Or Notes Mcom

8/17/2019 Or Notes Mcom

1/80

OPERATIONS RESEARCH

Describe in details the OR approach of problem solving. What are the limitations of

the Operations Research?

Answer:

OR approach of problem solving:

Optimization is the act of obtaining the best result under any given

circumstance. In various practical problems we may have to take many technical

or managerial decisions at several stages. The ultimate goal of all such decisions

is to either maximize the desired benefit or minimize the effort reuired. !e make

decisions in our every day life without even noticing them. "ecision#making is

one of the main activities of a manager or executive.

In simple situations decisions are taken simply by common sense$ sound

%udgment and expertise without using any mathematics. &ut here the decisions we

are concerned with are rather complex and heavily loaded with responsibility.

'xamples of such decision are finding the appropriate product mix when there are

large numbers of products with different profit contributions and production

reuirement or planning public transportation network in a town having its own

layout of factories$ apartments$ blocks etc. (ertainly in such situations also

decision may be arrived at intuitively from experience and common sense$ yet

they are more %udicious if backed up by mathematical reasoning.

The search of a decision may also be done by trial and error but such a search

may be cumbersome and costly. )reparative calculations may avoid long and

costly research. "oing preparative calculations is the purpose of Operations

research. Operations research does mathematical scoring of conseuences of a

decision with the aim of optimizing the use of time$ efforts and resources and

avoiding blunders.

The application of Operations research methods helps in making decisions in such

complicated situations. 'vidently the main ob%ective of Operations research is to

provide a scientific basis to the decision#makers for solving the problems

involving the interaction of various components of organization$ by employing a

team of scientists from different disciplines$ all working together for finding a

*


2/80

solution which is the best in the interest of the organization as a whole. The

solution thus obtained is known as optimal decision.

+eatures of Operations Research:

• It is ,ystem oriented:

OR studies the problem from over all points of view of organizations or

situations since optimum result of one part of the system may not be optimum for

some other part.

-It imbibes Inter disciplinary team approach:

,ince no single individual can have a thorough knowledge of all fast

developing scientific know#how$ personalities from different scientific and

managerial cadre form a team to solve the problem.

-It makes use of ,cientific methods to solve problems.

-OR increases the effectiveness of a management "ecision making ability.

-It makes use of computer to solve large and complex problems.

-It gives /uantitative solution.

-It considers the human factors also.

The first and the most important reuirement is that the root problem should be

identified and understood. The problem should be identified properly$ this indicates

three ma%or aspects:

• 0 description of the goal or the ob%ective of the study.

• 0n identification of the decision alternative to the system.

• Recognition of the limitations$ restrictions and reuirements of the system.

1imitations of OR:

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

factors.

• 2agnitude of computation: 2odern problem involve large number of variables

and hence to find interrelationship$ among makes it difficult.

• 3on uantitative factors and 4uman emotional factor cannot be taken into

account.

5


3/80

• There is a wide gap between the managers and the operation researches.

• Time and 2oney factors when the basic data is sub%ected to freuent changes then

incorporation of them into OR models are a costly affair.

• Implementation of decisions involves human relations and behavior

Describe in details the OR approach of problem solving. What are the

limitations of the Operations Research?

Answer:

OR approach of problem solvingOptimization is the act of obtaining the best result under any given circumstance.

In various practical problems we may have to take many technical or managerial

decisions at several stages. The ultimate goal of all such decisions is to either

maximize the desired benefit or minimize the effort reuired. !e make decisionsin our every day life without even noticing them. "ecision#making is one of the

main activity of a manager or executive. In simple situations decisions are taken

simply by common sense$ sound %udgment and expertise without using anymathematics. &ut here the decisions we are concerned with are rather complex

and heavily loaded with responsibility. 'xamples of such decision are finding the

appropriate product mix when there are large numbers of products with different profit contributions and productional reuirement or planning public

transportation network in a town having its own layout of factories$ apartments$

blocks etc. (ertainly in such situations also decision may be arrived at intuitivelyfrom experience and common sense$ yet they are more %udicious if backed up by

mathematical reasoning. The search of a decision may also be done by trial anderror but such a search may be cumbersome and costly. )reparative calculations

may avoid long and costly research. "oing preparative calculations is the purposeof Operations research. Operations research does mathematical scoring of

conseuences of a decision with the aim of optimizing the use of time$ efforts and

resources and avoiding blunders.

The application of Operations research methods helps in making decisions in such

complicated situations. 'vidently the main ob%ective of Operations research is to provide a scientific basis to the decision#makers for solving the problems

involving the interaction of various components of organization$ by employing a

team of scientists from different disciplines$ all working together for finding asolution which is the best in the interest of the organization as a whole.

The solution thus obtained is known as optimal decision. The main features

of OR are:

6


4/80

-It is ,ystem oriented: OR studies the problem from over all points of view of

organizations or situations since optimum result of one part of the system may not

be optimum for some other part.-It imbibes Inter disciplinary team approach. ,ince no single individual can

have a thorough knowledge of all fast developing scientific know#how$

personalities from different scientific and managerial cadre form a team to solvethe problem.

-It makes use of ,cientific methods to solve problems.

-OR increases the effectiveness of a management "ecision making ability.-It makes use of computer to solve large and complex problems.

-It gives /uantitative solution.

-It considers the human factors also.

The first and the most important reuirement is that the root problem should be

identified and understood. The problem should be identified properly$ this

indicates three ma%or aspects:

7*8 0 description of the goal or the ob%ective of the study$758 0n identification of the decision alternative to the system$ and

768 0 recognition of the limitations$ restrictions and reuirements of the system.

imitations of OR

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

money factors.

-2agnitude of computation: 2odern problem involve large number of variables

and hence to find interrelationship$ among makes it difficult.-3on uantitative factors and 4uman emotional factor cannot be taken into

account.

-There is a wide gap between the managers and the operation researches.-Time and 2oney factors when the basic data is sub%ected to freuent changes

then incorporation of them into OR models are a costly affair.

-Implementation of decisions involves human relations and behaviour

Operation Research is an aid for the e!ecutive in making his decisions b" providing

him the needed #uantitative information$ based on scientific method anal"sis.

9


5/80

Discuss.

Answer:# The Operations Research may be regarded as a tool which is utilized to

increase the effectiveness of management decisions. In fact$ OR is the ob%ective feeling of the administrator 7decision#maker8. ,cientific method of OR is used to understand and

describe the phenomena of operating system.

OR models explain these phenomena as to what changes take place under altered

conditions$ and control these predictions against new observations. +or example$ OR maysuggest the best locations for factories$ warehouses as well as the most economical means

of transportation. In marketing$ OR may help in indicating the most profitable type$ use

and size of advertising campaigns sub%ect to the final limitations.

The advantages of OR study approach in business and management decision making may

be classified as follows:

%. &etter 'ontrol. The management of big concerns finds it much costly to provide

continuous executive supervisions over routine decisions. 0n OR approach directs theexecutives to devote their attention to more pressing matters. +or example$ OR approach

deals with production scheduling and inventory control.

5. &etter 'o(ordination. ,ome times OR has been very useful in maintaining the law

and order situation out of chaos. +or example$ an OR based planning model becomes avehicle for coordinating marketing decisions with the limitation imposed on

manufacturing capabilities.

). &etter *"stem. OR study is also initiated to analyses a particular problem of decision

making such as establishing a new warehouse. 1ater$ OR approach can be further

developed into a system to be employed repeatedly. (onseuently$ the cost ofundertaking the first application may improve the profits.

+. &etter Decisions. OR models freuently yield actions that do improve an intuitive

decision making. ,ometimes$ a situation may be so complicated that the human mind cannever hope to assimilate all the important factors without the help of OR and computer

analysis.

,-AT/TAT/01 T1'2/,-1* O3 OR:

0 brief account of some of the important OR models providing needed uantitative

information base on scientific method analysis are given below:

%. Distribution 4Allocation5 6odels: "istribution models are concerned with the

allotment of available resources so as to minimize cost or maximize profit sub%ect to

prescribed restrictions. 2ethods for solving such type of problems are known asmathematical programming techniques. !e distinguish between liner and non#liner

programming problems on the basis of linearity and non#linearity of the ob%ective

function andor constraints respectively. In linear programming problems$ the ob%ective

;


6/80

function is linear and constraints are also linear ineualitieseuations. Transportation and

0ssignment models can be viewed as special cases of linear programming. These can be

solved by specially devised procedures called Transportation and Assignment

Techniques.

In case the decision variables in a linear programming problem are restricted to eitherinteger or zero#one value$ it is known as Integer and


7/80

. Replacement 6odels: The models deal with the determination of optimum

replacement policy in situations that arise when some items or machinery needreplacement by a new one. Individual and group replacement polices can be used in the

case of such euipments that fail completely and instantaneously.

. *imulation 6odels: ,imulation is a very powerful techniue for solving muchcomplex models which cannot be solved otherwise and thus it is being extensively

applied to solve to solve a variety of problems. This techniue is more useful when

following two types of difficulties may arise:

4i5 The number of variables and constraint relationships may be so large that it is notcomputationally feasible to pursue such analysis.

4ii5 ,econdly$ the model may be much away from the reality that no confidence can be

placed on the computational results.

In fact$ such models are solved by simulation techniues where no other method is

available for its solution.

"efinition and 'xplanation:Linear programming is a mathematical technique which permits determinationof the best use of available resources.

It is a valuable aid to management because it provides a systematic and efficientprocedure which can be used as a guide in decision making. The heart of

management's responsibility is the best or optimum use of limited resources that

include money, personnel, materials, facilities, and time.

&asic (oncept of 1inear )rogramming )roblemThe Ob%ective +unction is a linear function of variables which is to be optimised i.e.$

maximised or minimised. e.g.$ profit function$ cost function etc. The ob%ective function

may be expressed as a linear expression

(onstraints

A


8/80

0 linear euation represents a straight line. 1imited time$ labour etc. may be expressed as

linear ineuations or euations and are called constraints.

e.g.$ If 5 tables and 6 chairs are to be made in not more than *B hours and * table is madein x hours while * chair is made in y hours$ then this constraint may be written as

0 linear euation is also called a linear constraint as it restricts the freedom of choice ofthe values of x and y.

Optimisation

0 decision which is considered the best one$ taking into consideration all the

circumstances is called an optimal decision. The process of getting the best possibleoutcome is called optimisation. The best profit is the maximum profit. 4ence optizmizing

the profit would mean maximising the profit. Optimising the cost would mean

minimising the cost as this would be most favourable.

,olution of a 1))

0 set of values of the variables x*$ x5$C.xn which satisfy all the constraints is called thesolution of the 1)).

+easible ,olution

0 set of values of the variables x*$ x5$ x6$C.$xn which satisfy all the constraints and also

the non#negativity conditions is called the feasible solution of the 1)).

Optimal ,olution

The feasible solution$ which optimises 7i.e.$ maximizes or minimizes as the case may be8the ob%ective function is called the optimal solutio

Simplex method is considered one of the basic techniques from which many linear

programming techniques are directly or indirectly derived. The simplex method is aniterative, stepwise process which approaches an optimum solution in order to reach

an objective function of maximiation or minimiation.

inear 8rogramming 8roblems @ 3ormulation

1inear )rogramming is a mathematical techniue for optimum allocation of limited or

scarce resources$ such as labour$ material$ machine$ money$ energy and so on $ to severalcompeting activities such as products$ services$ %obs and so on$ on the basis of a given

criteria of optimality.

The term inearB is used to describe the proportionate relationship of two or morevariables in a model. The given change in one variable will always cause a resulting

proportional change in another variable.

D


9/80

The word $ programmingB is used to specify a sort of planning that involves the

economic allocation of limited resources by adopting a particular course of action or

strategy among various alternatives strategies to achieve the desired ob%ective.

4ence$ inear 8rogramming is a mathematical techniue for optimum allocation of

limited or scarce resources$ such as labour$ material$ machine$ money energy etc.

*tructure of inear 8rogramming model.

The general structure of the 1inear )rogramming model essentially consistsof three components.

i8 The activities 7variables8 and their relationships

ii8 The ob%ective function and

iii8 The constraints

The activities are represented by E*$ E5$ E6 CC..En.

These are known as "ecision variables.

The ob%ective function of an 1)) 71inear )rogramming )roblem8 is a mathematical

representation of the ob%ective in terms a measurable uantity such as profit$ cost$revenue$ etc.

Optimize 72aximize or 2inimize8


10/80

variable respectively. The ob%ective function is the direct sum of the individual

contributions of the different variables

inearit"

0ll relationships in the 1) model 7i.e. in both ob%ective function and constraints8 must be

linear.

Ceneral 6athematical 6odel of an 88

Optimize 72aximize or 2inimize8


11/80

1et the primal problem be

2aximize < F (* E* G (5 E5 G CC G (nEn

,ub%ect to constraints$a**E*G a *5E5GCCCCCCG a *nEn $ b*

a5*E*G a 55E5GCCCCCCG a 5nEn $b5

a6*E*G a 65E5GCCCCCCG a 6nEn $b6am* E*G a m5E5GCCCCCCG a mnEn $bm

and E*$ E5 C.En J B

Then its "ual is

2inimize ? F b*!* G b5!5G b6!6 G .......G bm!m

,ub%ect to constraints$

a**! * G a5*!5 G a6*!6 G................G am*!m J (*a*5! * G a55!5 G a65!6 G................G am5!m J (5

a*6! * G a56!5 G a66!6 G................G am6!m J (6

a*n! * G a5n!5 G a6n!6 G................G amn!m J (n

! *$ !5$ !6 ......!m J B

'xample.*!rite the "ual of the following 1))

2in < F 5E5G ;E6

E*G E5 J 55E*G E5 G @E6 @

E*# E5 G6E6 F 9

and E*$ E5$E6 J B

Rearrange the constraints into a standard form$ we get

2in < F BE* G 5E5G ;E6

,ub%ect to constraints$E*G E5 G BE6J 5

#5E*# E5 # @E6 J #@

E*# E5 G6E6 J 9#E* G E5 #6E6 J #9

and E*$ E5$E6 J B

The "ual of the above primal is as follows2ax.? F 5!* #@!5G 9!6 9!9


! * #5!5 G !6 !9 B! * # !5 # !6 G !9 5

B! * # @!5 G 6!6# 6!9 ;

! *$ !5$ !6$!9 J B

2ax.? F 5!* #@!5G 97!6 !98


! * #5!5 G 7!6 !9 8 B

**


12/80

! * # !5 # !6 G !9 5

B! * # @!5 67!6# !9 8 ;

! *$ !5$ !6$!9 J B

2ax.? F 5!* #@!5G 9!;

,ub%ect to constraints$! * #5!5 G !; B

! * # !5 !; 5

B! * # @!5 6!; ;! *$ !5$ J B $ !; is unrestricted in sign

'xample.5

!rite the "ual of the following 1))2in < F 9E* G ;E5# 6E6


E*G E5 G E6 F 55

6E*G ;E5 # 5E6 @;E*G AE5 G9E6 J *5B

E* $ E5 J B and E6 is unrestricted

,ince E6 is Knrestricted$ replace E6 with 7E9 # E; 8 and

bring the problem into standard form2in < F 9E* G ;E5# 67E9 # E;8


E*G E5 G 7E9 # E;8 J55

#E*# E5 # 7E9 # E;8 J# 55#6E*# ;E5 G 57E9 # E;8 J #@;

E*G AE5 G97E9 # E;8 J *5B

E* $ E5 $ E9 $E; J B

The "ual of the above primal is as follows

2ax.? F 557!* #!58# @;!6 G *5B!9,ub%ect to constraints$

! * #!5 # 6!6 G!9 9

! * # !5 # ;!6 G A!9 ;

! * # !5 G 5!6G 9!9 #6#! * G !5 # 5!6# 9!9 6

! *$ !5$ !6$!9 J B

2ax.? F 55!; # @;!6 G *5B!9


! ;# 6!6 G!9 9!; #;!6 G A!9 ;

! * # !5 G 5!6G 9!9 #6

#! * G !5 # 5!6# 9!9 6

! *$ !5$ !6$!9 J B

*5


13/80

What are the characteristics of the standard form of .8.8.? What is the standard

form of .8.8.? *tate the fundamental theorem of .8.8?

Answer:

Introduction:

In mathematics$ linear programming 71)8 is a techniue for optimization of a

linear ob%ective function$ sub%ect to linear euality and linear ineuality constraints.Informally$ linear programming determines the way to achieve the best outcome 7such as

maximum profit or lowest cost8 in a given mathematical model and given some list of

reuirements represented as linear euations.

2ore formally$ given a polyhedron 7for example$ a polygon8$ and a real#valuedaffine function defined on this polyhedron$ a linear programming method will find

a point on the polyhedron where this function has the smallest 7or largest8 value if such point exists$ by searching through the polyhedron vertices.

1inear programs are problems that can be expressed in canonical form:

2aximize

,ub%ect to

Represents the vector of variables 7to be determined8$ while and are vectors of 7known8 coefficients and is a 7known8 matrix of coefficients. The expression to be

maximized or minimized is called the ob%ective function 7 in this case8. The

euations are the constraints which specify a convex polytope over which theob%ective function is to be optimized.

1inear programming can be applied to various fields of study. 2ost extensively it

is used in business and economic situations$ but can also be utilized for some engineering

problems. ,ome industries that use linear programming models include transportation$energy$ telecommunications$ and manufacturing. It has proved useful in modeling diverse

types of problems in planning$ routing$ scheduling$ assignment$ and design.

Kses:

*6

http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Optimization_(mathematics)http://en.wikipedia.org/wiki/Linearhttp://en.wikipedia.org/wiki/Linearhttp://en.wikipedia.org/wiki/Objective_functionhttp://en.wikipedia.org/wiki/Linear_equalityhttp://en.wikipedia.org/wiki/Linear_inequalityhttp://en.wikipedia.org/wiki/Constraint_(mathematics)http://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Polyhedronhttp://en.wikipedia.org/wiki/Polyhedronhttp://en.wikipedia.org/wiki/Polygonhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Affine_functionhttp://en.wikipedia.org/wiki/Canonical_formhttp://en.wikipedia.org/wiki/Canonical_formhttp://en.wikipedia.org/wiki/Convex_sethttp://en.wikipedia.org/wiki/Polytopehttp://en.wikipedia.org/wiki/Optimization_(mathematics)http://en.wikipedia.org/wiki/Linearhttp://en.wikipedia.org/wiki/Objective_functionhttp://en.wikipedia.org/wiki/Linear_equalityhttp://en.wikipedia.org/wiki/Linear_inequalityhttp://en.wikipedia.org/wiki/Constraint_(mathematics)http://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Polyhedronhttp://en.wikipedia.org/wiki/Polygonhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Affine_functionhttp://en.wikipedia.org/wiki/Canonical_formhttp://en.wikipedia.org/wiki/Convex_sethttp://en.wikipedia.org/wiki/Polytopehttp://en.wikipedia.org/wiki/Mathematics


14/80

1inear programming is a considerable field of optimization for several reasons. 2any

practical problems in operations research can be expressed as linear programming

problems. (ertain special cases of linear programming$ such as network flow problemsand multi commodity flow problems are considered important enough to have generated

much research on specialized algorithms for their solution. 0 number of algorithms for

other types of optimization problems work by solving 1) problems as sub#problems.4istorically$ ideas from linear programming have inspired many of the central concepts

of optimization theory$ such as duality, decomosition, and the importance of conve!ity

and its generalizations. 1ikewise$ linear programming is heavily used in microeconomicsand company management$ such as planning$ production$ transportation$ technology and

other issues. 0lthough the modern management issues are ever#changing$ most

companies would like to maximize profits or minimize costs with limited resources.

Therefore$ many issues can boil down to linear programming problems.

,tandard form:

"tandard form is the usual and most intuitive form of describing a linear

programming problem. It consists of the following three parts:

• 0 linear function to be ma!imied

'.g. maximize

• 8roblem constraints of the following form

e.g.

• on(negative variables

e.g.

PERT#$PM 81RT @ 8rogram 1valuation E Review Techni#ue @ It is generally used for those

pro%ects where time reuired to complete various activities are not known as a riori. It is probabilistic model L is primarily concerned for evaluation of time. It is event oriented.

'86 @ 'ritical 8ath Anal"sis @ It is a commonly used for those pro%ects which are

repetitive in nature L where one has prior experience of handling similar pro%ects. It is a

deterministic model L places emphasis on time L cost for activities of a pro%ect.,teps for drawing ()2)'RT network:

*. 0nalyze L break up of the entire pro%ect into smaller systems i.e. specific

activities andor

*9

http://en.wikipedia.org/wiki/Operations_researchhttp://en.wikipedia.org/wiki/Microeconomicshttp://en.wikipedia.org/wiki/Operations_researchhttp://en.wikipedia.org/wiki/Microeconomics


15/80

events.

5. "etermine the interdependence L seuence of those activities.

6. "raw a network diagram. 9. 'stimate the completion time$ cost$ etc. for each activity.

;. Identify the critical path 7longest path through the network8.

@. Kpdate the ()2)'RT diagram as the pro%ect progresses.

etwork Representation:

'ach activity of the pro%ect is represented by arrow pointing in direction of progress of pro%ect. The events of thenetwork establish the precedence relationship among different

activities.

Three rules are available for constructing the network.

Rule %. Each activity is reresented %y one & only one, arrow'

Rule 7. Each activity must %e identified %y two distinct events & o two or more

activities can have the same tail and head events'

+ollowing figure shows how a dummy activity can be used to represent two concurrent

activities$ 0 L &. &y definition$ a dummy activity$ which normally is depicted by adashed arrow$ consumes no time or resources.

"ummy activity is a hypothetical activity which takes no resource or time to complete. Itis represented by broken arrowed line L is used for either distinguishing activities having

common starting L finishing events or to identify L maintain proper precedence

relationship between activities that are not connected by events.

*;


16/80

Inserting dummy activity in one four ways in the figure$ we maintain the concurrence of

0 L &$ and provide uniue end events for the two activities 7to satisfy Rule 58.Rule ). To maintain correct recedence relationshi, the followin questions must %e

answered as each activity is added to the network*(a) +hat activities must %e immediately recede the current activity

(%) +hat activities must follow the current activity

(c) +hat activities must occur concurrently with the current activity

The answers to these questions may require the use of dummy activities to ensure correct

recedences amon the activities' -or e!amle, consider the followin sement of a

roject*

*. 0ctivity ( starts immediately after 0 and & have been completed.5. 0ctivity ' starts only after & has been completed.

*@


17/80

)art 7a8 of the figure above$ shows the incorrect representation of the precedence

relationship because it reuires both 0 L & to be completed before ' can start. In part 7b8

the use of dummy rectifies situation.

umbering the 1vents 43ulkersonBs Rule5

*.The initial event which has all outgoing arrows with no incoming arrow is numberedM*N.

5. "elete all the arrows coming out from node M*N. This will convert some more nodes

into initial events. 3umber these events as 5$ 6$ 9$ C.6. "elete all the arrows going out from these numbered events to create more initial

events. 0ssign the next numbers to these events.

9. (ontinue until the final or terminal node$ which has all arrows coming in with no

arrow going out is numbered.

Determination of time to complete each activit":

The ()2 system of networks omits the probabilistic consideration and is based on a

,ingle Time 'stimate of the average time reuired to execute the activity.

In )'RT analysis$ there is always a great deal of uncertainty associated with the activitydurations of any pro%ect. Therefore$ te estimated time is better described by a probability

distribution than by a single estimate. Three time estimates 7 from %eta ro%a%ilitydistri%ution8 are made as follows:

*8 The Optimistic Time 'stimate 7to8: ,hortest possible time in which an activity can be

completed in ideal conditions. 3o provisions are made for delays or setbacks whileestimating this time.

58 The 2ost 1ikely Time 7tm8: It assumes that things go in normal way with few

setbacks.

68 The )essimistic Time 7tp8: The max. possible time if everything go wrong L abnormalsituations prevailed. 4owever$ ma%or catastrophes such as earthuakes$ labour troubles$

etc. are not taken into account.

The expected time 7mean time8 for each activity can be approximated using the weightedaverage i.e.

1!pected Time 4te5 F 7to G 9tm G tp8@

3orward 8ass 'omputation: It is the process of tracing the network from ,T0RT to'3". It gives the earliest start L finish times for each activity.

1arliest event time 41F5: The time that event j will occur if the preceding activities are

started as early as possible. '% is the ma!. of the sums 'i G ti% involving each

immediately precedent event i L intervening event ij'

&ackward 8ass 'omputation: It is the process of tracing the network starting from

10,T node L moving backward.

atest event time 4F5: The latest time that event i can occur without delayingcompletion of beyond its earliest time. 1i is the min. of the differences 1i # ti% involving

each immediately precedent event j L intervening event ij'

The critical path can be identified by determining the following four parameters for each

activity:- 1*T # earliest start time: the earliest time at which the activity can start given that all its

precedent activities must be completed first F 1i

*A


18/80

- 13T # earliest finish time$ eual to the earliest start time for the activity plus the time

reuired to complete the activity F 1*T4i(F5 G tiF

- 3T # latest finish time: the latest time at which the activity can be completed withoutdelaying 7beyond its targeted completion time8 the pro%ect F F- *T # latest start time$ eual to the latest finish time minus the time reuired to

complete the activity H 3T4i(F5 ( tiFCRITICAL PATH: The critical path is the path through the pro%ect network in which

none of the activities have float 7total float is .ero8 i.e. 0 critical path satisfies following 6

conditions:- ',T F1,T

- '+TF1+T

- '% 'i F 1% 1i F ti%

The duration of pro%ect is fixed by the time taken to complete the path through thenetwork with the greatest total duration. This path is known as critical path L activities

on it are known as critical activities. 0 delay in the critical path delays the pro%ect.

,imilarly$ to accelerate the pro%ect it is necessary to reduce the total time reuired for the

activities in the critical path.

The total float time for an activity is the time between its earliest and latest start time$ or between its earliest and latest finish time. It is the amount of time that an activity can be

delayed past its earliest start or earliest finish without delaying the pro%ect. F *T(1*Tor 3T(13T F 1+T#',T#ti% F 1+T# 7',TGti%8The slack time or slack of an event in a network is the difference the latest event time L

earliest event time i.e. i( 1iThe free float time of an activity is eual to the amount by which its duration can be

increased without affecting either the pro%ect time or the time available for the subseuentactivities. It indicates the value by which an activity can be delayed beyond the earliest

starting point without affecting the earliest start$ L therefore$ the total float of

the activities following it. H Total 3loatiF @ 4*lack of event 5The independent float time of an activity is the amount by which the duration of an

activity could be extended without affecting the total pro%ect time$ the time available for

subseuent activities or the time available for the preceding activities. H I3ree 3loatiF @

4*lack of event i5J or K1RO$ whichever is higher. 0lso ',T of following

activity 1+T of preceding activity "uration of current activity or


19/80

8roFect 'rashing: There are usually compelling reasons to complete the pro%ect earlier

than the originally estimated duration of critical path computed on the normal basis of a

new pro%ect. !irect Cost : This is the cost of the materials$ euipment and labour reuired to perform

the activity. !hen the time duration is reduced the pro%ect direct cost increases.

0ctivity (ost ,lope F 7(c# 3c873t#(t8!here$ (c F (rash (ost F "irect cost that is anticipated in completing an activity within

crash time.

3c F 3ormal (ost F This is the lowest possible direct cost reuired to complete an

activity 3t F 3ormal Time F 2in. time reuired to complete an activity at normal cost.

(t F (rash Time F 2in. time reuired to complete an activity.

/ndirect $ost* It consists of two parts: fixed cost and variable cost. The fixed cost is dueto general and administrative expenses$ insurance$ etc. Pariable indirect cost consists of

supervision$ interest on capital$ etc.

The total pro%ect cost is the sum of the direct L the indirect costs.Optimum duration is the pro%ect duration at which total pro%ect cost is lowest.

8RO=1'T *'21D-/C: 81RT9'86• 0 pro%ect is made up of a series of tasks called activities

• ,ome activities must be completed before other activities can be started

o ,ome activities 7say &8 that must be started immediatel" before another

activity 7say "8 that is$ no other activities must be done after & and before " 0ctivity & is an immediate predecessor for 0ctivity "

*H


20/80

0ctivity " is an immediate successor to 0ctivity &

o 0 network can be drawn that has nodes representing the activities 7with their

estimated completion times8 and arcs showing the precedence relations.

• The goal of many pro%ect scheduling models is to complete the pro%ect in

minimum timeo )'RT()2 7)rogram 'valuation and Review Techniue(ritical )ath

2ethod8 is a solution approach for solving for this minimum time It determines a set of earliest start 7',8 and earliest finish 7'+8

times by making a forward ass through the network

• +or activities with no predecessors: ', F B

• +or all other activities: ', F max 7'+ of immediate predecessors8

• +or all activities: '+ F ', G time to do the activity

• The expected pro%ect completion time '7T8 F max '+

It determines a set of latest finish 71+8 and latest start 71,8 times

by making a %ackwards ass through the network • +or activities with no successors: 1+ F '7T8

• +or all other activities: 1+ F min 71, of immediate succesors8

• +or all activities: 1, F 1+ time to do the activity

*lack time for an activity F 1, ', or 1+ '+

• If an activity is delayed by more than its slack time$ the pro%ect

is delayed by the difference between the delay and the slack

• 0ctivities with slack F B cannot be delayed without delaying

the pro%ect and are called critical activities The set of critical activities forms a critical path through the network

• 0 delay in any activity on the critical path delays the pro%ect

• The sum of the completion times on the critical path is the

expected pro%ect completion time F '7T8

• 0 linear program can be solved to determine the completion time for the pro%ect:

2I3 Time to finish )ro%ect

s.t. 0ctivities cannot start before their immediate predecessors are completed 0ll times Q B

Lou should know how to use the 81RT9'86 template for proFects

whose activit" completion times are known with certaint".

5B


21/80

The 81RT9'86 Three(Time 1stimate Approach

0ctivity completion times are rarely 7if ever8 known with certainty$ so a probability

approach is more realistic in evaluating a pro%ects expected completion time. 0

three#time estimate approach allows for such probability analyses

Three time estimates are determined 7by studies$ guesses$ etc.8 for each activity

o a F an optimistic completion time 7the chance of finishing in a is very small8

o m F a most likely completion time 7this is the mode8

o b F a pessimistic completion time 7the chance of finishing in J b is very small8

0ctivity approximations

o 0n approximation for the distribution of an activitys completion time is a

#$TA distributiono 0n approximation for the mean completion time for an activity is a weighted

average 7*@$ 9@$ *@8 of the three completion timesS so it is @

b9ma ++

o 0n approximation for the standard deviation for the completion time for an

activity is its Range@ or

−

@

a b

o The variance of an activitys completion time is the suare of the standard

deviation F

5

@

a b

−

)ro%ect assumptions

*. The distribution of the pro%ect completion time is determined by the critical pathusing the mean activity completion times

5. The activity completion times are independent %ust because one activity takes

longer or shorter than expected does not affect another activitys time.6. There are enough activities on the critical path so that the central limit can be used

to determine the distribution$ mean$ variance and standard deviation of the pro%ect

)ro%ect distribution

?iven the above assumption this means

o The pro%ect completion time distribution is normal

o The mean 4e!pected5 completion time$ M$ of the pro%ect is the sum of the

expected completion times along the critical path

o The variance of the completion time$ N7 $ of the pro%ect is the sum of the

variance in completion times along the critical path

o The standard deviation of the completion time$ N$ of the pro%ect is the suare

root of the variance of the completion time of the pro%ect

The probability of completing by a certain date t$ can now be found by finding the

)7E t8 from a normal distribution with mean and standard deviation U

5*


22/80

e" terms

Activit" P 0 task to be completed.

'86 4'ritical 8ath 6ethod5 P

0 pro%ect management techniue based on a pro%ect networkS the focus of ()2 is pro%ect planning$ with the critical path defining those activities into which

additional resources might be poured to accelerate the schedule.

'ritical path P The path through a pro%ect network that links the critical events that must begin

on time and the critical activities that must reuire no more than their estimated

duration if the pro%ect is to be completed on time.

Diverge P To split a single input path into multiple paths.

Dumm" activit" P

0n activity that links parallel events$ but consumes neither time nor resources.

Duration P The elapsed time reuired to complete an activity.

1arliest event time 411T5 P The earliest time the event can possibly begin.

1vent P

The beginning or end of an activity.

atest event time 41T5 P

The latest time an event can occur without impacting the pro%ect schedule.

6erge P

To combine two or more input paths into a single output path.

81RT 48rogram 1valuation and Review Techni#ue5 P

0 pro%ect management techniue based on a pro%ect networkS with )'RT$ thecritical path is the primary focus of management control and monitoring thecritical events provides an early warning if estimates are inaccurate.

8roFect network P

0 bubble chart that graphically depicts activities$ their starting and completiontimes$ and their interrelationships.

*lack P

The maximum time an activity can slip without affecting the pro%ect schedule.

'ompare and contrast '86 and 81RT. -nder what conditions would "ou

recommend scheduling b" 81RT? =ustif" "our answer with reasons.

Answer :# )ro%ect management has evolved as a new field with the development of two

analytic techniues for planning$ scheduling and controlling pro%ects. These are the(ritical )ath 2ethod 7()28 and the )ro%ect 'valuation and Review Techniue 7)'RT8.

)'RT and ()2 are basically time#oriented methods in the sense that they both lead to

the determination of a time schedule.

55


23/80

&asic Difference between 81RT and '86

Though there are no essential differences between )'RT and ()2 as both of them share

in common the determination of a critical path. &oth are based on the networkrepresentation of activities and their scheduling that determines the most critical activities

to be controlled so as to meet the completion date of the pro%ect.

81RT

,ome key points about )'RT are as follows:

• )'RT was developed in connection with an RL" work. Therefore$ it had to cope

with the uncertainties that are associated with RL" activities. In )'RT$ the total

pro%ect duration is regarded as a random variable. Therefore$ associated

probabilities are calculated so as to characterise it.

•

It is an event#oriented network because in the analysis of a network$ emphasis isgiven on the important stages of completion of a task rather than the activitiesreuired to be performed to reach a particular event or task.

• )'RT is normally used for pro%ects involving activities of non#repetitive nature in

which time estimates are uncertain.

• It helps in pinpointing critical areas in a pro%ect so that necessary ad%ustment can

be made to meet the scheduled completion date of the pro%ect.

'86

• ()2 was developed in connection with a construction pro%ect$ which consisted of

routine tasks whose resource reuirements and duration were known with

certainty. Therefore$ it is basically deterministic.

• ()2 is suitable for establishing a trade#off for optimum balancing between

schedule time and cost of the pro%ect.

• ()2 is used for pro%ects involving activities of repetitive nature.

8RO=1'T *'21D-/C &L 81RT('86

It consists of three basic phases: planning$ scheduling and controlling.

8hases of 81RT('86

*. 8roFect 8lanning: In the pro%ect planning phase$ you need to perform the followingactivities:

56


24/80

• /dentif" various tasks or work elements to be performed in the pro%ect.

• Determine reuirement of resources$ such as men$ materials$ and machines$ for

carrying out activities listed above.

•

1stimate costs and time for various activities.

• *pecif" the inter#relationship among various activities.

• Develop a network diagram showing the seuential inter#relationships between

the various activities.

5. 8roFect *cheduling: Once the planning phase is over$ scheduling of the pro%ect is

when each of the activities reuired to be performed$ is taken up. The various steps

involved during this phase are listed below:

•

'stimate the durations of activities. Take into account the resources reuired forthese execution in the most economic manner.

• &ased on the above time estimates$ prepare a time chart showing the start and

finish times for each activity. Kse the time chart for the following exercises.

V To calculate the total pro%ect duration by applying network analysis techniues$ such asforward 7backward8 pass and floats calculation

V To identify the critical path

V To carry out resource smoothing 7or levelling8 exercises for critical or scarce resourcesincluding re#costing of the schedule taking into account resource constraints

6. 8roFect 'ontrol: )ro%ect control refers to comparing the actual progress against theestimated schedule. If significant differences are observed then you need to re#schedule

the pro%ect to update or revise the uncompleted part of the pro%ect.

*/6-AT/O

• 0 simulation is the imitation of the operation of a real#world process or system

over time. ,teps include

o ?enerating an artificial history of a system

o Observing the behavior of that artificial history

59


25/80

o "rawing inferences concerning the operating characteristics of the real

system

• Kse the operation of a bank as an example:

o (ounting how many people come to the bankS how many tellers$ how longeach customer is in serviceS etc.

o 'stablishing a model and its corresponding computer program.

o 'xecuting the program$ varying parameters 7number of tellers$ service

time$ arrival intervals8 and observing the behavior of the system.

o "rawing conclusions: increasing number of tellersS reducing service timeS

changing ueueing strategiesS etc.

• The behavior of a system as it evolves over time is studied by developing asimulation model .

• 0 model is a set of entities and the relationshi among them.

+or the bank example: entities would include customers$ tellers$ and ueues.

Relations would include customers entering a ueueS tellers serving the customerS

customers leaving the bank.

• Once developped$ a model has to be validated. There are many different ways to

validate a model: observation 7measurement8S analytical model comparison

7analysis8.

*"stems and *"stem 1nvironment

• 0 system is a group of ob%ects that are %oined together in some regular interaction

or interdependence toward the accomplishment of some purpose.

• 0 system is often affected by changes occurring outside the system. ,uch changes

are said to occur in the system environment . In modeling a system$ it is necessary

to decide on the %oundary between the system and its environment.

'.g. !hen studying cache memory using simulation$ one has to decide where isthe boundary of the system. It can be simply the ()K and cache$ or it can include

main memory$ disk$ O.,.$ compiler$ or even user programs.

5;


26/80

'omponents of a *"stem

• 0n entity is an ob%ect of interest in the system. '.g. customers in a bank.

•

0n attri%ute is a property of an entity. '.g. the checking account balance of thecustomer.

• 0n activity represents a time period of specified length. 4ere the time period is

emphasized because often the simulation involves time. '.g. deposit money into

the checking account at a specified date and time.

• The state of a system is defined to be that collection of variables necessary to

describe the system at any time$ relative to the ob%ectives of the study. '.g.

number of busy tellers$ number of customers waiting in line.

•

0n event is defined as an instantaneous occurrence that may change the state ofthe system. '.g. customer arrival$ addition of a new teller$ customer departure.

Discrete and 'ontinuous *"stems

• 0 discrete system is one in which the state variable7s8 change only at a discrete set

of points in time. '.g. customers arrive at 6:*;$ 6:56$ 9:B*$ etc.

• 0 continuous system is one in which the state variable7s8 change continuously

over time. '.g. the amount of water flow over a dam.

2odel of a ,ystem

0 model is defined as a representation of a system for the purpose of studying the system.

2any times one canWt experiment with a real system such as a bank$ or a highway system.

One has to expriment with a model of the real system. 0 model is often not exactly thesame as the real system it presents. Rather$ it includes a few 7or ma%ority of8 key aspects

of the real system. It is an a%straction of the real system.

T"pes of 6odels

• ,tatic vs. dynamic: 0 static simulation model$ sometimes called 2onte (arlo

simulation$ represents a system at particular point in time. 0 dynamic simulation

model represents systems as they change over time.

• "eterministic vs. stochastic: 0 deterministic simulation contains no random

variable7s8. e.g. patients arrvie in a doctorWs office at a pre#scheduled time. 0 stochastic simulation involves one or more randome variables as input.

5@


27/80

• "iscrete vs. continuous: 7already discussed8.

*teps in a *imulation *tud"• 8roblem formulation

(learly state the problem.• *etting of obFectives and overall proFect plan

4ow we should approach the problem.

• 6odel conceptualiation

'stablish a reasonable model.

• Data collection

(ollect the data necessary to run the simulation 7such as arrival rate$ arrival process$ service discipline$ service rate etc.8.

• 6odel translation

(onvert the model into a programming language.

• 0erification

Perify the model by checking if the program works properly. Kse common sense.

• 0alidation

(heck if the system accurately represent the real system.

• 1!perimental design

4ow many runs> +or how long> !hat kind of input variations>

• 8roduction runs and anal"sis

0ctual running the simulation$ collect and analize the output.

• Repeatition

Repeat the experiments if necessary.

• Document and report

"ocument and report the results.

Advantages and Disadvantages

2ain advantages of simulation include:

• ,tudy the behavior of a system without building it.

• Results are accurate in general$ compared to analytical model.

• 4elp to find un#expected phenomenon$ behavior of the system.

• 'asy to perform XX!hat#IfWW analysis.

2ain disadvantages of simulation include:

5A


28/80

• 'xpensive to build a simulation model.

• 'xpensive to conduct simulation.

• ,ometimes it is difficult to interpret the simulation results.

*tate two maFor reasons for using simulation. 1!plain the basic steps of 6onte(

'arlo simulation. &riefl" describe the application in finance E Accounting.

Answer: @ ma%or reasons for using simulation

,imulation is also called experimentation in the management laboratory. !hile dealing

with business problems$ simulation is often referred to as Y2onte (arlo 0nalysis. Two

0merican mathematicians$ Pon 3eumann and Klan$ in the late *H9Bs found a problem inthe field of nuclear physics too complex for analytical solution and too dangerous for

actual experimentation. They arrived at an approximate solution by sampling. The

method they used had resemblance to the gamblers betting systems on the roulette tableShence the name Y2onte (arlo has stuck.

Imagine a betting game where the stakes are based on correct prediction of the number of

heads$ which occur when five coins are tossed. If it were only a uestion of one coinS

most people know that there is an eual likelihood of a head or a tail occurring$ that is the probability of a head is Z. 4owever$ without the application of probability theory$ it

would be difficult to predict the chances of getting various numbers of heads$ when five

coins are tossed. In this kind of situation simulation plays an important role.

6OT1 'ARO */6-AT/O

2onte (arlo simulation is useful when same elements of a system$ such as arrival of parts to a machine$ etc.$ exhibit a chance factor in their behavior. 'xperimentation on

probability distribution for these elements is done through random sampling. +ollowing

five steps are followed in the 2onte (arlo simulation:

)rocedure of 2onte (arlo ,imulation:

*. "ecide the probability distribution of important variables for the stochastic

process.

5. (alculate the cumulative probability distributing for each variable in ,tep *

6. "ecide an interval of random numbers for each variable.

9. ?enerate random numbers.

;. ,imulate a series of trials and determine simulated value of the actual randomvariables.

5D


29/80

0pplication of ,imulation in finance L 0ccounting.

The range of application of simulation in business is extremely wide. Knlike other

mathematical models$ simulation can be easily understood by the users and therebyfacilitates their active involvement. This makes the results more reliable and also ensures

easy acceptance for implementation. The degree to which a simulation model can bemade close to reality is dependent upon the ingenuity of the OR team who identifies the

relevant variables as well as their behavior.

.It can also be employed for a wide variety of problems encountered in production

systems the policy for optimal maintenance in terms of freuency of replacement of

spares or preventive maintenance$ number of maintenance crews$ number of euipmentfor handling materials$ %ob shop scheduling$ routing problems$ stock control and so forth.

The other areas of application include dock facilities$ facilities at airports to minimize

congestion$ hospital appointment systems and even management games.

In case of other OR models$ simulation helps the manager to strike a balance betweenopposing costs of providing facilities 7usually meaning long term commitment of funds8

and the opportunity and costs of not providing them.

A**/C61T AD TRA*8ORTAT/O 8RO&16

What is an Assignment 8roblem?

• The assignment problem can be stated as a problem where different %obs are to be

assigned to different machines on the basis of the cost of doing these %obs. The

ob%ective is to minimize the total cost of doing all the %obs on different machines

• The peculiarity of the assignment problem is only one %ob can be assigned to one

machine i.e.$ it should be a one#to#one assignment

• The cost data is given as a matrix where rows correspond to %obs and columns to

machines and there are as many rows as the number of columns i.e. the number of

%obs and number of 2achines should be eual

• This can be compared to demand euals supply condition in a balanced

transportation problem. In the optimal solution there should be only one

assignment in each row and columns of the given assignment table. one can

5H


30/80

observe various situations where assignment problem can exist e.g.$ assignment

of workers to %obs like assigning clerks to different counters in a bank or salesman

to different areas for sales$ different contracts to bidders.

• 0ssignment becomes a problem because each %ob reuires different skills and the

capacity or efficiency of each person with respect to these %obs can be different.

This gives rise to cost differences. If each person is able to do all %obs eually

efficiently then all costs will be the same and each %ob can be assigned to any

person.

• !hen assignment is a problem it becomes a typical optimization problem it can

therefore be compared to a transportation problem. The cost elements are given

and is a suare matrix and reuirement at each destination is one and availability

at each origin is also one.

• In addition we have number of origins which euals the number of destinations

hence the total demand euals total supply . There is only one assignment in each

row and each column .4owever If we compare this to a transportation problem we

find that a general transportation problem does not have the above mentioned

limitations. These limitations are peculiar to assignment problem only.

75 What is a &alanced and -nbalanced Assignment 8roblem?

A balanced assignment problem is one where the number of rows F the number of

columns 7comparable to a balanced transportation problem where total demand Ftotal

supply8

&alanced assignment problem: no of rows F no of columns

6B


31/80


32/80

D/331R1T 61T2OD* O3 A**/C61T 8RO&16

2ungarian method

,implex 2ethod

Transportation

)roblem(omplete 'numeration

)5 What is a 8rohibited Assignment 8roblem?

0 usual assignment problem presumes that all %obs can be performed by

all individuals there can be a free or unrestricted assignment of %obs and individuals. Aprohibited assignment problem occurs when a machine ma" not be in$ a position to

perform a particular %ob as there be some technical difficulties in using a certain

machine for a certain %ob. In such cases the assignment is constrained by given facts.

To solve this type problem of restriction on %ob assignment we will have

to assign a very high cost 2 This ensures that restricted or impractical combination does

not enter the optimal assignment plan which aims at minimization of total cost.

+5 What are the methods to solve an Assignment 8roblem 42ungarian 6ethod5?

There are different methods of solving an assignment problem:

65


33/80

%5'omplete 1numeration 6ethod: This method can be used in case of assignment.

problems of small size. In such cases a complete enumeration and evaluation of all

combinations of persons and %obs is possible.

One can select the optimal combination. !e may also come across more than one optimal

combination The number of combinations increases manifold as the size of the problem

increases as the total number of possible combinations depends on the number of say$

%obs and machines. 4ence the use of enumeration method is not feasible in real world

cases.

75 *imple! 6ethod: The assignment problem can be formulated as a linear programming

problem and hence can be solved by using simplex method.4owever solving the problem

using simplex method can be a tedious %ob.

)5Transportation 6ethod: The assignment problem is comparable to a transportation

problem hence transportation method of solution can be used to find optimum allocation.

4owver the ma%or problem is that allocation degenerate as the allocation is on basis one

to one per person per person per %ob 4ence we need a method specially designed to solve

assignment problems.

+ 52ungarian Assignment 6ethod 42A65:

This method is based on the concept of opportunity cost and is more efficient in solving

assignment problems.

6ethod in case of a minimiation problem.

0s we are using the concept Mopportunity this means that the cost of any opportunity that

is lost while taking a particular decision or action is taken into account while making

assignment. ?iven below are the steps involved to solve an assignment problem by using

4ungarian method.

66


34/80

*tep %:

"etermine the opportunity cost table

*tep 7:

"etermine the possibility of an optimal assignment

*tep )

2odify the second reduced cost table

*tep +:

2ake the optimum assignment

*tep %:

"etermine the opportunity cost tableI

1ocate the smallest cost in each row and subtract it from each cost figure in that

row. This would result in at least one zero in each row. The new table is called

reduced cost table.

• 1ocate the lowest cost in each column of the reduced cost table subtract this

figure from each cost figure in that column. This would result in at least one zero

in each row and each column$ in the second reduced cost table.

*tep 7:

Determine the possibilit" of an optimal assignment:

69


35/80

• To make an optimal assignment in a say 6 x 6 table. !e should be in a position to

locate 6 zeros in the table. ,uch that 6 %obs are assigned to 6 persons and the total

opportunity cost is zero .0 very convenient way to determine such an optimal

assignment is as follows:

• "raw minimum number of straight lines vertical and horizontal$ to cover all the

zero elements in the second reduced cost table. One cannot draw a diagonal

straight line. The aim is that the number of lines 738 to cover all the zero

elements should be minimum. If the number of lines is eual to the number of

rows 7or columns8 7n8 i.e 3Fn it is possible to find optimal assignment .

• 1!ample :for a 6 x 6 assignment table we need 6 straight lines which cover all the

zero elements in the second reduced cost table. If the number of lines is less than

the number of rows 7columns8 3 n optimum assignment cannot be made. we

then move to the next step.

*tep ):

2odify the second reduced cost table:

• ,elect the smallest number in the table which is not covered by the lines. ,ubtract

this number from all uncovered numbers aswell as from itself.

• 0dd this number to the element which is at the intersection of any vertical and

horizontal lines.

• "raw minimum number of lines to cover all the zeros in the revised opportunity

cost table.

• If the number of straight lines at least euals number of rows 7columns8 an

optimum assignment is possible.

6;


36/80

*tep +:

2ake the optimum assignment:

If the assignment table is small in size it is easy to make assignment after step 6.

4owever$ in case of large tables it is necessary to make the assignments systematically.

,o that the total cost is minimum. To decide optimum allocation.

• ,elect a row or column in which there is only one zero element and encircle it

0ssign the %ob corresponding to the zero element i.e. assign the %ob to the circle

with zero element. 2ark a E in the cells of all other zeros lying in the column

7row8 of the encircled zero. ,o that these zeros cannot be considered for next

assignment.

• 0gain select a row with one zero element from the remaining rows or columns.

2ake the next assignment continue in this manner for all the rows.

• Repeat the process till all the assignments are made i.e. no unmarked zero is left.

• now we will have one encircled zero in each row and each column of the cost

matrix. The assignment made in this manner is Optimal.

(alculate the total cost of assignment from the original given cost table.

6a!imiation method

In order to solve a maximization type problem we find the regret values instead of

opportunity cost. the problem can be solved in two ways

6@


37/80

• The first method is by putting a negative sign before the values in the assignment

matrix and then solves the sum as a minimization case using 4ungarian methods

as shown above.

• ,econd method is to locate the largest value in the given matrix and subtract each

element in the matrix from this value. Then one can solve this problem as a

minimization case using the new modified matrix.

4ence there are mainly four methods to solve assignment problem but the most efficient

and most widely used method is the 4ungarian method

, 5 ote on Traveling *alesmen problem.

Traveling salesman problem is a routine problem. It can be considered as a typical

assignment problem with certain restrictions. (onsider a salesman who is assigned the

%ob of visiting n different cities. 4e knows 7is given8 the distances between all pairs of

cities. 4e is asked to visit each of the cities only once. The trip should be continuous and

he should come back to the city from where he started using the shortest route. It does not

matter$ from which city he starts. These restrictions imply.

7*8 3o assignment should be made along the diagonal.

758 3o city should be included on the route more than once

This type of problem is uite simple but there is no general algorithm available for its

solution. The problem is usually solved by enumeration method$ where the number of

enumerations is very large.

3or e!ample for a salesman who is instructed to visit five cities we shall have to consider

more than *BB possible routes. The method is therefore impractical for large size

problems and it also implies approximations in finding route with minimum distance.

6A


38/80

The peculiar nature of the problem and the various restrictions imposed on resulting

solution indicate that the method of solution to a traveling salesman problem should

include:

7*8 0ssigning an infinitely large element 2 in the diagonal of the distance matrix.

758 ,olve the problem using 4ungarian 2ethod as it gives shortcut route but$

768 Test the solution for feasibility whether it satisfies the condition of a continues route

without visiting a city more than once.

If the route is not feasible$ make ad%ustments with minimum increase in the total distance

traveled by the salesman. This is how one can solve traveling sales man problem

;5 What is a Transportation 8roblem?

• 0 transportation problem is concerned with transportation methods or selecting

routes in a product distribution network among the manufacturing plants and

distribution warehouses situated in different regions or local outlets.

• In applying the transportation method$ management is searching for a distribution

route$ which can lead to minimization of transportation cost or maximization of

profit.

• The problem involved belongs to a family of specially structured 1)) called

network flow problems.


39/80

0ggregate supply exceeds the aggregate demand 0ggregate demand exceeds the aggregate supply

7 8ossibilities That 6ake The 8roblem -nbalanced

7i8 0ggregate supply exceeds the aggregate demand or

7ii8 0ggregate demand exceeds the aggregate supply.

,uch problems are called unbalanced problems. It is necessary to balance them before

they are solved.

&alancing the transportation problem

• Where total *uppl" e!ceeds total demand.

In such a case the excess supply is$ assumed to go to inventory and costs nothing for

shipping7transporting8. This type of problem is balanced by creating a fictitious

destination. This serves the same purpose as the slack variable in the simplexmethod 0

column of slack variables is added to the transportation tableau which represents a

dummy destination with a reuirement eual to the amount of excess supply and the

transportation cost eual to zero. This problem can now be solved using the usual

transportation methods.

When aggregate demand e!ceeds aggregate suppl" in a transportation

problem

6H


40/80

R1A*O* 3OR -A0A/A&//TL O3 RO-T1*

,trikes in certain region Knfavorable weather conditions on a particular route 'ntry restriction.

!hen aggregate demand exceeds aggregate supply in a transportation problem a dummy

row is added to restore the balance. This row has an availability eual to the excess

demand and each cell of this row has a zero transportation cost per unit. Once the

problem is balanced it can be solved by the procedures normally used to solve a

transportation problem.

>5 What is a 8rohibited Transportation 8roblem?

,ometimes in a given transportation problem some route7s8 may not be available.

This could be due to a variet" of reasons like(

7i8 ,trikes in certain region

7ii8 Knfavorable weather conditions on a particular route

7iii8 'ntry restriction.

In such situations there is a restriction on the routes available for transportation. To

overcome this difficulty we assign a very large cost 2 or infinity to such routes. !hen a

large cost is added to these routes they are automatically eliminated form the solution.

The problem then can be solved using usual methods.

5 What is Degenerac" in a Transportation 8roblem?

9B


41/80

The initial basic feasible solution to a transportation problem should have a total number

of occupied cell 7stone suares8 which is eual to the total number of rim re#uirements

minus one i.e. m G n P %. When this rule is not met the solution is degenerate.

Degenerac" ma" occur(

/f the number of occupied cells is more than m G n P %.

This type of degeneracy arises only in developing the initial solution. It is caused by

an improper assignment of freuencies or an error in formulating the problem. In such

cases one must modify the initial solution in order to get a solution which satisfies the

rule m G n=i.

The problem becomes degenerate at the(

7i8 /nitial stage

!hen in the initial solution the number of occupied cells is less than m G n = *

7rim reuirements minus *8 i.e. the number of stone suares in insufficient

4ii5 When two or more cells are vacated simultaneousl"

"egeneracy may appear subseuently when two or more cells are vacated

simultaneously in the process of transferring the units$ along the closed loop to

obtain an optimal solution.

When transportation problem becomes degenerate

!hen transportation problem becomes degenerate it cannot be tested for

optimality because it is impossible to compute u and$ v values with 2O"I method. To

overcome the problem of insufficient number of occupied cell we proceed by assigning

an infinitesimally small amount 7close to zero8 to one or more 7if needed8 empty cell and

treat that cell as occupied cell. This amount is represented by the ?reek letter ' 7epsilon8.

It is an insignificant value and does not affect the total cost. &ut it is appreciable enough

to be considered a basic variable. !hen the initial basic solution is degenerate$ we assign

c to an independent empty cell. 0n independent cell is one from which a closed loop

9*


42/80

*tep %:

(heck whether the given T.). is balanced or not

*tep 7:

"evelop initial feasible solution by any of the five methods

cannot be traced. It is preferably assigned to a cell which has minimum per unit cost.

0fter introducing e we solve the problem using usual methods of solution.

%Q5 *teps to solve a Transportation 8roblem.

A transportation problem can be solved in 7 phases

82A*1 /

*tep %:

(heck whether the given T.). is balanced or not. If it is unbalanced then balance it by

adding a row or a column.

*tep 7:

"evelop initial feasible solution by any of the five methods:

a8 3orth !est (orner Rule 73!(R8 or ,outh !est (orner Rule 7,!(R8

b8 Row 2inima 2ethod 7R228

c8 (olumn 2inima 2ethod 7(228

d8 2atrix 2inima 2ethod 72228

e8 Pogels 0pproximation 2ethod 7P028

95


43/80

!e discuss here the two commonly used methods to make initial assignments

7*8 3orthwest corner rule 758 Pogels 0pproximation 2ethod 7P028

4%5 orthwest 'orner Rule:

,tart with the northwest corner of the transportation tableau and consider the cell in the

first column and first row. !e have values a* and b* at the end on the first row and

column i.e. the availability at row one is a* and reuirement of column * is b*.

7i8 If al J b* assign uantity b* in the cell$ i.e. x* b*. Then proceed horizontally to the

next column in the first row until a* is exhausted i.e. assign the remaining number a* # b*

in the next column.

7ii8 If al b* then put El al and then proceed vertically down to the next row until b* is

satisfied. i.e. assign b* a* in the next row.

7iii8 If a* F b* then put EII F a* and proceed diagonally to the next cell or suare

determined by next row and next column.

In this way move horizontally until a supply source is exhausted$ and vertically down

until destination demand is completed and diagonally when a* F b*$ until the south#east

corner of the table is reached.

475 0ogelBs Appro!imation 6ethod 40A65:

The north#west corner rule for initial allocation considers only the reuirements and

availability of the goods. It does not take into account shipping costs given in the tableau.

It is therefore$ not a very sound method as it ignores the important factor$ namely cost

whi[h we seek to minimize. The P02$ on the other hand considers the cost in each cell

while making the allocations we explain below this method.

96


44/80

7i8 (onsider each row of the cost matrix individually and find the difference between two

least cost cells in it. Then repeat this exercise for each column. Identify the row or

column with the 1argest difference 7select any one in case of a tie8.

7ii8 3ow consider the cell with minimum cost in that column 7or row8 and assign the

maximum units possible to that cell.

7iii8 "elete the rowcolumn that is satisfied.

7iv8 0gain find out the differences and proceed in the same manner as stated in earlier

paragraph and continue until all units have been assigned.

82A*1 // @ T1*T 3OR O8T/6A/TL

&efore we enter phase II$ the following two conditions should be fulfilled in that order.

7i8 Obtaining a basic feasible solution implies finding a minimum number of i% values.

This minimum number is m G n ( %. !here m is the number of origins n is the number of

destinations. Thus initial assignment should occupy m G n # I cells i.e. reuirements of

demand and supply cells minus = *.

7ii8 These i% should be at independent positions

These reuirements are called R/6 re#uirements.

Test for optimalit" 4or improvement5:

0fter obtaining the initial feasible solution$ the next step is to test whether it is optimal or

not.

99


45/80

!e explain here the 6odified Distribution 46OD/5 6ethod for testing the optimality.

If the solution is non#optimal as found from 2O"I method then we improve the solution

by exchanging non#basic variable for a basic variable. In other words we rearrange the

allocation by transferring units from an occupied cell to an empty cell that has the largest

net cost change or improvement index$ and then shift the units from other related cells so

that all the rim 7supply$ demand8 reuirements are satisfied. This is done by tracing a

closed path or closed loop.

*tep /:

0dd a column u to the R4, of the transportation tableau and a row v at the bottom of the

tableau.

*tep %:

0dd a column u to the R4, of the transportation tableau and a

row v at the bottom of the tableau.

*tep 7:

0ssign$ arbitrarily$ any value to u or v generally u F B.

*tep )

4aving determined u* and v calculate i% F 7u* G v*8 = for every

unoccupied cell.

*tep +:

If the solution is not optimal select the cell with largest positive

improvement index.

*tep :

Test the solution again for optimality and improve fit if

9;


46/80

*tep 7:

0ssign$ arbitrarily$ any value to u or v generally u F B. This method of assigning values

to u* and v* is workable only if the initial solution is non#degenerate i.e.$ for a table there

are exactly m G n #* occupied cells.

*tep ):

4aving determined u* and v calculate i% F 7u* G v*8 = for every unoccupied cell. This

represents the net cost change or improvement index of these cells

7*8 If all the empty cells have negative net cost change i%$ the solution is optimal and

uniue

758 If an empty cell has a zero Ei% and all other empty cells have negative Ei% the solution

is optimal but not uniue.

768 If the solution has positive 0i for one or more empty cells the solution is not optimal.

*tep +:

If the solution is not optimal select the cell with largest positive improvement index.

Then trace a closed loop and transfer the units along the route.

Tracing loop 4closed path5:

*8 (hoose the unused suare to be evaluated.

758 &eginning with the selected unused suare trace a loop via used suares back to the

original unused suares. Only one loop exists for any unused suare in a given solution.

768 0ssign 7G8 and 7=8 signs alternately at each suare of the loop beginning with a plus

sign at the unused suare. 0ssign these sign in clockwise or anticlockwise direction.

These signs indicate addition or subtraction of units to a suare.

9@


47/80

798 "etermine the per unit net change in cost as a result of the changes made in tracing

the loop. (ompare the addition to the decrease in cost. It will give the improvement

index. 7It is euivalent to % in a 1))8.

7;8 "etermine the improvement index for each unused suare.

7@8 In a minimization case. If all the indices are greater than or eual to zero$ the solution

is optimal. If not optimal$ we should find a better solution.

We ma" also note the following points:

7i8 0n even number of at least four cells participate in a closed loop. 0n occupied cell can

be considered only once.

7ii8 If there exists a basic feasible solution with m G n = * positive variables$ then there

would be one and only one closed loop for each cell.

7iii8 0ll cells that receive a plus or minus sign except the starting empty cell$ must be the

occupied cells.

7iv8 (losed loops may or may not be suare or rectangular in shape. They may have

peculiar configurations and a loop may cross over itself.

*tep :

Test the solution again for optimality and improve fit if necessary. Repeat the process

until an optimum solution is obtained.

%%5 What are the differences between assignment problem and transportation

problem>

9A


48/80

The differences between A8 and T8 are the following:

*. T) has supply and demand constraints while 0) does not have the same.

5. The optimal test for T) is when all cell evaluation \s are greater than or eual to

zero whereas in 0) the number of lines must be eual to the size of matrix.

6. 0 T) sum is balanced when demand is eual to supply and an 0) sum is balanced

when number of rows are eual to the number of columns.

9. for 0). !e use 4ungarian method and for transportation we use 2O"I method

;. In 0). !e have to assign different %obs to different entities while in transportation

we have to find optimum transportation cost.

,%75 What are the advantages and disadvantages of 88?

0 1)) is concerned with the use of allocation of resources such as time$ capital$

materials$ etc.

T21 AD0ATAC1* 88 AR1:

*. It helps the sale manager to negotiate prices with customers. 4e can price on the basis

of customer demand and price on the basis of supply and demand of them market.

5. )roduction manager can formulate optimal maximum product mix.

6. it helps manager improve his decision making abilities.

9. it helps make the best decision for cost minimization and profit maximization.

T21 D/*AD0ATAC1* O3 88 AR1:

*. 0 primary reuirement for 1)) is the ob%ective function and every consistent must be

linear. In practical situation it is not possible to state all coefficients in the ob%ective

function and constraints with certainty.

5. There is no guarantee that 1)) will give an integer value solution.

eg. a solution may call for H.6 trucks or D.A units of product.

6. It does not take into consideration the effect of time and uncertainty.

9. There may be cases of infeasibility and unbounded ness.

9D


49/80

What do "ou understand b" the transportation problem? What is the basic

assumption behind the transportation problem? Describe the 6OD/ method of

solving transportation problem.

Answer:

Transportation )roblem L its basic assumption:

This model studies the minimization of the cost of transporting a commodity from

a number of sources to several destinations. The supply at each source and the demand at

each destination are known. The transportation problem involves m sources$ each of

which has available an i 7i F *$ 5$ m8 units of homogeneous product and n destinations$

each of which reuires b% 7% F *$ 5C.$ n8 units of products. 4ere a

i and b% are positive integers. The cost ci% of transporting one unit of the product from theith source to the

%th destination is given for each i and %. The ob%ective is to develop an integral

transportation schedule that meets all demands from the inventory at a minimum total

transportation cost. It is assumed that the total supply and the total demand are eual i.e.

(ondition 7*8 The condition 7*8 is guaranteed by creating either a fictitious

destination with a demand eual to the surplus if total demand is less than the total supply

or a 7dummy8 source with a supply eual to the shortage if total demand exceeds total

supply. The cost of transportation from the fictitious destination to all sources and from

all destinations to the fictitious sources are assumed to be zero so that total cost of

transportation will remain the same.

+ormulation of Transportation )roblem:

The standard mathematical model for the transportation problem is as follows. 1et

xi% be number of units of the homogenous product to be transported from source i to thedestination %. Then ob%ective is to#

9H


50/80

Theorem:

0 necessary and sufficient condition for the existence of a feasible solution to the

transportation problem 758 is that

The Transportation 0lgorithm 72O"I 2ethod8:

The first approximation to 758 is always integral and therefore always a

feasible solution. Rather than determining a first approximation by a direct

application of the simplex method it is more efficient to work with the table given

below called the transportation table. The transportation algorithm is the simplex

method specialized to the format of table it involves: i. finding an integral basic

feasible solution ii. testing the solution for optimality iii. improving the solution$

when it is not optimal iv. Repeating steps 7ii8 and 7iii8 until the optimal solution is

obtained.

The solution to T.) is obtained in two stages. In the first stage we find &asic

feasible solution by any one of the following methods a8 3orth#west corner rule b8 2atrix

2inima 2ethod or least cost method c8 Pogels approximation method. In the second

;B


51/80

stage we test the &.+s for its optimality either by 2O"I method or by stepping stone

method.

/.;: Describe the orth(West 'orner rule for finding the initial basic feasible

solution in the transportation problem.

Answer:

;*


52/80

The Initial basic +easible solution using 3orth#!est corner rule:

1et us consider a T.) involving m#origins and n#destinations. ,ince the sum of

origin capacities euals the sum of destination reuirements$ a feasible solution

always exists. 0ny feasible solution satisfying m G n * of the m G n constraintsis a redundant one and hence can be deleted. This also means that a feasible

solution to a T.) can have at the most only m G n * strictly positive component$

otherwise the solution will degenerate.

It is always possible to assign an initial feasible solution to a T.). in such a

manner that the rim reuirements are satisfied. This can be achieved either by

inspection or by following some simple rules. !e begin by imagining that the

transportation table is blank i.e. initially all xi% F B. The simplest procedures for

initial allocation discussed in the following section.

3orth !est (orner Rule:

*tep%:

a. The first assignment is made in the cell occupying the upper left hand

73orth !est8 corner of the transportation table.

b. The maximum feasible amount is allocated there$ that is x** F min 7a*$

b*8 ,o that either the capacity of origin O* is used up or the reuirement

at destination "* is satisfied or both.

c. This value of x** is entered in the upper left hand corner 7,mall ,uare8

of cell 7*$ *8 in the transportation table.

*tep 7:

a. If b* J a* the capacity of origin O$ is exhausted but the reuirement at

destination "* is still not satisfied $ so that at least one more other variable

in the first column will have to take on a positive value.

;5

Or Notes Mcom

Documents