-INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR

Date ofExamination: ...... -4-2011, FN/AN, Time: 3 Hours, Full Marks: 50 Number of Students: 70, End-Spring Semester 2011, Department of Mathematics,

Subject No: MA30014, Subject Name : Operations Research 3rd Yr. B. Tech/ M.Sc, Branch: All Department

Instructions: Answer any Ten Questions. Questions are of equal value If needed Graph Paper will be supplied in the Examination Hall

Ql. (a) State the basic difference between the Two- Phase Simplex Method and the Big-M method. Why do we use Two- Phase Simplex method and the Big- M method ?

(b) Solve the given LPP by Two- Phase Simplex Method:

Subject to Min: Z= 3x1 + 4x2 + 5x3

x1 + x2 - x3 ~50

x 2 + x3 ~50

2x1 + 2x2 ~ 105

xl>x2 ,x3 ~ 0 (2+3 marks)

Q2. (a) Prove that optimal value oflinear function Z = cr X, defined over a convex region AX= b,X ~ 0 if exists to be found at the extreme point(s) of the convex regiOn. (b) Solve the given LPP by Primal Simplex Method:

Min: Z= x1 - 3x2 + 2x3

Subject to 3x1 -x2 +2x3 ~ 7

- 4x1 + 3x2 + 8x3 ~ 10

-X1 +2X2 ~ 6

xpx2 ,x3 ~ 0 (2+3 marks)

Q3.(a) Prove that for any LPP, Dual of a Dual Program is the Primal Program. (b) Establish whether the feasible solution vector ( 0, 20, 0, 0 ) is an optimal

solution of the given LPP or not. For this feasible solution of the LPP and find the Dual variables.

Min: Z= - x1 - 2x2 - 3x3

Subject to x1 + 2x2 + x3 + x4 ~ 40

2x1 + 3x2 + 4x3 + x4 ~ 60

XpX2 ,x3 ,x4 ~ 0 (2+3 marks) Q4. (a) Present the Steps ofthe Dual Simplex Method.

(b) Solve the given LPP by Dual Simplex Method: Min: Z= 3 x1 + 2x2 + 3x3

Subject to 4x1 + 4x2 + 8x3 ~ 540

3x1 + 5x2 + 6x3 ~ 480,

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( 2+3 marks)

Q5.(a) Present the steps of the Branch and Bound method to solve an Integer Programming Problem.

(b) Solve the given Integer Programming Problem by Branch and Bound method: Max:Z=XI-X2

Subject to XI + 2X2 ~ 4

6XI +2X2 ~9 XI. X 2 = 0,1,2, .... (2+3 marks)

Q6.(a) Present the steps of the Gomory's Cutting Plane method to solve an Integer Programming Problem.

(b) Solve the given Integer Programming Problem by Gomory's Cutting Plane method:

Max : Z = XI + 4X 2

Subject to 2XI + 4X2 ~ 7

5XI +3X2 ~ 15

XpX2 = 0,1,2,.... (2+3 marks) Q7.(a) Establish the Dual of a balanced Transportation Problem if it has Minimization type objective function with m number.of sources and n number of destinations. (b) Find the Phase- I solution of the given transportation problem by Vogel's Approximation Method ( V AM):

To Destination Supply From

2 4 6 5 7 400 Source 7 6 4 2 4 600

8 7 4 2 5 600 2 7 6 4 8 400

Demand 400 400 200 500 500

(2+3 marks) Q8(a)

To Destination Supply From

22 26 34 28 2500 Source 32 36 28 20 3000

42 48 26 20 4000 Demand 2000 2250 2750 2500

Find the Phase- I solution of the given transportation problem by Least Cost Method . (b) Find the Phase- II solution of the above Transportation Problem by MODI

method. (2+3 marks)

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Q9.(a) Prove that if we add or subtract a constant K to a row or column of an Assign­ment Matrix of a given Assignment Problem, optimal assignment remains unchanged.

(b) Solve the given Assignment Problem by Hungarian Method.

M1 M2 M3 M4 Job 1 100 140 280 70 Job2 130 160 200 60 Job 3 80 130 300 90 Job4 150 110 250 50

(2+3 marks)

Q10.(a) Discuss the method ofDominance to solve Two Person Zero Sum unstable game. (b) Apply the method of Dominance to solve the following unstable game:

Player B 40 20 20 20

A 20 30 50 40 30 20 10 10 20 10 20 10

(2+3 marks)

Q1l.(a) Present the LP method for a m x n Two-Person Zero-Sum unstable game.

( b) Solve the matrix game by any suitable method:

Player B 20 -30 40

A -30 40 -50 40 -50 60

(2+3 marks)

End of Paper

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