Download DU (University of Delhi) B-Tech 3rd Semester 1677 Introduction to Operational Research and Linear Programming Question Paper

Download DU (University of Delhi) B-Tech (Bachelor of Technology) 3rd Semester 1677 Introduction to Operational Research and Linear Programming Question Paper

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This question paper contains 8 printed pages]
Roll No.
S. No. onuestion Paper : 1677
Unique Paper Code : 2362301 F?3
Name of the Paper : Introduction?to OperationalResearch & Linear Programming
Namc'ofthc Course : B.Tech. (Computer Science) Allied Courses
Semester : [11
Duration : 3 Hours - ? . Maximum Marks : 75
(Write your 'Roll No_. on the top immediately on receipt of this question paper.)
All questions carry equal marks.
There are three section?s in the paper.
All sectigns are compulsory.
Attempt any ?i?e questions ?omeach Section.
Use of simple calculator -is ?alloiwed.
' Section A
1. ,?D?e?ne th? term 0'.R. ancilfwrite. its a?piica?qns in different m of real life. _
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2. , Explain anytw?q ofthe follbwing: ? _ _ .1
'(a) ?Smckyariablos
(b) Az??ci?al variablw
(C) 13,1181 Pricszs.
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do you mean by extreme point
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r 77 ( 3 ) 1677
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A Iher (a) Dctemtine all the basic solutions of the problem? and classify them as feasible and
? 3? infeasible.
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.- ; (b) Verify graphically that the solution obtained in (a) is the optimum LP solution. Hence,
A ? conclude that the optimum solution can be determined algebraically by considering the
? in - . basic feasible solutions only. ? .
-
Section B
7. Day Trader wants to inyeet a?sum of money that would ?generate an annual~yield of at least
$ 10000. Tvtro stocks groups are avilable :?blue chips and high tech, with average annual yields
' , ,4- of? 10% and 25%, respeetisfely. Though high-tech stocks tarovide higher yielct, they ate more
' risky, and Trader wants to timit the amount invested in these stocks to no more than 60%
of the total investment. Formulate the abeve problem as a LP_P to maximize an annual
'yield - I
s. ?Solve the given LPP graphiCally :
MaximizeZ=x1+2rQ
Subject to :
_xl"x251
xl'? XZ Z 0.
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9. Consider the following LP :
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Maximize Z = xI
Subject to :
5x] + x2 =
6x1 + x3 =
4
8
3xl+x'4=3
0
x1, x2, x3, x4 _>_
(a) Solve" the problem By inspection (do not use the Gauss-Jordan row operations), and
justify the answer in terms of the basic solutions of me Simplex method.
(b) Repeat (a) assuming that the objeCtive function calls for minimizing z 3- x1.
10. Considcrfthe ?tWO-dimensi?nal s'olution space in ?gure given below. ?
Suppose thatmerobjectiv? function is given as ? ?
Maximize. z = 6x1?+ 3x2. . ?
If the sim?lex iterati?dns s?tatt? at- point A,
identify the path to the optimum -'
. pointD. ' v . '
Determine me entering va'?able, the muesponding ratios of thc-feasibility condition, and the '
changeinthevalttcofz.? ' ? r ' .

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l 1. For the following LP, identify three allemative optimal basic solutions and then write a general
expression for all the non?basic alternative optima :
Maximize Z = 2xl + 4x2 V
Subject to _:
x] + sz 5 S?
x] + x2 5 4
x1, x2 > 0
12, Use Big-Mvmethod to solv-?e?: ?
Max1mxzeZ= 12.xl + 20x2
Subject. to z.
6x, + $22100? "
7x, .+ '12:;2'3120- .
~ x], X2 Z 0.? ?1:
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( 6 ) 1677
Section C
Comment on the future of arti?cial variables at the optimal table of phase 1.
Write the dual of the following LPP :
Minimize Z?= x] + x2 + x3
Subject to z
x] ? 3x2 + 4x3. = 5
? 2x2 5 3
2x2 ? x3 2 4,
x1, x2 2 o and x3 iSunrestx-ictpd. .'
What do you understand by feasibility and optimality ranges of the van'ables 1n LPP ? -
Consider the following LP model
Maxirhist=4xltlox2
_Subje'ctto: _ ' _ . .1 - 1:
Z"1"??V255Ogh? 7"?. . ? . , _ '. f
2r1+5x2?lOO:
2x,+3$2590
x1: x2 2 0
4? ???_?r?- ?7?

( 7 ) 1677
Check the optimality and feasibility of each 01: the following basic solution :
5/8 ?1/8 0
Basic variables = (x1, x2, x5), Inverse = ?1/4 1/4 0
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-l/2 -1/2 1
17. Use dual simplex method to solve the given LPP : _
MinimiZe Z = 3x1 + x2 7
Subject to :? ?
5:1 + x2 2? 1
2x] + 3x2 2 2
x1; x2 2 0.
Maximize 2% 5x1 + l?xz + 4x3
?ubject to 2
x4 '+?2x2 .+' x: s 10
v
.0
5?1: x2: x3 ?
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( 8 ) 1677
Using x4 and A as starting variables, the optimal tableau is given as :
Basic xl x2 x3 x4 A Solution
Z 0 0 . 3/5 . 29/5 ?2/5+M 274/5
x2 0 . 1 ~1/5 2/5 ' ?1/5 12/5
x, 1 ,0 7/5 1/5; ? 2/5 26/5
Write the associated dual problem and detem?ne its optimal solution.
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1
x
1,200