201502-Semester II-MB0048-Operations Research-DE.docx

8/19/2019 201502-Semester II-MB0048-Operations Research-DE.docx

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MB0048 - OPERATIONS RESEARCH

ROLL NO.: 1408011810

Ques. 1 Discuss the methodology of Operations Research. Explain in brief the phases of Operations Research.

Answer :

Meaning of Operations Research

Operations research OR! is an analytical method of problem-solving and decision-making that is usefulin the management of organizations. In operations research, problems are broken down into basiccomponents and then solved in defined steps by mathematical analysis.

Methoo!og" of Operations Research #

1. Operational Research echni!ues. "ome methodological aspects of operational research, andsome of the main OR techni!ues, including# $ritical %ath &nalysis, %roduction, "cheduling, 'arkov$hains, (ueueing heory.

). 'athematical %rogramming. *inear programming# from the most basic introduction to sufficient

conditions for optimality+ duality+ sensitivity of the solution+ discovery of the solution to smallproblems by graphical methods, and proof of optimality by testing the sufficient conditions.

Phases of Operations Research

"tep #: Defining the $roblem

It is often found that the real-world problems are described in very broad terms. y problem definition, we

mean to gather the following necessary information concerning the problem#

• Recognition that a problem eists

• etermining its magnitude

• efining it precisely

• /oting what symptoms are

"tep %: De&eloping the 'odel

• ecision 0controllable variable or simply variable variables.


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• 2ncontrollable variables

• Result variables

"tep (: Obtaining #nput Data

Once the model has been developed, the net step is to obtain the 0input data that are to used in themodel. ata collection is a critical part of an OR study, and therefore must be checked with care and

diligence. ata can be collected from several sources#

• 3rom organisation records and documents.

• 3rom interviews with employees or other persons related to the organisation and the study at hand.

• 3rom research studies and 4ournals.

"tep ): "ol&ing the 'odel

&fter constructing the re!uired model and collecting the necessary input data, we are ready now to solvethe model.

"tep *: +esting and Analysing the "olution

&fter the solution has been obtained, it should be completely tested. "ince accuracy of the solutiondepends on the accuracy of the input data and the model, it re!uires testing. Inaccurate data will lead to anincorrect solution. It is said that garbage-in, garbage-out.

"tep ,: #mplementing the Results

In simple words, implementation can be defined as putting a solution to work. Implementation is the mostrewarding aspect of an OR pro4ect, but it can also be difficult. here is no golden rule of implementation, buteperience has shown that certain conditions, such as those given below, foster a situation that facilitatesimplementation#

• "upport from top management in the organisation.

• &vailability of accurate and ade!uate data.

• &vailability of sufficient time to analyse a real problem with a sophisticated approach.

%rovision of accurate and timely information to the decision maker.


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Ques. % a. Explain the graphical method of sol&ing -inear $rogramming $roblem.

Answer :

$raphica! %etho of so!&ing !inear progra%%ing pro'!e%

1 Initially we draw the coordinate system correlating to an ais the variable , and the other ais tovariable y, as can see in the figure.

) 5e mark, in these ais, a numerical scale appropriate to the values it can take the variablesaccording to the constraints of the problem. o do this work, for each constraint we must to void allvariables ecept the related to a certain ais, so we establishing the right value for such ais. hisprocess must be done for every ais.

6 3ollowing, we represent all constraints. 5e take the first one and we draw the line that is obtainedby considering the constraint as an e!uality. In the figure, this is represented with the &- edge, andthe region that defines this constraint is shown in 78**O5 color. 5e repeat the process with theother restrictions, limiting *28 and R8 regions for the second and third constraint respectively.

9 3easible region is determined for the intersection of every region defined by the constraints and thenon-negativity condition of each variable, that is, both ais. his feasible region is represented bythe O-3-:-;-$ polygon, in


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O (0,0) 0

C (0,14) 28

G (3,12) 33

H (6,6) 30

F (8,0) 24

b. A paper mill produces two grades of paper &i./ 0 and . 2ecause of raw material

restrictions/ it cannot produce more than )33 tons of grade 0 paper and (33 tons of grade

paper in a wee4. +here are 1,3 production hours in a wee4. #t re5uires 3.%3 and 3.)3 hours

to produce a ton of grade 0 and papers. +he mill earns a profit of Rs. %33 and Rs. *33 per

ton of grade 0 and paper respecti&ely. 6ormulate this as a -inear $rogramming $roblem.

Answer :

(or%)!ation of *PP

he procedure for mathematical formulation of a linear programming problem consists of the followingma4or steps#

"tep 1: he key decision is to determine the weekly rate of production for the three types of clothes.

"tep %: *et us designate the weekly production of suiting, shirting, and woollens as 1 meters, )

meters, and 6 meters respectively.


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"tep (: &s it is not possible to produce negative !uantities, feasible alternatives are sets of values

of 1, ), and 6 satisfying 1, ), and 6 @ A.

"tep ): he constraints are the limited availability of three operational departments. One meter of

suiting re!uires 6 minutes of weaving. he !uantity being 1 meters, the re!uirement for suitingalone will be 61 units. "imilarly, ) meters of shirting and 6 meters of woollen will re!uire 9) and66 minutes respectively. hus, the total re!uirement of weaving will be 6 1=9)=66, which should

not eceed the available 6BAA minutes. herefore, the labour constraint becomes 61=9)=66C6BAA.

"imilarly, the constraints for the processing department and packing departments are )1=)=66C )9AAand 1=6)=66C 9DAA respectively.

"tep *: he ob4ective is to maimise the total profit from sales. &ssuming that whatever is produced

is sold in the market, the total profit is given by the linear relation z? )1=9)=66.

he linear programming problem can thus be put in the following algebraic format#

3ind 1, ), and 6 so as to maimize

Ques. ( a. Explain how to sol&e the degeneracy in transportation problems.

Answer :

+egenerac" In Transportation Pro'!e%

& basic solution to an m-origin, n destination transportation problem can have at the most m=n-1 positive

basic variables 0non-zero, otherwise the basic solution degenerates. It follows that whenever the number of basic cells is less than m = n E 1, the transportation problem is a degenerate one. he degeneracy candevelop in two ways#

7ase 1 8 he degeneracy develops while determining an initial assignment via any one of the initial

assignment methods discussed earlier. o resolve degeneracy, you must augment the positivevariables by as many zero-valued variables as is necessary to complete the re!uired m = n E 1basic variable. hese zero-valued variables are selected in such a manner that the resulting m = n E1 variable constitutes a basic solution. he selected zero valued variables are designated byallocating an etremely small positive value F to each one of them. he cells containing theseetremely small allocations are then treated like any other basic cells.

he FGs are kept in the transportation table until temporary degeneracy is removed or until the optimumsolution is attained, whichever occurs first. &t that point, we set each F ? A.


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7ase % 8 he degeneracy develops at the iteration stage. his happens when the selection of the

entering variable results in the simultaneous drive to zero of two or more current 0pre-iteration basicvariables. o resolve degeneracy, the positive variables are augmented by as many zero-valuedvariables as it is necessary to complete m=n-1 basic variables. hese zero-valued variables areselected from among those current basic variables, which are simultaneously driven to zero. herest of the procedure is eactly the same as discussed in case 1.

9ote 8 he etremely small value F is infinitely small and it never affects the value it is added to or subtracted from. Introduce H∈G in unallocated minimum cost cell to avoid forming a loop.

b. Explain the procedure of 'OD# method of finding solution through optimality test.

Ans.

Transportation A!gorith% ,MO+I Metho

& feasible solution has to be found always. Rather than determining a first approimation by a directapplication of the simple method, it is more efficient to work with the transportation table. hetransportation algorithm is the simple method specialised to the format of table involving the followingsteps#

i 3inding an initial basic feasible solution

ii esting the solution for optimality

iii Improving the solution, when it is not optimal

iv Repeating steps 0ii and 0iii until the optimal solution is obtained

The so!)tion to transportation pro'!e% is o'taine in t.o stages

In the first stage, we find the basic feasible solution using any of the following methods#

1 /orth-west corner rule

) 'atri minima method or least cost method

6


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Ques. ) a. Explain the steps in&ol&ed in ungarian method of sol&ing Assignment problems.

Answer :

H)ngarian Metho A!gorith%

:ungarian method algorithm is based on the concept of opportunity cost and is more efficient in solvingassignment problems. he following steps are adopted to solve an &% using the :ungarian methodalgorithm.

• "tep 1: %repare row ruled matri by selecting the minimum values for each row and subtract it from

the other elements of the row.

• "tep %: %repare column-reduced matri by subtracting minimum value of the column from the other

values of that column.

• "tep (: &ssign zero row-wise if there is only one zero in the row and cross 0 or cancel other zeros

in that column.

• "tep )# &ssign column wise if there is only one zero in that column and cross other zeros in that

row.

• "tep *: Repeat steps 6 and 9 till all zeros are either assigned or crossed. If the number of

assignments is e!ual to number of rows present, you have arrived at an optimal solution, if not,proceed to step B.

• "tep ,: 'ark 0 the unassigned rows. *ook for crossed zero in that row. 'ark the column

containing the crossed zero. *ook for assigned zero in that column. 'ark the row containingassigned zero. Repeat this process till all the makings are done.

• "tep ;: raw a straight line through unmarked rows and marked column. he number of straight

line drawn will be e!ual to the number of assignments made.

• "tep


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b! 6ind an optimal solution to an assignment problem with the following cost matrix

Answer :

&ssuming each machine '2" be in production simultaneously 0and can only do one task at a time, I

drew a new table which listed the 4obs each machine could perform in order of efficiency#

'₁L 9, 6, ), 1

'₂L 1, ), 6, 9

'₃L )M6, 1M9

'₄L ), 1, 6, 9

he principle I am going by is that if all machines must be running, the most efficient settings would be

those that Nminimize inefficiencyN.

1 '₁ does ₄ most efficiently.

) '₂ does ₁ most efficiently.

6 /ote that '₃ and '₄ can both do ₂ most efficiently, but they canPt both do the same 4ob at the same

time. "ince '₃ does ₃ e!ually as well as ₂, we will let '₃ do ₃...

9 ...and let '₄ do ₂.

otal $ost then is

$ ? '₁₄ = '₂₁ = '₃₃ = '₄₂

$ ? J = 6 = 1 = 6

C = 14


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Ques. * a. Explain 'onte 7arlo "imulation.

Answer :

Monte-Car!o Si%)!ation

he 'onte-$arlo method is a simulation techni!ue in which statistical distribution functions are created byusing a series of random numbers. his approach has the ability to develop many months or years of datain a matter of few minutes on a digital computer.

he method is generally used to solve the problems that cannot be ade!uately represented bymathematical models or where solution of the model is not possible by analytical method.

he 'onte-$arlo simulation procedure can be summarised in the steps depicted in below figure#

6ig. : 'onte87arlo "imulation $rocedure

"tep 1: efine the problem#

a Identify the ob4ectives of the problem.

b Identify the main factors that have the greatest effect on the ob4ectives of the problem.

"tep %: $onstruct an appropriate model#

a "pecify the variables and parameters of the model.

b 3ormulate the appropriate decision rules, i.e., state the conditions under which the eperiment is to beperformed.


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c Identity the type of distribution that will be used. 'odels use either theoretical distributions or empiricaldistributions to state the patterns of occurrence associated with the variables.

d "pecify the manner in which time will change.

e efine the relationship between the variables and parameters.

"tep (: %repare the model for eperimentation#

a efine the starting conditions for the simulation.

b "pecify the number of runs of simulation to be made.

"tep ): 2sing steps 1 to 6, eperiment with the model#

a efine a coding system that will correlate the factors defined in step 1 with the random numbers to begenerated for the simulation.

b "elect a random number generator and create the random numbers to be used in the simulation.

c &ssociate the generated random numbers with the factors identified in step1 and coded in step 90a.

"tep *: "ummarise and eamine the results obtained in step 9.

"tep ,: 8valuate the results of the simulation.

"tep ;: 3ormulate proposals for advice to management on the course of action to be adopted and

modify the model, if necessary.

b. A 7ompany produces 1*3 cars. 2ut the production rate &aries with the distribution.

&t present the track will hold 1KA cars. 2sing the following random numbers determine the average number of cars waiting for shipment in the company and average number of empty space in the truck. Random/umbers D), K9, KA, QB, DK, 69, 6A, A), B9, 9J.

Answer :

Be!o. Ta'!e +epicts The Pro)ction Rate An Pro'a'i!it"


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Ques. , a. Explain the dominance principle in game theory.

Answer :

+o%inance

In a rectangular game, the pay-off matri of player & is pay-off in one specific row ( r row ) th eceeding thecorresponding pay-off in another specific row( s row ) th . his means that whatever course of action isadopted by player , for &, the course of action Ar yields greater gains than the course of action As .herefore, Ar is a better strategy than As irrespective of Gs strategy. :ence, you can say that Ar dominates As .

&lternatively, if each pay-off in a specific column ( p column ) th is less than the corresponding pay-off in

another specific column( q column ) th , it means strategy B p offers minor loss than strategy Bq irrespective

of &Gs strategy. :ence, you can say that B p dominatesBq . herefore, you can say that#

a In the pay-off matri, if each pay-off in r row th is greater than 0or e!ual to the correspondingpay-off in the s row th , &r dominates &s .

b In the pay-off matri, if each pay-off in p column th is less than 0or e!ual to the

corresponding pay-off in the q column th , B p dominates Bq .

&t times, a conve combination of two or more courses of action may dominate another course of action.

5henever a course of action 0say As or Bq is dominated by others, then that course of action 0 As or Bq can be deleted from the pay-off matri. "uch a deletion will not affect the choice of the solution, but itreduces the order of the pay-off matri. "uccessive reduction of the order using dominance property helps

in solving games.

b. Describe the 7onstituents of a Queuing "ystem.

Answer :

Constit)ents of a /)e)ing S"ste%

he constituents of a !ueuing system include arrival pattern, service facility and !ueue discipline.

Arri&al pattern: It is the average rate at which the customers arrive.

"er&ice facility: 8amining the number of customers served at a time and the statistical pattern of

time taken for service at the service facility.

Queue discipline# he common method of choosing a customer for service amongst those waiting

for service is H3irst $ome 3irst "erveG.


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c. Differentiate between $ER+ and 7$'

Answer :

+ifference 'et.een PERT an CPM

here are no essential differences between %8R and $%' as both of them share in common thedetermination of a critical path. oth are based on the network representation of activities and their scheduling, which determines the most critical activities to be controlled in order to meet the completiondate of the pro4ect.

PERT

"ome key points of %8R are as follows#

%8R was developed in connection with an Research and evelopment 0R work. herefore, it

had to cope with the uncertainties that are associated with R activities. In %8R, the total pro4ectduration is regarded as a random variable. herefore, associated probabilities are calculated inorder to characterise it.

It is an event-oriented network as in the analysis of a network, emphasis is given on the important

stages of completion of a task rather than the activities re!uired to be performed to reach aparticular event or task.

%8R is normally used for pro4ects involving activities of non-repetitive nature in which timeestimates are uncertain.

It helps in pinpointing critical areas in a pro4ect, so that necessary ad4ustment can be made to meet

the scheduled completion date of the pro4ect.

CPM

$%' was developed in connection with a construction pro4ect, which consisted of routine tasks

whose resource re!uirements and duration were known with certainty. herefore, it is basicallydeterministic.

$%' is suitable for establishing a trade-off for optimum balancing between schedule time and cost

of the pro4ect.


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$%' is used for pro4ects involving activities of repetitive nature.

Documents

201502-Semester II-MB0048-Operations Research-DE.docx