Chapter 15
Capital Budgeting
Principles of Engineering Economic Analysis, 5th edition
The Classical Capital
Budgeting Problem
Independent and
Indivisible Investments
Principles of Engineering Economic Analysis, 5th edition
Systematic Economic Analysis Technique
1. Identify the investment alternatives
2. Define the planning horizon
3. Specify the discount rate
4. Estimate the cash flows
5. Compare the alternatives
6. Perform supplementary analyses
7. Select the preferred investment
Principles of Engineering Economic Analysis, 5th edition
When deciding which investment
opportunities to fund wholly (versus not
at all), the optimum portfolio can be
obtained by solving a binary linear
programming problem with an objective
of maximizing the present worth of the
portfolio
Principles of Engineering Economic Analysis, 5th edition
Mathematical Programming Formulation of
the Capital Budgeting Problem
Maximize
subject to
PW1x1 + PW2x2 + ... + PWn-1 xn-1 + PWn xn
c1x1 + c2x2 + ... + cn-1 xn-1 + cn xn < C
xj = (0,1)
j = 1, ..., n
(15.1)
(15.2)
(15.3)
Establish an investment portfolio that maximizes the present worth of the
portfolio without exceeding a constraint on the amount of investment
capital available. The investment opportunities are independent and nondivisible, i.e., either the investment is pursued in total or not at all – no
partial investments.
Principles of Engineering Economic Analysis, 5th edition
Example 15.1
• Recall the IRR example from Chapter 8
which includes 5 mutually exclusive
investment alternatives, each of which
returns the initial investment at any time
the investor desires.
• Suppose each investment lasts for exactly
10 years and the investor can choose as
many of the investment options as she or
he wants, so long as no more than the total
invested does not exceed $100,000.
• Which ones should be chosen? (Cannot
choose multiples of the same investment.)
Principles of Engineering Economic Analysis, 5th edition
Data for Example 15.1
Investment Opportunity
Initial Investment
Annual Return
Salvage Value
Present Worth
Internal Rate of Return
1
2
3
$15,000.00 $25,000.00 $40,000.00
$3,750.00
$5,000.00
$9,250.00
$15,000.00 $25,000.00 $40,000.00
$4,718.79
$2,247.04
$9,212.88
25.00%
20.00%
23.13%
Capital available: $100,000
MARR: 18%
Principles of Engineering Economic Analysis, 5th edition
4
5
$50,000.00 $70,000.00
$11,250.00 $14,250.00
$50,000.00 $70,000.00
$10,111.69
$7,415.24
22.50%
20.36%
Mathematical Programming Formulation for
Example 15.1
Maximize $4,718.79x1 + $2,247.00x2 + $9,212.88x3 + $10,111.69x4 + $7,415.24x5
subject to $15,000x1 + $25,000x2 + $40,000x3 + $50,000x4 + $70,000x5 < $100,000
xj = (0,1)
j = 1, ..., 5
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Solving a BLP Using Enumeration
Recall, in Chapter 1 (Example 1.5), we
enumerated all possible investment
alternatives when there were 3 investments
available. Specifically, with m investment
proposals there are 2m possible mutually
exclusive investment alternatives, including
the “Do Nothing” alternative. In Example 1.5,
m = 3; therefore, there were 8 possible
alternatives.
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.1 with
Enumeration
With m = 5, there are 25 = 32 possible
investment alternatives. Shown below is a
binary table, similar to Table 1.1, giving all
possible investment alternatives.
Investment alternatives that violate the
capital constraint of $100,000 are eliminated,
as shown. (Half of the possible investment
alternatives are eliminated.)
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.1 with
Enumeration
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
x1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x3
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x4
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Principles of Engineering Economic Analysis, 5th edition
Cum Inv Cum Ret
PW
$0
$0
$0.00
$70,000 $14,250 $7,415.24
$50,000 $11,250 $10,111.69
$120,000 $25,500 $17,526.94
$40,000
$9,250 $9,212.88
$110,000 $23,500 $16,628.12
$90,000 $20,500 $19,324.57
$160,000 $34,750 $26,739.81
$25,000
$5,000 $2,247.04
$95,000 $19,250 $9,662.29
$75,000 $16,250 $12,358.74
$145,000 $30,500 $19,773.98
$65,000 $14,250 $11,459.92
$135,000 $28,500 $18,875.16
$115,000 $25,500 $21,571.61
$185,000 $39,750 $28,986.86
Solving Example 15.1 with
Enumeration
Combination
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x2
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x3
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x4
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Principles of Engineering Economic Analysis, 5th edition
Cum Inv Cum Ret
$15,000
$3,750
$85,000 $18,000
$65,000 $15,000
$135,000 $29,250
$55,000 $13,000
$125,000 $27,250
$105,000 $24,250
$175,000 $38,500
$40,000
$8,750
$110,000 $23,000
$90,000 $20,000
$160,000 $34,250
$80,000 $18,000
$150,000 $32,250
$130,000 $29,250
$200,000 $43,500
PW
$4,718.79
$12,134.03
$14,830.48
$22,245.73
$13,931.67
$21,346.91
$24,043.36
$31,458.60
$6,965.83
$14,381.08
$17,077.53
$24,492.77
$16,178.71
$23,593.95
$26,290.40
$33,705.65
Solving Example 15.1 with
Enumeration
Of the 16 feasible investment alternatives,
combination 7 has the greatest present
worth ($19,324.57). Investments 3 and 4 are
to be made.
The same solution was obtained using
Excel® SOLVER tool to solve the BLP
problem.
Principles of Engineering Economic Analysis, 5th edition
Adding Constraints
Mutually Exclusive
Contingent
Principles of Engineering Economic Analysis, 5th edition
Example 15.2
• Suppose the investment portfolio to be
optimized consists of a mixture of
independent and dependent investments.
• In particular, in the previous example,
suppose investment 3 is contingent on
investment 2 being selected (in other
words, you cannot choose 3 without
choosing 2).
• To solve the linear programming problem,
a further constraint is required, x3 < x2, or
D7 < C7.
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.2 with
Enumeration
Given the reduced binary table from
Example 5.1, we now eliminate investment
alternatives that violate the contingency
constraint. Blue lines are used to show the
alternatives that are eliminated.
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.2 with
Enumeration
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
x1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x3
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x4
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Principles of Engineering Economic Analysis, 5th edition
Cum Inv Cum Ret
PW
$0
$0
$0.00
$70,000 $14,250 $7,415.24
$50,000 $11,250 $10,111.69
$120,000 $25,500 $17,526.94
$40,000
$9,250 $9,212.88
$110,000 $23,500 $16,628.12
$90,000 $20,500 $19,324.57
$160,000 $34,750 $26,739.81
$25,000
$5,000 $2,247.04
$95,000 $19,250 $9,662.29
$75,000 $16,250 $12,358.74
$145,000 $30,500 $19,773.98
$65,000 $14,250 $11,459.92
$135,000 $28,500 $18,875.16
$115,000 $25,500 $21,571.61
$185,000 $39,750 $28,986.86
Solving Example 15.2 with
Enumeration
Combination
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x2
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x3
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x4
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Principles of Engineering Economic Analysis, 5th edition
Cum Inv Cum Ret
$15,000
$3,750
$85,000 $18,000
$65,000 $15,000
$135,000 $29,250
$55,000 $13,000
$125,000 $27,250
$105,000 $24,250
$175,000 $38,500
$40,000
$8,750
$110,000 $23,000
$90,000 $20,000
$160,000 $34,250
$80,000 $18,000
$150,000 $32,250
$130,000 $29,250
$200,000 $43,500
PW
$4,718.79
$12,134.03
$14,830.48
$22,245.73
$13,931.67
$21,346.91
$24,043.36
$31,458.60
$6,965.83
$14,381.08
$17,077.53
$24,492.77
$16,178.71
$23,593.95
$26,290.40
$33,705.65
Solving Example 15.2 with
Enumeration
Of the 14 feasible investment alternatives,
combination 27 has the greatest present
worth ($17,077.53). Investments 1, 2, and 4
are to be made.
The same solution was obtained using
Excel® SOLVER tool to solve the BLP
problem.
Principles of Engineering Economic Analysis, 5th edition
Example 15.3
• Extending the previous example, suppose
investments 2 and 4 are mutually
exclusive.
• To add a mutually exclusive constraint, it
is necessary to ensure that either the
product of x2 and x4 equals zero or their
sum is less than or equal to 1.0
• As shown on the following slide, the sum
of x2 and x4 is entered in cell E11 and a
constraint is added that E11 < 1.
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.3 with
Enumeration
Given the reduced binary table from
Example 5.2 we now eliminate investment
alternatives that violate the mutually
exclusive constraint. Black lines are used to
show the alternatives that are eliminated.
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.3 with
Enumeration
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
x1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x3
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x4
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Principles of Engineering Economic Analysis, 5th edition
Cum Inv Cum Ret
PW
$0
$0
$0.00
$70,000 $14,250 $7,415.24
$50,000 $11,250 $10,111.69
$120,000 $25,500 $17,526.94
$40,000
$9,250 $9,212.88
$110,000 $23,500 $16,628.12
$90,000 $20,500 $19,324.57
$160,000 $34,750 $26,739.81
$25,000
$5,000 $2,247.04
$95,000 $19,250 $9,662.29
$75,000 $16,250 $12,358.74
$145,000 $30,500 $19,773.98
$65,000 $14,250 $11,459.92
$135,000 $28,500 $18,875.16
$115,000 $25,500 $21,571.61
$185,000 $39,750 $28,986.86
Solving Example 15.3 with
Enumeration
Combination
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x2
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x3
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x4
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Principles of Engineering Economic Analysis, 5th edition
Cum Inv Cum Ret
$15,000
$3,750
$85,000 $18,000
$65,000 $15,000
$135,000 $29,250
$55,000 $13,000
$125,000 $27,250
$105,000 $24,250
$175,000 $38,500
$40,000
$8,750
$110,000 $23,000
$90,000 $20,000
$160,000 $34,250
$80,000 $18,000
$150,000 $32,250
$130,000 $29,250
$200,000 $43,500
PW
$4,718.79
$12,134.03
$14,830.48
$22,245.73
$13,931.67
$21,346.91
$24,043.36
$31,458.60
$6,965.83
$14,381.08
$17,077.53
$24,492.77
$16,178.71
$23,593.95
$26,290.40
$33,705.65
Solving Example 15.3 with
Enumeration
Of the 12 feasible investment alternatives,
combination 29 has the greatest present
worth ($16,178.71). Investments 1, 2, and 3
are to be made.
The same solution was obtained using
Excel® SOLVER tool to solve the BLP
problem.
Principles of Engineering Economic Analysis, 5th edition
Example 15.4
• Now, consider 6 investment opportunities, with MARR =
10%, C = $100,000, and the data shown below.
EOY
0
1
2
3
4
5
CF(1)
CF(2)
CF(3)
CF(4)]
CF(5)
CF(6)
-$15,000.00 -$18,000.00 -$20,000.00 -$25,000.00 -$30,000.00 -$40,000.00
$4,500.00
$3,000.00
$4,000.00
$4,500.00
$6,000.00 $15,000.00
$4,500.00
$4,500.00
$5,000.00
$4,500.00
$9,000.00 $15,000.00
$4,500.00
$6,000.00
$6,000.00
$4,500.00 $12,000.00 $25,000.00
$4,500.00
$7,500.00
$7,000.00
$4,500.00 $15,000.00
$0.00
$4,500.00
$9,000.00
$8,000.00
$4,500.00
$0.00
$0.00
• Investments 1 and 2 are mutually exclusive and
investment 6 is contingent on either or both of
investments 3 and 4 being funded.
• To add an “either/or” contingent constraint, we set D14
equal to the sum of D9 and E9 and add the constraint:
G9 <= D14, which is the same as
x6 < x3 + x4.
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.4 with
Enumeration
With m = 6, there are 26 = 64 possible investment
alternatives. Shown below is a binary table listing
all possible investment alternatives.
Investment alternatives that violate the capital
constraint of $100,000 are eliminated using red
lines. Of the remaining investments, those that
violate the mutually exclusive constraint are
eliminated using blue lines. Of those that remain,
those that violate the either/or constraint are
eliminated using black lines.
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.4 with
Enumeration
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
x1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
Principles of Engineering Economic Analysis, 5th edition
x5
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x6
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Cum Inv
PW
$0
$0.00
$40,000 $4,815.93
$30,000 $2,153.54
$70,000 $6,969.47
$25,000 $3,365.64
$65,000 $8,181.57
$55,000 $5,519.18
$95,000 $10,335.11
$20,000 $2,024.95
$60,000 $6,840.88
$50,000 $4,178.49
$90,000 $8,994.42
$45,000 $5,390.59
$85,000 $10,206.52
$75,000 $7,544.13
$115,000 $12,360.06
Solving Example 15.4 with
Enumeration
Combination
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
x1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
Principles of Engineering Economic Analysis, 5th edition
x5
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x6
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Cum Inv
$18,000
$58,000
$48,000
$88,000
$43,000
$83,000
$73,000
$113,000
$38,000
$78,000
$68,000
$108,000
$63,000
$103,000
$93,000
$133,000
PW
$3,665.06
$8,480.99
$5,818.60
$10,634.53
$7,030.70
$11,846.63
$9,184.25
$14,000.17
$5,690.01
$10,505.94
$7,843.55
$12,659.48
$9,055.65
$13,871.58
$11,209.19
$16,025.12
Solving Example 15.4 with
Enumeration
Combination
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x5
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
Principles of Engineering Economic Analysis, 5th edition
x6
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Cum Inv
$15,000
$55,000
$45,000
$85,000
$40,000
$80,000
$70,000
$110,000
$35,000
$75,000
$65,000
$105,000
$60,000
$100,000
$90,000
$130,000
PW
$2,058.54
$6,874.47
$4,212.08
$9,028.01
$5,424.18
$10,240.11
$7,577.72
$12,393.65
$4,083.49
$8,899.42
$6,237.03
$11,052.96
$7,449.13
$12,265.06
$9,602.67
$14,418.60
Solving Example 15.4 with
Enumeration
Combination
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x5
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
Principles of Engineering Economic Analysis, 5th edition
x6
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Cum Inv
$33,000
$73,000
$63,000
$103,000
$58,000
$98,000
$88,000
$128,000
$53,000
$93,000
$83,000
$123,000
$78,000
$118,000
$108,000
$148,000
PW
$5,723.60
$10,539.53
$7,877.14
$12,693.07
$9,089.25
$13,905.17
$11,242.79
$16,058.71
$7,748.55
$12,564.48
$9,902.09
$14,718.02
$11,114.19
$15,930.12
$13,267.74
$18,083.66
Solving Example 15.4 with
Enumeration
Of the 34 feasible investment alternatives,
combination 46 has the greatest present
worth ($12,265.06). Investments 1, 3, 4, and
6 are to be made.
The same solution was obtained using
Excel® SOLVER tool to solve the BLP
problem.
Principles of Engineering Economic Analysis, 5th edition
Example 15.5
• In the previous example, suppose at most 3 and
at least 2 investments must be made.
EOY
0
1
2
3
4
5
CF(1)
CF(2)
CF(3)
CF(4)]
CF(5)
CF(6)
-$15,000.00 -$18,000.00 -$20,000.00 -$25,000.00 -$30,000.00 -$40,000.00
$4,500.00
$3,000.00
$4,000.00
$4,500.00
$6,000.00 $15,000.00
$4,500.00
$4,500.00
$5,000.00
$4,500.00
$9,000.00 $15,000.00
$4,500.00
$6,000.00
$6,000.00
$4,500.00 $12,000.00 $25,000.00
$4,500.00
$7,500.00
$7,000.00
$4,500.00 $15,000.00
$0.00
$4,500.00
$9,000.00
$8,000.00
$4,500.00
$0.00
$0.00
n
• “at most” implies  x j  3 or H9 <=3
j 1
• “at least” implies H9 >=2
• As shown in Figure 15.6, the optimum investment
portfolio is {2,4,6} with PW = $11,846.63 and IRR
= 15.70%
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.5 with
Enumeration
Given the reduced binary table from Example 5.4
we now eliminate investment alternatives that
violate the constraint that at most 3 and at least 2
investments must be made. Green lines are used
to show the alternatives that are eliminated.
Principles of Engineering Economic Analysis, 5th edition
Solving Example 15.4 with
Enumeration
Combination
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
x1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
Principles of Engineering Economic Analysis, 5th edition
x5
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x6
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Cum Inv
PW
$0
$0.00
$40,000 $4,815.93
$30,000 $2,153.54
$70,000 $6,969.47
$25,000 $3,365.64
$65,000 $8,181.57
$55,000 $5,519.18
$95,000 $10,335.11
$20,000 $2,024.95
$60,000 $6,840.88
$50,000 $4,178.49
$90,000 $8,994.42
$45,000 $5,390.59
$85,000 $10,206.52
$75,000 $7,544.13
$115,000 $12,360.06
Solving Example 15.4 with
Enumeration
Combination
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
x1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
Principles of Engineering Economic Analysis, 5th edition
x5
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
x6
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Cum Inv
$18,000
$58,000
$48,000
$88,000
$43,000
$83,000
$73,000
$113,000
$38,000
$78,000
$68,000
$108,000
$63,000
$103,000
$93,000
$133,000
PW
$3,665.06
$8,480.99
$5,818.60
$10,634.53
$7,030.70
$11,846.63
$9,184.25
$14,000.17
$5,690.01
$10,505.94
$7,843.55
$12,659.48
$9,055.65
$13,871.58
$11,209.19
$16,025.12
Solving Example 15.4 with
Enumeration
Combination
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
x3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x5
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
Principles of Engineering Economic Analysis, 5th edition
x6
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Cum Inv
$15,000
$55,000
$45,000
$85,000
$40,000
$80,000
$70,000
$110,000
$35,000
$75,000
$65,000
$105,000
$60,000
$100,000
$90,000
$130,000
PW
$2,058.54
$6,874.47
$4,212.08
$9,028.01
$5,424.18
$10,240.11
$7,577.72
$12,393.65
$4,083.49
$8,899.42
$6,237.03
$11,052.96
$7,449.13
$12,265.06
$9,602.67
$14,418.60
Solving Example 15.4 with
Enumeration
Combination
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
x1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
x4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
x5
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
Principles of Engineering Economic Analysis, 5th edition
x6
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Cum Inv
$33,000
$73,000
$63,000
$103,000
$58,000
$98,000
$88,000
$128,000
$53,000
$93,000
$83,000
$123,000
$78,000
$118,000
$108,000
$148,000
PW
$5,723.60
$10,539.53
$7,877.14
$12,693.07
$9,089.25
$13,905.17
$11,242.79
$16,058.71
$7,748.55
$12,564.48
$9,902.09
$14,718.02
$11,114.19
$15,930.12
$13,267.74
$18,083.66
Solving Example 15.4 with
Enumeration
Of the 25 feasible investment alternatives,
combination 22 has the greatest present
worth ($11,846.63). Investments 2, 4, and 6
are to be made.
The same solution was obtained using
Excel® SOLVER tool to solve the BLP
problem. (For m > 6, enumeration is not
reasonable. Use SOLVER.)
Principles of Engineering Economic Analysis, 5th edition
Sensitivity Analysis
Capital Available
MARR
Principles of Engineering Economic Analysis, 5th edition
Example 15.6
• Recall Example 15.1.
• What effect does the capital limit have on
the optimum investment portfolio?
• For example, what if the capital limit is
raised to $105,000.
• What would be the impact on PW and IRR?
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
IRR = ($3,750 + $9,250 + $11,250)/$105,000 = 0.2310 or 23.10%
Principles of Engineering Economic Analysis, 5th edition
Sensitivity Analysis of the Optimum Investment Portfolio
Capital
Available Optimum
for
Portfolio
Investment
$5,000
$10,000
$15,000
$20,000
$25,000
$30,000
$35,000
$40,000
$45,000
$50,000
$55,000
$60,000
$65,000
$70,000
$75,000
$80,000
$85,000
$90,000
$95,000
$100,000
{Ø}
{Ø}
{1}
{1}
{1}
{1}
{1}
{3}
{3}
{4}
{1,3}
{1,3}
{1,4}
{1,4}
{1,4}
{1,2,3}
{1,2,3}
{3,4}
{3,4}
{3,4}
Portfolio
PW
$0.00
$0.00
$4,718.79
$4,718.79
$4,718.79
$4,718.79
$4,718.79
$9,212.88
$9,212.88
$10,111.69
$13,931.67
$13,932.67
$14,830.48
$14,831.48
$14,832.48
$16,178.71
$16,179.71
$19,324.57
$19,325.57
$19,326.57
Capital
Portfolio Available Optimum
IRR
for
Portfolio
Investment
0.00%
0.00%
25.00%
25.00%
25.00%
25.00%
25.00%
23.13%
23.13%
22.50%
23.64%
23.64%
23.08%
23.08%
23.08%
22.50%
22.50%
22.78%
22.78%
22.78%
$105,000
$110,000
$115,000
$120,000
$125,000
$130,000
$135,000
$140,000
$145,000
$150,000
$155,000
$160,000
$165,000
$170,000
$175,000
$180,000
$185,000
$190,000
$195,000
$200,000
Principles of Engineering Economic Analysis, 5th edition
{1,3,4}
{1,3,4}
{1,3,4}
{1,3,4}
{1,3,4}
{1,2,3,4}
{1,2,3,4}
{1,2,3,4}
{1,2,3,4}
{1,2,3,4}
{1,2,3,4}
{3,4,5}
{3,4,5}
{3,4,5}
{1,3,4,5}
{1,3,4,5}
{1,3,4,5}
{1,3,4,5}
{1,3,4,5}
{1,2,3,4,5}
Portfolio
PW
Portfolio
IRR
$24,043.36
$24,044.36
$24,045.36
$24,046.36
$24,047.36
$26,290.40
$26,291.40
$26,292.40
$26,293.40
$26,294.40
$26,295.40
$26,739.81
$26,740.81
$26,741.81
$31,458.60
$31,459.60
$31,460.60
$31,461.60
$31,462.60
$33,705.65
23.10%
23.10%
23.10%
23.10%
23.10%
22.50%
22.50%
22.50%
22.50%
22.50%
22.50%
21.72%
21.72%
21.72%
22.00%
22.00%
22.00%
22.00%
22.00%
21.75%
Example 15.7
• Consider the cash flow profiles given below, with a
MARR of 10% and capital limit of $100,000.
EOY
0
1
2
3
4
5
CF(1)
CF(2)
CF(3)
CF(4)]
CF(5)
CF(6)
-$15,000.00 -$18,000.00 -$20,000.00 -$22,000.00 -$35,000.00 -$40,000.00
$4,500.00 $10,000.00
$4,000.00
$6,500.00
$6,000.00 $10,000.00
$4,500.00
$7,500.00
$5,000.00
$6,000.00
$6,900.00 $10,000.00
$4,500.00
$5,000.00
$6,000.00
$5,500.00
$7,935.00 $10,000.00
$4,500.00
$2,500.00
$7,000.00
$5,000.00
$9,125.25 $10,000.00
$4,500.00
$5,000.00
$8,000.00
$8,500.00 $15,494.04 $15,000.00
• How sensitive is the optimum portfolio to MARR values
in the interval [0%,26%]?
Optimum
Portfolio
[0.00%,8.10%]
{2,3,4,6}
[8.11%,8.64%]
{1,2,3,6}
[8.65%,12.89%]
{1,2,3,4}
[12.90%,13.45%]
{1,2,3}
[13.46%,15.23%]
{1,2}
[15.24%,25.07%]
{2}
MARR Range
IRR
14.01%
14.48%
15.98%
17.30%
20.17%
25.07%
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Sensitivity Analysis of Present Worth
$15,000.00
Present Worth
$10,000.00
$5,000.00
$0.00
-$5,000.00
Minimum Attractive Rate of Return
-$10,000.00
PW(1)
PW(2)
PW(3)
PW(4)
-$15,000.00
Principles of Engineering Economic Analysis, 5th edition
PW(5)
PW(6)
The Capital Budgeting
Problem
Independent and
Divisible Investments
Principles of Engineering Economic Analysis, 5th edition
Mathematical Programming Formulation of
the Capital Budgeting Problem with Divisible
Investments
Maximize
subject to
PW1p1 + PW2p2 + ... + PWn-1pn-1 + PWnpn
c1p1 + c2p2 + ... + cn-1pn-1 + cnpn < C
0 < pj < 1
j = 1, ..., n
(15.1)
(15.2)
(15.3)
Establish an investment portfolio that maximizes the present worth of the
portfolio without exceeding a constraint on the amount of investment
capital available. The investment opportunities are independent and
divisible, i.e., a percentage of an investment can be pursued.
Principles of Engineering Economic Analysis, 5th edition
when partial funding of investments is
allowed, to obtain the optimum
investment portfolio, (a) rank the
investment opportunities on their
internal rates of return, and (b) form the
portfolio by “filling the investment
bucket,” starting with the opportunity
having the greatest internal rate of return
and proceeding sequentially until the
“bucket” is full.
Principles of Engineering Economic Analysis, 5th edition
Example 15.8
• Recall Example 15.1.
• Now, suppose the investments are
divisible, i.e., you can choose to make
fractional investments.
• When investments are independent and
divisible, the optimum investment
portfolio is obtained by rank ordering the
investments based on IRR and investing,
beginning with the investment having the
greatest IRR, until “your money runs out,”
as shown on the following chart.
Principles of Engineering Economic Analysis, 5th edition
Economically Viable Investments
1
2
3
4
5
IRR
25.00%
20.00%
23.13%
22.50%
20.36%
Annual Return $3,750.00
$5,000.00
$9,250.00 $11,250.00 $14,250.00
Investment
$15,000.00 $25,000.00 $40,000.00 $50,000.00 $70,000.00
Economically Viable Investments Sorted by IRR
1
3
4
5
2
IRR
25.00%
23.13%
22.50%
20.36%
20.00%
Annual Return $3,750.00
$9,250.00 $11,250.00 $14,250.00
$5,000.00
Investment
$15,000.00 $40,000.00 $50,000.00 $70,000.00 $25,000.00
Investment Cumulative
Portfolio Funds Req'd
{B}
$15,000.00
{B,D}
$55,000.00
{B,D,E}
$105,000.00
{B,D,E,F} $175,000.00
{F,D,E,F,C} $200,000.00
Principles of Engineering Economic Analysis, 5th edition
Optimum Divisible Portfolio
$250,000
$200,000
$150,000
250000
Make
100% investments in 1 & 3 and
Make 100% investments in 1 & 3 and
make
a 90% investment in in
4 for 4
an for an
make
a
90%
investment
200000
overall return of 23.13% and a present
$23,032.19
overall returnworth
ofof23.13%
and a
150000
present worth of $23,032.19
100000
$100,000
investment capital available
50000 capital available
investment
0
{1}
{1,3}
{1,3,4}
{1,3,4,5}
{1,2,3,4,5}
$50,000
$0
{1}
{1,3}
{1,3,4}
Principles of Engineering Economic Analysis, 5th edition
{1,3,4,5}
{1,2,3,4,5}
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Two Observations
• With divisible investments, the investments
are rank ordered on IRR, which is contrary
to everything we learned regarding mutually
exclusive investment alternatives and nondivisible, independent investments.
• Notice, rank ordering the investments on
PW, which we do with mutually exclusive
investment alternatives and non-divisible,
independent investments, will not yield the
optimum portfolio—as shown on the
following charts.
Principles of Engineering Economic Analysis, 5th edition
Economically Viable Investments
C
D
E
$2,247.04
$9,212.88 $10,111.69
$5,000.00
$9,250.00 $11,250.00
$25,000.00 $40,000.00 $50,000.00
Present Worth
Annual Return
Investment
B
$4,718.79
$3,750.00
$15,000.00
Present Worth
Annual Return
Investment
Economically Viable Investments Sorted by Present Worth
E
D
F
B
C
$10,111.69
$9,212.88
$7,415.24
$4,718.79
$2,247.04
$11,250.00
$9,250.00 $14,250.00
$3,750.00
$5,000.00
$50,000.00 $40,000.00 $70,000.00 $15,000.00 $25,000.00
Investment Cumulative
Portfolio Funds Req'd
{E}
$50,000.00
{E,D}
$90,000.00
{E,D,F}
$160,000.00
{E,D,F,B} $175,000.00
{E,D,F,B,C} $200,000.00
Principles of Engineering Economic Analysis, 5th edition
F
$7,415.24
$14,250.00
$70,000.00
Suboptimum Divisible Portfolio
$250,000.00
$200,000.00
Making 100% investments in E & D
and investing $10,000 in F yields an
overall return of 22.54% and a
present worth of $20,383.89
$150,000.00
$100,000.00
investment capital available
$50,000.00
$0.00
{E}
{E,D}
{E,D,F}
Principles of Engineering Economic Analysis, 5th edition
{E,D,F,B} {E,D,F,B,C}
A Third Observation
• The IRR for each investment alternative is
independent of the MARR. Hence, the only
effect a change in the MARR has on the
optimum investment portfolio is the
elimination of alternatives with an IRR less
than the MARR. If all investment alternatives
have IRR values greater than the MARR,
then the optimum investment portfolio will
be unchanged. However, the PW of the
optimum investment portfolio will change
with changes in the MARR.
Principles of Engineering Economic Analysis, 5th edition
Example 15.11
In Example 15.10, suppose investments 1 and 2
are mutually exclusive and investments 3 and 4
are mutually exclusive. Further, suppose
investment 5 is contingent on investment 3 being
funded. Show how SOLVER can be used to
determine the optimum investment portfolio.
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Principles of Engineering Economic Analysis, 5th edition
Question
• With indivisible investments, how would you
incorporate into SOLVER the following
constraint: investment 4 is contingent on
either investment 3 or investment 5?
• Add a constraint to SOLVER for x3 to be less
than or equal to the value of a cell in which
you have entered x3 + x5
• For our example: E5 <= F8 when D5+F5 has
been entered in cell F8
Principles of Engineering Economic Analysis, 5th edition
Question
• With indivisible investments, how would you
incorporate into SOLVER the following
constraint: investment 4 is contingent on
either investment 3 or investment 5?
• x4 < x3 + x5
• Add a constraint to SOLVER for x3 to be less
than or equal to the value of a cell in which
you have entered x3 + x5
• For our example: E5 <= F8 when D5+F5 has
been entered in cell F8
Principles of Engineering Economic Analysis, 5th edition
Question
• With indivisible investments, how would you
incorporate into SOLVER the following
constraint: investment 4 is contingent on
either investment 3 or investment 5?
• x4 < x3 + x5
• Add a constraint to SOLVER for x4 to be less
than or equal to the value of a cell in which
you have entered x3 + x5
• For our example: E5 <= F8 when D5+F5 has
been entered in cell F8
Principles of Engineering Economic Analysis, 5th edition
Another Question
• How would you incorporate into SOLVER
the following constraint: at least two of the
indivisible investments 1, 2, or 3 must be
included in the investment portfolio?
• x1 + x2 + x3 > 2
• After entering B5+C5+D5 in F8, add a
constraint to SOLVER: F8>=2
Principles of Engineering Economic Analysis, 5th edition
Another Question
• How would you incorporate into SOLVER
the following constraint: at least two of the
indivisible investments 1, 2, or 3 must be
included in the investment portfolio?
• x1 + x2 + x3 > 2
• After entering B5+C5+D5 in F8, add a
constraint to SOLVER: F8>=2
Principles of Engineering Economic Analysis, 5th edition
Another Question
• How would you incorporate into SOLVER
the following constraint: at least two of the
indivisible investments 1, 2, or 3 must be
included in the investment portfolio?
• x1 + x2 + x3 > 2
• After entering B5+C5+D5 in F8, add a
constraint to SOLVER: F8>=2
Principles of Engineering Economic Analysis, 5th edition
Pit Stop #15—The Finish Line Is In Sight!
1.
2.
3.
4.
5.
True or False: When several independent, indivisible
investments are available, form the investment portfolio
so that the present worth of the portfolio is maximized.
True
True or False: If independent, indivisible investments 3
and 4 are mutually exclusive, then x3 + x4 < 1 is added as
a constraint to the BLP formulation. True
True or False: If indivisible investment 2 is contingent on
indivisible investment 1 being funded, then x2 - x1 < 0 is
added as a constraint to the BLP formulation. True
True or False: When multiple independent, divisible
investments are available, form the investment portfolio
so that the internal rate of return is maximized. False
True or False: When multiple independent divisible
investments are available, choose investments to add to
the portfolio on the basis of their IRR and move from the
largest to the smallest IRR until the capacity limit is
reached. True
Principles of Engineering Economic Analysis, 5th edition