Solutions to LP Practice Problems1
1. Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each chair
uses 3 units of wood. A desk contributes $40 to profit, and a chair contributes $25.
Marketing restrictions require that the number of chairs produced be at least twice the
number of desks produced. There are 20 units of wood available.
Using the graph below, determine a production plan that maximizes Furnco’s profit.
(a) Draw isoprofit lines where the total profit equals 125, 150, 175, and 200.
Here are the points where the isoprofit lines cross the axes:
z = 125
z = 150
z = 175
z = 200
X1
0
3.125
0
3.75
0
4.375
0
5
X2
5
0
6
0
7
0
8
0
Your isoprofit lines ought to look like this:
FurnCo
10
9
8
7
Chairs
6
5
4
3
2
1
0
0
1
2
3
4
5
6
Desks
Based on problems 3.3, 3.5, 4.20, and 4.27 (pp. 89-144) in Practical Management Science, by Wayne Winston
and S. Christian Albright. Solutions by David C. Juran.
1
Shade in the feasible region.
(b) Determine a daily production plan that maximizes total profit.
There are three critical points: Point A (at the origin), Point B (where the wood constraint
crosses the non-negativity constraint on desks), and Point C (where the wood constraint
line crosses the marketing constraint).
The best solution is at Point C (2, 4). Produce 2 desks and 4 chairs.
(c) What is the optimal total profit?
Point C is (2, 4), where the total profit is $180.
Operations Management
2
Prof. Juran
2. A farmer in Iowa owns 45 acres of land. She is going to plant each acre with wheat or
corn. Each acre planted with wheat yields $200 profit; each with corn yields $300 profit.
The labor and fertilizer used for each acre are given in the table below. 100 workers and
120 tons of fertilizer are available.
Labor
Fertilizer
Wheat
3 workers
2 tons
Corn
2 workers
4 tons
Using the graph below, determine the planting scheme that will maximize profit for the
farmer.
(a) Draw isoprofit lines where the total profit equals $6,000, $8,000, $10,000, and
$12,000.
Here are the points where the isoprofit lines cross the axes:
X1 X2
0
20
30
0
z = 8,000 0 26.67
40
0
z = 10,000 0 33.33
50
0
z = 12,000 0
40
60
0
z = 6,000
Your isoprofit lines ought to look like this:
Iowa Farmer
50
40
Corn
30
20
10
0
0
10
20
30
40
50
60
Wheat
Operations Management
3
Prof. Juran
(b) Shade in the feasible region.
(c) Determine the planting scheme that maximizes total profit.
There are four critical points: Point A (at the origin), Point B (where the fertilizer constraint
crosses the non-negativity constraint on wheat), Point C (where the fertilizer constraint
line crosses the labor constraint), and Point D, where the labor constraint crosses the nonnegativity constraint on corn).
The best solution is at Point C (20, 20). Plant 20 acres of wheat and 20 acres of corn.
(d) What is the optimal total profit?
The optimal total profit is $10,000.
Operations Management
4
Prof. Juran
3. A bank is attempting to determine where its assets should be invested during the
current year. At present, $500,000 is available for investment in bonds, home loans, auto
loans, and personal loans. The annual rates of return on each type of investment are
known to be the following: bonds, 10%; home loans, 16%; auto loans, 13%; and personal
loans, 20%. To ensure that the bank’s portfolio is not too risky, the bank’s investment
manager has placed the following three restrictions on the bank’s portfolio:
 The amount invested in personal loans cannot exceed the amount invested in
bonds.
 The amount invested in home loans cannot exceed the amount invested in auto
loans.
 No more than 25% of the total amount invested may be in personal loans.
Below are various elements of the Excel model used to solve the problem: the spreadsheet
model, the Solver parameters, the Solver options, the answer report and the sensitivity
report.
A
B
C
D
E
F
1
Bonds Home Loans Auto Loans Personal Loans
2
25.0%
25.0%
25.0%
25.0%
3
Investment Returns
10.0%
16.0%
13.0%
20.0%
4
=E2
5
6
7
Bonds Home Loans Auto Loans Personal Loans
8 Total Funds = 100% 25.0%
25.0%
25.0%
25.0%
100.0%
9
Personal <= Bonds -100.0%
0.0%
0.0%
100.0%
0.0%
10
Home <= Auto
0.0%
100.0%
-100.0%
0.0%
0.0%
11 Personal Loans Limit
0.0%
0.0%
0.0%
100.0%
25.0%
Operations Management
5
G
H
14.75%
I
J
objective function:
=SUMPRODUCT(B2:E2,B3:E3)
=SUM(B8:E8)
=SUMPRODUCT(B9:E9,B2:E2)
<=
<=
<=
<=
100.0%
0.0%
0.0%
25.0%
Prof. Juran
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A
B
C
D
Microsoft Excel 9.0 Answer Report
Worksheet: [spract-lp4.xls]Bank Portfolio
Report Created: 11/29/01 5:06:50 PM
E
Target Cell (Max)
Cell
Name
$G$2
Original Value
15.0%
Final Value
14.8%
Adjustable Cells
Cell
Name
$B$2 Bonds
$C$2 Home Loans
$D$2 Auto Loans
$E$2 Personal Loans
Original Value
50.0%
0.0%
0.0%
50.0%
Final Value
25.0%
25.0%
25.0%
25.0%
Constraints
Cell
Name
$F$8 Total Funds = 100%
$F$9 Personal <= Bonds
$F$10 Home <= Auto
$F$11 Personal Loans Limit
Operations Management
Cell Value
100.0%
0.0%
0.0%
25.0%
6
Formula
$F$8<=$H$8
$F$9<=$H$9
$F$10<=$H$10
$F$11<=$H$11
F
G
Status Slack
Binding 0.0%
Binding 0.0%
Binding 0.0%
Binding 0.0%
Prof. Juran
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
A
B
C
D
Microsoft Excel 9.0 Sensitivity Report
Worksheet: [spract-lp4.xls]Bank Portfolio
Report Created: 11/29/01 5:06:50 PM
E
F
G
H
Adjustable Cells
Cell
$B$2
$C$2
$D$2
$E$2
Name
Bonds
Home Loans
Auto Loans
Personal Loans
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
25.0%
0.0%
10.0%
4.5%
1.0%
25.0%
0.0%
16.0%
1.0%
3.0%
25.0%
0.0%
13.0%
1.0%
9.0%
25.0%
0.0%
20.0%
1E+30
1.0%
Constraints
Cell
$F$8
$F$9
$F$10
$F$11
Name
Total Funds = 100%
Personal <= Bonds
Home <= Auto
Personal Loans Limit
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side
Increase Decrease
100.0%
14.5%
100.0%
1E+30
50.0%
0.0%
4.5%
0.0%
25.0%
50.0%
0.0%
1.5%
0.0%
50.0%
50.0%
25.0%
1.0%
25.0%
25.0%
25.0%
(a) What is the optimal allocation of funds to the various investment types?
Cells E13:E16 in the answer report indicate that we should invest 25% of the funds in each
of the four types.
(b) What is the expected return on investment from the optimal portfolio?
From cell E8 in the answer report, the optimal portfolio will return 14.75% ($73,750).
(c) What would be the improvement in the return on investment if the limit on the total
amount invested in personal loans were increased to 30%?
We look at cell E20 in the Sensitivity Report, and see that the shadow price is 1%. That
means that for every unit of increase in this constraint’s right-hand side, we will realize a
1% improvement in the objective function. If we change the right-hand side of this
constraint from 25% to 30% (a change that is within the allowable increase shown in cell
G20 of the Sensitivity Report), then the objective function will increase by 0.05 * 0.01 =
0.0005. Our portfolio return would go from 0.1475 to 0.1480 (from 14.75% to 14.80%).
(d) If the return on bonds increases from 10% to 13%, what will happen to the optimal
allocation of funds?
Cell G9 in the Sensitivity Report indicates that the bond return would have to increase by
at least 4.5% before the optimal investment mix would change. Since this is only a 3%
increase, the portfolio would not change (although it would become more profitable).
Operations Management
7
Prof. Juran
4. Sunco Oil manufactures three types of gasoline (gas 1, 2, and 3). Each type is produced
by blending three types of crude oil (crude 1, crude 2, and crude 3). Sunco can purchase up
to 5,000 barrels of each type of crude daily.
The selling price per barrel of gasoline and the purchase price per barrel of crude oil are
given in Table 4.1 below.
Gas 1
Gas 2
Gas 3
Selling Price per Barrel
$70
$60
$50
Crude 1
Crude 2
Crude 3
Purchase Price per Barrel
$45
$35
$25
Table 4.1
The three types of gasoline differ in their octane rating and their sulfur content. The crude
oil blended to form gas 1 must have an average octane rating of at least 10 and contain at
most 1% sulfur. The crude oil blended to form gas 2 must have an average octane rating of
at least 8 and contain at most 2% sulfur. The crude oil blended to form gas 3 must have an
average octane rating of at least 6 and contain at most 1% sulfur. The octane rating and
sulfur content of the three types of crude oil are given in Table 4.2 below.
Crude 1
Crude 2
Crude 3
Octane Rating
12
6
8
Sulfur Content
0.5%
2.0%
3.0%
Table 4.2
It costs $4 to transform 1 barrel of oil into 1 barrel of gasoline, and Sunco’s refinery can
produce up to 14,000 barrels of gasoline daily. Sunco’s customers require the following
amounts of each gasoline: gas 1, 3000 barrels per day; gas 2, 2000 barrels per day; gas 3,
1000 barrels per day. The company considers it an obligation to meet these demands.
Sunco also has the option of advertising to stimulate demand for its products. Each dollar
spent daily in advertising a particular type of gas increases the daily demand for that type
of gas by 10 barrels. For example, if Sunco decides to spend $20 daily in advertising gas 2,
the daily demand for gas 2 will increase by 200 barrels. Below are the answer report and
the sensitivity report from a Solver optimal solution to this problem.
Operations Management
8
Prof. Juran
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
A
Purchase prices per barrel of crude
Crude 1
Crude 2
Crude 3
B
C
D
$45
$35
$25
E
F
Purchase costs
Production costs
Advertising costs
Sales revenue
Profit
G
$487,500
$54,000
$750
$830,000
$287,750
H
I
J
K
L
Cost to transform
$4
Purchase/production plan
10
Crude 1
Crude 2
Crude 3
Increase in demand from advertising
Gas 1
2088.89
777.78
133.33
Gas 2
Gas 3 Total purchased
Max Available
2111.11 800.00
5000
<=
5000
4222.22
0.00
5000
<=
5000
3166.67 200.00
3500
<=
5000
Requirements for gasolines
Minimum octane
Maximum Sulfur
Gas 1
10
1%
Gas 2
8
2%
Gas 3
6
1%
Total produced
13500
<=
Max Capacity
14000
Gas 1
30800
>=
30000
Gas 2
76000
>=
76000
Gas 3
11200
>=
6000
Gas 1
30
<=
30
Gas 2
190
<=
190
Gas 3
10
<=
10
Gas 1
$70
Gas 2
$60
Gas 3
$50
Demand for gasolines
Gas 1
3000
Gas 2
2000
Gas 3
1000
Amount produced
$0
0
3000
=
3000
$750
7500
9500
=
9500
$0
0
1000
=
1000
Sulfur content
Crude 1
Crude 2
Crude 3
0.5%
2.0%
3.0%
Octane ratings
Crude 1
Crude 2
Crude 3
12
6
8
Original demand
Constraint on total production
Octane constraints
Actual total octane
Required
Sulfur constraints
Actual total Sulfur
Required
Advertising
Extra demand
Total demand
Sale price per barrel of gasoline
Operations Management
9
Prof. Juran
1
2
3
4
5
6
7
8
9
10
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17
18
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20
21
22
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24
25
26
27
28
29
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31
32
33
34
35
36
37
38
39
40
41
1
2
3
4
5
6
7
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37
A
B
C
Microsoft Excel 9.0 Answer Report
Worksheet: [spract-lp5.xls]Sheet1
Report Created: 12/3/01 2:54:08 PM
Target Cell (Max)
Cell
$G$5 Profit
Name
Adjustable Cells
Cell
Name
$G$9 Crude 1 Gas 1
$H$9 Crude 1 Gas 2
$I$9
Crude 1 Gas 3
$G$10 Crude 2 Gas 1
$H$10 Crude 2 Gas 2
$I$10 Crude 2 Gas 3
$G$11 Crude 3 Gas 1
$H$11 Crude 3 Gas 2
$I$11 Crude 3 Gas 3
$G$17 Advertising Gas 1
$H$17 Advertising Gas 2
$I$17 Advertising Gas 3
D
F
Original Value
$287,750
Final Value
$287,750
Original Value
2088.89
2111.11
800.00
777.78
4222.22
0.00
133.33
3166.67
200.00
$0
$750
$0
Final Value
2088.89
2111.11
800.00
777.78
4222.22
0.00
133.33
3166.67
200.00
$0
$750
$0
Constraints
Cell
Name
$B$19 Total produced
$B$22 Actual total octane Gas 1
$C$22 Actual total octane Gas 2
$D$22 Actual total octane Gas 3
$B$27 Actual total Sulfur Gas 1
$C$27 Actual total Sulfur Gas 2
$D$27 Actual total Sulfur Gas 3
$G$19 Total demand Gas 1
$H$19 Total demand Gas 2
$I$19 Total demand Gas 3
$J$9 Crude 1 Total purchased
$J$10 Crude 2 Total purchased
$J$11 Crude 3 Total purchased
A
B
C
Microsoft Excel 9.0 Sensitivity Report
Worksheet: [spract-lp5.xls]Sheet1
Report Created: 12/3/01 2:54:09 PM
E
Cell Value
13500
30800
76000
11200
30
190
10
3000
9500
1000
5000
5000
3500
D
E
Formula
$B$19<=$D$19
$B$22>=$B$24
$C$22>=$C$24
$D$22>=$D$24
$B$27<=$B$29
$C$27<=$C$29
$D$27<=$D$29
$G$19=$G$21
$H$19=$H$21
$I$19=$I$21
$J$9<=$L$9
$J$10<=$L$10
$J$11<=$L$11
F
G
Status
Slack
Not Binding
500
Not Binding
800
Binding
0
Not Binding 5200
Binding
0
Binding
0
Binding
0
Binding
0
Binding
0
Binding
0
Binding
0
Binding
0
Not Binding 1500
G
H
Adjustable Cells
Cell
$G$9
$H$9
$I$9
$G$10
$H$10
$I$10
$G$11
$H$11
$I$11
$G$17
$H$17
$I$17
Name
Crude 1 Gas 1
Crude 1 Gas 2
Crude 1 Gas 3
Crude 2 Gas 1
Crude 2 Gas 2
Crude 2 Gas 3
Crude 3 Gas 1
Crude 3 Gas 2
Crude 3 Gas 3
Advertising Gas 1
Advertising Gas 2
Advertising Gas 3
Final
Value
2088.89
2111.11
800.00
777.78
4222.22
0.00
133.33
3166.67
200.00
$0
$750
$0
Reduced
Objective
Cost
Coefficient
0.00
21
0.00
11
0.00
1
0.00
31
0.00
21
0.00
11
0.00
41
0.00
31
0.00
21
($209)
-1
$0
-1
($409)
-1
Allowable
Increase
0
3.19744E-14
51.125
1.27898E-14
1E+30
0
0
2.13163E-14
204.5
209
1E+30
409
Allowable
Decrease
3.19744E-14
47.025
0
0
1.27898E-14
1E+30
2.13163E-14
30.9
0
1E+30
104.5
1E+30
Allowable
Increase
1E+30
800
800
5200
5
5
5
500
1E+30
500
200
400
1E+30
Allowable
Decrease
500
1E+30
2800
1E+30
2
31.66666667
5
400
7500
250
200
1400
1500
Constraints
Operations Management
Cell
$B$19
$B$22
$C$22
$D$22
$B$27
$C$27
$D$27
$G$19
$H$19
$I$19
$J$9
$J$10
$J$11
Name
Total produced
Actual total octane Gas 1
Actual total octane Gas 2
Actual total octane Gas 3
Actual total Sulfur Gas 1
Actual total Sulfur Gas 2
Actual total Sulfur Gas 3
Total demand Gas 1
Total demand Gas 2
Total demand Gas 3
Crude 1 Total purchased
Crude 2 Total purchased
Crude 3 Total purchased
Final
Value
13500
30800
76000
11200
30
190
10
3000
9500
1000
5000
5000
3500
Shadow
Price
0
0
-3.55271E-15
0
3090
3090
3090
20.8
-0.1
40.8
57.25
20.9
0
10
Constraint
R.H. Side
14000
0
0
0
0
0
0
0
0
0
5000
5000
5000
Prof. Juran
(a) What is the optimal amount of profit for Sunco?
Cell E8 of the Answer Report indicates that the optimal profit is $287,750.
(b) How much of Crude Oil 2 should Sunco purchase?
Cell D40 of the Answer Report indicates that Sunco should purchase 5,000 barrels of
Crude Oil 2.
(c) How much of the Crude Oil 2 will be used to make Gas 3?
Cell D18 of the Answer Report indicates that none of the Crude Oil 2 will be used to make
Gas 3.
(d) The marketing director insists that Sunco needs to spend $500 advertising Gas 1.
What will this do to the net profit?
The optimal decision is to spend $0.00 advertising Gas 1. Any other decision would be less
profitable (unless something important has been left out of the model).
(e) 500 barrels of Crude Oil 2 are available on the spot market. How much should
Sunco offer per barrel for this commodity?
According to the Sensitivity Report (cell E36), Sunco should offer no more than $20.90 per
barrel.
(f) Assuming that the seller agrees to Sunco’s price, how many barrels of Crude Oil 2
should they buy?
The allowable increase (G36) is 400 barrels. We might want more than 400 more barrels,
but we can’t tell without re-running Solver.
(g) 500 barrels of Crude Oil 3 are available on the spot market. How much should
Sunco offer per barrel for this commodity?
None. (See cell E37 of the Sensitivity Report.)
(h) Assuming that the seller agrees to Sunco’s price, how many barrels of Crude Oil 3
should they buy?
None. In fact, Sunco is not even buying all of the 5,000 barrels of Crude Oil 3 that were
originally available (see D41 in the Answer Report).
Operations Management
11
Prof. Juran