Download DU (University of Delhi) B-Tech 3rd Semester Introduction of Operational Research and Linear Programming Question Paper

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[This question paper contains 6 printed pages.]
Sr. No. of Question Paper : 1509 F-7 Your Roll No ............... .
Unique Paper Code : 2362301
Name of the Paper : Introduction of Operational Research and Linear

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Programming
Name of the Course : B.Tech Computer S?cience (Erstwhile FYUP) Allied
.53 Coufse "
4. . Semester " : HI
Diiiration : 3 Hours Maximum Marks : 75

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Instructions for Candidates
1. Write your Roll No. on the top immediateiy on the receipt of this question paper.
2. Answer ?fteen questions in all.
3. All questions carry eq.ual marks.
4. Simple calculators are allowed.

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1. Explain the importarice of operational research in decision making. (5)
2. Find all possible basic solutions to the following set of linear equations.
Q31
2xl+3x2+x3=6
xl + 3x2+ 5x3 = 5

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Also check whether any of the above solutions is degenerate or not. (5)
3. De?ne the basis and dimension of a vector space. Test whether the set of
vectors a1 = [1,1,0], a2= [3, 0,1], a3 = [5,2,1] form a basis of R3 ?
(5)
P.ITO.

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1509 2
4. De?ne a convex set. Examine the convexity of the following set :
S = {(x1, x2): x1 + x2 S l, Vxl, x2 6 R} (5)
5. A ?rm produces three products A, B and C. It uses two type of raw material I

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and II of which 5,000 and 7,500 units available. The raw material requirements
, per unit of the products are given below :
Rew Material Requirement per unit of product ?
e M
B C

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I 4 5
II 3 5
The labqur time for each unit of prhduct A is twice that of product B and three
times that of product C. The entire labour force of the ?rm can produce the
equivalent of 3,000 units. The minimum demand of the-three produots is 600, 650

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and 500 units respectively. Formulate the problem as a linear programming problem
(LPP) that will maximize the pro?t. (5)
6. Consider the following LPP :
Maximize Z = 5xl + 3x2
Subject to: 3xl + 5x2 <= 15 - I 5.

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5x1 + 2x2 <= 10 ?
x1, x2 >= 0
(a) Determine all basic solutions, of the problem and classify them as ?feasible
and infeasible. ? (3)
(b) Show how the infeasible basic solutions are represented on the graphical

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solution space. (2)

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7. Use graphical method to solve the following LPP :
Minimise Z = 3x1 + 4x2

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Subject to 3xl + 4x2 >= 240
2x1 + x2 >= 100
5x1 +.3x2 >= 120
x1, xz>= 0 , (5)
r

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I
I
8. Use Big M method to solve the following LPP
Maximize Z = 10xl + 20x2
Subject to 2xl + 4x2>= 16

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x1 + 5x2 >= 15
x1, x2 >= 0 (5)
9. How do you identify following in the optimal simplex table ?
(a) Alternate solution ? (2)
(b) Unbounded solution (2)

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(c) Infeasible solution (1)
'10. Solve the following LPP by; dual simplex mgthod
Minimize; Z = 3111 ? 232 + x3 I
Subject to 3x1 + x2+ x3 >= 3
?3x1 + 3X2 + 'x3 >= 6

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x1 + x2 + x3 <= 6
x1, x2, x3, x4 >= 0 (5)
13.110.

1509 4

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11. Consider the following LPP
Maximize Z = 3xI + 4x2 + x3+ 7x4
Subject to 8xl + 3x2+ 4x3 + x4 <= 7
2x1 + 6x2 + x3+ 5x4 <='3
x1+ 4x2 + 5x3;+ 2x4 <= 8

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x1, x2, x3, x4 >= 0
1'
Its associated optimal simplex table is given as :
Basic x1 x2 x3 x4 x5 x6 x7 Solution
Z 0 169/38 1/2 0 1/38 53/38 0 83/19

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x1 1 9/38 ?/2 0 5/38 5/38 0 16/19
x4 0 21/19 0 1 ?1/19 ?1/19 0 5/19
x7 . 0 159/38 9/2 0 ?1/38 ?1/38 1 126/19
Obtain the variations in cost coefficients Which are permitted without changing
the optimal solutions. - (5)

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12. Obtain the dual for the following primal problem : 1
?Minimize Z = 5x] + 6x2+ x3
Subject to x] + 2x2 + x3 = 15
?x1 + 5x2 <= 18
4x1 + 7x2 <= 20

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x1, x2 >= 0, x3 unrestricted (5)

1509 5
13. Consider the following LPP :
Maximize Z = 4x? + 14x2

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Subject to 2xl + 7x2 + x3 = 21
7x1+ 2x2+ x4= 21
x1, x2, x3,~x4 >= 0
? Check'rthe optimality and feasibility of the following basic solution :
17 0

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Basic variables = (x2, x4), Inverse of the basis matrix = (497 1] (5)
14. Find any three alternate optimal solution (if they exist) for the following
LPP: '
Maximize Z = 2xl + 4x2
Subject to x1 + 2x2 <= 5

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x1 + x2 <= 4
x >= 0 (5)
1,, 15. Consider the LPP :
' Maximize Z = le + 4x2 + 4x3 ? 31K4
Subject to x1 + x2+ x3 = 4

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x1 + 4x2 + x?= 8
xl, x2, x3, x4 >= 0
By using x3 and x4 as the starting variables, the optimal table is given
by
P. T.O.

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1509 6
Basic , x1 x2 x3 x4 Solution
Z 2 0 O 3 1 6
x3 3/4 0 1 ?l/4 2

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x2 1/4 1 0 1/4 "2
Write the associated dga] problem, and determine its optimal solution in tWo
ways.
,
I

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I
16. Solve the following LPP by Two Phase method :
Maximize Z = 3xl + 2x2
? ? Subject to 2xl + x2 <= 2
x1, x2 >= 0

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3xl + 4x2 =>=12
(5)
(5)
(100)