o+t19os+.

Code: ME 11714

NATIONAL INSTITUTE OF TECHNOLOGY, RAIPUR (C.G)

DEP,ARTMENT OF MECHANICAL ENGINEERING

B.Tech. (?th Sernester) Examination, Nov * Dec 20LL

Sub: C)peration Research

Time:3 hrs Maximum Marks: 80

Note: Answer ANY FIVE questions. All questions carty equal marks. Normal distribution tablesare allowed.

Q1. A local travel agent is planning a chafter trip to a major sea resofi. The eight daylseven

night package includes the fare for round-trip travel, sruface transportation, board and lodging

and selected tour options. The charter trip is restricted to 200 persons and past experience

indicates that there will not be any problem for getting 200 persons. The problem for the travel

offer for

this charter" These three plans differ according to seating and service for the flight, quality of

accornmodations, meal plans and tour options. The following table summarizes the estirnated

prices for the three packages and the corresponding expenses for the travel agent. The travel

agent has hired an aircraft for the flat fee of Rs. 2, 00,000 for the entire trip.

Tour Plan Prices and costs for tour packages per person

Prices (Rs.) Hotel costs (Rs.) Meals & other expenses (Rs.)

Deluxe 10,000 3,000 4,754

Standard 7,0Cr0 2,200 2,500

Economy 6,500 1,900 2,240

In planning the trip, the following considerations must be taken into account:

(i) At least 10 % of the packages must be deluxe type.

(ii) At least 35 % but not more than 70 %o must be of the standard type.

(ii| At least 30 o/o must be of the economy type.

(iv) The maximum number of deluxe packages available in any airuaft is restricted to 60.

(v) The hotel desires that at least 120 of the tourists should be on the deluxe and standard

packages together.

The travel agent wishes to determine the number of packages to olfer in each type so as to

agent is to determine the number of Delu

Page lof4

maximize the total profit.

(a) Formulate the above as a linear programming problem.

(b) Restate the above linear programming problem in terms of two decision variable taking

advantage of the fact that 200 packages will be sold.

(c) Find the optimum soiution using graphical method for the restated linear programming

problem and interpret results.

Q2. Solve the following L.P. problem by simplex method:

Minimize Z = xr- 3x2 + lxr,

.subject to 3xt - xz + 2xt =

7 ,

2xt+ 4xz > -72,

- 4x1 + 3x2 + 8x3 < 10,

Xto X2, x: ) 0.

g{ ni"a the optimum solution to the following transportation problem in which the cells

contain the transportation cost in rupees'

Q4. Solve the following assignment problem:

Q5. Arrival rate of telephone calls at a telephone booth are according to Poisson dislribtrtion,

with an average time of 9 rninutes between two consecutive arrivals. The length of

telephone call is assumed to be exponentially distributed, with mean 3 minutes.

(a) Detennine the probability that aperson arriving at the booth will have to wait.

(b) Find the average queue length that is formed from time to time.

Wr Wz Wt LYd Ws AvailableF1 7 6 4 5 9 40

Fz 8 5 6 7 8 .30Ft 6 I 9 6 5 2A

Fq 5 7 7 8 6 l0Required 30 30 15 20 5 100 (Toral)

I il UI IV f/v

111 l7 8 t6 20

", 9.| t2 6 15

3 13 16 15 12 16

4 2l 24 l7 28 265 t4 10 t2 11 13

Page 2of 4

6Q*csvrtJAloY-DGC- >6tI

(c) The telephone company will install a second booth when convinced that an arrival would

expect to have to wait at least four minutes for the phone. Find the increase in flou'rate

of arrivals which will justify a second booth.

(d) What is the probability that an arrival will have to wait for more than 10 minutes before

the phone is free?

(e) What is the probability that he will have to wait for more than 10 minutes before the

phone is available and the call is also complete?

(fl Find the fraction of a day that the phone will be in use.

gffi civil engineering firm has to bid or the construction of a dam. The activities and their time

estimates are given belor.v:

The policy of the firm with respect to submitting bids is to bid the minimum amount that

will provide a95sA of probability of at best breaking-even. The fixed costs for the proiect

are eight lakhs and the variable costs are 9,000 every day spent working on the project. The

duration is in days and the costs are in rupees. What amount should the firm bid under this

policy? (Calculations on time should be performed upto two decimal placesJ.

QZ-Solve the following garne whose payoff matrix is given below:

Page 3of4

Activity t-2 2-3 2-4 2-8 -AJ-+ 3-5 4-6 5-7 5-9 6-7 6-8 7-9 8-9

Optimistictirne

l4 T4 13 16 0 15taLJ 0 14 0 0 t6 14

Mosttime

likely t7 18 i5 t9 0 18 T7 0 18 o 0 20 16

Pessimistic

time

25 21 18 28 0 27 21 0 20 0 0 4I 22

Player

APlayer B

B1 82 B3 B4 B5 B6

A1 4 2 0 2 i 1

A2 4 J I aJ 2 2

AJ 4 3 -5 I 2

A4 4aJ 4 I 2 2

A5 4 aJ J

.)2 2

Production/dayProbability

Q8, A company manufactures around 200 mopeds. Depending

materials and other conditions, the daily production has been

204 mopeds, whose probability distribution is as given below:

upon the availability of raw

varying from 196 mopeds to

The finished mopeds are transported in a specially designed three-storeyed logy that canaccommodate only 200 rnopeds. Using the following 15 random numbers g2, gg, :lg, 24, 53 ,

6l, 1 8, 4,5,,,04, 23, 50, 77, 27, 54 and, I0, simulate the process to find out:(j) what will be the average number of moppds rvaiting in the faclory?(ii) What will be the number of empty spaces in the lorry?

Page 4of 4

i

31 9 t 1" (,: Lt9

Code: ME 10714

NATIONAL TNSTITUTn OF TECHNOLOGY, RAIPUR (C.G)

DEPARTMENT OF MECHANICAL ENGINEEzuNG

B.Tech. (7th semester) Examination Nov - Dec 2010

Subject: Operations Research

Time: 3 hrs. Maximum marks: 80

Note: Answer ANY FIVE questions. All questions carry equal marks. Normal distributiantables are allowed.

Qt. A manufacturer produces two different models * X and Y * of the same product.

Model X makes a contribution of Rs 50 per unit and model Y, Rs 30 per unit, towardstotal profit. Raw materials rl and ftalf,E required for production. At least 18 kg of 11

and 12 kg or 12 must be used daily. Also at most 34 hours of labour are to be utilized.A quantity of 2 kg of rr is needed for model X and 1 kg of rr for model Y. For each ofX and Y, 1 kg of 12 is required. it takes 3 hours to mariufacture model X and 2 hours

to manufactue model Y. Formulate the LPP and determine how many units of each

model should be produced. (Use graphical method)

Q2. Solve the following LPP using Big-M method.

Minimize Z:3P + 8C

Subjectedto P+C:200

P<80,C>60

And PZ0, C>0

Find the optimum solution to the following transportation problern in which the cellscontain the transportation cost in rupees.

Q3.

W1 W2 W3 W4 W5 AvailableFI 7 6 4 5 9 40F2 I 5 6 7 I 30F3 6 8 9 6 5 2A

F4 5 1 7 8 6 l0Reauired 30 30 15 20 5

F"p tdb 3

P o R .s TA 10 16 t2 18 IB 24 22 20 25 26C 30 28 JL JZ 34D t5 t2 10 T4 l6

Q4,

Q5.,+

Four wagons are availablestations P, Q, R, S and T.table below. How shouldmileage covered?

.;+"N w-f

In a large maintenance department, fitters draw parts from the parts-stores which is atpresent staffed by one storeman. The maintenance foreman is concerned about thetime spent by fitters getting parts and wants to know if the employment of a stores

labourer would be worthwhile. On investigation it is found that

a) A single queue situation exists

b) Fitters cost Rs. 2.50 per hour,

c) The storeman costs Rs. 2 per hour and can deal, on average, with l0 fitters perhour,

d) A labour could be employed at Rs. 1.75 per hour and would increase the servicecapacity of the stores to 12 per hour,

e) On average 8 fitters visit the stores each hour.

A civil engineering frm has to bid for the construction of a dam. The activity detailsare given below.

The policy of the firm with respect to submitting bids is to bid the minimum amountthat will provide a95o/o of probability of at best breaking-even. The fixed costs for theproject are 8 lakhs and the variable costs are 9,000 every day. Duration is in days and

the costs are in rupees. What amount should the firm bid under this policy?(Calculations on tirne should be performed up to tow decimal places).

at stations A, B, C and D. These are required at fiveThe mileages between various stations are given by thetlre wagons be transported so as to minimize the total

d^

.*

Q6.

Activity A B C D E F G H I

Predecessors A A A B B,C E E,F D,F

Optimistictime

t4 t4 13 16 l5 13 14 l6 t4

Most likelyTime

17 18 15 t9 18 t7 18 20 16

Pessimistictime

25 2I 18 28 27 2t 20 41 22

Pory Z -'b 3

Q7. Solve the game whose payoffmatrix is given below:

,.Q8. A bakery keeps stock of a popular brand of cake. Daily demand based on pastexperience is given below.

Daily demand 0 15 )5 35 45 50Probabilitv 0.01 0.15 0.20 0.50 0.r2 4.02Consider the following sequence of random numbers:

48, 79, 09, 51, 56, 77, 15, 14, 69 and 09.

(i) using the sequence, sirnulate the demand for the next l0 days.(it) Find the stock situation if the owner of the bakery decides to make 35 cakes

every day.

Also estimate the daily average demand for the cakes on the basis of the simulateddata.

Player A Player BB1 B2 B3 B4

AI J 2 4 0A2 a

J 4 2 4A3 4 2 4 0A4 0 4 0 8

P""fr 3 db 3

CSVTU

Time: Three Hours

Note: 1) Attempt all questions.2) All parts of each question should be answered together.

Code: ME 09714B. Tech. (Seventh Semester) Examination Nov-Dec,2009

Subject: Operations ResearchBRANCH: MECHANICAL ENGINEERING

Max Marks: 80Min Marks: 28

1 (A) Old hens can be brought at Rs.2 each and young one at Rs 5 each. The old henslay 3 eggs per week and young one lay 5 eggs per week, each egg being worth 30paise .A hen costs Rs 1 per week to feed. Mr. Amit has only Rs 80to spend forhens .how many of each kind should Mr. Amit buy to give a profit of at least Rs 6per week. Assuming that Mr. Amit cannot have more than 20 hens. Formulate itsLPP.

!P) Solve the following by Simplex method.' Max Z:l5xr + 45xzSubiectto x1 * 16x2 <- 240

5x1 + 2x2 <= 162

x2 <: 50Whereas Xr axz ): 0i) Find the optimal solution to the Lpp.

05

10+5

l2+3

Solve any Two PartsFine jobs are to be processed and fine machines are available. Any machine can 10

.esI-

ii) Discuss the effect of changing availability from (240,1 Gp50) to {250,150,52)

(C) Solve the following by Simplex

Max. Z:4xy * 3xzSubject to X1 * x2 1: 50

xt*2xz)=803xt+2xz>:140x1 ,x2 >: 0

ORmethod, and also comment on its result.

Jobs

A B C D EI 32 38 4A 28 402 40 24 28 21 36a) 41 27 JJ 30 )t :

4 22 38 41 36 36i.22 35 ?q:

(B) Prachi Airlines that operates severl days a week has a timetable shown below.Crew must have a minimum layover of 6 hours between flights. Obtain the 10

pairing of flights that minimizes layover times away from home for any given

irins the crew will oe based at the city that resrilts in the smaller Icrew wr DE fesults m ver.Flight No. Chennai

DepartMumbaiArrival

Flight No. MumbaiDenart

Chennaiarrival

7AM 9AM 101 9AM 11 AM2 9AM I AM t02 IO AM 12 NOONJ l:30 PM 3:30 PM 103 3:30 PM 5:30 PM

4 7:30 PM 9:30 PM 104 8PM IO PM

National Oil Co. has 3 refineries and & 4 depots. Transportation cost per ton &ts are siven below:

3 (A)

Determine optimal allocation of output using Vogals approximation method and

check the result by Modified distribution method.

Define Traffic intensity or utilization rate and idle rate.

At a cycle repair shop on an average a customer arrives every five minutes and onan average the service time is 4 minutes per customer. Find Traffic intensity, Idletime, and number of customers in system and queue length.

(B) Construct A Network & Find The Minimum Time Required For The CompletionOf,The Task.Also Find Lst, Lft, Est, EftA<D,E; B,D<F; C<G; B,D<H; F,G<I.Timc TakenA:23 Days B:8 Days C:20 Days D:i6 Days E:24Days F:18 DaysG:l9days H:4 Days I:10 Days

OR(c) Draw the network for followins oroiw rlu

Activitv Preceded By Duration (Weeks)

a l0b a 9

c a 7

d b 6

e b t2f c 6o c 8

h f 8

I d 4g,h ll

k e 5

I a

10(c)

15

15

ulremen are v9nDr Dz D3 D4 Caoacitv

Pr 5 7 l3 l0 700

P, 8 6 14 l3 400P" t2 10 9 l1 800

uirement 300 600 100 400

(B)

NOV'D%" zoo9

Solve anY Two Parts(A) What is bimulation? What are the steps involved in simulation? Discuss Monte

Carlo Simulation'

Brite bakery keeps stock of a popular brand of cake. Privious experience

ndicates the da lv demand as given here.

Dailv Demand 0 l0 20 30 40 50

Probability 0.01 0.20 0.1 5 0.50 0.12 0.02

Consider the following sequence of random numbers'

R.No 48,78,19,5 1,5 6,7 7,1 5,I 4,68,09

Using this sequence, calculate the demand for next 10 days' Find the stock

situation if the owner of the bakery decides to make 30 cakes every day. Also

estimate dauly avercge demand for cake s on the basis of simulated data'

Define Game value, Saddle point with relevant example'

10

10

10(c)Usins Dominance principle. Solve for probability o{rglggtiqg rlglg s.

Firm A

Br B2 Br Ba

Ar 35 6s 25 05

Az 30 20, l5 00

Ar 40 50 00 l0Aa 55 60 l0 15

csyru

Time: 3 Hours

Note: Attempt allplace in the

. .. r_._ , .. ode No. M E ab7l4B.Tech. Seventh Semester Examination April'2009

(Mechanical Engineering Branch) i

OPERATIONS RESEARCH

Max Marks : g0

Min Pass Marks :ZB

the questions. All parts of a question should be attempted in continuation at aanswer copy.

Is this solLition optimum? ---------------(2)

Are there more than one optinial solution? -------------(2)

Is this solr:tion degenerate? ----------------(2)

is this solution feasible? ------(2)

If Sl is slack in machine A and 52 is slack in machine B, which of these

machines is being used to the full capaeity when producing aecording to this

I The simplex table for a maximization problem of linear programming is given belou,:

Here, Xl &.X2 are ijec!sion variatllesand Sl & 52 are slackvariables. Answeithe fblloii,ins

questions, giving reasons irr brief:

(i)'(ii)-(rrr)

(iv)

(u)

TD

solution?

A customer would like to have one unit of produet Xl and is willing to pay in

elcess of normal price in order to get it. How much sllould the price be

increased in order to ensure no reduction of profits? -------------(4)

Machine A (associated with slack Sl, in hrs./week; has to be shuf down for

repairs for 2 hours next week. What will be the effect on profit? --------(4)

(vi)

C; Easis ttt,AI X2 S1 S2 b;,5 I

I I 0 r00 51., I n I -)

LI 4 ) 0 05 5 5 0

Ci-Zi -l 0 -5 0

?7-o

(a)

Part (a) is compulsory. Attempt one more part from the remairiing.

Name any two methods used to determine starting solution ofTransportation Problem --------(2)

It is necessary that Assignment Matrix must always be a Square Matrix.

State true or false. ----------(l)

when does a transportation problem have a unique solution? state in one

sentence. ---------(2)

Four new machines Ml, M2, M3 & M4 are to be installed in a machine shop. There are

five vacant places A,B,C,D & E. Because of limited space, machine M2 can not be

placed at C and machine M3 can not be placed at A Assignment costs are tabulated

be low: ---( I 5)

(i,)--

(iD\

.....(nr)

.t

,'&{"'-.oh

d1*60

Find optimal assignment sehedule

(c) A company is manufaeturing air-coolers has trvo plants locaied at Bombay and Caicutta

with a capacity of 20C units and 100 units per week respectively. T'he companl,supplies

the air-coolers to its four siroivrooms situated at Ranchi. Delhi, Lucknow and Kanpur.

lvhich has maximum demand of 75,100,100 and 30 units respectively. Profit per unit is

tabulated below. -----(l5)

Plan the production program so as to maximize the profit. The company may have its production

capacity at both plants partly or wholly unused.

-l (a) anr*r. following each of I mark. 1X5 : -5 marks.

A E C D EM1 4 (t t0 5 6M2 7 4 * r'x c

J 4M3 -{4 6 9 6 L

M4 9 3 7 2 3

Rancla! Delhi Lucknow KannunEonsbav 90 90 r00 il0Calcutta 50 70 130 B5

PTO

I G),

\ (ttr

-\tiiitJ.. .

/ ltv)- (v)

ftP F* z'e-c' )

_ LSvTL)

Saddle Point is the point of intersection of pure strategies. True or False?

A pure strategy game is one in which each player has only one optimal strategy. - True orFalse.

When do you say that game is stable?

Define value of game.

When do players apply mixed strategy? Apply when there is no saddle point.

A small retailer has studied the weekly receipts and payments over the past 200 weeks

and has developed the following set of information. --------- ------( I 5)

(b)

ps

300050007000r 2000

0.200.300.400. l0

Weekly payments

4000600080001 0000

Probability

0.300.400.200.10

pattern of receipts aird

is Rs. 8000, u,hat is the

highest weekly balance

:(

lo

Using follorving set of random nuntbers, simulate the rveekly

payments for l2 weeks. Assuming that beginning bank balance

estimated balance at the end of l2 weekly periods? What is the

durins the ouarter?a+3

R-andom Nos.

Fo r rece i pt s : Q3,,, !. I, l-B-5 5, I 7.4 6,3243,69*7 2,2 4,ZZ

For Fayments: 5 1,963A 32,A3,88,48, 28,88, 18,7 1,99

Solve either part (a) or (b)(a) A small maintenance project consist of the following jcbs wliose preeedence

relatir:nship is given trelow:

(i) Draw an arrow diagram representing the project.

(ii) Find the total float for each activity.

(iii) Find critical path and total project duration.

OR

(b) Solve Soth (i) and (ii)

Jotr 1-2 1-3 2-3 2-5 3-4 3-6 4-5 4-6 5-5 tj- I

Duratiom(Davsi

t5 l5 3 5 8 t2 l4 3 14

.,Y: T'-(

(i) Explain Kendall's notation. ----------------(5)

(ii) Customers arrive at a one window drive in bank accoding to Poisson distribution

with mean l0 per hour. Service time per customer is exponential with mean 5 minutes.

The space in front of the window including that for the serviced car can accommodate a

maximum of 3 cars.. Others san wait outside the space. -------------(15)

r What is the probability that an arriving customer can drive directly to the space in front ofwindow.

e What is the probability that an aniving customer will have to wait outside the indicated

space.

c How long is an arriving customer expected to wait before starting service?

,a' ,l ('l (r

Give linear prc$alnming form of assignment problem '---(5)

- 1,^,r'-) \.i

Code No' ME08?14

C 5 VTU Nov-Dec 2oo8

B"Teeh' Seventh SemesterU:uryYlon I\\'r .-"- - -

(MechanicJ fnginoe'ring Branch)' **"nj?Yfr,J#ffiu"o*

Max Marku t 19

Time: 3 Hours ffiPass Marks :28

Note: Attempt arl tlee questions. A* parts of a question should be attempted at one plaee of the' '

, ,-*rT:":"oll***,*r,* two sentences) anv 15 from the foilowing' ----'xtr5= r'i5 rnarks)

/ n Basic solution and Basic feasible solution?t

-X What isthe difference betwee:

.")4Definer:.rn";':,mr;:,?1$'fi?rcatetha1,!1r,?Phasunboundedsolution?

Wffi:1fi::.frq:l'"li#r**:*#T'JJili# methocr nor be used ro soive

\'/>i Why "ut the transportatron Iesluuqrv "'

/ ' ett:ignment problern?

on the oritical path of a projeot network?

, X( C*-u durnmv activitv'appearr:il;"fil;, tt'z*'o?

W, m:l 1;

ffiff^$J:?;i?i'lTJ t';;t"ing even'[s?

- lx. What is 'il;"* ir' Q1euins theory?

X' What is *xPielsivc s1-ater

"X{ Whenis gen:e iriir?

xJr. what tn t?* ,o,r*uer of non basic variabres in balanced transportation p'obienr having 4

"'7, . '"*1",,* 5 eulun"lns' ,,-1 to every cost element in the transportation problenr table'

tr'(: tmffif;,;i;?;Jli,l"tl#;ii;h""se'.is iitnre or rarse' - rumn or assignment

w,/ XIV' r we aJJ "r-

subtraot * ""#;*t fr; af eiements of a row or co

'"**;1'1."";H#;ffi ;'ilH"i"?tffi '-;il"'dJ:f S;;'J":T;*"crassignment

\./ cost rnatrix, rhen opri*.r* iuil" *iir u* alter.e$. trs it true or false?

;Nr.*n*'l'l''i*#;i"b'bliwfii'iu*ionusedinPERr?Xvr. what

':]]ffi ;;-'h *oai in Queuins theory?

j;;i;:'. rvi,;" il;" * tn" a salne is stable?

t;f

V-TC

2Pafi(a)iscornpulsory.AttemptC}neff}orepartftomtheremaining.

,.16' Solve ttrliowing LPP using Eig M rnethod and comment on obtainecl sclution (if any)

MaxZ=A1"l+jxz

Sub. To : Xi+X2 d 50

x1+2X2>/ 80

3Xr+2X2>148XL,Nz ?A

(b) A dairy firm has two milk plants with daily floilk production of 6 rrritrlion triters and 9

miltion lit"rs'r[ieolir"iv. bacn dav fit* ryu* n!1 ttt" needs of three dis*ibution

centers with milk requirement ii l..s and 3 million liters respeotivetry' cost of

shipping t *iition liier from *utn gf*t P *.utl distribution center is given in

hr:ndreds "f ;;;;; below. ronnuiate'LPP involving only two decision variables to

minimize transportation coast' ----------(8)

Oistribution Cenlgll

Period

).A

2

Aa

5

6

A haspitai administrator iras the tbtrlorn'ing rninimal daily requirem€nts 6f nursing

personnel. "-,---"-(8)

Clock Tire Minimal No'.gf Nurses Requirssi

667A'60' .-

50-24,30'

6AfuT-1OAM10Ah4-2 PM2PM-6PM6PM-10PIyI-lOPIVT-zAM-2AM-6AM

Nurses repod to the hospital wards at tho beginning of each period and work for 8

eonsecutive hours. The hospital wants tcrn rleterrnine minimum no. of nr.lrses to empioy

so that there will be sufflcient no. of nursing personnel available for eaeh period' Only

formulate this as a LFF.

UA'J{'

N 6V*D Gc*zoaY'-sJrr)

_i

(a)dt--''

Fart {a) is oompulsor-v". Soive any one from remaining two parts.

A caterer undertakes to organize special dinner for a week and decides that he willneed a suppiy of fiesh napkins daily. A new napkin costs Rs. 20/- each. He ean alsouse soiled napkins a{ler washing. A laurldry provides regular service for Rs. ::i pernapkin and retumls the napkin after two days" There is also aq express scrvice e"t Rs.10 in whicil naptr<in is made available next day. The caterer will need 300, 600 250,30A,275 and 45S napkins from Monday to Saturday respectiveiy. Deterrnine how he

should buy and send for laundering so as to niinimize total cost. ---------------(10)

A firrn produeing a single product has 3 plants and 4 customers. Three plants produce30,50,40 r.l;rits respectively. The co. has committed to supply 25 units to customer 1,

20 units tc *us'ioiner 3 and at least 10 units to customer 2" Both customer 2 & 4 are

willing tr- p*t any ntrmber of units frorn the co. 'fhe profit associated is shown in thetable beio'w. Workout the optirnal distribution pattem.

(b)

(10)

CustoroerSPPiants Y

I ) 3 4

1 2l t6 25 l82 l7 t7 15 20aJ 32 27 l8 30

/^(cX Five mechanics are availabie to work on five rnaehines; the respective cost of man\./ machine eorurbination is given below. A sixth rnachine is available to replace one of

the existing imachivres and associated costs are given below. Detemine: -*-------(10)

(i) Whetleer ner i rna*hine can be accepted? If,yes then which one shouid be lepiaced?(ii) The optimax assignment and associated costs

Part (a) is cornpulsory. Attempt one more part from the remaining.

Explain Kendail's notation

The whetlier at a particular location satisfies the following description.The probability of rain tomonow is 0.6 if it is raining today. The probability of itsbeing clear (no rain) tomorrow is 0.8 if it is clear today. Using following random

4

@{\./(b)

MacEaime

FrleehanicI 2 3 4 5 6

A t9 t5 16 13 22B 13 t{ 2l t4C 15 l7 t9 20 t2 18

* 20 )) t6 l8 17t'in t6 l4 t9 18 15

?trt

nurxlbers sirnulate evolution of weather for 10 days, begiruring with a day after clear

duy.0.3CI" r].03, 0.38, 0.60, 0.22, A.42, CI.37, {}.79, 0.23' 0.01 *----(15)

,/'(c/ .A, oroiect consists of series of tasks A,B, .".. H, I with the following relationships.

,,/ (W<X,Y means X & Y cannot qtart until W,is fi1ished. X, Y<W means W cannot" start until both X & Y have, finished") With these not4tions, construct network

d-iagrarn having following constraints. -------- -----(15)

A<D,H; B,il<F; C<G; B,D,G<FI; F,G < {

Identify critical path and also find minimum time of cornpletion of project when the time

in days of completion of each task is as follows:

,qBCDEFGHI23 I 20 L6 24 l"E 19 4 10

B.Tech. Seventh senrester Examination, Nov.'(N{echaniczrl Engineering Btanch)

AICTE Scheme

OPERATIONS RESEAI{CT{

Llode l.Jo. (, t,'1 e Z LS

2008

Tiure: 3 Houls

Note: Attempt all tiie questions. Allplace in answer copy.

Max iWarksMin Pass Marks

: 100

:l -5

parts of a question shoulcl be atte'rpteci in continuation at a

1 Part (a) is compulsory. Attempt one more pari from the remaining.

(a) Explain tlie following tenns giving suitable example and physical significance. (anytt'") &/rtificiai Variable

-------(15)

-{ Redundant Constraint-/Deger,trracy in Lpp

,gr"-'Selsiti vity A nalysis

€,Si :.,: l, ;.,;, iliic ;

(b) Us;e Sinrplex method io sioive:

N4ax Z:6Xl i 4X2

(1 0)

s.1o.

2X1+3a25 393X1+2X2<24x I Fx2>-i

x1, x2>0Find alternate soiution (if any).

(c) Solve by Big Nf rnerhocl"

l"iirr Z- i?,i i ij;i2

Subjeci 1o: 2X I +5X2> I 50

3Xt+X2>120x1, x2>0

P; C

i, .{$''""":1\ )-2 i'ait i.r; is ctltitpulsol'1r. r\'uo-',rpt olto mole Il:trt fi'or1 the reuraiui'g

;@f-I'r'ipti Aillines that operates seven days a rveek has the tirnetable given belon,. Llrew mi,rsthave a nrinirnunt lal,over of 6 hours between flights. Obtain the paring o1'llights thatminimizes layover- tirne away from honie. For any given paring the crerv u,ili be bascd atthe city tliat results in srnaller layover. For each pair also mention the citl' where crervshould be base$. ------:------

Flight No. ChennaiDenarturc

Munrb*iArrival

Flight No. MumbaiIlenarture

CheirnaiArrival

I 7am., 9 anr t0t 9 arn 11 am2 9am ll am 102 i0 am i2 noonn-) .1.30 am 3.30 prn 103 3.30 pm 5.30 pm4 7,30 pr:i 9.30 104 8pm l0 prn

There are four factolies and three sales depot. Profit associatecl for transporling one unitis indicated in the table below. Determine the optimal transportation. ---l---------1ro;

lt

{ l.D

lltl

S1To

Il2F3

. (cy/ A Ccuri;any has a team of fbur saleslnen and there are

.:,.Jhr 1i'''11{s to starl L'usiness.'l"he co. eslinraies that profit per-\ tp-OJ" in caclt iiisii'it,ri is as 1bllorvs. Irincl the allocation of

rnaxirriize thc plofii -----------(10)

four districts r,vhele 1lie companyday, i1-r rullees foe each salesmensaiesinen to various clistricts to

AT1

C i5f) t2

(a) is oompulsory. Attempt ()ne morrr part from the remaining.

(irve trvo practical examples o1 each rf the following situation. ---,---(10)il:-iiii arrivai .'' sirrgie server!nfirritc Sev,-'i

Last come lirst serve.Arrival from llnite popuialion

I

!jt+1(tJ

3,t Part

(a)

r)i-

S2

.--o

s3 Capacity

t0-) -4 150

?(2 503 100

120 lsc I

:i F,j, L_

Ft 6t/ * Lbo INLCTE

(b) '1.'irr: owii:i cl'a cir:rir oj' jagi 1b,lci re sta rr.ant is consi,:iciirrg a ltcw conrprutt:r ,sysl.em fbraccounting anc-i inveri,.;);-r/ .:01!t1.ol. A cc;nl;uter coutpany ,",,t tlic filloriing inibrmationabout the systern instailaricu.------------ -___(15)

Activity

A

Activiry Ii

S"t.t CB Design Int

Desigir ltl. -evs1

D

--lrAlsel&lu

Develor-r IF Devuicp lrrp

rorrtiqII

Create DlInstall S

I Test &. in

laldrvare j Bl)

---------(1 s)

Pess int ist ic

I

2518

(i)(i i)

roul lnc I I

__Sryq! pli;bise-l_ r*jr L* _f_ ^ _ _l___p-ffi*'j-iiffi

Construct alt aric\,\, diagrarn ibl.th" pr;G;.

-

-Ddter'rine criiir:.,r path a'cr ."1r...'r,i project col,preticn time,

'::)" A repairmall is to be itrrctl to ''c|art'lnachines that breakilorl'n lollorving a iloiss{rr.i jrrocessrvitlt a'averaile Latc o1'lour lrci'lLour. fhe cosr of non-p.oa.,"i;;;|..];;,r*"iu mu. I n.,.hour- f'he colllpaily iias iirc opl;on olchcc:;ing eithel'r r"r, ol a s:loq, rcpair'ra'" .1.5e fhstrepainlan charges I{s. 6 per hcur and u'ill repair machinel; at aii arrerage rate .i. 7 perhour' T'he sior'v i'epairntatr charges Rs. 3 pel hour and u,ill repaii' machines a-t an a'eragerate of 5 i:rcl i:oui. \x/irich rci1;i:-it-:riir shcrilrj !.-e }:i,,::C.i -_ _---ai5) __;_

Part (a) is cornprrisory. Alieiltj)1, luc urcre lrari fi-om..he rernaining.

rjtc iri blielabriu: iir tcilo.,vill ."- __(10)(a) w7.,

/,\,,lL-/\ 'l. lt-@/r

6 FiL

seuclo I{and,;r:r idui'r -1.

imitatior, ui Sinl rl::l i,r,v(; Persorr lel,,_' Srit;t

ttl llrc r alu'; of il irv/ t) ,

- -t- ,r'

,c'i' l-l ti 5:;; 'l'cciriiitluc g:" L r--te_J-L, t gt-rfl.,i ir,trrc

luZ. ' ir1 p yaL+- v

,r-lrcii^i' liN fl"o'r the enclosed table and :;iurilla.iing 20 timcs

(c) Deilnc followilig lcrntr, iir casc of {iamr: theory;

n \, Piaycl

" \ Strategv

,l.--- r1::i

Descriptiori

\-_r _ii.'uff Sti.ates!, u.,

" itvi;*ea strrl"gy" |Rectangular Carrrcs

" /eaycffy Saddlelloijtt

-i- (,i

a r!,1.!. ( .1ji :.rrrjlrin N rrlr bei-s .fable

?'181e22te6 20i19l]i9s4 c,?62130tr,t 837485601q ,i 'r r28r05582 72t)sat,t5. ) q:oiir iis: );,.;;;;;; -rrj j765742271i2071:;s6 t!ia,;.,.i.:1i.;,,tAzit;isZ gf:igiign; ;Jfiit;:;d55747?,.68 5o::ll!ls" s;aesie,, i2ei/6s.rja )4ree52r,r58 30?,1?1lqs?2 liis\siiu'r1;;;,u;i ;A?liil;;j3s4s174t40 2zr,:t.lyt sitaiigirs ;e;e;ir6i llrnrurno,,4e420382 ,r83:g1ggl2 loiossiess s43zr ii6i0 duuoonr,rn,er031610i1 ?4r36865ee ligiiiilc, u4r42e44jo e:0764238ei4 1669:?7?s,? iia\6ili r,z ;ii3il;d;,ri1?;Z!;:tr34e3e6es25 02t?112ry? oisiresila ggss6sssZ| il.ooru*r,,j2e20544t>6 0j0001462e itZgiisitg iloBil;ffi ,uirorr.rno

43e14261)7 e89l9ll:g qc\iiiiiis stzzsiliTz ioztstqsoi3E07e505'e est:gqiis iriio'tt6io rsliozissX r!,uurrr,,l;fi'fffi: ;il3ijiSlS ;*njj:1 ?y3_7el:t16 so,r:ros+q05eo{s344:r anorpsq'ia ;;e';;;;i;; ;il;;liiiX ;i;gfi;f;se5665543i{ 5585:g:11: ;,r.;o;;;; ;;;6;;;;;; rr},rroorou,751s11628,1 en2g:1]1s \iiiliiiiq ooiigii66i ilrrrrrro,B-s0877ren 40355s+32r 6;;;i;;;;; y:??:?rOd iitsezzs,z6r)7002458ti s3241326s6 iiaiiz\i6i 442s14318s :iioos:zsss7eeeelt8:. 0r3s:g!:3e )e;;;;ii;; rssro:ii03 jdl,rru,r,o

iir,i{:!e)i :i?;?t?lt{ trjjtjfjtj 112:i1;i!i iit,*,,,,,riisq6r;:z<c 47651llbil n,.;.itli.li ;l#;;fffi t ji?il;;:q1i055505'' 515st1t,d-.,l:;,,;;,;; tJEr675b-j9 8907:ybn97;:;i,3il!3 l#iij;'lii l;;; ll::i s,,uoszszo ,i'u,,n,.,,

t;i;il?;y,i #,;;iilj ;i;i:t'i*) diifiiijl :ns,i;;;?#t:,z,]tri ltliiiitii iiiii,#ii! fitririiri iwf;;=,!,i'::?,li?,1 l'=;1tr1, :r; ir ryiii;: i;li6ii'tti iiff i;?:S' ;. ;,;,,;:1jl .. ;;ii',;;i;li,'' i;ri,r-;ur; ,1tfi61;i U;;lllXr9.- trlIS I L)jt,.r,,1 1,: :,.r]lr;i.i, orEjBT-Zlr7 :ournu.O:C"'l i /5r,:,5JJ lnl.lI'rr', ri,() rlJ.,J--1 ll;lj007s] rlr;.il.ts:;-i471561,]2:\ -tj,14()j j4,n t1i,,,ii:tirq ::oe.16iogt i.in*ur,,n,,857,|:6?n9? 2i\094s6-/61 jsot;rr*iil E22-1980i42 -iz:SO:rS::ri6i122at1 0146s808e2 sil:,,iaii;i g]jl:rqe&e szo+.r,"rosc(.1 rx5tBr)7ii 3rLt!()c UQrrT ili:']Sr:lq {Jg,9{Fjj30 2roi>ijrrOtrij:j4 i8230s 6?a j4763e.:, iqi;it:tair; 3-i4j2s0te5 :eq+ci-:6sslit94'1.2-l'/.2 ?.c],1.1'.:'l:tr.ti f;.,,l.,,tiili:i l{,1t20l0gt 60,?.112145{;ii)74}7},1}i oct-)(;.t-i-l-tlr, :Zl,91,i9i.,:; .Ji9-igg4.24(t

g?.24i2g9rR

AICTI.:B.'fech. Seventh Senrester l:xarnination Apr.il,0g

(Mechanical Lingineering Branch) I'1,'),

OPI'RATI OT{S RESI,ARCH

Tinte: 3 Hours Max N4arks

Min Pass h4arks

ccrcle No c'J/021--\

: 100:35

Note: Attenipt all the questions' All parts of a qlrestion should be atter'ptcd in continuation at aplace ir the answel.copy. v(v\r rrr

I Part (a) is cornpulsory. Atle'rpt one nror-e par.r ri.urn rrre r.enrairing.

(a) Explain the following terms giving suitable example antl physical significance. (anyfive)Siack variable

-----(15)

SurplLrs VariableRedundant ConstrainlDegeneracy in LIipSensitivity ArralysisFeasible regionCanonical lorm of constraints

Soive iiriiorving Uy graphicai nretliocl, incijcate tbasibleReciundanf constraint, if an1,. -__-____--(10)

MinZ,:3Xi+ 5X2

S.to.: -3Xl+4X2{2X1+3X2>2X1-XZ: -2xl< 4. X2>xl,x2 >0

(c) Mininiize Z: 5XlSubject to:

2 Part (a) ls compuisory. Attempt

(a) A coSP$/ is having a problem of scheduling its monthly production for the next fourmonlhs' The production capacity and unit cosiof production is gi,r"n in the table belor.v.

1)

t2

2

+ B X2 _ _- ____r t0;x2>2xl <4X1+;12:.5v -r \,21- ,l.,\irnz 1f

one trrore part fi'om the remaining.

region arrd inciicale

-\qr.-.*-(\\>{(= j>J(€)

E

I'TO

'i'he ploduct prorlucr:tl ilestirtr:rteci slor:tsc ,trt_ci rl i

sufilcient rir-rarrtit). Ii-i lui..1prodr.rclioi; ite sclteiii; jerl :;, ;

'ill,: ir.t(ri1lh tr-r.r'. l,r-'

i{ ,' i ,rgr' 1;11rr ir,,'1.tlri' ;,',rr1r.."luai Sa je-;

its fo rninintizr: toial

li,. lcl lbi'sir jc irr llri,_r rirttiiilrs, tnrl a1 airttrottt[r. 'I ltl r,r.rrn1':irr.,, r]rrrr,l itrgduce i'in flte ncx{ l(ti_r ntor.}llrr ]iol, shoulcl thecost'/ -----_. -_-_ _-{ I.51

!o) J soli'e fbllowing assiglllllerlt Ptci.,lelr. li'he ciata relbrs to the

'roducti.n irr u^its"---(10)

h{achinesA:-T- n 'ff.- -\l

_t__l-_L_r--; tr!\

assrgurrrent probleur.-ri 0\

I

7

4

five jotrs to five ilei'sorrs. The

| - 4-ld r9

5

It-,l

ol assign\

3w-"

(a)

lart (a) is co'rpLrlsoll'. 1lt'Lempt orre 'roLe

pa-ri li,,rr the re'rai'irrg.

Give tu'o Practical e;xnmi.'les o['each of-the ,llorving sitLrarj3n. _-_ ___(l(]t- Bulk ar r-ir a j. sirrgle server.- tlLrltiple parallei facilities .,vith ntultiple ciueues- Llst cr.-lttte lirst st'l've.- An"i..,:ri il.r:,1 li;rii: pt.riruiatiofi

(t') The orvtler of a chalil of fast food resJaurant is ionsiclering a rlew courplr1er sl,stem lbr.accor-rnting and inventory control. A conrputrrr co,-rlrrur,i ,i,t tfr* foiin*ilrg infoi"matiorr' about the systern installaiion

erafor

{''rc

APl,-nfrf -e'g/41 c'r e

Activity Activilv Descliption InrrrecliatePreclecessor Optinristic Most

LikelvPess irnist"ic

A Select (lonr zl 6 8I)L)

-eDesign input/Output A 5 1 t5Design N4onitoring

SvstenrA 4 O n-

qE

Assenrble Hardware R l5 2A 25DeveloD Progranr B 10 l8 26

F Develop Input/Outputroutine

0() o l6

C Cleate Database E 4 B t2H Install Svstem D.F 2 3I Test & hnplernent GH 6 1 3

(i)(ii)

ConstructDetermine

au arrow dia.gram for tl-re probleln.cf itical path and expected project completion time.

(c) A lepairman is to be hired to repair machilles that breakdown follon'ing a poissol processwith an a\/erage rate of fbur per hor:r. The cost of non-productive mzrchine is lts. 9 perhour. The colnpany has the option of choosing either a fast or a slow rel)airr]larl. The iastrepairman charges lts. 6 irer hour and will repair ilachines at an avefage fatr-: of 7 perhour. Tire slo*'reltait'trian ciiarges I{s. 3 per hour ancl will repair nracllises i:r1 ijr a\/eragerate o1'5 per irour. wiricli rep.i'nan sho'lcr be hirecl? --- i15)

3-+frt(a) is cornpulsorl,. Atternpt olle lllore part fi'om tlle reinaining.

(a) Write in brief abour the follo.;virlg: ---------_--------------i l !i)

' Pseuiio Random Nuurberu Discrete Simulation N4oclels* Trvo Ferson Zero Sum Ciatle

q, \r *t*w\\r t**1,'{'if\h''''l *,' ' 1n'

(b) 7'O"t'er 100 days period, tlie claily dernaud of a comnroclity shons fbllorviug li-equericl,\'r' distribution pattern. Sirnulate a ten clay seQrjslloer of clenrancl values, - - ( I 5)

DailyDemandNo. of l)a

Use follorving set of l{andom numbers.

67 .84.01 ,77 ,90.14,15.7 4,44,1J

l"'"{'o

(c) in a cedaiu gltl.lA has thrce possible ch.,.i,:es I-,1r4 ancl I.{ r,,,hile pia.,,e. Ll has tu,ochoices P and Q' Paynrents argto be nrad' ,l""ordir.g 1o follcx.r,ing orrung.-r'e't: ---(10)

N4P

MQNPNQ

'what are the best strategies fol player A ancl B intlrc ganre'/ \\/hat is the valtre crf garrrc,/

LP A pays B l{.c. 3LQ B pays A R,;. 3

A pays B I{r. -5

B pays A R:;. 2B pays A Rs. fB pays A [.].., I

AICl'Il .: ,;.1,., Citrtlc:o

l\{irr l,ass N{ark:35 ,,

-1 .r .': ,:ri: ' :utt. l.:.1#,,tr"V*,B.Tcc'.(Sevc' r * Sc nr csi", ; nir'nri n;iiout u*.-"rn,, ;ffi ff

Subjcct:-Opcra{ions Rcscarctr ' -",'1if." ';

:'

Time:ThreChoUr.s i i : ,,i,1:[Sfii:.r.:fii:,j

i\lax M:rrk:I00,

il,:i,'J'i;::'11,,:ft:,:];]:''" "' '\il l''ts.l':r t1'r'sri,,r) sl*rrir,i r)(.i,rrrr)rrlrctr *r trric jrr.ce

,li l:l i-()lt:j'rlilSr,.,. ,\ilutiii;,1 r)l)c lii{.),.d pliil i),)tr} iii; li;i;l:'tiii.,r.

i'.xplli,: tlre iirjlorviirl ttnlts giving: :;tutirlrjt. rr;iillul)lc i.luti i,lr).sicirls i r-l rr i ll r:ir n i:c. (li rr1, iir i .,rJ __ -- _-_ -_ _( I t l;'=i* Ll'f iiii'-i r'jtl,r- j,i*cirr l'rograrrr'ir.g r)r.tlbre'r.s.*,,t' Slacii \,:rr iableve-'. Ireasibic Sloiuticin

r""Fl r\ I rernirr e ()pti nral Solutiorr\r" " ii crir-rrt, iilu, eut.]itr.lr : jli.

.-1.-iS c ns iti v i i;, z\.na I ys i s,r'/

'{ r\ ( rrp11'1111)'Jrl')!ltrees llrrr''' t1'Ie5, t,1- plotlrr..l:; }i \r' riir*.r ,1 irrri rrsi.s ij}r.,-,r

,tU5;,ttnrrtt'r.iul liI,liL,ancJ lil.,l'iie lbilou,ing,,llrrr ait:*irl,r,, -_____--,-__(151

" 1- l''i., ':iri I il. i ;,,.inL.rrt r,j pr()(ir;rr, (,,," l,9u r lvlrrr. t)rrrr,. -l

i ,,- I , j _\'_)__1, lr\viiilrrlric II ;;i _l _ I _ I q- _ r- r- .--;; --]l-':--, , -l i l-, I fr. t

l - j1. l- , 1.- 1 ,?,,-l-I_ l- " i., __ll rrr.ti 'icn i :,iii, ] ,ru.0g_11-[0 j I

l

i).,,.,'.I (ilLr.,i:i

.:

.ttl) ) ")/r.-g 'a fti1 lf1 UE- L

i:-?f

l

i

't

- ..: :fl

. :.:.1: .':

'-:::a'::: !'1"'..,:l:i:i1.1:::, Ia':a:t:.l:.i.:-. .. _

.-:r-i:.,.rr-i'..

:. :. ii.,lt i -

rrltB /-l.z ur/ ! !a-tr.\..

! )'lY

t i ', t

'13 z'l!t

).'J,ii

r

Solr,; lll': lrioblcnr lo rrr.rx jiiii:r.c 1-rrttlit. .-(_;,-.,-.z\ --

-9 .x .t 3'z -'-------------i I

'L=SXl i2X2Ft0Xl

Xl-X3q t0x2-x3 ) 10

Xl+x2r,.i3 u toxi, x2, x3 > ()

(9, I Niux

S.to:I

i

l) l',-'\

r:i:l'v

(a)

6$*r\\

l)

"9--"

<$\-(*.

d-lo

l:ii'c ti illt'rciti jirtrs irrc t, l-.e assigncd to six rjifli,rcnl rnecitincs:. I'irc rssociatc..jc('its .iit ilii'etr bcltrt',. l-)':tc;ttiittc vi'lricir rnirclrilc shouicl nt.rl be assigrtcc! alii, joliLr, rtiiitillli..,-,: tirc ttttai cosl / -*----------l i 5)

---.. - .i!

I.-.,,-lI

i

I

t

j.l j

i)lr:.t ( f.iicr rlri ()nc nt0lc jr.li i |c nlal t) t n tl

(.i] l..rjii,'ltrilrL.,i-ilj c\ irril;ir,.of iblilrilili!t tit:ji':j _-_" (f)

' i _:isi lOt1ii,j, i il st serrYc

A clrild care shop dealing rvith children's requirements has one cashier v,,1o

, handles all custontet's plt-r'ttrcttts. Thc ciisirie r rlikcs on an,average 4 nrinutcs pelcustoilltli'. Custotrtcrs conte 1o cashier in a rlnCorli nlilruter but i:rr an rverage of l0Peop'1. Ftr jt,lttr. Ir'.llrtlget:ierii u'arlts to iir','cl':titlile 'tirc fbllorriirg: -,, ---.--- -( li)

' .\r li,'rrrrri ,rt1ir rt1/\r-, ,...,-, i.. ..- .r "i.-" ']li|t (a) 1S C()ntl)LtlS()l\'...\ltrlt)l)l ()lle !lt're |J!.t l|il111 111., tei{jilillltlg. .i., , ;; ;:

\\/ritc ceneral Nlathcnraticar N,fudcl of l ransporrarion problern. I-lo*, f,i-' ;:: ,H1;

lransportation problerns arc diffcrent than I-PP? ------ -!:':'j,'i: J..-' .,,.:lC

.,t firL' producing a singrc producr 'as

.l pra'rs and 4;;;;,;r;:'1ffi, ;,'*froCuce 30,50,40 uttils tcsPectii'c11". l-hc c:i'r. hirs conrrnittc:cl to suppll,Zs urrir{'iii( ils-[()l]le l 1. 2(J rrttits l() ( Lls[()l]t('r i rtrtrl il! lr';r\i l0 Luirt:; t() ,-jrislo,,,cr :. tlttill(tlStolilcr l & -i ltrr ri,illjriti to iru\ iuir irrrrlthi,l,iil rrrrits llclrrr tir. crl. I1l.. lrri,titllr:s{-)cialc'i is sltr;iitt iii tltt'llt[llc 1,,-l.ri'. \\''rrt'litrirr llic rilllirrrlrl rlistiiirtrli.rr p,,,,arn. .__-( i i )

Pp!;

./._.-'(c )

(b)

Ir'I - --lbii il-la-ri0I ()

,i:...i-t:: ,

.lB',,r:lli I.

+,.",-,€$:::

iig:':fai. i:ii-:l ii:1li:,

=:;{ i

;i-,ii':;:ini .

ll--

{-,

I

o

l_5

lrlzi1

ll

+'.., /{,: -. .,

"1'

12

i-t.'lcr rrrri,', r iiic.ri |lrtlt l)ei.t:r rtrinc crtr.l\, sllrrt.lrrr,,lirri' '- --(i.rt

Dtre -J /+N * 26 a'/ *ag

Ar c.re

I i I I,-: s I :: I lt r'i

&r.

(i) t'hat is the average iengrh of rvaiting time to be expectetj'? i:': ' : '1t:'':i

(ii) \\'hat portion of his time cashier is expected to be icile?(iii) \\'lrirt is tlte avcragc' lcrrgth of (inrc that a custorner *,ould bc cxpected to n,ai{

to pay fbr his purchase?(iv) \\/hat is the probability that a crlslomer has to wait atleast l2 minutes?'

llt r' r ri-'t't I 'll,'rviltg jrtl()l,t) ltir,tr.ltr'lirerr

. Aetir iie:,,'\.lJ.lrrrtl l: rrlr" ire st;rrt,,'tl lrl tlrc beginnitrri ol'llrt'Iroiect" \\ihuir lciivitv l3 ls cortrl;lcte l) lnci (j nray start.u \\']1.-tr llctir itr . , IS 1.,t,,1,.1e .. ; itlitl ..iilt't.. Cl clurr,trr \\lrrrn [)otl] l:unti i) iri'e eonrplclc lrrd is Irurl lrcti\rt\* Il car :teirI aliel C is cornplctc arrcl is llnalacti\it),,e K nrllst loliorv ihe finish of E.

" l- clin llot stiut Lintilbolh (llrrrcl K lric contplctc.. .l clin:;llLt't r.,'lrerr l. is crr;nplt:tc liriti rs iinal activll),.

Ac:t:r'tlr

l)t:irv tirt r 't,. . , rr i

tLrrislr tiLr, li,i ,rr,

l:r,l tlrtrati,lrr - - l1r"a T- Nlrrr f 't.lt rr*

-T P.,.'s",rir1"'r*" ]lalsl_l -8 ---l l(r -

ii{, 1811.0 I- -l ,- i 1--lt- j --l-r*t-l---r:----]

i .rr t (;r) i

Write irr bricf about thc follo*'ing: ---------( l0)

' Pseudo Random Nuntber* D]scrctc Sinrrrlation lr,locir:1su 'itt,o l'crsorr Z-clo Suin (liriirc

.9'l-/

t\&

rr0

Vlr) A largc rr'urkshop is undcrtuking servicing jobs. 'lhe rlcchanics ob,lain theirtools fronr a central store. N'lanager:.'of tlre workshop is worriecl:d'bout' the

u,aiting tinre of tire mechanic and is in the process of detennining"r\'lrelhcr

nlore altendants be hired for the store for raising the levei of service. -l-he

follou'inp (lata on Lhc tinrcs trctu'e.ctt successive arrivals and service tirncs lorllt.' rrrrcllrlrics lrirr,.' irccrr rrirtltirtccl lltltn tlrc plrst 20() trltscrvirticttts tna<ls t,tt the

S\:ilctli. - ---( l5)

'*.E,.€

i!4

EH

!i

l)i L,il'rttt,,1 tLl Lril:

'l'inic 1i\'1inut.e s

I)isti iirrrlioir ()l .scr'\'lce lrlr)e

r.,1 1ir,1i 1i1rrt.

lrrg1gq!15_i

t2I

i.:

)0i1

l1

l Li llttir-'

8

_i0

8E.1 6

0

J

6

9

tzr5

l'robatril4l

0.06 .

1) 11 1)

0l50370.160.07

Ilrrh$i1ilr

(i. I t)0.lli/l.1 1

0.24

]Ullg iN1r,lrrrt.

.1

(r

tl

I ('

t2

it if i l,,l_rl]l.lt LaI Sir,\_!!!- iilr-L-t.,-

l.isirr' lgllt-irling rattrloirt ti'.ttii1-'.'i. detet'tnini: ilvcragc tinl': of tltc lllcChallic ltrlti

i1\,,rt;tgc rlll(rue ielrgtli t,') 'trrilLrllitirrg' lor l5 rLttls'

:

ii..ir;,l,rtp-pt,, ltlgfiLhil lil'.'l'l.l-1 (r9,i5,55,i-)()'90,66 'i'tr,39,15,90,98'j'9

g'i l, (j.28,97,69,37.52,:)2, I 5,8 5.4 1,82,98,99.21

er ll--i --l- -- 2 .*-,

-j--'- -(l- "-'.'- --'-'-' '- - i

_6:4

_lt;

II

--li

i,i:: 'lli ':- j

,',,:fi,;l!

\r:i:,' \li',"i

1

url

,W (a)

j."

,i', ','il;r .i

: ;rll., r. ri .,'I I ri i:'.'l.t ,:.i:.i ii:ii

:':i': : r'l: I

- ': i:ri:,.1i;,,, ,':;

::in ,, r

'ljl ":ll::'l ..i.: 'l :.L

,i ,'' ;:'lf;:;;';

'11il"' '' '"... ir ."j ".:ii,!' I

,,1I

l1: , ,, :ri., :

;;.:;,' ril,l', i1-:'..1

:,, :..,1:..:.;it :..tI

:r : .11

'...t...i

"i: ll

T'iu.ic: 3llr;ilrs

e nrle h.ln . t\A:i^11 c

B . E. Sevenrh S e r rr e.srer Ex:m i rrirj irn a p r- 1,t,,,:z0ttdrie.rNo

u6 i r : t - ;

(Mechanicll,f n3::rccring ii rirnch)

OPERAIIIONS RESEA }{ CI]

, il,Iax ir4arls : 100

lr,lin lrrsi h{erk. :35

Note: ,iilt.*t-r;1.;t all.the questions. All Ilatls tjf a quesrjoflaliswer cOpy.

;iii-:iild be aiiernpred at One piace 0f the

i

[':Lr.l (a) is compulsor;', Atternpt .ne morc pan irom tire rernaini'g.

l:llt't the following-teTrs glving suitablc e)iiunDlc anrl physical significance, (anv

I I V{'i

r e,,-r,,^ \,r^-:^Lr- ; ;,, " - -- -----:--------$----(15)

" lurylus Variable *fl-y{i'/fc,,\q[ 9,t,-uAa,_/rtc. ;" Redundant Consrrainl

4 U' -- \I Degeneracy ix LPP . al 6 &g( itu sa N{ultiple Optirnum Soiurionr Feasible Solution

J Canonical fonn of corurrainrs

! Shadow price:

Sr,.il.,iiollo1yingb1,8raltricatmeLhorl,indicatefeaslblgregj0Ilanoina.l.atl:r'dunia;:r consrdinr, it'any. _-_.,_- _ - _-gr;;^l

^-" !{rru "'v'/efe

Min Z =3Xt +5Xz

.:'l

S.rq: -3Xt #X2g12

a ! :::

' : ".

r .. :l', '

';:).

li.

(b)

: "':i

":'"iiti!:

,: 2X1 EX2>t2 :

xl\<4, x2>2 / **-Xl Xt -n

/ -\(cj trnntmtze7-. = 5 XI +g X2Subjecr to:, XJ>z

' xt+x? -{ :

Y] Yl .J^...1,,_.-J

Pan.(e) is c-ornBulsory'. Afiempt .ne m're pan fr'm the Lemainine i

. (a) wrlle Generar }{athernaticar }Lfocler orrra'sportatro,r"trror,tem. --__-_-__---_ ._(r0)

\{ir{

t

,l

.ii.i

. "i','iJi i., :

":l:Ts.*o !.- , :.' i;: ' ,'.1i;i;;ii;i]',, ' '., .rt .i:....11.:.ir,.r;-il.r.,. . .:

li .' ::,':,i:,:,.....il, rl -.,::,ll,i :: :i; I . .,r;:l:i:: .

.i;:

in) f rve diffcrent jobs ai.e rc bir :ssigneij roi:0sLs are given beiolv. Dercrmiqc ivhichro nrjnimize rhe total cosr? _--_-__-_ ,ij;;

slx diFferent rnachines, .I,he associateci

rnachine shoulcl n0r be assigneO.aa, joU

1.

3

lz )r5

8

I

i i\4.5

| \,16\

(c) soive r,e lbllorving assignmenr problem. Thetaa relers ro pooor.,lon ,oil

oA

10

11

J:-*]=--4l'l

L--7

D

8

9 10a

1

0 !5

I

.l

I1

3 Par-r (a) ts compulsory'. Atrempt one mcre part ir:om the rernaining

11

-(15)

(a) Gil'e 111L. t . . rf sel-ir-c r{ic^i-r;-^ ... . ..' i:ll:;::ff*iTffi;;;#;inn; r;n .,., c er f , i 0p, f5,1

iJr:lut\,S:rl<.ronrviihrnore,flrni*..Urlr,,.,uo. fi_.r-r l,g$r )ig*t"

strop inside a r;;';; u,*r.l'r, m:rchincs or rhar.r.L : ui Ula[ plrircrt lar hclory onlt,

JJ,,|,'l.:'il:';r'J:'iTJ'T,i,.|'::'::store, rvlrere service nrec;,nics,rijvc io,,oo .^",-se^,cti 0,, u iil-s'basis. The srore is ,,1r-.r':n,.,

;;;';'il;:Jiit;:Xi:f:,t:l'Jr*nrecita:rics irr an .lrc t:]::

li' o,ib attenianr ri,[6 .^, ui.rj'.inr,,1,.11

.",,;j# il,l';: il,?JJ,:i #;.::l::r rates or irre"iechunics 0,.,,g,s 6 i,,se^'iu- ri,ri is c;i,onenriar o.i,iJur,.o,'.il:ffi: 'il:li l: _::ll]uf ,lt"nor;;"#

r*J

jiL.il[:fflitilqilruiru;r:rxl.---.----...----(l0)1r') Expecred number.i;;;;#i:u,fi*ourr..

:'i:' ": 'j

:;rt:::l

-MAb\w

(c) (U(it

4 j)u (a) is

.) \\/n,

What do you undersand by Free Fbat and Ildepcndenr Floar ___._.r5; ,

\Jr\.1. pmcrical example of butk ,*;;i ,"0 ilr,",* server. _,___ --_f.)fconrpulsory,. Attempt one ntorc pall j.ron tlte rernaininr.

I,

APp*nnf-2-eoT ,i

ArcrG ,, .j\,,1"\, ,i: :i :il

\,ir

\

::it .' t

1,.'.i. ..

:l.il(..

1 ' Pserido Random Number'' Q Discrete Siniulation Models

' Two Person Zero Sum Game

I :t-i.i .'+

' .; ,,;l

: :::1.:ji+ir,i!-: ' ;i.-:!i

.,: 'r:..:rt:t. : : ,.a...,;;11

li r::i:.ei,:+.€' ', i .i,':, :l;

,, ,,:':: rl :l:i iill'r+'ri;rjJs

, :.t

(b) For a i00 day penod, the daily demancj oi a cerlain commoiliry shorvs loliowingftequency distnburion,': :

DaiiyDemand 0 1 2 3 4 5

':lr:'':'.:'.::. .i

;'j

rii

.:l

j

:t l.lI rtl'': ii'' rri

No. ofdays 102A402054

(c)

using the raxdom n0.67' 84,01,7?,g0.14,1i,74, 44",77 simurare a ten irey.\,rjruence 0fdemand value and,:detemrile average denand.

using dominance pr.pefty, obtain the 'ptunar

srrategies for boih rhe playen anddetermine rhe value of rhe_game. The payoffmatrix for ptlyer a-i, gnou, " -.ii1'.'" """

iPla

Jt..4

, :rt , l,i:i;)',,ir/ ':.i''., rl.{

'': . " 1

. . ti.',.1

rt .,, ii:ri: :i .:;r.lrr 'l .1' l

'.: . .i .:i,'::l, : ,J

:.. .:: .ti1r-i;."1

l

:''''].,::'i:t:i

rI iI n/1Y

tll: r4 t3 8irll :<. 6 3 7 0m i'6 I

l1 i9 6 ,'iln/1Y .4 .) I 4

r{r:'0T!l_i

Code No.067021 - $'4

Pl6Y'pec

B.E. Seventh Semester Examination 2006

(Mechanical Engineering B ranch)

OPERATIONS RESEARCH

Time: 3 Hours Max Marks : 100

h{in Pass Marks :35

Note: Attempt all the questions. Al1 parts of a question should be attempted at one place of the

answer copy'

lPar1(a)iscompulsory.Attemptonemofepartfrorrrtheremaining.

1"'-'''; +,i-i# The rolis of ctottr oi size 55mX180cms are slit into narrorver widths so as to satisfy' L-f

,, ".i.i,i \4'

the following dernand.'i

. ; i'. {''- Widtii 80cms 45cms 27ems.,-,

No. of roils 200 IZE 130

.= ru_, O6tain the trinear programarring fonni-rlagion of slitting p;:(}gram so as to rninimize lltc

\ u\)v- ;;;"g*, d*ring uUttil"g arid also to satisfy tire ai-rove demand' ------------"(10)t1 € /

n=u H .1'u {qX A Z4-h0ur supenaarket has thre fotrlc.,a'ing miirimal reqliirements for eashier'e,'/:'J'.:".-;\l , r:--..'r.-=-{-.;----l-, -ffi

Feriod I 2 3 4 5 G

Tirne af theday(24 hr ciock)

3-/ 7-X 1 r 1-15 r5-i9 L-a-23 z3-3

MinimumNo.

a 20 L4 20 10 5

(dt Explain the following tems giving suitabie example and physical significanee' (any

\,:\. five) .r'l) -82 Artificial Vari,ablt'

{irjl""1,*.:-u u!-1.,- V-thadow Pnce ;--a

".1.' i,*\''J.jttt'' * ,!,,i-iil* The rc,lls cf cloth of size 55mX1

pefiod i follows immediately after period 6. A cashier works.eight consecutive hours,

starting at the beginning of tne or the.six periodt tr"gs11t gitJl:J-"T*i1$':-;1t' : n.nnruit*ioe protle,n to determine a daity employbe worksheet' whtch satlslles ine

ieast number of Personnel'.-- -{ *qurffi*ffi;fi't number of personnel' --------(10)

i "Lt' ,ot,iu, , P-Tai !-.r, : -3' ivlr.'- - ev\ i".-. !

' I t'lttt--,ti.

2 Part (a) is compr:isory' Attempt one more part ti'om the remalnlng

,kl\TiiptiAirlinesthatoperatessevendaysaweekhasthetimetablegivenbelow.CrewmustVf r-.lo c rninimuffr lavover of e ftou" between flights' Obtain the paring.:l fltll:"filil'fJ'1il'#ffJH"li*1"'?,i q #ii"T{{'":.',':I.-o:i:,T"'*'.:::TtJ,1 llt#:.f::have a mintmum l?vovcr ur u 'vsr" lring the crew wili be based at

minimizes layover time away from home F"1 Ty-g^t'":l.L -o-ri^. the citv where crew

il:":TfHjit':ffiJ'il"#;il^il;;;; F"' "'n'p* 1T-: T"""',lhe citv where crew

should be based' -------- ---7 Vx

IV{umbaiArrival

Flight No. MumbaiDeparture

ChennaiArrivalFlight I\o" Chennai

Departure101 9am 11 am

7am oI

102 10 am 1? nnon?. 9am 11am 5iOnm

103 3.30 pm1.30 am ?iOnm

10 pm104 8pm

4 7.30 pm OlOnm

rs. Three Plants Produce(b) A firm produ

"ingKngl.,pr*u:r^has 3 piants and 4 customers"lhree plants pI{''uuLtr

3 0,s 0,4 0 units re spe ctiv e rv' *: :': lii:;::l l" ;:* t;, 3,t";:',t; t tj,:STii tl';i:

iliil;i'"Jlliil""i:H;iJJ;;''il;t",r::3::"* customer 2 & 4 are wl'ing to

buy any nurnber "f ;;;"i; the co' rttt pi"nt associated is shown in the table below'

Workout the optimal distribution pattem' -:--- -- -------(10)

{lustomers -w

Plants

1 2 J 4

16 25 t81

1

zlt7 17 15 .LU

27 18 30J

'52

/.' - '; c c^---,^r. ^.-*h. nsgiened to five different rnactrines' Tire profit associated(*f A bateh of tour.loos can ."" i:"::":-:,::;;;;t*e is shown in rhe table below' I)etei

r"n;n cach i"b:i';;;-a'sign*a to a given tnachirie 1t,;;.^- ^^,,i^ he the ext.ett'r- oI max!**'::S,$:*""'.'*";l'ffi .*r;;'l?:H*fi lffiH*"'ft.'tr-'?:::'*Lii;H:trff

|;H,:fi -i"*:#;l"fi -fi

:ffi #q1;;;;111:ttgg1ff rheext€ct-ef rnaxjsusl--

# lx'^'T:"ffl$d;';;ruru et'---- *- --- --------(i0)

*Vun(a) is eompuXsory' Attempt one rnore part from the remainin'g'

i;l

;s1

35

I

1t:

,*, r{, \h/' !!

r "s4.i1r#fi\ )i"

_/. \.). - ,-, ".t,t JJt

'L!r" ! -:

--n +r,ri'' -.\ -Ifi:6

!"1;'r - r':I . \{ \'+ .._-'-.:'" !

.t 2 3 4 Io6

tl L 2A IU

7t--:--t-)

113

14 1211It i0B 1- 14

6 aI

t-t*lt)

10"7

lf t1

. d.ia * F/ ov * D6-c * L{z'("

O oi', fl*Hz, 27CTGI * "WU-"

(,r)."' ,A;r"ro, Iinu,n of fast food restaurant is considering a new computer system for

:? "r/";';;";il;

and inventory control. A computer company sent the following information'-

about the iystem installaiion ----(15)

(i)(ii)

Activity Activity DescriPtion ImmediatePredecessor

TimeOptimistic Most

LikeiyPessimistic

A Select ComPuter 4 6 8

B Design InPutlOutPut A 5 7 15

C Design MonitoringSystem

A 4 I 12

D Assemble Hardware B 15 zCI 25

E Develop Program B 10 18 26

F Develop InPut/OutPutroutine

C 8 9 16

\t Create Database E 4 8 I2

H Install Svstem D,F I 2 3

I Test & Implement G.H 6 'l 8

Construct an affow diagram for the problem'

Determine critical path and expected project completion time'

A repainnan is to be hired to repair machinos that breakdown following a Foissoll piocess

with an average late of four per hour. The eost of non-produetive maehine is Rs. I pei'

hour. The company has the opti*n of ctroosing either a fast or a slow repainnan. 'fhe fast

repairman charges Rs. 6 per hour and will repair machines at an average rate of 7 per

hour. The slow repairman charges Rs. 3 per hour and will repalir machines at an average

rate of 5 per hour. Which repairman should be hired? ---{15}

(a) is compulsory. Attemirt cne nlore part fiom the remairiing.

Write in brief abcut the following:

Fseudo R.andom NumberDiscrete S imulation N4odels

Two FersonZera Sum Game

Pa_r{

\{e)

4

-.(")

B

B

' /tti €

$N % large workstrop is undefiaking servieing jobs. The rnecharries obtain their toois

frorn'a central u*r*. h4anager or tn* workshop is woffiecl atrout the waiting time of'

the mechanie and is in the process of detennining whether more attendants be hired

for the store for raising the l-evel of service. The following data o!1 the times iletween

successive arrivals anl service times for the rcechanics have been obtained from the

past200observationsmadeonthesystem.-.;---.---:--

?'T 0

Distribution of Inier arrival time:

f ime (Minutes

0J

6

9

12

15

Ti$s(Mirytes

468

10L2

Thc cost to the eom1lany

Find suitable strategY.

Distribution of senrice time:

FrequencJ

12,.

18

507432L4

Probability

0.050.090.254370.160.07

Frobabilify

0.040.i00.180.44a.z4

Frecuencv

I2A368846

Using foliowing random numbet, deierirrine avel?ge tinre of tlre meehanic and average

querre length by simulating for 15 runs'

.

5 9,4'7,23' ;69$ ;ss'og tecipti S x 3*;tsf8'98#+' - "'

Rarccmno.rorSer.vicetin:gt'7,3g.i]8,}7.59,8.7.::,2,52,1,5,q5,41,82,g8,gg,23

, l4 A cor'pany'S man*gement.and laborunion are negotiaf-iirg a new three years setttrement'

-. . ijach partY has four strategtes:r";\:-'

"it q* '- '

r,-- , .c th) - '' slFl" I : Hard and aggressive bargaining approaeh

IL: R"easo'ning arrd L-ogical apploach

IIX: lrgalistie aPProaeh

iV : ConeiliatorY aPProaeh

for every pair of strategic choiees are given il,t}.* table beicv":

------(15)

Union StrategY eo*puqy JqutggyI i EI w

t 20 15 12 35

u 25AA1+ I 10

ili 4A 2 10 J

w -5 4 11 0

-.J:

$+#

qb",+ ' . .l:'.'"'..',