Additional LP Problems:
1. A company wants to determine how many units of each of two products, A and B, they
should produce. The profit on product A is $50 and the profit on product B is
$45. Applying linear programming to this problem, which of the following is the
objective function if the firm wants to make as much money as possible?
A) Minimize Z = 50 A + 45 B
B) Maximize Z = 50 A + 45 B
C) Maximize Z = A + B
D) Minimize Z = A + B
E) Maximize Z = A/45B + B/50A
Answer: B
2. An agribusiness company mixes and sells chicken feed to farmers. The costs of the
chicken feed ingredients vary throughout the chicken feeding season but the
selling price of chicken feed is independent of the ingredients. On August 1,
management needs to know how many units of each of three grains (Q, R, and S)
should be included in their chicken feed in order to produce the product most
economically. The cost of each grain is, for a unit of Q, $30; for a unit of R, $37;
and for a unit of S, $78. Applying linear programming to this problem, which of
the following is the objective function?
A) Minimize Z = 30 Q + 37 R + 78 S
B) Maximize Z = 30 Q + 37 R + 78 S
C) Minimize Z = (Q x R x S)/3
D) Minimize Z = Q + R + S
E) Maximize Z = Q + R + S
Answer: A
3. Apply linear programming to this problem. A firm wants to determine how many units
of each of two products (products D and E) they should produce to make the most
money. The profit in the manufacture of a unit of product D is $100 and the profit
in the manufacture of a unit of product E is $87. The firm is limited by its total
available labor hours and total available machine hours. The total labor hours per
week are 4,000. Product D takes 5 hours per unit of labor and product E takes 7
hours per unit. The total machine hours are 5,000 per week. Product D takes 9
hours per unit of machine time and product E takes 3 hours per unit. Which of the
following is one of the constraints for this linear program?
A) 5 D + 7 E =< 5,000
B) 9 D + 3 E => 4,000
C) 5 D + 7 E = 4,000
D) 5 D + 9 E =< 5,000
E) 9 D + 3 E =< 5,000
Answer: E
4. Apply linear programming to this problem. A one-airplane airline wants to determine
the best mix of passengers to serve each day. The airplane seats 25 people and
flies 8 one-way segments per day. There are two types of passengers: first class
(F) and coach (C). The cost to serve each first class passenger is $15 per segment
and the cost to serve each coach passenger is $10 per segment. The marketing
objectives of the airplane owner are to carry at least 13 first class passengersegments and 67 coach passenger-segments each day. In addition, in order to
break even, they must at least carry a minimum of 110 total passenger segments
each day. Which of the following is one of the constraints for this linear
program?
A) 15 F + 10 C => 110
B) 1 F + 1 C => 80
C) 13 F + 67 C => 110
D) 1 F => 13
E) 13 F + 67 C =< (80/200)
Answer: D
5. A firm wants to determine how many units of each of two products (products D and E)
they should produce to make the most money. The profit in the manufacture of a
unit of product D is $100 and the profit in the manufacture of a unit of product E
is $87. Although the firm can readily sell any amount of either product, it is
limited by its total labor hours and total machine hours available. The total labor
hours per week are 4,000. Product D takes 5 hours of labor per unit and product
E takes 7 hours of labor per unit. The total machine hours are 5,000 per week.
Product D takes 9 hours of machine time per unit and product E takes 3 hours of
machine time per unit.
a. What is the LP model for this problem?
b. How many units of each product should be produced?
c. What is the total profit contribution?
d. Which constraints are active?
e. Is there any slack/surplus value for any constraint?
6. Suppose that an optimal corner point to a problem is (6, 10), and that the following
constraint is part of the problem formulation;
8X1 + 12X2 < 175
Which of the following statements about the amount of slack in this constraint at
the optimal solution is true?
a. Slack = 0
b. Slack < 3
c. Slack < 5
d. Slack < 7
Answer: d
7. What is the meaning of a slack or surplus variable?
The amount by which the left-hand side falls short of the right-hand side is the slack
variable. The amount by which the left-hand side exceeds the right-hand side is the
surplus variable.
8. Briefly describe the meaning of a shadow price.
The shadow price is the marginal improvement in Z caused by relaxing the
constraint by one unit.
9. Consider the following linear programming formulation:
Maximize:
20X1 + 100X2
Subject to:
X1 +
X2 < 80
6X1 + 10X2 > 600
4X1 +
> 80
X1, X2 > 0
a. Solve the problem using graphical solution procedure.
b. What is the objective function value?
c. Is there any slack/surplus value?
10. Consider the following linear programming formulation:
Minimize:
2X1 + 4X2
Subject to:
4X1 + 6X2 < 120
2X1 +6X2 > 72
X2 > 10
X1, X2 > 0
d. Solve the problem using graphical solution procedure.
e. What is the objective function value?
f. Is there any slack/surplus value?