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SPECIAL CASES 249

3 Consider the following linear programme:

Max 5x 1 þ 9x 2
s:t:
0:5x 1 þ 1x 2 8
1x 1 þ 1x 2 10
0:25x 1 þ 1:5x 2 6
x 1 ; x 2 0
a. Write the problem in standard form.
b. How many variables will be set equal to zero in a basic solution for this problem? Explain.
c. Find the basic solution that corresponds to s 1 and s 2 equal to zero.
d. Find the basic solution that corresponds to x 1 and s 3 equaltozero.
e. Are your solutions for parts (c) and (d) basic feasible solutions? Extreme-point solutions?
Explain.
f. Use the graphical approach to identify the solutions found in parts (c) and (d). Do the graphical
results agree with your answer to part (e)? Explain.
4 Consider the following linear programming problem:
Max 60x 1 þ 90x 2
s:t:
15x 1 þ 45x 2 90
5x 1 þ 5x 2 20
x 1 ; x 2 0
a. Write the problem in standard form.
b. Develop the portion of the simplex tableau involving the objective function coefficients, the
coefficients of the variables in the constraints and the constants for the right-hand sides.
5 A partially completed initial simplex tableau is given:

x 1 x 2 s 1 s 2
Basis c B 5 9 0 0
0 10 9 1 0 90
s 1
s 2 0 5 3 0 1 15
z j
c j – z j

a. Complete the initial tableau.
b. Which variable would be brought into solution at the first iteration?
c. Write the original linear programme.
6 The following partial initial simplex tableau is given:

x 1 x 2 x 3 s 1 s 2 s 3
Basis c B 5 20 25 0 0 0

2 1 0 1 0 0 40
0 2 1 0 1 0 30
3 0 0.5 0 0 1 15

z j
c j – z j

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