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LPP 1

This document is a question paper for a linear programming and theory of games exam containing 5 questions with multiple parts each. It provides instructions for candidates to write their roll number and attempt any 2 parts of each question. The questions cover topics like solving linear programming problems using various methods, duality theory, transportation problems, assignment problems, and solving games.

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Shubham Agarwal
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423 views4 pages

LPP 1

This document is a question paper for a linear programming and theory of games exam containing 5 questions with multiple parts each. It provides instructions for candidates to write their roll number and attempt any 2 parts of each question. The questions cover topics like solving linear programming problems using various methods, duality theory, transportation problems, assignment problems, and solving games.

Uploaded by

Shubham Agarwal
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 4

[This question paper contains 4 printed pages.

Sr. No. of Question Paper 6620 D Your Roll No ............... .

Unique Paper Code 235505

N arne of the Course B.Sc. (Hons.) MATHEMATICS

N arne of the Paper Linear Programming and Theory of Games (Paper V.4)

Semester v
Duration : 3 Hours Maximum Marks : 7 5

Instructions for Candidates

1. Write your Roll No. on the top immediately on receipt of this question paper.
2. Attempt any two parts of each questions.
3. All questions carry equal marks.

1. (a) Prove that to every extreme point ofthe feasible region, there corresponds
a basic feasible solution of the linear programming problem

Minimize z = ex

subject to Ax = b, x ::::: 0.

(b) Solve the following linear programming problem by the simplex method starting
with the basic feasible solution (x"x) = ( 4,0)
Maximize -x 1 + 2x 2
subject to 3x 1 + 4x 2 = 12
2x 1 - x2 ::::; 12

(c) x 1 = 1, x 2 = 1, x 3 = 1 is a feasible solution of the system of equations


X
1
+ X
2
+ 2 X3 = 4
2x I -X
2
+X
3
= 2

Is this solution basic feasible ? If not, reduce it to a basic feasible solution.

PT.O.

-------r
6620 2

2. (a) Using two phase method, solve the linear programming problem

Minimize z = 3x 1 + 2x 2 + X
3

subject to

x 1 is unrestricted, x 2 2:: 0, x 3 2:: 0.

(b) Using simplex method, solve the system of equations


3x 1 + 2x2 = 4
4x I - x 2 = 6

Also, find inverse of the coefficient matrix

(c) Using big-M method, solve the linear programming problem

Minimize z = -x I - 3x 2 + x 3
subject to

3. (a) (i) State and prove the Weak Duality Theorem.

(ii) Verify that the dual of the dual is primal.

Maximize z = -8x I + 3x 2

subject to

l
6620 3

(b) Obtain the dual of the following primal problem :

Maximize

subject to

-2x I - 8x 3 = 2

x 1 ~ 0, x2 ~ 0, x 3 is unrestricted.

(c) Solve the following linear programming problem using duality:

Minimize

subject to

" 4. (a) Solve the following transportation problem:

o, 02 03 D4 Supply
o, IO 2 20 II 15
02 I2 7 9 20 25
o, 4 14 16 18 10
Demand 5 I5 15 I5

(b) Solve the following cost minimization assignment problem:

I II III IV v
A 11 6 14 16 17
B 7 13 22 7 10
c IO 7 2 2 2
0 4 10 8 6 11

E 13 I5 I6 10 18

P.T.O.
6620 4

(c) (i) Find the saddle point for the game having the following pay-off matrix :

[ -~ ~ ~l
1 0 -2J

(ii) Determine the range of values of p and q that will render (2,2) a
saddle point for the game

5. (a) Use dominance relation to reduce the following game to a 2 X 2 game, and
hence find the optimum strategies and value of the game

(b) Solve graphically the rectangular game whose pay-off matrix is :

2 2 3 -1]
[4 3 2 6

(c) Reduce the following game to a linear programming problem and then solve
by simplex method.

(2800)

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