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Summer 2015

Exam Oct/Nov 2015

Masters of Business Administration

MBA Semester 2

MB0048 –OE!A"#ONS !ESEA!$%Assi&nments

'1( )*at are t*e features of o+erations researc*, )*at are t*e

-imitations of o+erations researc*,

Ans.er

ist and ex+-ain an ve features of o+erations researc*

• OR is system-oriented. OR scrutinizes the problem from an

organization’s perspective. The results can be optimal for one part of 

the system, while the same can be unfavorable for another part of the

system.

• OR imbibes an inter–disciplinary team approach. ince no single

individual can have a thorough !nowledge of all the fast developing

scienti"c !now-how, personalities from di#erent scienti"c and

managerial cadre form a team to solve the problem.

• OR uses scienti"c methods to solve problems.

• OR increases e#ectiveness of the management’s decision-ma!ing

ability.

 

OR uses computers to solve large and comple$ problems.

ist and ex+-ain t*e ve -imitations of o+erations researc*

 The limitations are more related to the problems of model building, time, and

money factors. The limitations are%

• Ma&nitude of com+utation – &odern problems involve a large

number of variables. The magnitude of computation ma!es it di'cult

to "nd the interrelationship.

• #ntan&i3-e factors – (on–)uantitative factors and human emotional*

factors cannot be ta!en into account.

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• $ommunication &a+ – There is a wide gap between the e$pectations

of managers and the aim of research professionals.

• "ime and mone factors – /hen you sub0ect the basic data to

fre)uent changes then incorporating them into OR models becomes a

costly a#air.

• %uman factor – 1mplementation of decisions involves human relations

and behaviour.

'2( a( Ex+-ain t*e &ra+*ica- met*od of so-vin& inear ro&rammin&

ro3-em(

3( A furniture manufacturin& com+an +-ans to mae t.o +roducts

c*airs and ta3-es from its avai-a3-e resources6 .*ic* consists of 

400 3oard feet of ma*o&an tim3er and 450 man*ours of -a3or( #t

no.s t*at to mae a c*air re7uires 5 3oard feet and 10 man*oursand ie-ds a +rot of !s(45 .*i-e eac* ta3-e uses 20 3oard feet and

15 man*ours and *as a +rot of !s(80(

ormu-ate t*e to maximi9e t*e +rot(

Ans.er a:

;ra+*ica- Met*ods to So-ve

/hile obtaining the optimal solution to an 2 by the graphical method, thestatement of the following theorems of linear programming is used%

 The collection of all feasible solutions to an 2 constitutes a conve$

set whose e$treme points correspond to the basic feasible solutions.  There are a "nite number of basic feasible regions within the feasible

solution space. 1f the conve$ set of the feasible solutions of the system of 

simultaneous e)uation is a conve$ polyhedron, then at least one of the

e$treme points gives an optimal solution. 1f the optimal solution occurs at more than one e$treme point, the

value of the ob0ective function will be the same for all conve$

combination of these e$treme points.

)orin& ru-e

 The method of solving an 2 on the basis of the above analysis is !nown as

the graphical method. The wor!ing rule for the method is as follows.

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Ste+ 1 ormu-ate the problem in terms of a series of mathematical

e)uations representing ob0ective function and constraints of 2.

Ste+ 2 -ot each of the constraints e)uation graphically. Replace the

ine)uality constraint e)uation to form a linear e)uation. lot the e)uations

on the planar graph with each a$is representing respective variables.

Ste+

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chairs and tables, the total re)uirement of raw material will be :$0 A 3;$ 3 ,

which should not e$ceed the available raw material of o&e-?s A++roximation Met*od @>AM:(

Ans.er

a

>o&e-?s a++roximation met*od

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 The Iogel’s appro$imation method 5I8&6 ta!es into account not only the

least cost ci0, but also the cost that 0ust e$ceeds ci0. The steps of the method

are given as follows%

Ste+ 1 7or each row of the transportation table, identify the smallest and

the ne$t to smallest costs. Jetermine the di#erence between them for eachrow.

Ste+ 2 1dentify the row or column with the largest di#erence among all the

rows and columns. 1f a tie occurs, use any arbitrary tie brea!ing choice. 2et

the greatest di#erence correspond to the ith row and let 9i0 be the smallest

cost in the ith row. 8llocate the ma$imum feasible amount $i0 G min 5ai, b06 in

the 5i, 06th cell and cross o# the ith row or the 0th column in the usual

manner.

Ste+ < Recomputed the column and row di#erences for the reduced

transportation table and go to step 3. Repeat the procedure until all the rim

re)uirements are satis"ed.

3:

 

So-ution

 The steps 1nvolved in determining an initial solution are as follows%

5i6 9alculate the di#erence between the two lowest transportation costs for

each row and column. These are written by the side of each row and column

and arc !nown’s row and column penalties.

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5ii6 elect the row or column with the largest penalty and circle this value. 1n

case of a tie, select that row or column that allows the greatest movement of 

units.

5iii6 8ssign the largest possible allocation within the restrictions of the row

and column re)uirements to the lowest cost cell for the row or columnselected in step 5ii6.

5iv6 9ross out any column or row satis"ed by the assignment made in the

prior step.

5v6 Repeat the steps 5r6 to 5iv6 until all allocations have been made.

Ksing the above steps, the inital solution to the problem of erfect

manufacturing 9ompany is as follows%

 Thus, the initial solution is

 Transport from factory 8 to warehouse :%=;; units

 Transport from factory L to warehouse %

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and the corresponding cost of transportation is ;;;;.

'(4

a( )*at is #nte&er ro&rammin& ro3-em @#: ,

3( Ex+-ain ;omor?s a-- inte&er a-&orit*m of an #(

Ans.er

=ene #nte&er ro&rammin& ro3-em

 The 1 is a special case of 2inear rogramming roblem 526, where all or

some variables are constrained to assume non-negative integer values. 1n

2, the decision variables as well as slac! or surplus variables were allowedto ta!e any real or fractional value. Nowever, there are certain real life

problems in which the fractional value of the decision variables has no

signi"cance. 7or e$ample, it does not ma!e sense saying .: men wor!ing on

a pro0ect or .M machines in a wor!shop. The integer solution to a problem

can, however, be obtained by rounding o# the optimum value of the

variables to the nearest integer value. This approach can be easy in terms of 

economy of e#ort, time and cost that might be re)uired to derive an integer

solution but this solution may not satisfy all the given constraints. econdly,

the value of the ob0ective function so obtained may not be an optimal value.

1nteger programming techni)ues come to our rescue during such scenarios.

"+es of #nte&er ro&rammin& ro3-ems

2inear integer programming problems can be classi"ed into three categories%

. ure @a-- inte&er +ro&rammin& +ro3-ems: Nere, all decision

variables are restricted to integer values.

3. Mixed inte&er +ro&rammin& +ro3-ems Nere some, but not all, of the

decision variables are restricted to integer values.

>. Cero – one inte&er +ro&rammin& +ro3-ems Nere, all decision

variables are restricted to integer values of ; and .

A-- # A-&orit*m

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 The iterative procedure for the solution of integer programming problem is

as follows%

7igure depicts the iterative procedure of 1.

Ste+ 1 9onvert the minimization 1 into ma$imization form. 1gnore the

integrality condition.

Ste+ 2 1ntroduce the slac! or surplus variables, if needed to convert the ine)uations into e)uations and obtain the optimum solution of the given 2

by using simple$ algorithm.

Ste+

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b6 1f the optimum solution does not include all integer values, then proceed

to the ne$t step.

Ste+ 4 P$amine the constraint e)uations corresponding to the current

optimum solution.

Ste+ 5 P$press the negative fractions if any, in the !th row of the optimum

simple$ table as the sum of a negative integer and a non-negative fraction.

Ste+ D 7ind the Qomorian constraint

Ste+ tart with a new set of e)uation constraints. 7ind the new optimum

solution by dual simple$ algorithm, that is, choose a variable to enter into

the new solution having the smallest ratio% H590– 406 yi0C yi0D;S so that Qsla

56 is the initial leaving basic variable.

Ste+ 8 1f the new optimum solution for the modi"ed 2 is an integer

solution, it is also feasible and optimum for the given 1. 1f it is not an

integer solution, then return to step < and repeat the process until an

optimum feasible integer solution is obtained.

'5( A: Ex+-ain Monte $ar-o Simu-ations(

Ans.er 

&onte 9arlo simulations, a statistical techni)ue used to model probabilistic

5or stochasticU6 systems and establish the odds for a variety of outcomes.

 The concept was "rst popularized right after /orld /ar 11, to study nuclear

"ssionC mathematician tanislaw Klam coined the term in reference to an

uncle who loved playing the odds at the &onte 9arlo casino 5then a world

symbol of gambling, li!e 2as Iegas today6. Today there are multiple types of 

&onte 9arlo simulations, used in "elds from particle physics to engineering,"nance and more.

 To get a handle on a &onte 9arlo simulation, "rst consider a scenario where

we do not need one% to predict events in a simple, linear system. 1f you !now

the precise direction and velocity at which a shot put leaves an Olympic

athlete’s hand, you can use a linear e)uation to accurately forecast how far

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it will Wy. This case is a deterministic one, in which identical initial conditions

will always lead to the same outcome.

B: A $om+an +roduces 150 cars( But t*e +roduction rate varies.it* t*e distri3ution(

roduct

ion rate

14 148 14F 150 151 152 15<

ro3a3i

-it

0(05 0(10 0(15 0(20 0(

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3 :< :; - -> :; :; - -< VM :> > -: =: :3 3 -M >< :; - -

>; :; - -= ;3 V M< : -;

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the solution, but it reduces the order of the pay-o# matri$. uccessive

reduction of the order using dominance property helps in solving games.

3(

$onstituents of a 'ueuin& Sstem

1n the previous section, you learnt the operating characteristics of a )ueuing

system. Xou will now learn the constituents of a )ueuing system. The

constituents of a )ueuing system include arrival pattern, service facility and

)ueue discipline.

• Arriva- +attern 1t is the average rate at which the customers arrive.

• Service faci-it P$amining the number of customers served at a time

and the statistical pattern of time ta!en for service at the service

facility.

• 'ueue disci+-ine% The common method of choosing a customer for

service amongst those waiting for service is Y7irst 9ome 7irst erve’.

c(

E!" and $M 

Loth are based on the networ! representation of activities and their

scheduling, which determines the most critical activities to be controlled in

order to meet the completion date of the pro0ect.

E!"

• PRT was developed in connection with an Research and Jevelopment

5RZJ6 wor!. Therefore, it had to cope with the uncertainties that are

associated with RZJ activities. 1n PRT, the total pro0ect duration is

regarded as a random variable. Therefore, associated probabilities are

calculated in order to characterize it.

• 1t is an event-oriented networ! as in the analysis of a networ!,

emphasis is given on the important stages of completion of a tas!

rather than the activities re)uired to be performed to reach a particular

event or tas!.

$M

• 9& was developed in connection with a construction pro0ect, which

consisted of routine tas!s whose resource re)uirements and duration

were !nown with certainty. Therefore, it is basically deterministic.

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• 9& is suitable for establishing a trade-o# for optimum balancing

between schedule time and cost of the pro0ect.

• 9& is used for pro0ects involving activities of repetitive nature.

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