ITM 501: Decision Analysis
Homework Assignment 2
ITM 501
Assignment 2
Due date: 2018/03/23 12.00 PM
- This assignment must be submitted through D2L Course shell.
Questions:
1. Modem Corporation of America (MCA) is the world’s largest producer of modem
communication devices for microcomputers. MCA sold 9,000 of the regular model and
10,400 of the smart (“intelligent”) model this September. Its income statement for the
month is shown in the table on this page. Costs presented are typical of prior months
and are expected to remain at the same levels in the near future. The firm is facing
several constraints as it prepares its November production plan. First, it has
experienced a tremendous demand and has been unable to keep any significant
inventory in stock. This situation is not expected to change. Second, the firm is located
in a small Iowa town from which additional labor is not readily available. Workers can
be shifted from production of one modem to another, however. To produce the 9,000
regular modems in September required 5,000 direct labor hours. The 10,400
intelligent modems absorbed 10,400 direct labor hours.
MCA Income Statement Month Ended September 30
REGULAR
MODEMS
$450,000
INTELLIGENT
MODEMS
$640,000
Less: Discounts
Returns
Warranty replacements
Net sales
Sales costs
Direct labor
Indirect labor
Materials cost
Depreciation
Cost of sales
Gross profit
Selling and general expenses
General expenses—variable
General expenses—fixed
Advertising
Sales commissions
Total operating cost
Pretax income
Income taxes (25%)
10,000
12,000
4,000
$424,000
15,000
9,500
2,500
$613,000
60,000 0
9,000
90,000
40,000
$199,000
$225,000
76,80
11,520
128,000
50,800
$267,120
$345,880
30,000
36,000
28,000
31,000
$125,000
$100,000
25,000
35,000
40,000
25,000
60,000
$160,000
$185,880
46,470
Net income
$ 75,000
$139,41
Sales
Third, MCA is experiencing a problem affecting the intelligent modems model. Its
component supplier is able to guarantee only 8,000 microprocessors for November
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ITM 501: Decision Analysis
Homework Assignment 2
delivery. Each intelligent modem requires one of these specially made microprocessors.
Alternative suppliers are not available on short notice.
MCA wants to plan the optimal mix of the two modem models to produce in
November to maximize profits for MCA.
(a) Formulate, using September’s data, MCA’s problem as a linear program.
(b) Solve the problem graphically.
(c) Discuss the implications of your recommended solution.
a.
Let: X1  number of MCA regular modems made and sold in November
X2  number of MCA intelligent modems made and sold in November
Data needed for variable costs and contribution margin are in the table.
Hours needed to produce each modem:
MCA regular 
5, 000 hours
 0.555 hour/modem
9, 000 modems
MCA intelligent 
10, 400 hours
 1.0 hour/modem
10, 400 modems
Maximize profit  $22.67X1  $29.01X2
subject to 0.555X1  1.0X2  15,400 (direct labor hours)
X2  8,000 (intelligent modems)
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ITM 501: Decision Analysis
Homework Assignment 2
Table for Problem 7-43(a)
Net sales
MCA REGULAR MODEM
MCA INTELLIGENT MODEM
Total
Total
Per Unit
$424,000
$47.11
$613,000
$58.94
60,000
6.67
76,800
7.38
9,000
1.00
11,520
1.11
Materials
90,000
10.00
128,000
12.31
General
expenses
30,000
3.33
35,000
3.37
$31,000
$3.44
$60,000
$5.76
$220,000
$24.44
$311,320
$29.93
Variable costs
a
Direct labor
Indirect labor
Sales
commissions
Total
costs
variable
Contribution
margin
a
Per Unit
$204,000
$22.67
$301,680
$29.01
Depreciation, fixed general expense, and advertising are excluded from the calculations.
b.
c. The optimal solution suggests making all MCA regular modems. Students should discuss
the implications of shipping no MCA intelligent modems.
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ITM 501: Decision Analysis
Homework Assignment 2
2. Graph the following LP problem and indicate the optimal solution point:
Maximize profit = $3X + $2Y
subject to
2X + Y <= 150
2X + 3Y <= 300
(a) Does the optimal solution change if the profit per unit of X changes to $4.50?
(b) What happens if the profit function should have been$3X + $3Y?
Using the isoprofit line or corner point method, we see that point b (where X  37.5 and Y  75)
is optimal if the profit  $3X  $2Y. If the profit changes to $4.50 per unit of X, the optimal solution
shifts to point c. If the objective function becomes P  $3X  $3Y, the corner point b remains
optimal.
3X +
4.50X + 2Y
Y
3X +
X
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ITM 501: Decision Analysis
Homework Assignment 2
3. Margaret Black’s family owns five parcels of farmland broken into a southeast
sector, north sector, northwest sector, west sector, and southwest sector. Margaret
is involved primarily in growing wheat, alfalfa, and barley crops and is currently
preparing her production plan for next year. The Pennsylvania Water Authority has
just announced its yearly water allotment, with the Black farm receiving 7,400 acrefeet. Each parcel can only tolerate a specified amount of irrigation per growing
season, as specified in the following table:
PARCEL
AREA (ACRES)
Southeast
North
Northwest
West
Southwest
2,000
2,300
600
1,100
500
WATER IRRIGATION
LIMIT (ACRE-FEET)
3,200
3,400
800
500
600
Each of Margaret’s crops needs a minimum amount of water per acre, and there is a
projected limit on sales of each crop. Crop data follow:
CROP
MAXIMUM SALES
Wheat
Alfalfa
Barley
110,000 bushels
1,800 tons
2,200 tons
WATER NEEDED PER
ACRE (ACRE-FEET)
1.6
2.9
3.5
Margaret’s best estimate is that she can sell wheat at a net profit of $2 per bushel,
alfalfa at $40 per ton, and barley at $50 per ton. One acre of land yields an average of
1.5 tons of alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels
per acre.
(a) Formulate Margaret’s production plan.
(b) What should the crop plan be, and what profit will it yield?
(c) The Water Authority informs Margaret that for a special fee of $6,000 this year, her
farm will qualify for an additional allotment of 600 acre-feet of water. How should she
respond?
a. Let Xij = acres of crop i planted on parcel j
where
i = 1 for wheat, 2 for alfalfa, 3 for barley
j = 1 to 5 for SE, N, NW, W, and SW parcels
Irrigation limits:
1.6X11 + 2.9X21 + 3.5X31
1.6X12 + 2.9X22 + 3.5X32
1.6X13 + 2.9X23 + 3.5X33
1.6X14 + 2.9X24 + 3.5X34
1.6X15 + 2.9X25 + 3.5X35
 3,200 acre-feet in SE
 3,400 acre-feet in N
 800 acre-feet in NW
 500 acre-feet in W
 600 acre-feet in SW
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ITM 501: Decision Analysis
Homework Assignment 2
5
1.6 X
j 1
5
5
j 1
j 1
1, j   2.9 X 2, j   3.5 X 3, j  7, 400
water acre-feet total
Sales limits:
X11 + X12 + X13 + X14 + X15  2,200 wheat in acres (= 110,000 bushels)
X21 + X22 + X23 + X24 + X25  1,200 alfalfa in acres (= 1,800 tons)
X31 + X32 + X33 + X34 + X35  1,000 barley in acres (= 2,200 tons)
Acreage availability:
X11 + X21 + X31  2,000 acres in SE parcel
X12 + X22 + X32  2,300 acres in N parcel
X13 + X23 + X33  600 acres in NW parcel
X14 + X24 + X34  1,100 acres in W parcel
X15 + X25 + X35  500 acres in SW parcel
Objective function:
5
5
5
j 1
j 1
j 1
maximize profit   $2  50 bushels  X1, j   $40 1.5 tons  X 2, j    $50  2.2 tons  X 3, j
b. The solution is to plant
X12 = 1,250 acres of wheat in N parcel
X13 = 500 acres of wheat in NW parcel
X14 = 312 acres of wheat in W parcel
X15 = 137 acres of wheat in SW parcel
X25 = 131 acres of alfalfa in SW parcel
X31 = 600 acres of barley in SE parcel
X32 = 400 acres of barley in N parcel
Profit will be $337,862.10. Multiple optimal solutions exist.
c. Yes, need only 500 more water-feet.
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ITM 501: Decision Analysis
Homework Assignment 2
4. Triangle Utilities provides electricity for three cities. The company has four electric
generators that are used to provide electricity. The main generator operates 24 hours
per day, with an occasional shutdown for routine maintenance. Three other generators
(1, 2, and 3) are available to provide additional power when needed. A start-up cost is
incurred each time one of these generators is started. The start-up costs are $6,000 for
1, $5,000 for 2, and $4,000 for 3. These generators are used in the following ways: A
generator may be started at 6:00 A.M. and run for either 8 hours or 16 hours, or it may
be started at 2:00 P.M. and run for 8 hours (until 10:00 P.M.). All generators except the
main generator are shut down at 10:00 P.M. Forecasts indicate the need for 3,200
megawatts more than provided by the main generator before 2:00 P.M., and this need
goes up to 5,700 megawatts between 2:00 and 10:00 P.M. Generator 1 may provide up
to 2,400 megawatts, generator 2 may provide up to 2,100 megawatts, and generator 3
may provide up to 3,300 megawatts. The cost per megawatt used per eight-hour period
is $8 for 1, $9 for 2, and $7 for 3.
(a) Formulate this problem as an integer programming problem to determine the leastcost way to meet the needs of the area.
(b) Solve using computer software.
a. Let Xij = 1 if generator i is functioning during time period j, and 0 otherwise; where
i = 1, 2, 3 and j = 1 for 6–2 time period; j = 2 for 2–10 time period; j = 3 for 6–10 time period.
Let Yij = megawatts produced by generator i in time period j, where i = 1, 2, 3 and j = 1 for
6–2 time period; j = 2 for 2–10 time period.
Minimize cost = 6,000(X11 + X12 + X13) + 5,000(X21 + X22 + X23) + 4,000(X31 + X32 + X33) +
8(Y11 + Y12) + 9(Y21 + Y22) + 7(Y31 + Y32)
Subject to:
Y11 + Y21 + Y31  3,200
megawatts requirements from 6–2
Y12 + Y22 + Y32  5,700
megawatts requirements from 2–10
Y11  2,400(X11 + X13)
maximum megawatts from #1 from 6–2
Y12  2,400(X12 + X13)
maximum megawatts from #1 from 2–10
Y21  2,100(X21 + X23)
maximum megawatts from #2 from 6–2
Y22  2,100(X22 + X23)
maximum megawatts from #2 from 2–10
Y31  3,300(X31 + X33)
maximum megawatts from #3 from 6–2
Y32  3,300(X32 + X33)
maximum megawatts from #3 from 2–10
X11 + X12 + X13  1
generator #1 starts up at most once
X21 + X22 + X23  1
generator #2 starts up at most once
X31 + X32 + X33  1
generator #3 starts up at most once
Xij = 0 or 1 for all i, j
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ITM 501: Decision Analysis
Homework Assignment 2
Yij  0 for all i, j
b. The solution is: X12 = 1, X33 = 1, Y12 = 2,400, Y31 = 3,200, Y32 = 3,300, total cost = $74,700.
Thus, generator #1 will be utilized in the period 2–10 and will generate 2,400 megawatts of
electricity. Generator #3 will be started at 6 and utilized for the entire 16 hours. It will generate
3,200 megawatts during the 6–2 time period, and 3,300 megawatts during the 2–10 time period.
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ITM 501: Decision Analysis
Homework Assignment 2
5. Complete and solve the following NLP models using Excel solver.
a)
Maximize profit = 3X1 + 4X2
subject to:
X12 - 5X2 >= 8
3X1 + 4X2 >= 12
b)
Minimize cost = 18X1 + 5X2 + X22
subject to:
4X1 - 3X2 >= 8
X1 + X2 >= 18
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