Solutions to LP Practice Problems[footnoteRef:1] [1: 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. 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 Furncos 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:
X1
X2
z = 125
0
5
3.125
0
z = 150
0
6
3.75
0
z = 175
0
7
4.375
0
z = 200
0
8
5
0
Your isoprofit lines ought to look like this:
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.
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.
Wheat
Corn
Labor
3 workers
2 workers
Fertilizer
2 tons
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
z = 6,000
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
Your isoprofit lines ought to look like this:
(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 non-negativity 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.
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 banks portfolio is
not too risky, the banks investment manager has placed the
following three restrictions on the banks 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) 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 constraints 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).
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.
Selling Price per Barrel
Purchase Price per Barrel
Gas 1
$70
Crude 1
$45
Gas 2
$60
Crude 2
$35
Gas 3
$50
Crude 3
$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.
Octane Rating
Sulfur Content
Crude 1
12
0.5%
Crude 2
6
2.0%
Crude 3
8
3.0%
Table 4.2
It costs $4 to transform 1 barrel of oil into 1 barrel of
gasoline, and Suncos refinery can produce up to 14,000 barrels of
gasoline daily. Suncos 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.
(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 Suncos 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 cant 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 Suncos 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).
1
11
Operations ManagementProf. Juran
1
2
3
4
5
6
7
8
9
10
11
A
B
C
D
E
F
G
H
I
J
Bonds
Home Loans
Auto Loans
Personal Loans
25.0%
25.0%
25.0%
25.0%
14.75%
Investment Returns
10.0%
16.0%
13.0%
20.0%
Bonds
Home Loans
Auto Loans
Personal Loans
Total Funds = 100%
25.0%
25.0%
25.0%
25.0%
100.0%
