Download DU (University of Delhi) B-Tech 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

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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.)

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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.

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' 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

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'(a) ?Smckyariablos
(b) Az?ci?al variablw
(C) 13,1181 Pricszs.
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What ? ' ' ?
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

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? 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

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' , ,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 :

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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 =

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4
8
3xl+x'4=3
0
x1, x2, x3, x4 _>_

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(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 ? ?

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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 '

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changeinthevalttcofz.? ' ? r ' .

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l 1. For the following LP, identify three allemative optimal basic solutions and then write a general

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expression for all the non?basic alternative optima :
Maximize Z = 2xl + 4x2 V
Subject to _:
x] + sz 5 S?
x] + x2 5 4

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x1, x2 > 0
12, Use Big-Mvmethod to solv-?e?: ?
Max1mxzeZ= 12.xl + 20x2
Subject. to z.
6x, + $22100? "

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7x, .+ '12:;2'3120- .
~ x], X2 Z 0.? ?1:
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16.

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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

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Subject to z
x] ? 3x2 + 4x3. = 5
? 2x2 5 3
2x2 ? x3 2 4,
x1, x2 2 o and x3 iSunrestx-ictpd. .'

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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

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2r1+5x2?lOO:
2x,+3$2590
x1: x2 2 0
4? ?_?r?- ?7?

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( 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

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2x] + 3x2 2 2
x1; x2 2 0.
Maximize 2% 5x1 + l?xz + 4x3
?ubject to 2
x4 '+?2x2 .+' x: s 10

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v
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5?1: x2: x3 ?
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1;

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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

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x, 1 ,0 7/5 1/5; ? 2/5 26/5
Write the associated dual problem and detem?ne its optimal solution.
I
1
x

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1,200