below.
Total
Initial
Total
Initial
Projects
Investment
Projects
Investment
A $- 10,000
AD
$-
25
,000
B
- 5,000
BC
- 13,000
C
- 8,000
BD
- 20,000
D
-15
,000
CD
-23
,000
AB
-15
,000
ABC
-23
,000
AC
- 18,000
Do nothing
0
SECTION
12.2
Capital Rationing Using PW Analysis
of
Equal-Life Projects
The
procedure to solve a capital budgeting problem using
PW
analysis is as
follows:
1.
Develop a
ll
mutually exclusive bundles with a total initial investment that
does not exceed the capital limit
b.
2.
Sum the net cash flows NCF
j,
for all projects in each bundle} and each year
t from ] to the expected project life
nj'
Refer to the initial investment
of
bundle j at time t = 0 as
NCFjO'
3.
Compute the present worth value
PW
j
for each bundle at the
MARR.
PW
j
=
PW
of
bundle net cash flows - initial investment
1=llj
PW
j
= I
NCFiP
j
F,i,t)
-
NCF
jo
[12.1]
1=1
4.
Select the bundle with the (numerically) largest
PW
j
value.
Selecting the maximum
PW
j
means that this bundle produces a return larger than
any other bundle. Any bundle with
PW
j
< 0 is discarded, because it does not
produce a return
of
MARR.
EXAMPLE
12.1
The projects review committee
of
Microsoft has $20 million to allocate next year to
new software product development.
Any
or
all
of
five projects
in
Table 12-1 may be
accepted.
All
amounts are
in
$1000 units. Each project has
an
expected life
of
9 years.
Select the project if a
15
% return
is
expected.
TABLE
12-1
Five Equal-Life Independent Projects ($1000 Units)
Initial
Annual
Net
Project
Project
Investment
Cash Flow Life, Years
A
$-10,000
$2,870
9
B
-
15
,000
2,930
9
C
- 8,000 2,680
9
D
- 6,000
2,540
9
E
- 21,000
9,500
9
Solution
Use the procedure above with b = $20,000 to select one bundle that maximizes present
worth. Remember the units are
in
$1000.
I. There are 2
5
= 32 possible bundles. The eight bundles which require no more than
$20,000 initial investments are described
in
columns 2 and 3
of
Table]
2-2
. The
$21
,000 investment for E eliminates it from all bundles.
427
428
CHAPTER
12
Selection from Independent Projects Under Budget Limitation
TABLE
12-2
Summary
of
Present Worth Analysis
of
Equal-Life
Independent Projects
($1000 Units)
Initial Annual
Net
Present
Bundle
Projects
Investment
Cash Flow
Worth
j
Included
NCF
jo
NCF
j
PW
j
(1
)
(2) (3)
(4)
(5)
A
$-10,000
$2,870
$
+3,694
2 B
-15,000
2,930
-1,019
3 C
-8,000
2,680 +4,788
4 D
-6,000
2,540 +6,120
5 AC
-18,000
5,550 +8,482
6 AD
-16,000
5,410 +9,814
7 CD
-14,000
5,220 + 10,908
8
Do nothing
0
0 0
2.
The bundle net cash flows, column 4, are the sum
of
individual project net cash
flows.
3.
Use Equation [12.1] to compute the present worth for each bundle. Since the
annual NCF and life estimates are the same for a bundle,
PW
j
reduces
to
PW
j
=
NCF/P/A,15%,9)
-
NCF
jO
4.
Column 5
of
Table 12-2 summarizes the PW
j
values at i = 15%. Bundle 2 does
not return 15%, since
PW
2
<
O.
The largest
is
PW
7
= $10,908; therefore, invest
$14 million
in
C and D. This leaves
$6
million uncommitted.
Comment
This analysis assumes that the $6 million not used in this initial investment will
return the MARR by placing it in some other, unspecified investment opportunity.
The return on bundle 7 exceeds 15% per year. The actual rate
of
return, using the
relation
0 =
-14,000
+
5220(P/A,i*,9)
is
i*
= 34.8%, which significantly exceeds
MARR
= 15%.
12
.3 CAPITAL RATIONING USING PW ANALYSIS
OF
UNEQUAL-LIFE PROJECTS
Usually independent projects do not have the same expected life. As stated in
Section 12.1, the
PW
method for solution
of
the capital budgeting problem as-
sumes that each project will last for the period
of
the longest-lived project n
L
.
Additionally, reinvestment
of
any positive
net
cash flows is assumed to be
at
the
MARR from the time they are realized until the end
of
the longest-lived project,
SECTION
12.
3 Capital Rationing Using PW Analysis
of
Unequal-Life Projects
that
is,
from year
nj
through year n
L
•
Therefore, the use
of
the least
common
multiple
of
lives is not necessary, and it is correct to use
Equation
[12.1] to
select bundles
of
unequal-life projects
by
PW
analysis using the procedure
of
the previous section.
EXAMPLE
12.2
.~
:
For a MARR =
15%
per year and b = $20,000, select from the following independent
projects. Solve by hand and by computer.
Initial Annual
Net
Project Life,
Project
Investment
Cash
Flow
Years
A
$ - 8,000
$3,870
6
B
-
15
,0
00 2,930
9
C
- 8,000
2,680
5
D
- 8,000 2,540 4
Solution by Hand
The unequal-life values make the net cash flows vary over a bundle's life, but the selection
procedure is the same
as
above.
Of
the
24
=
16
bundles, 8 are economically feasible.
Their
PW values
by
Equation [12.1] are summarized
in
Table 12-3.
As
an
illustration, for
bundle
7:
PW
7
=
-1
6,000 + 5220(P/A,15%,4) + 2680(P/F,15%,5) = $235
Select bundle 5 (projects A and C) for a $16,000 investment.
TABLE
12-3
Present Worth Analysis for Unequal-Life Independent
Projects, Example
12
.2
Initial Present
Bundle
Investment,
Net
Cash Flows
Worth
j
Project
NCF
jO
Year t NCF
jt
PW
j
(1
)
(2) (3)
(4)
(5)
(6)
A
$ - 8,000
1-6
$3,870 $+6,646
2
B
- 15,000
1-9 2,930
-1,019
3 C
-8,0
00
1- 5 2,680 +984
4
D
-8
,000
1-4
2,540
-748
5
AC - 16,000 1- 5 6,550 +7,630
6 3,870
6
AD
16,000
1-4
6,410
+5,898
5-6
3,870
7
CD
- 16,000 1-4 5,220
+235
5
2,680
8
Do nothing
0
0 0
429
430
~
E-Solve
CHAPTER 12
Selection from Independent
Projects Under Budget Limitation
Solution
by
Computer
Figure
12
-2
presents a spreadsheet with the same information as
in
Table 12-3. It
is
necessary to initially develop the mutually exclusive bundles and total net cash
flow
s each
year. Bundle 5 (projects A and
C) has the largest PW value (row 16 cells). The NPY function
is
used
to
determine PW for each bundle j over its respective
Ufe,
using the format
NPV(MARR,NCF
-yea
r_I
: NCF _year_n) + investment.
X Microsoft Excel - Example 12.2 I!lrill3
fi,
AlB
A
3
Bundle 1
2
3 4
5 6
7 8
A B C D AC AD
CD
Do nothin
Net cash flows NCF
t
6 0 -$8.000
-$15.000 -$8,000 -$8,000 -$16,000 -$16,000
0
7 $3,870 $2,930 $2,680 $2,540 $6,550 $6,410 0
8
2 $3,870
$2,930
$2,680 $2,540 $6,550 $6,410 0
9 3 $3,870 $2,930 $2,680
$2,540
$6,550
$6,410
0
10 4 $3,870 $2,930 $2,680 $2,540 $6,550 $6,410 0
11
5
$3,870 $2,930 $2,680
$6,550
$3,870
0
12 6 $3,8
70
$2,930 $3,870 $3,870 0
13
7 $2,930 0
8
$2,930
0
$2,930 0
-$1,019
$984 -$748 $235 $0
= NPV($B$1 ,C7:CI5
)+
C6
=N
PV($B$ I,H7:H l5
)+
H6
.r
Figure
12-2
Spreadsheet analys
is
to
select from independent projects
of
unequal life using the PW method
of
capital
rationing, Example 12.2.
It
is
important to understand why solution
of
the capital budgeting problem by
PW evaluation using Equation [12.1]
is
correct.
The
following logic verifies the
assumption
of
reinvestment at the
MARR
for all net positive cash flows when
project lives are unequal. Refer to Figure
12-3, which uses the general layout
of
a two-project bundle. Assume each project has the same net cash flow each year.
SECTION
12
.3
Capital Rationing
Using
PW
Analysis
of
Unequal-Life Projects
P
~-
-
---
-
-------------------------1i
"
Project B
Investment
forB
PW
A
J
FWA
t
-------------------------------
Future worth i
f-------------
-
Period
of
reinvestment
Project A
Investment
for A
atMARR
Bundle PW = PW A + PW
B
The P / A factor is used for PW computation. Define n
L
as
the life
of
the longer
lived project. At the end
of
the shorter-lived project, the bundle has a total future
worth
of
NCF/F
/ A,
MARR,n)
as
determined for each project. Now, assume rein-
vestment at the MARR from year
n
j
+ I through year n
L
(a total
of
n
L
-
nj
years).
The assumption
of
the return at the MARR is important; this PW approach does
not necessarily select the correct projects
if
the return is not at the MARR. The
results are the two future worth arrows in year
n
L
in Figure 12-3. Finally, com-
pute the bundle
PW value in the initial year. This is the bundle PW = PW A +
PW B. In general form, the bundle j present worth is
PW
j
=
NCF/F/A
,
MARR,n)(F/P,MARR,nL-n)(P/F
,
MARR,n
L
) [12.2]
Substitute the symbol i for the MARR, and use the factor formulas
to
simplify.
(1
+
.)".
1 ·1
PW =
NCF
I J -
(1
+
iycnj
---
J J i
(1
+
iyL
=
NCFl
(1
+ i)"j - 1 ]
J i(1 + i)nj
[12.3]
=
NCF/P/A,i,n)
431
Figure
12-3
Representative cash flows
used to compute
PW for a
bundle
of
two indepen-
dent unequal-life projects
by Equation [12.1].
432
Figure
12-4
Initial
in
vestment a
nd
cash flows for bundle 7,
projects C and
D,
Example 17.2.
CHAPTER
12
Selection from Independent Projects Under Budget Limitation
FW = $57,
111
$5220
1
PW = $235 $2680
t
o 2 3 4 5 6 7 8 9
$
16
,000
Since the bracketed expression in Equation [12.3] is the
(PjA
,i
,n)
factor, com-
putation
of
PW
j
for
nj
years assumes reinvestment at the
MARR
of
all positive
net cash flows until the longest-lived project is completed
in
year
nL"
To demonstrate numerically, consider bundle j = 7 in Example
12
.2.
The
evaluation is
in
Table 12-3, and the net cash flow is pictured in Figure 12-4. At
15
% the future worth in year 9, life
of
B, the longest-lived project
of
the four, is
FW
= 5220(F
jA,
15
%,4)(Fj P
,1
5%,
5)
+ 2680(FjP,
15
%,4) = $57,111
The
present worth at the initial investment time is
PW
= - 16,000 +
57
,111(Pj F,15%,9) = $235
The
PW
value
is
the sa
me
as
PW
7
in Table 12-3 and Figure
12-
2.
This demon-
strates the reinvestment assumption for positive net cash flows.
If
this assump-
tion
is
not realistic, the
PW
analysis must be conducted using the
LCM
of
all
project
li
ves.
Project selection can also be accompljshed using the incremental rate
of
return
procedure.
Once
all viable, mutually exclusive bundles are developed, they are
ordered by increasing initial investment. Determine the incremental rate
of
return
on the first bundle relative to the do-nothing bundle, and the return for each in-
cremental investment and incremental net cash flow sequence on all other bun-
dles.
If
any bundle has an incremental return less than the
MARR,
it is removed.
The
last
ju
stified increment indicates the best bundle. This approach results in the
same
answer as the
PW
procedure. There are a number
of
incorrect ways to apply
the rate
of
return method, but the procedure
of
incremental analysis on mutually
exclusi ve bundles ensures a correct result, as in previous applications
of
incre-
mental rate
of
return.
12.4
CAPITAL BUDGETING PROBLEM FORMULATION
USING
LINEAR PROGRAMMING
The
capital budgeting problem can be stated in the form
of
the linear pro-
gramming model.
The
problem is formulated using the integer linear program-
ming
(ILP) model, which means simply that all relations are linear and that the
SECTION 12.4
Capital Budgeting
Problem Formulation Using Linear Programming
variable x can take on only integer values. In this case, the variables can only
take on the values
°
or
1,
which makes
it
a special case called the O-or-l ILP
model. The formulation
in
words follows.
Maximize:
Sum
of
PW
of
net cash flows
of
independent projects.
Constraints:
• Capital investment constraint is that the sum
of
initial investments must
not exceed a specified limit.
• Each project
is
completely selected
or
not selected.
For the math formulation, define b as the capital investment limit, and let
x
k
(k = 1 to
In
projects) be the variables to be determined.
If
x
k
=
1,
project k
is
completely selected;
if
x
k
= 0, project k
is
not selected. Note that the subsclipt k
represents each
independent project, not a mutually exclusive bundle.
If
the sum
of
PW
of
the net cash flows
is
Z,
the math programming formula-
tion is
Max
imize:
Co
nst
ra
ints:
k =111
I pWkX
k
= Z
k=l
k=ltl
I NCFkox
k
::::;
b
k=
1
X
k
= 0
or
1 for k = 1, 2, . . . , m
[12.4]
The PW k
of
each project
is
calculated using Equation [12.1] at
MARR
=
i.
PW k = PW
of
project net cash flows for n
k
years
1=
l1
k
= I NCFk
lP
/ F,i,t) - NCF
kO
[12.5]
1= 1
Computer solution is accomplished by a linear programming software package
which treats the ILP model. Also, Excel and its optimizing tool
SOLVER can
be used to develop the formulation and select the projects, as illustrated in
Example 12.3.
433
·.MiU·t*
_________________
........
;,;.;..
_
,.;;;;.;;
______
_
Review Example 12.2. (a) Formulate the capital budgeting problem using the math
programming model presented
in
Equation [12.4], and insert the solution into the model
to verify that
it
does indeed maximize present worth. (b) Set
up
and solve the problem
using Excel.
434
~
E-Solve
CHAPTER 12 Se
le
ction from Independent Projects Under Budget Limita
ti
on
Solution
(a) Define the subscript k = 1 through 4 for the four projects, which are relabeled
as
1,
2,3,
and
4.
The capital investment limit
is
b = $20,000
in
Equation
[J
2.4).
k= 4
Maximize:
I
pW
kX
k
= Z
k= 1
k= 4
Constraints:
)'
NCFkOx
k
:S
20,000
k = 1
X
k
= 0 or 1
for
k = 1 through 4
Calculate
thePW
k
forthe estimated net cash flows using i =
15
% and Equation [J2.5).
Net
Initial
Project
Cash
Flow
life
Factor Investment
Project
k NCF
kt
n
k
(P
/
A,15%A)
NCF
kO
PW
k
$3
,870
6
3.7845
$-8,000
$+6
,646
2
2,930
9
4.7716
-15
,000 - 1,019
3
2,680
5 3.3522 - 8,000
+984
4 2,540 4
2.8550
- 8,000 - 748
Now, substitute the
PW
k
values into the model, and put the initial investments
in
the budget constraint. Plus signs are used for
alJ
values in the capital investment
constraint.
We
ha
ve
the complete O-
or-IILP
formulation.
Maximize:
6646x
I
- 1019x
2
+ 984x3 - 748x
4
= Z
Constraints:
8000x
I
+ 15,000x2 + 8000x
3
+ 8000x
4
< 20,000
Solution
to
select projects 1 and 3 is written:
XI
= 1
for a PW value
of
$7630.
(b) Figure
12
- 5 presents a spreadsheet template developed
to
select from six or fewer in-
dependent projects with
12
years or less
of
net cash
flow
estimates per project. The
spreadsheet template can be expanded
in
either direction
if
needed. Figure 12-6
shows the
SOLVER parameters set
to
solve this example for four projects and an
investment limit
of$20,000. The descriptions below and the cell tags identify
the
con-
tents
of
the rows and cells in Figure 12-5, and their linkage
to
SOLVER parameters.
Rows 4 and
5:
Projects are identified
by
numbers
in
order to distinguish them
from spreadsheet column letters. Cell
15
is the expression for
Z,
the s
um
of
th
e PW values for the projects. This
is
the target cell for SOLVER
to
maxi-
mize (see Figure 12-6).
Rows 6
to
18
: These are initial investments and net cash
flow
estimates for each
project.
Zero values that occur after the life
of
a project need not be entered;
however, any
$0 estimates that occur during a project's life must be entered.
Row
19:
The entry in each cell is I for a selected project and 0 ifnot selected. These
are the changing cells for
SOLVER. Since each entry
mll
st be 0 or I, a binary