ASSIGNMENT 1(Based on BFS, Formulation of L.P.P. and Graphical Method)

Q.1. Two linearly independent equations with four variables are given below:

4x1 + 2x2 + 3x3 − 8x4 = 6

3x1 + 5x2 + 4x3 − 6x4 = 8.

How many basic solutions are there?

(a) 2 (b) 3 (c) 4 (d) 5

Q.2.

2x1 + 3x2 − x3 + 4x4 = 8

x1 − 2x2 + 6x3 − 7x4 = −3

x1, x2, x3, x4 ≥ 0.

Write down a basic feasible and a basic non-feasible solution of the above set of equations.

(a) (0, 0, 2, 1) and(0,−14

13, 0,

45

13

)(b) (1, 2, 0, 0) and

(−14

13, 0,

45

13, 0

)(c) (0, 0, 2, 1) and

(0,

45

13, 0,−14

13

)(d) (1, 2, 0, 0) and

(45

13, 0,−14

13, 0

)

Q.3. The two linearly independent equations with three variables are given below:

2x1 − 3x2 + 5x3 = 10

4x1 + x2 + 10x3 = 20.

Find, if possible, a basic solution with x2, a non-basic variable.

(a)[5

2, 0, 1

](b) [−5, 0, 4] (c)

[7

4, 0,

13

10

](d) Such a solution does not exist

Q.4. A factory is engaged in manufacturing three products A, B and C which involve lathe work, grind-ing and assembling. The cutting, grinding and assembling times required for one unit of A are 2, 1 and 1hours respectively. Similarly they are 3, 1 and 3 hours for one unit of B and 1, 3 and 1 hours for one unitof C. the profits on A, B and C are Rs. 2, Rs. 2 and Rs. 4 per unit respectively. Assuming that there areavailable 300 hours of lathe time, 300 hours of grinder time and 240 hours of assembly time, which of thefollowing represents the set of constraints for the equivalent L.P.P. formulated from the above data?

(a) 2x1 + x2 + x3 ≤ 300, 3x1 + x2 + 3x3 ≤ 300, x1 + 3x2 + x3 ≤ 240, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0(b) 2x1 + 3x2 + x3 ≤ 300, x1 + x2 + 3x3 ≤ 300, x1 + 3x2 + x3 ≤ 240, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0(c) 2x1 + x2 + x3 ≥ 300, 3x1 + x2 + 3x3 ≥ 300, x1 + 3x2 + x3 ≥ 240, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

1

(d) 2x1 + x2 + x3 ≥ 300, 3x1 + x2 + 3x3 ≥ 300, x1 + 3x2 + x3 ≥ 240, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

Q.5. Suppose in a hospital, it is decided that each patient should be given at least N1, N2, N3, N4 quantitiesof 4 nutrients daily. There are 5 such foods available which contain these nutrients in different proportions.Suppose aij (i=1,2,3,4; j=1,2,3,4,5) is the quantity of the ith nutrient in one unit of the jth food. The priceper unit of the jth food is cj . The problem is how to find a programme for the best diet, that is, the combi-nation of foods that can be supplied at minimum cost satisfying the daily requirements of the nutrients bythe patients.

Write down the mathematical formulation of the above mentioned problem as L.P.P.

(a) Min z =∑4

j=1 cjxj s.t.∑4

j=1 aijxj ≥ Ni ( for i=1, 2, 3, 4, 5), xj ≥ 0

(b) Min z =∑5

j=1 cjxj s.t.∑5

j=1 aijxj ≤ Ni ( for i=1, 2, 3, 4), xj ≥ 0

(c) Min z =∑5

j=1 cjxj s.t.∑5

j=1 aijxj ≥ Ni ( for i=1, 2, 3, 4), xj ≥ 0

(d) Min z =∑4

j=1 cjxj s.t.∑4

j=1 aijxj ≤ Ni ( for i=1, 2, 3, 4, 5), xj ≥ 0

Q.6. Solve the following L.P.P.Max z = 5x1 − 2x2

subject to 5x1 + 6x2 ≥ 30

9x1 − 2x2 = 72

x2 ≤ 9

x1, x2 ≥ 0

and choose the correct option denoting the solution to the problem.

(a) Unbounded solution (b) x1 = 8, x2 = 0 (c) x1 = 10, x2 = 9 (d) No feasible solution

Q.7. The optimal value of the objective function for the following L.P.P.

Max z = 4x1 + 3x2

subject to x1 + x2 ≤ 50

x1 + 2x2 ≤ 80

2x1 + x2 ≥ 20

x1, x2 ≥ 0

is

(a) 200 (b) 330 (c) 420 (d) 500

2

Q.8. How many basic solutions are there in the following linearly independent set of equations? Findall of them.

2x1 − x2 + 3x3 + x4 = 6

4x1 − 2x2 − x3 + 2x4 = 10.

(a) 2 basic solutions (b) 3 basic solutions (c) 4 basic solutions(d) 5 basic solutions

Q.9. Solve graphically :Minimize z = x1 + x2

subject to 5x1 + 9x2 ≤ 45

x1 + x2 ≥ 2

x2 ≤ 4

x1, x2 ≥ 0.

(a) zmin = 2 (b) zmin = 4 (c) zmin = 9 (d) zmin =29

5

Q.10. The manager of an oil refinery must decide on the optimal mix of two possible blending processes ofwhich the inputs and outputs per production run are as follows:

Process Input OutputCrude A Crude B Gasoline X Gasoline Y

1 5 3 5 8

2 4 5 4 4

The maximum amounts available of crudes A and B are 200 units and 150 units respectively. Market re-quirements show that at least 100 units of gasoline X and 80 units of gasoline Y must be produced. Theprofits per production run from process 1 and process 2 are Rs. 300 and Rs. 400 respectively. Formulate anL.P.P. for maximization of the profit and solve it graphically.

(a) Optimal solution is(400

13,150

13

)(b) Optimal solution is (40, 0)

(c) Unbounded solution (d) No feasible solution

Q.11.Maximize z = 3x1 + 4x2

3

subject to x1 − x2 ≤ −1

−x1 + x2 ≤ 0

x1, x2 ≥ 0.

(a) No feasible solution (b) Unbounded solution

Q.12. An agricultural farm has 180 tons of Nitrogen fertilizers, 250 tons of Phosphate and 220 tons ofPotash. It is able to sell 3:3:4 mixtures of these substances at a profit of Rs. 15 per ton and 2:4:2 mixture ata profit of Rs. 12 per ton respectively. Pose a linear programming problem to show how many tons of thesetwo mixtures should be prepared to obtain the maximum profit. (Let x1 and x2 tons of these two mixturesbe produced)

(a) Max z = 15x1 + 12x2 s.t. 3x1 + 2x2 ≤ 180, 3x1 + 4x2 ≤ 250, 4x1 + 2x2 ≤ 220, x1, x2 ≥ 0(b) Max z = 15x1 + 12x2 s.t. 5x1 + 6x2 ≤ 3600, 5x1 + 3x2 ≤ 2500, 5x1 + 8x2 ≤ 4400, x1, x2 ≥ 0(c) Max z = 15x1 + 12x2 s.t. 6x1 + 5x2 ≤ 3600, 3x1 + 5x2 ≤ 2500, 8x1 + 5x2 ≤ 4400, x1, x2 ≥ 0(d) Max z = 15x1 + 12x2 s.t. 2x1 + 3x2 ≤ 180, 4x1 + 3x2 ≤ 250, 2x1 + 4x2 ≤ 220, x1, x2 ≥ 0

Q.13. Solve graphically the L.P.P.Minimize z = −2x1 + x2

subject to x1 + x2 ≥ 6

3x1 + 2x2 ≥ 16

x2 ≤ 9

x1, x2 ≥ 0.

(a) (x1, x2) = (6, 0) (b) (x1, x2) = (0, 10) (c) No feasible solution(d) Unbounded solution

4

ANSWERS

Q.1. (d)

Q.2. (d)

Q.3. (d)

Q.4. (b)

Q.5. (c)

Q.6. (b)

Q.7. (a)

Q.8. (b)

Q.9. (a)

Q.10. (a)

Q.11. (a)

Q.12. (c)

Q.13. (d)

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