### Comparison of Alternatives Answers

P1

Let A: be automatic broom: PA = 35, ACA = 0, NA = 1
Let V be the vacuum cleaner: PV = 90, ACV = 10, NV = 4
Use study period of 4 years.

Present worth method:
Note that A requires investments at the beginning of the year.
NPWA = 35 + 35(P/A, 0.15, 3) = 35 + 35(2.283) = 114.91
NPWV = 90 + 10(P/A, 0.15, 4) = 118.55

Annual Cost method:
NACA = 35(A/P, 0.15, 1) = 35¥1.15 = 40.25.
NACV = 90(A/P, 0.15, 4)+ 10 = 90(0.3503) = 31.52 + 10 = 41.52
Both methods select the automatic broom.

P2

Comparing the NPW of costs:
NPW1 = 1100 + 200(P/A, 0.15, 10) - 100(P/F, 0.15, 10) = 2079
NPW2 = 1300 +152.3(P/A, 0.15, 10) - 100(P/F, 0.15, 10) = 2039

The second machine has the smallest net present worth of cost and should be chosen.

Comparing the NAW of costs or the NAC

NAC1 = 1100(A/P, 0.15, 10) + 200- 100(A/F, 0.15, 10) = 414.25
NAC2 = 1300(A/P, 0.15, 10) +152.3 - 100(A/F, 0.15, 10) = 406.40

The second machine has the smallest net annual cost and should be chosen.

P3

Compute the NAC of the first machine.
NAC = (P - S)(A/P, i, 10) + Si + A
= 1000(0.1992) + 100(0.15) + 500
= 199.2 + 15 + 500 = 714.2

Second machine
NAC = 1200(0.1992) + 100(0.15) + A = 714.2
A = 714.2 - 1200(0.1992) - 100(0.15) = 714.2 - 239.04 - 15 = 460.16
An operating cost of \$460 would make the two machines have approximately the same NAC of 714.

P4

Present Worth Cost Method

Type A
P = 500, A = 20, n = inf.
PW = 500 + 20(P/A, 0.06, inf.)
= 500 + 20/0.06 = 500 + 333.33
= 833.33
Type B
P = 700, A = 30 every two years, n = inf.
PW = 700 + 30(A/F,i,2)(P/A,i,inf.)
= 700 + 30*0.4854/0.06
= 700 + 242.72 = 942.72
Select type A with the lowest present worth of cost.

Annual Cost Method

NAC(A) = 500(A/P, 0.06, inf) + 20 = \$50/year

NAC(B) = 700(A/P, 0.06, inf) + 30(A/F, 0.06, 2) = \$56.56 per year

Select type A with lowest annual worth of cost.

Rate of Return Method

NAW(B-A) = NAW(B) - NAW(A)

= (-700i - 30(A/F, i, 2)) - (-500i - 20) = 0

By trial and error we find that ROR for B over A is a little more than 2%. The extra investment is not justified.

P5

PW Method (study period 10 yrs)
A: PW = 20 + 1.5(P/A, 0.12, 10) - 0.5(P/F, 0.12, 10) = 28.314
B: For 5 years
PW = 15 + 1(P/A, 0.12, 5) - 0.5(P/F, 0.12, 5) = 18.321
For 10 years
PW = 18.321 + 18.321(P/F, 0.12, 5) = 28.716

Annual Cost Method
A: NAC = 20(A/P, 0.12, 10) + 1.5 - 0.5(A/F, 0.12, 10) = 5.011
B: NAC = 15(A/P, 0.12, 5) + 1 - 0.5(A/F, 0.12, 5) = 5.082

Rate of Return Method

NAW(A - B) = NAW(A) - NAW(B) =
= -20(A/P, i, 10 - 1.5 + 0.5(A/F, i 10) - (-15(A/P, i, 5) - 1 + 0.5(A/F, i, 5)

By trial and error the ROR comes out to be slightly less than 14%. Accept the extra investment in the higher priced car.

All analyses indicate that the higher priced car should be chosen.

P6

Extra ceiling insulation
NPW = -500 + 100(P/A, 0.1, 7) = -500 + 486.84 = -13.16
NAW = -500(A/P, 0.1, 7) + 100 = -102.7 + 100 = -2.70
Extra insulation plus cooling fans
NPW = -1250 + 250(P/A, 0.1, 7) + 300(P/F, 0.1, 7) = 121
NAW = -1250(A/P, 0.1, 7) + 250 + 300(A/F, 0.1, 7) = 24.86.
The insulation alone does not have a 10% rate of return. The insulation plus the fans does provide the appropriate return, so invest in that alternative. The net annual worth of the savings is \$25 per year.

ROR return method

Rank alternatives: None (N), Insulation(I), Insulation plus fans (IF)

Compare I - N

NPW = -500 + 100(P/A, i, 7) = 0

ROR is between 9% and 10%. Reject I

Compare IF - N

NPW = -1250 + 250(P/A, i, 7) + 300(P/F, i, 7) = 0

ROR is between 12% and 13%, so accept extra investment.

Choose to install insulation plus fans.

P7

Using a NAC comparison
NAC(A) = 50(A/P, 0.12, 11) - 10(A/F, 0.12, 11) + 5 = 50*0.1684 - 10*0.0484 + 5
= 8.42 - 0.484 + 5 = 12.94/yr
NAC(B) = 40(A/P, 0.12, 10) + 2 = 40*0.1770 + 2 = 7.08 + 2 = 9.08/yr
Clearly B is better.

With the ROR method we would discover that the ROR(A - B) < 0%, so surely we would reject the extra investment.

P8

Present Worth Method

Use a study period of 18 years.
Machine A: PW Costs = 9000 + 5000(P/A, 0.1, 18) + 9000(P/F, 0.1, 6) + 9000(P/F, 0.1, 12)
PW Costs = 9000 + 5000(8.201) + 9000(0.5645) + 9000(0.3186) = 57,952.9

Machine B: PW Costs = 16000 + 4000(P/A, 0.1, 18) + 12000(P/F, 0.1, 9) - 4000(P/F, 0.1, 18)
PW Costs = 16000 + 4000(8.201) + 12000(0.4241) - 4000(0.1799) = 53,173.6

SELECT B.

Rate of Return Method

Machine A: NAW(A) = -9000(A/P, i, 6) - 5000

Machine B: NAW(B) = -16000(A/P, i, 9) - 4000 + 4000(F/A, i, 9)

To find ROR of the extra investment set NAW(B - A) = NAW(B) - NAW(A) = 0

By trial and error the interest rate or ROR is betwen 18 and 20%. Since the MARR is 10%, the extra investment is surely justified. Select B.

P9

Net Annual Cost Method

Manual: NAW = 90*700= 6300

Machine Tool: 10000*(A/P,15%,3) + 80*700 = \$60,379

Automated: 70000*(A/P,15%,5) + 60*700 = 62,882

Choose the Machine Tool with the smallest net annual cost.

Rate of Return Method

Ranking in order of investment: Manual, Machine Tool, Automated

Find ROR of: (Machine Tool) - (Manual)

NAW(Machine - Manual) = -10000(A/P, i, 3) - 56000 - (-63000)

or (A/P, i, 3) = 0.7

ROR = 49%, therefore accept machine tool over manual.

Find ROR of: (Machine Tool) - (Automated)

NAW(Automated - Machine) = -70000(A/P, i, 5) - 42000 - [-10000(A/P, i, 3) - 56000]

-70000(A/P, i, 5) + 10000(A/P, i, 3) + 14000 = 0

Trial and error
 i NAW(i) 0 3333 0.05 1503 0.08 348 0.09 -46 0.1 -445

Rate of return is between 9 and 9%. Since MARR is 15%, reject the extra investment. Choose the machine tool.