RTU 2016 Question Paper Semester VI Electronics and Communication Engineering Optimization Techniques

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RollNo. Total No of Pages: tr
CN 6E3093
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B. Tech. VI-Sem. (Otd Back) Exam., ApriUMay'2016
Electronics & Communication
rll 68C6.3 (O) Optimization Techniques
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Time: 3 Hours Maximum Marks: 80
Min. Passing Marks (Old Back): 24
I nstructions to Candidate s : -

Attempt any five questions, selecting one question from euch unit'
All
must be shown
Questions carry equal m-arks. Schematic diagrams
wherever necessary. Any data you feel missing may suitably be assumed
and stated clearlY.
Units of quantities used/ calculated must be stated clearly.

Use of foltowing supporting material is permitted during examination'
( M entione d in form N o -20 5 )

2. NIL
1. NIL

UNIT-I
Q.l (a) write in brief on historical development of optimization and mention a
few

applications of oPtimi zation t8l

(b) A coal grassfire can use three grades of coal to produce quality K and M of
producer gas. There are two processes (i.e. old and new) available to use
the

blended coal. For each production run the o1d process uses 10, 14 and 4 units
of

coal A, B and c to produce 12 units of K and 10 units of M' The new
process

6, 18 and 8 units of coal A, B and C to produce 10 units of quality K and
14
uses
at least
units of M. Due to prior commitments, the gassifire plant must produce
1g00 and 1600 units of K and M respectively for the next month.
It has available

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2000, 2500 and 1500 units of coal A, B and C respectively. For each unit of K, a
revenue of '.3000 and for each unit of M, '4000 are received. Formulate this as

LP problem so as to maximize the revenue. (Do not solve) t8l

OR

Q.l (a) Write in detail on classification of optimization problems, giving examples. t8l

(b) A closed cylindrical tank is to be designed to carry at least 5 m3 of chemicals.
Metal for the top and side costs < 200 per m2 but heavier metal of the base costs
< 800 pe. m2. Also the height of the tank cannot be more than twice its diameter.

Formulate the problem as a mathematical programming problem to minimize the
total cost incurred. (Do not solve it). Identify the decision variables and classify
the problem formulated. t8l

UNIT.II
Q.2 (a) Solve the following LPP using simplex method. t8l

MaxZ = Xr * 2x2+ 3x3

S.T. x1+'2x2+ 3x3 < 10
x1*x2(J
X1 S 1, x1, X2) 0
Find the alternate optimal solution if it exists.

(b) Solve the following LPP using revised simplex method. t8l

Max.Z=xr*Zxz

S.T. x1* x2( 3
\*2x24J
3x1+ x236, X1, x2 ) 0

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OR

Q.2(a)SolvethedualofthefollowingLPPbysimplexmethod,andfromthesolutionof
t8l
dual find opt. solution of the
primal'

Max.Z= 4xr +3xz

S.T. xr ( 6, x2 S 8,

x1* x237 , 3x1+ x2315'

X'22'L, x1' X2 2 0'

t8l
(b) Solve the following LPP using Big-M method'
Minimize Z= -xt* x2t X3

S.T. -2x1+ x2t x3)-)

x1- 2x2t 2x^ = 2

Xl, X2rX3 2 0

c2 (The coefficient of x2 in objective
using sensitivlty analysis, find how much
remain optimal'
function) can vary for the solution to

UNIT.III
-

Q.3(a)FourvillagoSDl,Dz,D:,D+areaffectedduetofloodsandfoodgrainistobe
dropped by three air craft's St' Sz'
Sr' in these villages'

one day
ai = number of trips Si can make in
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bj = number of trips required to Dj in one day, and

the amount of food grain that Si can carry to Dj in one trip are given in adjoining

table.

Solve the TP problem to find the number of trips that Si can make to Dj so that

the total quantity of food grain dropped in a day is maximum (use vAM)
tsl

Dl D2 D3 Da lai
108 6 3

8 105 3

25910

o) Find the minimum cost of transportation for adjoining Tp where the
transportation cost per unit, requirement (bj) at destination Dj and availability
(ai) at source Si are given. Interpret the result obtained.
t8l

t2t4
3321
4259

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OR

Qi (e) A company has four territories 1,2,3 and 4to be assigned to four salesman A, B,
C and D to promote their product. Based upon their capability the company

estimates the profit earned in thousands per day on assigning a territory to each

salesman is given in adjoining mqtrix. Find the assignment using Hungarian

method to have maximum profit.
t81

A t6 10 t4 11
B t4 11 15 15
C 15 15 13 t2
D 13 t2 14 15

O) Using Hungarian method, solve the adjoining problem of assigning tasks A, B,
C, ? t9 Pe{son l:2,32-4 tg mi-nimiz-q the total man Lr-oms-takenro-ftn-Lqhthe
*->'---r\=(t-=:-=:i---i:')-=-}->r-*+'-==;;:;rrffiEa--a
:-- +--4+ +- t: F> ='z<*e*;)?
-+ ->+-- - -=:i#
-

UNIT.IV
Q.4 Minimize f(x1, x2) - 2x: + 2x1x2+ x22 * Xr- Xz
starting from Xs = (0, 0) using,

(a) the method of steepest descent (complete only two iterations of the method). t8l
(b) the univariate method (complete only one cycle of movements parallel
to axes). t8l
OR

Q.4 (a) Using Zoutenditk's method, find the minimum of t8l

f (x) =f (x1, x2) =xrz +x22 -2x1-3x2+3,

subject to x1 + 2x2- 4< 0 with starting point X6 = (0, 0).

(b) By using exterior penalty method - t8l

Minimize x12 + x22 = f (xr, xz)

S.T. 5-xr-x2S0,

X2-X1(Q,

and find the solutions coresponding to r = 1, 10, "" (r being the penalty
parameter).

UNIT.V
Q.5 (a) State Belman's principle of optimality and using it solve the adjoining TSP
problem to find the shortest route from city 1 to city 7. I8l

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(b)Usedynamicprogrammingtofindthreenon-negativerealnumberssuchthat
-
sumofsquaresoftheseisminimumwiththerestrictionthattheirsumisnotless
t8l
than 30.

OR

4 units of weight is to be loaded with three items' Details
Q.5 (a) A vessel with capacity
aregivenintheadjoiningtable..Usingdynamicprogramming,findhowrnany

unitsofeachitemshouldbeloadedwithoutexceedingtheweightconstraint4to
t8l
of the loaded items'
have the maximum value

Item wt/unit value/unit

1 120
2 90

3 70

the following LPP t8l
(b) Use dynamic programming to solve

Max Z= 6xr + 5xz

S.T. x1S 2

xz(6
6x1 + 2x2< 18

x1, X2 2 0.

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