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72 CHAPTER 2 AN INTRODUCTION TO LINEAR PROGRAMMING

3 Show a separate graph of the constraint lines and feasible solutions for each of the
following constraints:
a. 3x 1 +2x 2 18
b. 12x 1 +8x 2 480
c. 5x 1 +10x 2 ¼ 200

4 Show a separate graph of the constraint lines and feasible solutions for each of the
following constraints:
a. x 1 0.25 (x 1 + x 2 )
b. x 2 0.10 (x 1 + x 2 )
c. x 1 0.50 (x 1 + x 2 )

5 Three objective functions for linear programming problems are 7x 1 +10x 2 ,6x 1 +4x 2
and 4x 1 +7x 2 . Determine the slope of each objective function. Show the graph of each
for objective function values equal to 420.
6 Identify the feasible region for the following set of constraints:
1 1
/ 2 x 1 þ / 4 x 2 30
1x 1 þ 5x 2 250
1 1
/ 4 x 1 þ / 2 x 2 50
x 1 ; x 2 0

7 For the linear programme:

Max 2x 1 þ 3x 2
s:t:
1x 1 þ 2x 2 6
5x 1 þ 3x 2 15
x 1 ; x 2 0
find the optimal solution using the graphical solution procedure. What is the value of the
objective function at the optimal solution?
8 Consider the following linear programme:
Min 3x 1 þ 4x 2
s:t:
1x 1 þ 3x 2 6
1x 1 þ 1x 2 4
x 1 , x 2 0

Identify the feasible region and find the optimal solution using the graphical solution
procedure. What is the value of the objective function?
9 Consider the following linear programme:

Max 1x 1 þ 2x 2
s:t:
1x 1 5
1x 2 4
2x 1 þ 2x 2 ¼ 12
x 1 ; x 2 0

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