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102 SUPPLEMENT 7 CAPAC I T Y A N D CO N S T R A I N T MA N A G E M E N T
ACTIVE MODEL S7.2: Break-even Analysis
1. Use the scrollbars to determine what happens to the break-even
point as the fixed costs increase? the variable costs increase? the
selling price increases?
by 10%. In this case, if the variable costs rise by 10%, then the
BEP rises by 15%. If the price per unit increases by 10%, then
would the fixed costs have to decrease? the variable costs? How
much would the selling price have to increase?
Fixed costs, $5,000; variable costs by $1.75 from $2.25
to $.50; the selling price would need to increase by the same
S7.8 Actual (or expected) output = (Effective capacity) (Efficiency)
(Equation (S7.3))
5.5 cars 0.880 = 4.84 cars.
the maximum of 10, 6, and 8.
S7.10 Process time of each machine = 60 min/hr ÷ 4 units/hr
S7.11 Converting each capacity to a process time, Station 1 = 60
min/hr ÷ 20 units/hr = 3 min/unit (for both Machine A and
Machine B); Station 2 = 60 min/hr ÷ 5 units/hr = 12 min/unit; and
Station 3 = 60 min/hr ÷ 12 units/hr = 5 min/unit
S7.14 (a) Workstation C is the bottleneck, at 20 min/unit
(A different Part 1 can be worked on by Workstation A and
Workstation B simultaneously.)
S7.15 (a) Converting each capacity to a process time,
Sawing = Sanding = 60 min/hr ÷ 6 units/hr = 10 min/unit;
Drilling = 60 min/hr ÷ 2.4 units/hr = 25 min/unit; Welding = 60
min/hr ÷ 2 units/hr = 30 min/unit; and each Assembly = 60
140.71) = 140.71
min
S7.16 Break-even:
(8) – 50,000 (10) – 70,000
(8) 20,000 (10)
20,000 10 – 8
10,000
XX
XX
XX
X
=
+=
=
=
==
A
20,000
(a) 1,667 pizzas;
14 – 2
30,000
BEP
S7.20
X = 13,333 pizzas
S7.21 Given:
Price ( ) = $8 unit
Variable cost ( ) = $4 unit
P
V
500 2,000 units
.75 .50
F
BEPxPV
= = =
−−
104 SUPPLEMENT 7 CAPAC I T Y A N D CO N S T R A I N T MA N A G E M E N T
= = = =
−
= = =
=
$0.01
0.05
15,000 15,000
(a) $18,750
1 – 0.2
1 – 1
15,000 15,000
(b) – 0.05 – 0.01 0.04
375,000 copies
V
P
F
BEP
F
BEPxPV
S7.23 Option A: Stay as is
Option B: add new equipment
=
=
==
$
50,000 2.50 $125,000 and
37,500
1–
37,500 $125,000
1.75
1– 2.50
BEP V
P
SUPPLEMENT 7 CA PA C I T Y A N D CO N S T R A I N T MA N A G E M E N T 105
(h) The manual process should be preferred over the mech-
anized process below 75,000 bags. The mechanized
process should be preferred over the manual process
above 75,000 bags.
S7.26 (a) Break-even volume:
Total fixed cost = $1,800 rent, utilities, etc.
106 SUPPLEMENT 7 CAPAC I T Y A N D CO N S T R A I N T MA N A G E M E N T
S7.27 (a) Break-even volume, where total fixed cost = labor
(at $250) + booth rental (at 5 $50) = $500.
Total sales at breakeven
(b) No. of wine servings × 25% of sales
=
at breakeven
986.19 × 0.25
= = 140.9 servings
$1.75
S7.28
Option A: EMV = (90,000 × .5) + (25,000 × .5) = 45,000 + 12,500
= $57,500
Option B: EMV = (80,000 × .4) + (70,000 × .6) = 32,000 + 42,000
= $74,000
S7.29
EMV for large line = [(Sales – Cost) × 2/3] + [(Sales – Cost) × 1/3]
= (200,000 × 2/3) + (–100,000 × 1/3) = $100,000
EMV for small line and no expansion = [(Sales – Cost) × 1/3]
+ [(Sales – Initial cost) × 2/3]
= [(300,000 – 300,000) × 1/3] + [(400,000 – 300,000) × 2/3]
= $0.0 + $66,666 = $66,666
Therefore, build a large line.
Decision tree solution:
S7.32
15
5600 5600 $1,765.35
3.17
(1 ) (1.08)
N
F
P
i
= = = =
+
or from Table S7.1:
NPV = F PVF8%, 15 = 5600 0.315 = $1,764
Item
(P)
Selling
Price
(V)
Variable
Cost
Var. Cost
Factor (%)
Total
Var. Cost
1– (V/P)
Estimated
Percent
Revenue
Contribution
Weighted
Revenue
Soft drinks
1.00
0.65
1.1
0.715
0.285
0.25
0.071
Wine
1.75
0.95
1.1
1.045
0.403
0.25
0.101
Coffee
1.00
0.30
1.1
0.330
0.670
0.30
0.201
Candy
1.00
0.30
1.1
0.330
0.670
0.20
0.134
Totals
1.00
0.507
Breakeven = TFC/wt contribution = 500/0.507 = $986.19
S7.33
Expense
Machine A
Machine B
Original cost
10,000
20,000
Labor per year
2,000
4,000
Maintenance per year
4,000
1,000
Salvage value
2,000
7,000
Year
Machine A
NPV Factor*
NPV
Now
Expense
10,000
1.000
–10,000
1
Expense
6,000
0.893
–5,358
2
Expense
6,000
0.797
–4,782
3
Expense
6,000
0.712
–4,272
–24,412
3
Salvage revenue
2,000
0.712
+1,424
–22,988
* NPV factor from Table S7.1.
(a) NPV of the three small ovens = –$8,511; NPV of the
two large ovens = –$5,855. Therefore, you should rec-
ommend that the firm purchase the two large ovens.
(b) The basic assumptions made with regard to the ovens
are:
◼ The ovens are of equal quality.
◼ The ovens are of equivalent production capacity.
(c) The basic assumptions made with regard to methodol-
ogy are:
◼ Future interest rates are known.
◼ Payments are made at the end of each time period.
S7.35 (a) Remember that Year 0 has no discounting.
Initial cost = $1,000,000 Yearly maintenance = $75,000
Salvage cost = $50,000 Yearly dues = $300,000
Interest rate = 0.10 No. of members = 500
108 SUPPLEMENT 7 CAPAC I T Y A N D CO N S T R A I N T MA N A G E M E N T
ADDITIONAL HOMEWORK PROBLEMS
S7.36 (a) Proposal A break even in units is:
Fixed cost 50,000 50,000 6,250 units
20 12 8SP VC = = =
−−
Fixed cost 70,000 70,000 7,000 units
20 10 10SP VC = = =
−−
S7.37 (a) Proposal A break even in dollars is:
12
20
Fixed cost 50,000 50,000 $125,000
0.40
11
VC
SP
===
−−
(b) Proposal B break even in dollars is:
$10
20
Fixed cost 70,000 70,000 $140,000
0.50
11
VC
SP
BEP = = = =
−−
S7.38 Set Proposal A = Proposal B; Solve for units
( ) ( )
(20 12) 50,000 (20 10) 70,000
(8) 50,000 (10) 70,000
(8) 20,000 (10)
20,000 10 8
20,000 2
10,000
A A A A B B B B
SP VC X F SP VC X F
XX
XX
XX
XX
X
X
− − = − −
− − = − −
− = −
+=
=−
=
=
S7.39 (a) Proposal A: Profit at 8,500 units
@ 8,500 for Proposal B:
(20 10)8,500 70,000 = 15,000
−−
Proposal A is best.
(b) Proposal B: Profit at 15,000 units
@ 15,000 units for Proposal A:
(20 12)15,000 50,000 = $70,000
@ 15,000 units for Proposal B:
(20 10)15,000 70,000 = $80,000
−−
−−
Proposal B is best.
S7.40 Investment A net income, using Table S7.2, 19,000
PVF9%, 7 – 61,000 = 19,000 4.486 – 61,000 = $24,234
Investment B Net Income
Year
NPV Factor*
NPV
Now
Expense
74,000
1.000
–74,000
1
Revenue
19,000
0.917
+17,423
2
Revenue
20,000
0.842
+16,840
3
Revenue
21,000
0.772
+16,212
4
Revenue
22,000
0.708
+15,576
5
Revenue
21,000
0.650
+13,650
6
Revenue
20,000
0.596
+11,920
7
Revenue
11,000
0.547
+6,017
23,638
* Table S7.1
Therefore, Investment A, with a payoff of $24,234, would be pre-
S7.41 Initial investment = $20,000
Cash Flows
NPV
Cash Flow 1
Cash Flow 2
Cash Flow 3
Year
Factor*
Cash
P
Cash
P
Cash
P
1
0.909
$1,000
$909
$7,000
$6,363
$10,000
$9,090
2
0.826
1,000
826
6,000
4,956
5,000
4,130
3
0.751
3,000
2,253
5,000
3,755
3,000
2,253
4
0.683
15,000
10,245
4,000
2,732
2,000
1,366
5
0.621
3,000
1,863
4,000
2,484
1,000
621
6
0.564
1,000
564
4,000
2,256
1,000
564
7
0.513
—
—
4,000
2,052
1,000
513
8
0.467
1,000
467
2,000
934
—
—
9
0.424
—
—
—
—
1,000
425
$17,127
$25,532
$18,962
*The NPV from Investment 2 is highest, at $5,532 (after the initial investment of $20,000 is subtracted).
102 SUPPLEMENT 7 CAPAC I T Y A N D CO N S T R A I N T MA N A G E M E N T
ACTIVE MODEL S7.2: Break-even Analysis
1. Use the scrollbars to determine what happens to the break-even
point as the fixed costs increase? the variable costs increase? the
selling price increases?
by 10%. In this case, if the variable costs rise by 10%, then the
BEP rises by 15%. If the price per unit increases by 10%, then
would the fixed costs have to decrease? the variable costs? How
much would the selling price have to increase?
Fixed costs, $5,000; variable costs by $1.75 from $2.25
to $.50; the selling price would need to increase by the same
S7.8 Actual (or expected) output = (Effective capacity) (Efficiency)
(Equation (S7.3))
5.5 cars 0.880 = 4.84 cars.
the maximum of 10, 6, and 8.
S7.10 Process time of each machine = 60 min/hr ÷ 4 units/hr
S7.11 Converting each capacity to a process time, Station 1 = 60
min/hr ÷ 20 units/hr = 3 min/unit (for both Machine A and
Machine B); Station 2 = 60 min/hr ÷ 5 units/hr = 12 min/unit; and
Station 3 = 60 min/hr ÷ 12 units/hr = 5 min/unit
S7.14 (a) Workstation C is the bottleneck, at 20 min/unit
(A different Part 1 can be worked on by Workstation A and
Workstation B simultaneously.)
S7.15 (a) Converting each capacity to a process time,
Sawing = Sanding = 60 min/hr ÷ 6 units/hr = 10 min/unit;
Drilling = 60 min/hr ÷ 2.4 units/hr = 25 min/unit; Welding = 60
min/hr ÷ 2 units/hr = 30 min/unit; and each Assembly = 60
140.71) = 140.71
min
S7.16 Break-even:
(8) – 50,000 (10) – 70,000
(8) 20,000 (10)
20,000 10 – 8
10,000
XX
XX
XX
X
=
+=
=
=
==
A
20,000
(a) 1,667 pizzas;
14 – 2
30,000
BEP
S7.20
X = 13,333 pizzas
S7.21 Given:
Price ( ) = $8 unit
Variable cost ( ) = $4 unit
P
V
500 2,000 units
.75 .50
F
BEPxPV
= = =
−−
104 SUPPLEMENT 7 CAPAC I T Y A N D CO N S T R A I N T MA N A G E M E N T
= = = =
−
= = =
=
$0.01
0.05
15,000 15,000
(a) $18,750
1 – 0.2
1 – 1
15,000 15,000
(b) – 0.05 – 0.01 0.04
375,000 copies
V
P
F
BEP
F
BEPxPV
S7.23 Option A: Stay as is
Option B: add new equipment
=
=
==
$
50,000 2.50 $125,000 and
37,500
1–
37,500 $125,000
1.75
1– 2.50
BEP V
P
SUPPLEMENT 7 CA PA C I T Y A N D CO N S T R A I N T MA N A G E M E N T 105
(h) The manual process should be preferred over the mech-
anized process below 75,000 bags. The mechanized
process should be preferred over the manual process
above 75,000 bags.
S7.26 (a) Break-even volume:
Total fixed cost = $1,800 rent, utilities, etc.
106 SUPPLEMENT 7 CAPAC I T Y A N D CO N S T R A I N T MA N A G E M E N T
S7.27 (a) Break-even volume, where total fixed cost = labor
(at $250) + booth rental (at 5 $50) = $500.
Total sales at breakeven
(b) No. of wine servings × 25% of sales
=
at breakeven
986.19 × 0.25
= = 140.9 servings
$1.75
S7.28
Option A: EMV = (90,000 × .5) + (25,000 × .5) = 45,000 + 12,500
= $57,500
Option B: EMV = (80,000 × .4) + (70,000 × .6) = 32,000 + 42,000
= $74,000
S7.29
EMV for large line = [(Sales – Cost) × 2/3] + [(Sales – Cost) × 1/3]
= (200,000 × 2/3) + (–100,000 × 1/3) = $100,000
EMV for small line and no expansion = [(Sales – Cost) × 1/3]
+ [(Sales – Initial cost) × 2/3]
= [(300,000 – 300,000) × 1/3] + [(400,000 – 300,000) × 2/3]
= $0.0 + $66,666 = $66,666
Therefore, build a large line.
Decision tree solution:
S7.32
15
5600 5600 $1,765.35
3.17
(1 ) (1.08)
N
F
P
i
= = = =
+
or from Table S7.1:
NPV = F PVF8%, 15 = 5600 0.315 = $1,764
Item
(P)
Selling
Price
(V)
Variable
Cost
Var. Cost
Factor (%)
Total
Var. Cost
1– (V/P)
Estimated
Percent
Revenue
Contribution
Weighted
Revenue
Soft drinks
1.00
0.65
1.1
0.715
0.285
0.25
0.071
Wine
1.75
0.95
1.1
1.045
0.403
0.25
0.101
Coffee
1.00
0.30
1.1
0.330
0.670
0.30
0.201
Candy
1.00
0.30
1.1
0.330
0.670
0.20
0.134
Totals
1.00
0.507
Breakeven = TFC/wt contribution = 500/0.507 = $986.19
S7.33
Expense
Machine A
Machine B
Original cost
10,000
20,000
Labor per year
2,000
4,000
Maintenance per year
4,000
1,000
Salvage value
2,000
7,000
Year
Machine A
NPV Factor*
NPV
Now
Expense
10,000
1.000
–10,000
1
Expense
6,000
0.893
–5,358
2
Expense
6,000
0.797
–4,782
3
Expense
6,000
0.712
–4,272
–24,412
3
Salvage revenue
2,000
0.712
+1,424
–22,988
* NPV factor from Table S7.1.
(a) NPV of the three small ovens = –$8,511; NPV of the
two large ovens = –$5,855. Therefore, you should rec-
ommend that the firm purchase the two large ovens.
(b) The basic assumptions made with regard to the ovens
are:
◼ The ovens are of equal quality.
◼ The ovens are of equivalent production capacity.
(c) The basic assumptions made with regard to methodol-
ogy are:
◼ Future interest rates are known.
◼ Payments are made at the end of each time period.
S7.35 (a) Remember that Year 0 has no discounting.
Initial cost = $1,000,000 Yearly maintenance = $75,000
Salvage cost = $50,000 Yearly dues = $300,000
Interest rate = 0.10 No. of members = 500
108 SUPPLEMENT 7 CAPAC I T Y A N D CO N S T R A I N T MA N A G E M E N T
ADDITIONAL HOMEWORK PROBLEMS
S7.36 (a) Proposal A break even in units is:
Fixed cost 50,000 50,000 6,250 units
20 12 8SP VC = = =
−−
Fixed cost 70,000 70,000 7,000 units
20 10 10SP VC = = =
−−
S7.37 (a) Proposal A break even in dollars is:
12
20
Fixed cost 50,000 50,000 $125,000
0.40
11
VC
SP
===
−−
(b) Proposal B break even in dollars is:
$10
20
Fixed cost 70,000 70,000 $140,000
0.50
11
VC
SP
BEP = = = =
−−
S7.38 Set Proposal A = Proposal B; Solve for units
( ) ( )
(20 12) 50,000 (20 10) 70,000
(8) 50,000 (10) 70,000
(8) 20,000 (10)
20,000 10 8
20,000 2
10,000
A A A A B B B B
SP VC X F SP VC X F
XX
XX
XX
XX
X
X
− − = − −
− − = − −
− = −
+=
=−
=
=
S7.39 (a) Proposal A: Profit at 8,500 units
@ 8,500 for Proposal B:
(20 10)8,500 70,000 = 15,000
−−
Proposal A is best.
(b) Proposal B: Profit at 15,000 units
@ 15,000 units for Proposal A:
(20 12)15,000 50,000 = $70,000
@ 15,000 units for Proposal B:
(20 10)15,000 70,000 = $80,000
−−
−−
Proposal B is best.
S7.40 Investment A net income, using Table S7.2, 19,000
PVF9%, 7 – 61,000 = 19,000 4.486 – 61,000 = $24,234
Investment B Net Income
Year
NPV Factor*
NPV
Now
Expense
74,000
1.000
–74,000
1
Revenue
19,000
0.917
+17,423
2
Revenue
20,000
0.842
+16,840
3
Revenue
21,000
0.772
+16,212
4
Revenue
22,000
0.708
+15,576
5
Revenue
21,000
0.650
+13,650
6
Revenue
20,000
0.596
+11,920
7
Revenue
11,000
0.547
+6,017
23,638
* Table S7.1
Therefore, Investment A, with a payoff of $24,234, would be pre-
S7.41 Initial investment = $20,000
Cash Flows
NPV
Cash Flow 1
Cash Flow 2
Cash Flow 3
Year
Factor*
Cash
P
Cash
P
Cash
P
1
0.909
$1,000
$909
$7,000
$6,363
$10,000
$9,090
2
0.826
1,000
826
6,000
4,956
5,000
4,130
3
0.751
3,000
2,253
5,000
3,755
3,000
2,253
4
0.683
15,000
10,245
4,000
2,732
2,000
1,366
5
0.621
3,000
1,863
4,000
2,484
1,000
621
6
0.564
1,000
564
4,000
2,256
1,000
564
7
0.513
—
—
4,000
2,052
1,000
513
8
0.467
1,000
467
2,000
934
—
—
9
0.424
—
—
—
—
1,000
425
$17,127
$25,532
$18,962
*The NPV from Investment 2 is highest, at $5,532 (after the initial investment of $20,000 is subtracted).
