ASSIGNMENT 5(Based on Dual Simplex method, IP)

Q.1. The following optimization problem is :

Maximize z = 2x− 9y

subject to 5x+ 7y ≤ 27

4x+ y ≤ 14

3x− 2y ≤ 9

x, y ≥ 0, x is integer .

(a) Mixed-Integer programming problem (b) Non-linear programming problem

(c) Quadratic programming problem (d) None of the above

Q.2. Use dual simplex method to solve the L.P.P. :

Maximize z = −2x1 − 3x2 − x3

subject to 2x1 + x2 + 2x3 ≥ 3

3x1 + 2x2 + x3 ≥ 4

x1, x2, x3 ≥ 0.

(a) zmax = −11

4(b) zmax =

11

4(c) Unbounded solution

(d) No feasible solution

Q.3. Solve the following L.P.P. by dual simplex method :

Minimize z = 10x1 + 6x2 + 2x3

subject to − x1 + x2 + x3 ≥ 1

3x1 + x2 − x3 ≥ 2

x1, x2, x3 ≥ 0.

(a) zmin = −10 (b) zmin = 10 (c) Unbounded solution

(d) No feasible solution

1

Q.4. Solve, if possible, the linear programming problem by using dual simplex method.

Minimize z = x1 + 5x2

subject to 3x1 + 4x2 ≤ 6

x1 + 3x2 ≥ 3

x1, x2 ≥ 0.

(a) No feasible solution (b) Unbounded solution (c) zmin = −21

5

(d) zmin =21

5

Q.5. Using dual simplex method, solve the following problem.

Minimize z = 7x1 + 3x2

subject to x1 − 3x2 ≥ 1

x1 + x2 ≥ 2

−2x1 + 2x2 ≥ 1

x1, x2 ≥ 0.

(a) zmin = −13 (b) zmin = 32 (c) Unbounded solution

(d) No feasible solution

Q.6. Solve, if possible, by dual simplex method the following problem.

Minimize z = 6x1 + 11x2

subject to x1 + x2 ≥ 11

2x1 + 5x2 ≥ 48

x1, x2 ≥ 0.

(a) zmin = 96 (b) zmin = 10 (c) Unbounded solution

(d) No feasible solution

Q.7 Solve the following L.P.P. using dual simplex algorithm :

Minimize z = 6x1 + 7x2 + 3x3 + 5x4

2

subject to 5x1 + 6x2 − 3x3 + 4x4 ≥ 12

x2 + 5x3 − 6x4 ≥ 10

2x1 + 5x2 + x3 + x4 ≥ 8

x1, x2, x3, x4 ≥ 0.

(a) No feasible solution (b) Unbounded solution (c) zmin =258

11

(d) zmin =127

11

Q.8. Solve the L.P.P. :Maximize z = 2x1 + 2x2

subject to 5x1 + 3x2 ≤ 8

x1 + 2x2 ≤ 4

x1, x2 ≥ 0 and are integers .

(a) zmax = −4 (b) zmax = 4 (c) Unbounded solution

(d) No feasible solution

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ANSWERS

Q.1. (a)

Q.2. (a)

Q.3. (b)

Q.4. (d)

Q.5. (d)

Q.6. (a)

Q.7. (c)

Q.8. (b)

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