Download - Operations Research
-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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