BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI - K.K. BIRLAG O A C A M P U S

FIRST SEMESTER 2011-2012

OptimizationTutorial–5

1. Consider the following LPP:Maximize z = 5x1 + 2x2

subject to4x1 + 2x2 ≤ 16, 3x1 + x2 ≤ 9, 3x1 − x2 ≤ 9, x1 ≥ 0 and x2 ≥ 0.

Show that the associated simplex iterations are temporarily degenerate and then, write the optimal solu-tion of the problem. Also, verify the result by solving the problem graphically.

2. Consider the following LPP:

Maximize z =34

x1 − 20x2 +12

x3 − 6x4

subject to

14

x1 − 8x2 − x3 + 9x4 ≤ 0

12

x1 − 12x2 −12

x3 + 3x4 ≤ 0

x3 ≤ 1

x1, x2, x3, x4 ≥ 0.

Use TORA to show that the simplex routine will cycle indefinitely.

3. Use Two-Phase method to solve the following LPP

Minimize z = 4x1 + 3x3

subject to

3x1 + 6x2 + 3x3 − 4x4 = 12,

2x1 + x3 = 4,

3x1 − 6x2 + 4x4 = 0,

x1, x2, x3, x4 ≥ 0.

Answer the following questions: (i) is the feasible region unbounded? (ii) is the optimal solution unique?(iii) If it is not unique, find another optimal solution.

4. For the following LPP, show that the optimal solution is degenerate and that there exist alternative solu-tions that are all non-basic.

Maximize z = 2x1 − x2

subject to

x1 − x2 ≤ 10,

2x1 − x2 ≤ 40,

x1, x2 ≥ 0.

5. Consider the following LPP model

Maximize z = 3x1 + 2x2 + 3x3

subject to

2x1 + x2 + x3 ≤ 2,

3x1 + 4x2 + 2x3 ≥ 8,

x1, x2, x3 ≥ 0.

Show by the two-phase method that the optimal solution includes an artificial basic variable. Use alternaterule to solve the problem.

6. For the following problem, show that in an unbounded feasible region, bounded optimal solutions exit.

Maximize z = 6x1 − 2x2

subject to2x1 − x2 ≤ 2 x1 ≤ 4 x1 ≥ 0 and x2 ≥ 0.

7. For the following problems find the optimal solution if exists.

(a)Maximize z = 20x1 + 30x2

subject to2x1 + x2 ≤ 40, 4x1 − x2 ≤ 20, x1 ≥ 30, x1 ≥ 0 and x2 ≥ 0.

(b)Maximize z = 10x1 + 20x2

subject to2x1 + 4x2 ≥ 16, x1 + 5x2 ≥ 15, x1 ≥ 0 and x2 ≥ 0.

8. Consider the following system of equations:1

2

3

x1 +

0

2

1

x2 +

1

4

2

x3 +

2

0

0

x4 =

3

4

2

.Determine if any of the following combinations forms a basis.

(a) (A1, A2, A3),

(b) (A1, A2, A4),

(c) (A2, A3, A4),

(d) (A1, A2, A3, A4).

Note: Students are supposed to revise Section number 7.1 and 7.1.1 from the text book themselves.

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