Ques 1.

Which of the following statements are true and which are false? Give reasons for your answer.

 a) In an LP model, the feasible solution space can be effected when redundant constraints are deleted.

b) If the primal LPP has an optimal solution, then the set of feasible solution to its dual is bounded.

c) In a simplex iteration, an artificial variable can be dropped all together from the simplex table once the variable becomes non basic.

d) The addition of a constant to all the elements of a payoff matrix in a two – person zero sum game can affect only the value of the game, not the optimal mix of the strategies.

e) There may be a balanced transportation problem without any feasible solution.

Ques 2.

a) A toy company manufactures two types of doll; a basic version-doll A and a deluxe version-doll B. Each doll of type B takes twice as long as to produce as one of type A, and the company would have time to make a maximum 2000 per day if it produce only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. The company makes profit of ₹3 and 5 per doll respectively on doll A and B. How many of each should be produced per day in order to maximize profit? Solve this problem by graphical method.a) A toy company manufactures two types of doll; a basic version-doll A and a deluxe version-doll B. Each doll of type B takes twice as long as to produce as one of type A, and the company would have time to make a maximum 2000 per day if it produce only the basic version. The supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). The deluxe version requires a fancy dress of which there are only 600 per day available. The company makes profit of 3 and 5 per doll respectively on doll A and B. How many of each should be produced per day in order to maximize profit? Solve this problem by graphical method.

b) Find all the basic solutions of the following system: 

Ques 3.

a) A marketing manager has 5 salespersons and 5 sales districts. Considering the capabilities of the salespersons and the nature of the districts, the marketing manager estimates the sales per month (in thousand  ) for each salesperson in each distinct as follows 

Salespersons

Find the assignment of sales persons to districts that will result in maximum sales.

Ques 4.

Find the maximum and minimax values of the following matrix game. 

Does the matrix have a saddle point. Justify your answer.

Ques 5.

The following table is obtained in the intermediate stage while solving an LPP by the simplex method.  

Discuss whether an optimal solution will exist or not.

Ques 6.

Solve by simplex method the following linear programming problem:   

Ques 7.

Show that the set of vectors 

from a basis for 

Ques 8.

Use the principle of dominance to reduce the size of the following game. Hence solve the game.

Ques 9.

i) Formulate the dual of the following problem:

ii) Check whether  is a feasible solution to the primal and  is a feasible solution to the dual.

iii) Use duality to check whether  is an optimal solution to the primal.

Ques 10.

Solve, graphically, the game whose pay-off matrix is: 

Ques 11.

Find an initial basic feasible solution for the following transportation problem using matrix-minima method. Also find the transportation cost. 

Ques 12.

Using the initial basic feasible solution for the transportation problem given below, find and optimal solution for the problem. 

Ques 13.

Test the following set for convexity. 

Ques 14.

Using the principle of dominance, solve the game whose pay-off matrix is given below:

Ques 15.

Without sketching the region, check whether  is in the convex hull of the points  and . If it is in the region, write P as convex combination of A,B and C.

Ques 16.

A businessman has to get 5 cabinets, 12 desks and 18 shelves cleaned. He has two part-time employees, Anjali and Arnav. Anjali can clean 1 cabinet, 3 desks and 3 shelves in a day, while Arnav can clean 1 cabinet, 2 desks and 3 shelves in a day. Arnav is paid 22 per day and Anjali is paid  25 per day. Formulate the problem of finding the number of days for which Anjali and Arnav have to be employed to get the cleaning done with minimum cost as a linear programming problem.

Ques 17.

For the following pay-off matrix, transform the zero-sum game into an equivalent linear programming problem:

Ques 18.

Solve the following LP problem by using two-phase simplex method:

Ques 19.

The initial basic feasible solution of a transportation problem is given below:

Check whether the given solution is optimal. If it is not, then find the optimal solution.