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UGC MINOR RESEARCH PROJECT [MATHEMATICS]

STUDY OF DECISION MAKING PROBLEMS

USING FUZZY THEORY

FINAL REPORT

UGC SANCTION NO: .F MRP – 4072/12 (/MRP/UGC-SERO) – P.No: 32

Dated 8th April, 2013

Submitted to

UNIVERSITY GRANTS COMMISSION

South- East Regional Office

Hyderabad.

Principal Investigator

P.BHARATHI M.Sc. M.Phil.,

Assistant Professor of Mathematics

Sri Sarada Niketan College For Women

Amaravathipudur,Sivagangai District

Tamilnadu.

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CHAPTER 1

INTRODUCTION

In this chapter, we give a concise account of preliminary definitions and results required

commonly for the forthcoming chapters. We start with basic definitions from decision

making theory. All the definitions on decision making theory given here are taken from

the standard texts on this subject.

1.1 DECISION MAKING THEORY

Making decisions is undoubtedly one of the most fundamental activities of human

beings. We all face in our daily life with varieties of alternative actions available to us

and, at least in some instances, we have to decide which of the available actions to take.

In the beginning, the decisions were made by the methods of election & social choice.

Since these initial studies, decision making has evolved into a respectable and rich field

of study. The current literature on decision making based largely on theories and methods

developed in this century is enormous. Decision making is a problem solving process by

which a method is selected among various methods in order to obtain an effective and

applicable result.

The subject of decision making is the study of how decisions are actually made and how

they can be made better or more successfully. Much of the focus in developing the field

has been in the area of management, in which the decision making process is of key

importance for functions such as inventory control , investment, personal actions, new

product development and allocation of resources as well as many others. Decision

making itself is broadly defined to include any choice or selection of alternatives and is

therefore of importance in many sciences and engineering.

Human performance with regard to decisions has been the subject of active research from

several perspectives:

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Psychological: examining individual decisions in the context of a set of needs,

preferences and values the individual has or seeks.

Cognitive: the decision-making process regarded as a continuous process

integrated in the interaction with the environment.

Normative: the analysis of individual decisions concerned with the logic of

decision-making and rationality and the invariant choice it leads to.

Decision making can be defined as a process of specifying a problem, identifying and

evaluating criteria or alternatives and selecting a preferred alternative among possible

ones defined in common terminologies in decision making as follows:

(a) Alternatives - a set of objects, products, actions or strategies.

b) Attributes - each alternative is defined by a set of characteristic.

c) Objectives - a collection of attributes selected by the decision maker(s) to be used as a

goal.

d) Weights - the relative importance of each attribute or the relative importance of an

instance of an attribute.

A decision making problem is the problem to find the best decision in the set of feasible

alternatives with respect to several criteria functions. When dealing with practical

decision making problems we often have to take into consideration uncertainty in the

problem data. Decision-making can be regarded as the cognitive process resulting in the

selection of a belief or a course of action among several alternative possibilities. Every

decision making process produces a final choice that may or may not prompt action.

Decision making is the study of identifying and choosing alternatives based on the values

and preferences of the decision maker. Decision making is one of the central activities of

management and is a huge part of any process of implementation.

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Decision analysis can be used to determine optimal strategies where a decision maker is

faced with several decision alternatives and an uncertain pattern of future events.

Decision theory comments with the assumption that regardless of the type of decision

involved, all the Decision Problems have certain common characteristics which are

enumerated below.

1. The Decision Maker

The decision maker refers to individual or a group of individuals responsible for making the choice of an appropriate course of action amongst the available course action.

2. Courses of Action

The courses of action or Strategies are the acts that are available to the decision maker .

3. States of Nature

The events identify the occurrences which are outside the decision maker’s control and which determine the level of success for a given act. These events are often called states of nature or outcomes.

4. Payoff

Each combination of a course of action and a state of nature is associated with a payoff, which measures the net benefit to the decision maker that accrues from a given combination of decision alternatives and events.

5. Payoff table

For a given problem, payoff table lists the states of nature which are mutually exclusive as well as collectively exhaustive and a set of given courses of action or strategies.

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General form of Payoff table

States of Nature Courses of Action

A1 A2 ………….. An

S1 P11 P12 …………. P1n

S2 P21 P22 ………….. P2n

…. ………. ……… ………. …………..

Sm Pm1 Pm2 …………. Pmn

Logical decision-making is an important part of all science-based professions, where

specialists apply their knowledge in a given area to make informed decisions. For

example, medical decision-making often involves a diagnosis and the selection of

appropriate treatment. Some research using naturalistic methods shows, however, that in

situations with higher time pressure, higher stakes, or increased ambiguities, experts use

intuitive decision-making rather than structured approaches following a recognition

primed decision that fits their experience and arrives at a course of action without

weighing alternatives. Recent robust decision research has formally integrated

uncertainty into its decision-making model. Decision analysis recognized and included

uncertainties in its theorizing since its conception in 1964.

With regard to management and decision-making, each level of management is

responsible for different things. Top level managers look at and create strategic plans

where the organization's vision, goals, and values are taken into account to create a plan

that is cohesive with the mission statement. For mid-level managers, tactical plans are

created with specific steps with actions that need to be executed to meet the strategic

objective. Finally, the front-line managers are responsible for creating and executing

operational plans. These plans include the policies, processes, and procedures of the

organization. Each must take into account the overall goals and processes of the

organization.

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The environment can also play a part in the decision making process. It is important to

know that environmental complexity is a factor that influences cognitive function and

well being. A complex environment is an environment with a large number of different

possible states which come and go over time. It is in different states at different times and

different in different places as opposed to the same all over. Peter Godfrey-Smith,

professor at Stamford University, states "whether a particular type of complexity is

relevant to an organism depends on what the organism is like- size, needs, habits and

physiology."

Classical decision making generally deals with a set of alternative states of

nature(outcomes, results), a set of alternative actions that are available to the decision

maker , a relation indicating the state or outcome to be expected from each alternative

action, and an objective function which orders the outcomes according to their

desirability. A decision is said to be made under conditions of certainty when the

outcome for each action can be determined and ordered precisely. In this case, the

decision making problem becomes an optimization problem, the problem of maximizing

the utility function. A Decision is made under conditions of risk on the other hand when

the only available knowledge concerning the outcomes consists of their conditional

probability distributions, one for each action. In this case the decision making problem

becomes an optimization problem of maximizing the expected utility. When probabilities

of the outcomes are not known, or may not even be relevant, and outcomes for each

action are characterized only approximately, we say that decisions are made under

uncertainty. This is the prime domain for fuzzy decision making.

Several classes of decision making problems are usually recognized. According to one

criterion , decision making problems are classified as those involving a single decision

maker and those which involve several decision makers. These problem classes are

referred to as individual decision making and multi person decision making, respectively.

According to another criterion, we distinguish decision making problems that involve a

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simple optimization of a utility function, an optimization under constraints, or an

optimization under multiple objective criteria.

A major part of decision-making involves the analysis of a finite set of alternatives

described in terms of evaluative criteria. Information overload occurs when there is a

substantial gap between the capacity of information and the ways in which people may or

can adapt. The overload of information can be related to problem processing and tasking,

which effects decision making. These criteria may benefit cost in nature. Then the

problem might be to rank these alternatives in terms of how attractive they are to the

decision maker(s) when all the criteria are considered simultaneously. Another goal

might be to just find the best alternative or to determine the relative total priority of each

alternative when all the criteria are considered simultaneously. Solving such problems is

the focus of multi-criteria decision analysis (MCDA), also known as multi-criteria

decision-making (MCDM). This area of decision-making, although very old, has

attracted the interest of many researchers and practitioners and is still highly debated as

there are many MCDA/MCDM methods which may yield very different results when

they are applied on exactly the same data. This leads to the formulation of a decision-

making paradox.

Group decision-making techniques

Consensus decision-making tries to avoid "winners" and "losers". Consensus

requires that a majority approve a given course of action, but that the minority

agrees to go along with the course of action. In other words, if the minority

opposes the course of action, consensus requires that the course of action be

modified to remove objectionable features.

Voting-based methods.

o Range voting lets each member score one or more of the available options.

The option with the highest average is chosen. This method has

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experimentally been shown to produce the lowest Bayesian regret among

common voting methods, even when voters are strategic.

o Majority requires support from more than 50% of the members of the

group. Thus, the bar for action is lower than with unanimity and a group of

"losers" is implicit to this rule.

o Plurality, where the largest block in a group decides, even if it falls short of

a majority.

Delphi method is structured communication technique for groups, originally

developed for collaborative forecasting but has also been used for policy making.

Democracy is a facilitation method that relies on the use of special forms called

Democracy Sheets to allow large groups to collectively brainstorm and recognize

agreement on an unlimited number of ideas they have authored.

Individual decision-making techniques

Pros and cons: listing the advantages and disadvantages of each option,

popularized by Plato and Benjamin Franklin.

Simple prioritization: choosing the alternative with the highest probability-

weighted utility for each alternative.

Satisfying: examining alternatives only until an acceptable one is found.

Contrasted with maximizing, in which many or all alternatives are examined in

order to find the best option.

Elimination by aspects: choosing between alternatives using Mathematical

psychology. The technique was introduced by Amos Tversky in 1972. It is a covert

elimination process that involves comparing all available alternatives by aspects.

The decision-maker chooses an aspect; any alternatives without that aspect are

then eliminated. The decision-maker repeats this process with as many aspects as

needed until there remains only one alternative.

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Preference trees: In 1979, Tversky and Shmuel Sattach updated the elimination by

aspects technique by presenting a more ordered and structured way of comparing

the available alternatives. This technique compared the alternatives by presenting

the aspects in a decided and sequential order. It became a more hierarchical system

in which the aspects are ordered from general to specific.

Acquiesce to a person in authority or an "expert"; "just following orders".

Flipism: flipping a coin, cutting a deck of playing cards, and other random or

coincidence methods[24]

Prayer, tarot cards, astrology, augurs, revelation, or other forms of divination.

Taking the most opposite action compared to the advice of mistrusted authorities

(parents, police officers, partners...)

Opportunity cost: calculating the opportunity cost of each options and decide the

decision.

Bureaucratic: set up criteria for automated decisions.

Political: negotiate choices among interest groups.

Participative decision-making (PDM): a methodology in which a single decision-

maker, in order to take advantage of additional input, opens up the decision-

making process to a group for a collaborative effort.

Use of a structured decision-making method.

Individual decision-making techniques can often be applied by a group as part of a group

decision-making technique. In Multi Criteria decision making, relevant alternatives are

evaluated according to a number of criteria. Each criterion includes a particular ordering

of the alternatives and we need a procedure by which to construct one over all preference

ordering.

An optimal decision is a decision such that no other available decision options will lead

to a better outcome. It is an important concept in decision theory. In order to compare the

different decision outcomes, one commonly assigns a relative utility to each of them. If

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there is uncertainty in what the outcome will be, the optimal decision maximizes the

expected utility (utility averaged over all possible outcomes of a decision).

Decision Making Steps

Each step in the decision-making process may include social, cognitive and cultural

obstacles to successfully negotiating dilemmas. It has been suggested that becoming more

aware of these obstacles allows one to better anticipate and overcome them.

1. Establishing community: creating and nurturing the relationships, norms, and

procedures that will influence how problems are understood and communicated.

This stage takes place prior to and during a moral dilemma.

2. Perception: recognizing that a problem exists.

3. Interpretation: identifying competing explanations for the problem, and evaluating

the drivers behind those interpretations.

4. Judgment: shifting through various possible actions or responses and determining

which is more justifiable.

5. Motivation: examining the competing commitments which may distract from a

more moral course of action and then prioritizing and committing to moral values

over other personal, institutional or social values.

6. Action: following through with action that supports the more justified decision.

Integrity is supported by the ability to overcome distractions and obstacles,

developing implementing skills, and ego strength.

7. Reflection in action.

8. Reflection on action.

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Other decision-making processes have also been proposed. One such process, proposed

by Pam Brown of Singleton Hospital in Swansea, Wales, breaks decision-making down

into seven steps:

1. Outline your goal and outcome.

2. Gather data.

3. Develop alternatives (i.e., brainstorming)

4. List pros and cons of each alternative.

5. Make the decision.

6. Immediately take action to implement it.

7. Learn from and reflect on the decision.

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1.2 FUZZY SET THEORY – INTRODUCTION & PRELIMINARIES

HISTORY AND DEVELOPMENT

The crisp set is defined in such a way as to dichotomize the elements in some given

universe of discourse into two groups: members, who certainly belong in the set and non-

members, those that certainly do not belong to the set. A sharp unambiguous distinction

exists between the members and non-members of the set. However, many classification

concepts we commonly employ and express in natural language describe sets that do not

exhibit this characteristic. Examples are the set of tall people, expensive cars, highly

contagious diseases, numbers much greater than one, modest profits, sunny days etc. We

perceive these sets as having imprecise boundaries that facilitate gradual transitions from

membership to non-membership and vice versa .There was no proper frame work to

describe study and formulate such problems in classical mathematics until the emergence of

fuzzy set theoretical approach.

Most of our traditional tools for formal modeling, reasoning and computing are crisp,

deterministic and precise in character. Also we always attempt to maximize its usefulness in

constructing modeling. This aim is closely connected with the relationship among three key

characteristics of every system model: complexity, credibility and uncertainty. Uncertainty

has a pivotal role in any efforts to maximize the usefulness of system models. All traditional

logic habitually assumes that precise symbols are being employed.

One of the meanings attributed to the term ‘uncertainty’ is ‘vagueness’. That is the difficulty

of making sharp. It is important to realize that this imprecision or vagueness that are

characteristic of natural language does not necessarily imply a loss of accuracy or

meaningfulness. In classical set theory, a crisp set can be defined as a collection of elements

where each element can either belong or not belong to the set. A classical set assigns a

membership of 1 to elements which are members of a set, and 0 to those which are not. So,

by using the classical way of evaluating brain performance of human, an individual can only

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be considered as 'intelligent' or 'not intelligent'. This situation shows that every set must be

precise and well-defined. However, words used in real life are not precise in nature and

carry a certain amount of fuzziness. When questions are asked, such as 'How intelligent

he/she is?" or "Intelligent in what sense?", the attribute 'intelligent' may refer to different

definition.

Let U be the universe of discourse or universal set, which contains all the possible elements

of concern in each particular context or application. A classical (crisp) set A, or simply a set

A, in the universe of discourse U can be defined by listing all of its members (the list

method) or by specifying the properties that must be satisfied by the members of the set (the

rule method). The list method can be used only for finite sets and is therefore of limited use.

The rule method is more general. In the rule method, a set A is represented as A = {x / x

meets some conditions} There is yet a third method to define a set A-the membership

method, which introduces a zero-one membership function (also called characteristic

function, discrimination function, or indicator function) for A, denoted by A(x), such that

the set A is mathematically equivalent to its membership function A(x).

A(x) = 1 0 . That is the characteristic function maps the elements of A to

the elements of the set [0,1]. So, Classical set theory requires that a set must have a well-

defined Property. To overcome this limitation of classical set theory, the concept of fuzzy

set was introduced. It turns out that this limitation is fundamental and a new theory is

needed-this is the fuzzy set theory.

A Mathematical frame work to describe this phenomenon was suggested by Lotfi. A. Zadeh

a professor at the university of california at Berkely in his paper “Fuzzy Sets”[13].In order

to deal with vagueness and ambiguity of human preferences in real life, he introduced the

fuzzy set theory by this definition. A fuzzy set is a pair (X, ) where X is a non-empty set

and :X→[0,1]. L. A. Zadeh introduced the notion of fuzzy set to describe vagueness

mathematically in its very abstractness and tried to solve such problems by assigning to

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each possible individual in the universe of discourse a value representing its grade of

membership in the fuzzy set. This grade corresponds to the degree to which that individual

is similar or compatible with the concept represented by the fuzzy set. Thus, individuals

may belong in the fuzzy set to a greater or lesser degree as indicated by a larger or smaller

membership grade. These membership grades are very often represented by real numbers

ranging in the closed interval between 0 and 1. The grade of membership of an element in a

given fuzzy set can be viewed as a degree of our certainty or degree of truth of the vague

notion. For instance, a fuzzy set representing our concept of sunny day might assign a

degree of membership of 1 to a cloud cover of 0%, .8 to a cloud cover of 20%, .4 to a cloud

cover of 30%, and 0 to a cloud cover of 80% and above. These grades signify the degree to

which each percentage of cloud cover approximates our subjective concept of ‘sunny’, and

the set itself models the semantic flexibility inherent in such a common linguistic term.

Because full membership and full non-membership in the fuzzy set can still be indicated by

the values of 1 and 0, respectively, we can consider the concept of a crisp set to be a

restricted case of the fuzzy set with the membership function as a characteristic function.

But in a fuzzy set, the transition from membership to non-membership is gradual rather than

abrupt. Thus fuzzy set theory can be considered as a generalization of classical set theory.

Because of this generalization, the theory of fuzzy sets has a wider scope of applicability

than classical set theory in solving various problems. Extensive applications of the fuzzy set

theory have been found in various fields such as Mathematics, Computer Science, Artificial

Intelligences, Medical Sciences ,Economics, Statistics, Neural networks etc. A fuzzy set in a

universe X is defined by membership function that maps X to the interval [0, 1] and

therefore implies a linear, i.e. total ordering of the elements of X, one could argue that this

makes them inadequate to deal with incomparable information . A possible solution,

however, was already implicit in Zadeh’s seminal paper in a footnote, he mentioned that “in

a more general setting, the range of the membership function can be taken to be a suitable

partially ordered set .”

A fuzzy set can be defined mathematically by assigning to each possible individual in the

universe of discourse a value representing its grade of membership in the fuzzy set.

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They are a further development of the mathematical concept of a set. Sets were first studied

formally by the German mathematician Georg Cantor (1845-1918). His theory of sets met

much resistance during his lifetime , but now a days most mathematicians believe it is

possible to express most if not all, of mathematics in the language of set theory. Many

researchers are looking at the consequences of ’fuzzifying’ set theory, and much

mathematical literature is the result. For control engineers, fuzzy logic and fuzzy relations

are the most important in order to understand how fuzzy rules work. Basically Fuzzy logic

is a multi-valued logic, that allows intermediate values to be defined between conventional

evaluations like true / false, yes / no, high / low etc. Fuzzy logic theory and applications

have a vast literature. With regards to documented literature, we can classify the

development in fuzzy theory and applications as having three phases;

Phase 1 (1965-1977) can be referred to as academic phase in which the concept of fuzzy

theory has been discussed in depth and accepted as a useful tool for decision making. The

phase 2 (1978-1988) can be called as transformation phase whereby significant advances in

fuzzy set theory and a few applications were developed. The period from 1989 onwards can

be the phase 3, the fuzzy boom period, in which tremendous application problems in

industrial and business are being tackled with remarkable success. Currently fuzzy

technique is very much applied in all fields.

Research on the theory of fuzzy sets has been witnessing an exponential growth; both within

mathematics and in its applications. This ranges from traditional mathematical subjects like

logic, topology, algebra, analysis etc. to pattern recognition, information theory, artificial

intelligence, operations research, neural networks, planning etc. Consequently, fuzzy set

theory has emerged as a potential area of interdisciplinary research. We now list below some

basic definitions and results in fuzzy set theory.

Fuzzy Set :

Let X be a non - empty set and let I be the unit interval [0, 1]. A fuzzy set in X is a function

with domain X and values in I that is, : X →[0,1].

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Fuzzy Relation:

A Fuzzy relation on a non empty V is a fuzzy subset of : V X V → [0,1].t Level Set :

Let S be a nonempty set and be a fuzzy subset of S. Then the set ’ = { x in S / (x)≥ t}

for all t in [0,1] is called the t level set. The set * = { x in S/ (x) > 0} is called the support

of . [ Supp(A) denotes the support of fuzzy set A]. If the support of a fuzzy set is empty, it

is called an empty fuzzy set. A fuzzy singleton is a fuzzy set whose support is a single point .

Height of a Fuzzy Set

The height of a fuzzy set is the largest membership value attained by any point.

The equality, containment, complement, union, and intersection of two fuzzy sets A and B

are defined as follows.

A and B are equal if and only if A(x) = B(x) for all x in U.

We say B contains A, denoted by A ⊏ B, if and only if A(x) ≤ B(x) for all x in U.

The complement of A is a fuzzy set in U whose membership function is defined as (x) = 1 - A(x).

The union of A and B is a fuzzy set in U, denoted by A U B, whose membership function is

defined as AUB (X) = Max { A(x) , B(x)}.

The intersection of A and B is a fuzzy set A ⋂ B in U with membership function is defined

as A⋂B (X) = Min{ A(x) , B(x)}.

The union of A and B is the smallest fuzzy set containing both A and B. More precisely, if C

is any fuzzy set that contains both A and B, then it also contains the union of A and B.

The De Morgan's Laws are true for fuzzy sets. That is, suppose A and B are fuzzy sets, then

= ⋂ .

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Applications of Fuzzy Systems

Fuzzy systems have been applied to a wide variety of fields ranging from control; signal

processing, communications, integrated circuit manufacturing, and expert systems to

business, medicine, psychology, etc.

The subject of decision making is the study of how decisions are actually made and how

they can be made better or successfully. Applications of fuzzy sets with in the field of

decision making have for the most part consisted of fuzzifications of the classical

theories of decision making.

While decision making under conditions of risk have been modeled by probabilistic

decision theories and game theories, fuzzy decision theories attempt to deal with the

vagueness and non specificity inherent in human formulation of preferences, constraints

and goals.

Fuzziness can be introduced into the existing models of decision models in various

ways. In the first paper on fuzzy decision making. Bellman and Zadeh [1970] suggest a

fuzzy model of decision making in which relevant go and constraints are expressed in

terms of fuzzy set and a decision is determined by an appropriate aggregation of fuzzy

sets.

In application of fuzzy set theory, fuzzy logic and fuzzy measure theory, the field of

engineering has undoubtedly been a leader. We can characterize some typical and

easily understandable engineering problems in the main engineering disciplines

in which the use of fuzzy set theory has already proven useful.

Also fuzzy set theory is the most important one in electrical engineering, mechanical

engineering and civil engineering. Fuzzy set theory is also becoming important in

computer engineering and knowledge engineering. Its role in computer engineering

which primarily involves the design of specialized hard ware for fuzzy logic.

Its role in knowledge engineering involves knowledge acquisition, knowledge

representation and human machine interaction. A membership function which doesn’t

assign to each element of the set one real number, but a closed interval of real numbers

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between the identified lower and upper bounds. Fuzzy sets defined by membership

function of this type are called interval valued fuzzy sets. These sets are defined

formally by functions of the form A : X ->(0, 1] where (0, 1) denotes the family of all

closed intervals is real numbers in [0,1]. Their advantage is that they allow us to express

our uncertainty in indentifying a particular membership function.

1.3 REVIEW OF LITERATURE

The British economist SHACKLE characterized the important category of

decision making problems in 1961.

In order to deal with the vagueness of human thought, the great mathematician

L.A.Zadeh (1965)[13] first introduced the fuzzy set theory. He gave a broad

overview of the important applications of fuzzy set theory in decision making in the

book “Fuzzy sets; Uncertainty and information”. Then the book was expanded to

reflect the tremendous advances that have taken place subsequently.

In the research paper Decision making in a Fuzzy Environment, Bellman and

Zadeh, May 1970 [1] suggest a fuzzy model of decision making in which relevant

goals and constraints are expressed in terms of fuzzy sets. Decision making in a fuzzy

environment is meant a decision process in which the goals and/or the constraints are

fuzzy in nature. This paper was published for NASA [National Aeronautics And

Space Administration ].

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Some of them have also applied the concept of linguistic variables proposed by

Zadeh(1975) [11]to handle linguistic terms and approximate reasoning in a group

decision making problem.

The study of new decision analysis field by using fuzzy theory was improved by

Lai and Hwang in 1996.

In 2006 Herrera and Martinez L published the articles on the topic “ Recent

advancements of fuzzy sets; Theory and Practice “. In that article they explained some

of the advanced concepts of fuzzy set theory.

Following this a lot of research has been conducted in the area of group decision

making under the application of fuzzy set theory by Sadi-Nezhad and Akhtari , 2008.

In 2008 Madavi I , Madavi-Amri N , Heidarzade A , Nourifar R [7] explained

some fuzzy decision making problems in the paper “Designing a model of fuzzy topics

in Multiple Criteria Decision making”.

In the research paper , “A Fuzzy Group decision making model for multiple

criteria based on borda count “ , Mr.Mohammad Anisseh and Rosnah(Feb 2011)[8]

proposed a new method for fuzzy decision making problems. In this Paper the group

decision making process was defined as a decision situation where there are two are

more individuals different preferences but the same access to information each

characterized by his own personality and all recognize the existence of a common

problem. There are two types of group decision making problems; Heterogeneous and

Homogeneous. The heterogeneous group decision making environment allows the

opinions of individuals to have different weights which are contrary to the homogeneous

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group with dissimilar individuals. Following these , a lot of research has been conducted

in the area of group decision making under the application of fuzzy set theory.

By this research work, I like to propose a fuzzy approach under the linguistic frame work

to obtain optimal solution for Multi Criteria Decision Making problems. To accomplish

this, an aggregate (Arithmetic mean, Geometric mean)-deviation method based on

triangular fuzzy numbers is proposed. . The purpose of this work is to enhance group

agreement on the group decision making outcomes

1.4 OBJECTIVES

The existence and development of MCDM methods play an important role in dealing with

decision making problems. However they involve lengthy computation. But by this

research work, a user-friendly fuzzy approach is adopted under the linguistic frame work

to obtain optimal solution for Multi Criteria Decision Making problems. This improvised

work will help decision makers and future researchers in handling Multi Criteria

Decision Making Problems efficiently. The main objectives of the research work are

(i) To Find out the steps to solve decision making problems [Multi Criteria Decision

Making problems] using fuzzy numbers.

(ii) Solving some real life numerical decision making problems [MCDM] to get the

optimal solution.

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1.5 SUMMARY OF THE THESIS

This dissertation consists of four chapters including this introductory one.

In the first chapter, all preliminaries related to the research work which includes

the classification of decision making problems are given .

Since our research work is mainly based on fuzzy numbers and their operators, in

the second chapter the preliminaries about the fuzzy numbers are given.

In the third chapter, the theory of Multi Criteria Decision Making Problems in

fuzzy environment is discussed.

In the fourth chapter, the proposed methods are introduced and some MCDM

problems are solved to get an optimal solution.

The subject matter of the project depends on the following papers:

“A Method to solve MCDM Problems using fuzzy numbers” presented in National

Conference RACMS’ 14. Dr.Umayal Ramanathan College for Women, Karaikudi

18.10.2014

“Applications of Fuzzy Logic in Decision Making Theory” Published in International Journal of

Computer Applications ISBN: 973-93-80878-95-9h Page No 22-25, December 2013

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CHAPTER 2

FUZZY NUMBERS & FUZZY AGGREGATION

Introduction – Fuzzy Numbers

A fuzzy number is a generalization of a regular real number in the sense that it does not

refer to one single value but rather to a connected set of possible values, where each

possible value has its own weight between 0 and 1. This weight is called the membership

function. A fuzzy number is thus a special case of a convex, normalized fuzzy set of the

real line.[1] Just like Fuzzy logic is an extension of Boolean logic (which uses absolute

truth and falsehood only, and nothing in between), fuzzy numbers are an extension of real

numbers. Calculations with fuzzy numbers allow the incorporation of uncertainty on

parameters, properties, geometry, initial conditions, etc. Fuzzy number is a basis for

fuzzy arithmetic, which can be viewed as an extension of interval arithmetic.

Among the various types of fuzzy sets of special significance fuzzy sets are defined on

the set of ℝ of real numbers. Membership functions of these sets, which have the form

A: ℝ→[0,1]. To qualify as a fuzzy number, a fuzzy set A on ℝ must possess at least the

following properties:

(i) A must be a normal fuzzy set.

(ii) A must be a closed interval for every in (0,1].

(iii) The Support of A must be bounded.

The fuzzy set must be normal since a set of real numbers close to r is fully satisfied by r

itself. Hence the membership grade of r in any fuzzy set that attempts to capture this

conception (i.e fuzzy number) must be 1. The bounded support of a fuzzy number and all

its -cuts for not equal to zero must be closed intervals to allow us to define

meaningful arithmetic operations on fuzzy numbers in terms of standard arithmetic

operations on closed intervals well established in classical interval analysis. Since -cuts

23

of any fuzzy number are required to be closed intervals for all in(0,1], every fuzzy

number is a convex fuzzy set. The inverse is not necessarily true, since -cuts of some

convex fuzzy sets may be open or half-open intervals. Fuzzy numbers are used in

statistics, computer programming, engineering (especially communications),

experimental science and also in mathematics. In all of these fuzzy types of presentation,

fuzzy numbers are most used as in linguistic, decision making, knowledge representation,

medical diagnosis, control systems, databases, and so forth.

Special cases of fuzzy numbers include ordinary real numbers and intervals of real

numbers. Furthermore, membership functions of fuzzy numbers need not be symmetric.

Among other applications fuzzy numbers are essential for expressing fuzzy cardinalities

and consequently fuzzy quantifiers.

Triangular Fuzzy Numbers

Triangular Fuzzy Numbers are extensively used in fuzzy applications owing to their

simplicity. TFNs are used in fuzzy applications where uncertainty exists on both sides of

a value or parameter. They are characterized by an ordered triplet of real numbers

<l, m, u>. Mathematically, a TFN is defined as follows:

Let l, m, u ∈R, l < m < u. The fuzzy number t: R → [0, 1] defined by

t =

⎩⎪⎨⎪⎧

0 <−− ≤ ≤−− ≤ ≤0 > ⎭⎪

⎬⎪⎫

where l and u correspond to the lower and upper bounds

of the fuzzy number t. The triangular fuzzy numbers is denoted by t = (l, m, u).

24

Triangular fuzzy numbers [1,2,3]

Positive triangular fuzzy number:

A positive triangular fuzzy number t is denoted as t = (a1, a2, a3) where all ai’s > 0

for all i=1, 2, 3.

Negative triangular fuzzy number:

A negative triangular fuzzy number t is denoted as t = (a1, a2, a3) where all ai’s < 0

for all i=1, 2, 3.

Equal Triangular fuzzy number:

Let t1 = (a1, a2, a3) and t2= (b1, b2, b3) be two triangular fuzzy numbers. If t1 is identically

equal to t2 if only if a1 = b1, a2 = b2 and a3 = b3.

The main operational laws for two triangular fuzzy numbers t1 and t2 are as follows:

If t1= (l1,m1,u1) and t2 = (l2, m2, u2) then

t1 + t2 = (l1 +l2, m1+m2, u1+u2)

t1 � t2 = (l1 X l2, m1 � m2, u1 X u2)

w X t1 = (wl1, wm1, wu1)

t1-1 = (1/l1, 1/m1, 1/u1)

Example: Let t1= (2, 4, 6) and t2=(1, 2, 3) be two fuzzy numbers. Then

i) t1+ t2 = (3, 6, 9) ii) t1� t2 = ( 2, 8, 18)

0 1 2 3 40

0 . 5

1

25

ii) (iii) 2 t1= (4, 8, 12) iv) t1-1 = (0.5, 0.25, 0.17)

Reasons to use fuzzy arithmetic

• Requires little data

• Applicable to all kinds of uncertainty

• Fully comprehensive

• Fast and easy to compute

Fuzzy Aggregation

Aggregation operations on fuzzy sets are operations by which several fuzzy sets are

combined to produce a single set. In general, any aggregation operation is defined by a

function h:[o,1]n→[0,1] for some n≥2. When applied to n fuzzy sets A1, A2,…An

defined on X, h produces an aggregate fuzzy set A by operating on the membership

grades of each x in X in the aggregated sets.

In order to qualify as an aggregation function, h must satisfy at least the following two

axiomatic requirements, which express the essence of the notion of aggregation.

Axiom 1[ Boundary Conditions]

h(0,0,…..0) = 0 and h(1,1,…..1)

Axiom 2

For any pair(ai such that I in ℕn) and (bi such that I in ℕn)where ai in [0,1] and bi in

[0,1], if ai≥bi for all i � ℕn then h(ai/ i � ℕn) ≥ h(bi / i � ℕn) that is h is monotonic

non decreasing in all its arguments.

Two additional more axioms are employed to characterize aggregation operations

despite the fact that they are not essential.

26

Axiom 3

h is continuous.

Axiom 4

h is a symmetric function in all its argument. That is h(ai / i � ℕn) = h(ap(i) / i � ℕn).

Aggregation operations on fuzzy numbers are operations by which several fuzzy

numbers are combined to produce a single fuzzy number.

Applications of Aggregation

The combination / aggregation/ fusion of information from different sources are at the

core of knowledge based systems. Applications include decision making, subjective

quality evaluation, information integration, multi-sensor data fusion, image processing,

pattern recognition , computational intelligence etc.

Aggregation Operators on triangular fuzzy numbers

(i)Arithmetic Mean

The Arithmetic mean operator defined on n triangular fuzzy numbers <l1,m1,u1> ,

<l2,m2,u2>…..<li,mi,ui>…..<ln,mn,un> produces the result < , , > where

= ∑ = ∑ and = ∑(ii)Geometric Mean

The geometric aggregation operator defined on n triangular fuzzy numbers <l1,m1,u1>,

<l2,m2,u2>…..<li,mi,ui>…..<ln,mn,un> produces the result < , , > where

= (∏ ) = (∏ ) and =(∏ )

27

Some more aggregation operations

= (∏ ) = (∏ ) -

(∏ ) and

= (∏ ) + (∏ )

Numerical Example

Consider the five triangular fuzzy numbers (5,7, 9) , (7,9,10), (3,5,7), (9,10,10) and

(3,5,7). The aggregated fuzzy number for the above five fuzzy numbers is < , , > where = (∏ ) = (7 � 9 � 5 � 10 �5)1/5 = 6.90965

= -(∏ )

=6.90965 -

( .9 97)( .9 97)(3.9 97)( 3.9 97)(3.9 9)5 = 6.567

= (∏ ) + (∏ )

=6.90965 + ( . )(4. )( . )(4. )( . 6 )5 = 10.1219

< , , > = ( . , . 0 , 10.121 )

28

CHAPTER 3

MULTI CRITERIA DECISION MAKING PROBLEMS IN

FUZZY ENVIRONMENT

Multi-criteria Decision Making (MCDM)

In real world, we deal with decision making cases which have different, antonyms and

multiple criteria. If we consider multiple qualitative and antonym elements in our

decision making process, we call this a multi-criteria decision making . Multi criteria

decision making have two models:

a) Multiple objective decisions making (MODM)

b) Multiple attribute decision making (MADM).

The first model is applied for design purposes whereas the latter is used for selecting top

options. As now a day’s systems benefit from expert employees on one hand and on the

other hand managers of these systems are in the same level, so it would be better to make

decisions with respect to the ideas of the whole group and the basic body of system's

decision makers. We call this kind of decision making; group decision making which can

be applied to multi-criteria conditions .Decision making is a complex and difficult

process due to various uncertainties and vagueness of information, mentalities and

linguistics. So, when we deal with uncertainty conditions in various concepts and

processes, we merge fuzzy sets with multi-criteria decision making. It is a technique

where alternatives or options are assessed based on a set of criteria and it is one of the

most widely used methods in decision making (Hwang & Yoon, 1981). MCDM methods

have been employed in many areas such as engineering, agricultural, banking, energy,

forestry, health services and education.

29

General form of MCDM problem with m alternatives and n criteria can be illustrated in

matrix format as follows:

/ 1 … …1. . .. . .Multi-criteria decision making (MCDM) represents an interest area of research since

most real-life problems have a set of conflict objectives. Many real-life problems have

been formulated as Fuzzy Multi Criteria Decision Making and have been solved by using

an appropriate technique. Some of these applications involved production,

manufacturing, location−allocation problems, environmental management, business,

marketing, agriculture economics, machine control, engineering applications and

regression modeling.

Multiple Criteria Decision Making was introduced as a promising and important field of

study in the early 1970’es. Since then the number of contributions to theories and models,

which could be used as a basis for more systematic and rational decision making with

multiple criteria have continued to grow at a steady rate. A number of surveys show the

vitality of the field and the multitude of methods which have been developed. When

Bellman and Zadeh, and a few years later Zimmermann, introduced fuzzy sets into the

field, they cleared the way for a new family of methods to deal with problems which had

been inaccessible to and unsolvable with standard MCDM techniques. Decision making

is a problem solving process by which a method is selected among various methods in

order to obtain an effective and applicable result .

In real life, decision makers often make evaluation based on a set of criteria which are

normally vague and imprecise. Decision Maker’s judgments are uncertain and cannot be

estimated by exact numerical values. Under many conditions, crisp data are inadequate to

model real-life situations. The classical multi decision making methods both

30

deterministic and random process cannot effectively handle Group Decision Making

problems with imprecise and linguistic information, therefore fuzzy MCDM methods

were developed. To deal with vagueness of human thought, fuzzy set was introduced

particularly in representing the vague information or criteria. Fuzzy set theory was first

utilized in solving decision making problem by Bellman and Zadeh in 1970. The key

concept of fuzzy set theory is that its elements have a varying grade of membership,

ranging from 0 to 1. The boundaries of these fuzzy sets are not sharp or imprecise. The

individual membership in a fuzzy set is represented by the degree of compatibility (Klir

et.al, 1997) and fuzzy sets are used to describe linguistic values for example "very good,"

"good," "fair," "poor," and "very poor". Instead of using exact numbers as input values,

fuzzy numbers were utilized in representing these linguistic terms. In multi-criteria

decision making method, the weight of elements and the estimated values are expressed

by fuzzy numbers or linguistic variables.

The introduction of fuzzy set theory also motivates many researchers in integrating the

theory with some of the classical MCDM methods. Pioneer work in incorporating fuzzy

element into decision making was done (Baas and Kwakernaak, 1977) by introducing an

algorithm for rating and ranking multiple aspects of alternatives using fuzzy sets. When

fuzzy set theory was introduced into MCDM research the methods were basically

developed along the same lines. There are a number of very good surveys of fuzzy

MCDM The concept of a fuzzy number plays a fundamental role in formulating

quantitative fuzzy variables. Sometimes it becomes a very difficult task to assess the

characteristics of some events through numerical formats. A useful tool which is

employed for this purpose is linguistic variables. They are variables in which their values

are sentences or words of natural or artificial languages. Decision makers' opinions can

be expressed in terms of linguistic variables.

31

Linguistic Variable:

A linguistic variable is a variable whose values are words or sentences in a natural or

artificial language. These linguistic variables can be expressed in positive triangular

fuzzy numbers. Each linguistic variable the states of which are expressed by linguistic

terms interpreted as specific fuzzy numbers is defined in terms of base variables, the

values of which are real numbers with in a specific range. A basic variable is a variable

in the classical sense exemplified by any physical variable (e.g. temperature, pressure,

speed, voltage, humidity, etc.) as well as any other numerical variable (e.g. Age, interest

rate, performance, salary, blood count, probability, reliability, etc.). In linguistic variable

linguistic terms representing approximate values of a base variable are captured by

appropriate fuzzy numbers. Each linguistic variable is fully characterized by a quintuple

(v, T, X, g, m) in which v is the name of the variable, T is the set of linguistic terms of v

that refers to a base variable whose values range over a universal set X, g is a syntactic

rule for generating linguistic terms and m is a semantic rule that assigns to each linguistic

term t in T. Each of the basic linguistic terms is assigned one of five fuzzy numbers by a

semantic rule. To deal with linguistic variable we need not only the various set theoretic

operations but also arithmetic operations on fuzzy numbers.

Examples:

Very Poor Poor Medium Poor Fair Medium Good Good Very Good

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6

32

Linguistic Variable

Linguistic Values

1.0

0 10 22.5 32.5 45 55 67.5 77.5 90 100

Linguistic variable can be represented by using the triangular membership function

as follows.

Very Poor VP (0,0,1)Poor P (0,1,3)Medium Poor MP (1,3,5)Fair F (3,5,7)Medium Good MG (5,7,9)Good G (7,9,10)Very Good VG (9,10,10)

PERFORMANCE

Very Small Small Medium Very LargeLarge

33

CHAPTER 4

SOLUTIONS OF MCDM PROBLEMS

The Classical Multi Criteria Decision Making methods, both deterministic and random

process, cannot effectively handle Group Decision Making problems with imprecise and

linguistic information, therefore fuzzy MCDM methods were developed.

A lot of research has been conducted in the area of group decision making under the

application of fuzzy set theory. Some of them have also applied the concept of linguistic

variables to handle linguistic terms and approximate reasoning in a group decision

making problems. Some researchers have been carried out in describing the uncertainty

of individual preferences for alternatives and aggregating these fuzzy individual

preferences into a group decision making. The “Method of marks” voting procedure

proposed by the French Scientist Jean-Charles de Borda (1733-1799) in paris represents

an important step in the development of modern electoral systems and indeed in the

theory of voting more generally. Borda rule is an appropriate procedure in multi-person

decision making when several alternatives are considered.

4.1 PROPOSED METHOD

The proposed method is a user-friendly fuzzy approach under the linguistic frame work

to obtain optimal solution for Multi Criteria Decision Making problems. To accomplish

this, an aggregate-deviation method based on fuzzy numbers is applied. A fuzzy decision

matrix plays an important role in our research problem .The purpose of this method is to

enhance group agreement on the group decision making outcomes.

Steps involved

Collect the evaluation of alternatives by expert decision makers with respect to all criteria

in terms of linguistic variables and we can form a decision matrix.

Replace each linguistic variable by corresponding fuzzy number.

Aggregate the fuzzy numbers in column wise based on criteria C1, C2 ,……..

34

Aggregate the fuzzy numbers column wise based on decision makers P1, P2 ,…..

Find the deviation in triangular fuzzy number.

The fuzzy number with minimum deviation comes first in ranking order

[ascending].

COMPUTATIONAL ASPECTS

Suppose group of expert decision makers want to select a most suitable candidate from

several alternatives based on some criteria.

STEP 1

Evaluation of alternatives by expert decision makers with respect to all criteria in terms

of linguistic variables.

Linguistic frame work[9].

Very Poor VP (0,0,1)

Poor P (0,1,3)

Medium Poor MP (1,3,5)

Fair F (3,5,7)

Medium Good MG (5,7,9)

Good G (7,9,10)

Very Good VG (9,10,10)

35

Evaluation table

C1 C2 …… …Cm

P1

A1 L1 L2 L3

… .. … .. .. .. .. .. ..

..

An

P2

A1

.. .. .. .. .. .. .. .. .. .. .. ..

….

An

.

..

Pk A1 .. ..

..

An

Here C1, C2,…. are the criteria. A1, A2, A3…. are the alternatives. P1,P2,…. are the

decision makers.L1, L2,… are the linguistic variables.

STEP 2

36

To construct a fuzzy decision matrix replace each linguistic variable by corresponding

fuzzy number.

STEP 3

Aggregate the fuzzy numbers in column wise based on criteria C1, C2,…….. by using the

formula Fag = (Lag , Mag, Uag)

where

Lag= ∑ ,

Mag = ∑ and Uag = ∑

for all fuzzy numbers (li,mi,ui).

We get,

P1 P2……….. …..Pk

A1 Fag 1 Fag 2………

.

.

.

.

.

.

Am …….. ………

Where Fag 1, Fag 2……… are fuzzy numbers.

STEP 4

Aggregate fuzzy numbers column wise using the same formula based on decision makers

P1, P2,…..

37

We get

A1 Fag 1

A2 Fag 2

…. ….

Am Fag m

STEP 5

Find the deviation in triangular fuzzy number by using the formula.

Df =( ) + −+

where (l,m,u) is triangular fuzzy number.

STEP 6 [CONCLUSION]

The fuzzy number with minimum deviation (Df) comes first in ranking order [ascending].

38

FLOW CHART PRESENTATION

Input evaluation of alternatives

Replace each linguistic variable by corresponding fuzzy number(li,mi,ui.)

Compute Fag = (Lag , Mag, Uag)

where Lag= Mag =

& Uag =

Compute the deviation

Df = +

Fuzzy number with minimum deviation (Df) comes first in ranking order [ascending].

Output the ranking order

39

NUMERICAL EXAMPLE

Here we work out a numerical example taken from (Chen, 2000) to illustrate the

proposed methods for decision making problems with fuzzy data. Suppose 3 expert

decision makers want to select a system analysis engineer for a software company from 3

alternatives based on 5 criteria which are attitude, communication skills, hardworking,

general knowledge, and programming knowledge. After preliminary screening, three

candidates (alternatives) A1, A2, A3 remain for further evaluation. A committee of three

decision makers P1, P2, P3 has been formed to conduct interview and select the most

suitable candidate. Five benefits criteria are considered:

C1: Attitude C2: Communication skills C3: Hardworking C4: General knowledge

C5: Programming knowledge.

Decision makers used the linguistic variables to evaluate the rating of alternatives with

respect to each criterion . Then the linguistic variables are converted into triangular fuzzy

numbers to construct the fuzzy decision matrix.

The following steps must be followed.

1. Identifying evaluation criteria

2. Generating alternatives.

3. Identifying weights of criteria.

4. Construction of fuzzy decision matrix.

5. Aggregation of fuzzy numbers.

6. Find out the deviation

7. Concluding the ranking order.

The hierarchical structure of the problem is shown below.

40

Hierarchical structure

The proposed method is currently applied to solve this problem and the computational

procedure is summarized as follows.

STEP 1

Collect the evaluation of alternatives by expert decision makers with respect to all criteria

in terms of linguistic variables and we can form a decision matrix.

C1 C2 C3 C4 C5

A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3

P1

MG G VG G VG MG F VG G VG VG VG F VG G

GOAL

Attitude Communication Skills

Hardworking

General knowledgeProgramming knowledge

A3 A2 A1

41

P2

G G G MG VG G G VG MG G VG VG F MG G

P3

MG MG F F VG VG G G VG VG VG MG F G MG

STEP 2

Replace each linguistic variable by corresponding fuzzy number.

FUZZY DECISION MATRIX

C1 C1 C3 C4 C5

P1 A1 (5,7,9) (7,9,10) (3,5,7) (9,10,10) (3,5,7)

A2 (7,9,10) (9,10,10) (9,10,10) (9,10,10) (9,10,10)

A3 9,10,10 (5,7,9) (7,9,10) (7,9,10) (7,9,10)

P2 A1 (7,9,10) (5,7,9) (7,9,10) (7,9,10) (3,5,7)

A2 (7,9,10) (9,10,10) (9,10,10) (9,10,10) (5,7,9)

A3 (7,9,10) (7,9,10) (5,7,9) (9,10,10) (7,9,10)

P3 A1 (5,7,9) (3,5,7) (7,9,10) (9,10,10) (3,5,7)

A2 (5,7,9) (9,10,10) (7,9,10) (9,10,10) (7,9,10)

A3 (3,5,7) (9,10,10) (9,10,10) (5,7,9) (5,7,9)

STEP 3

Aggregate the fuzzy numbers in column wise based on criteria C1, C2 ,……..

C1 C2 C3 C4 C5 Aggregation

P1 A1 (5,7,9) (7,9,10) (3,5,7) (9,10,10) (3,5,7) (5.4,7.2,8.6)

42

A2 (7,9,10) (9,10,10) (9,10,10) (9,10,10) (9,10,10) (8.6, 9.8, 10)

A3 9,10,10 (5,7,9) (7,9,10) (7,9,10) (7,9,10) (7,8.8,9.8)

P2 A1 (7,9,10) (5,7,9) (7,9,10) (7,9,10) (3,5,7) (5.8,7.8,9.2)

A2 (7,9,10) (9,10,10) (9,10,10) (9,10,10) (5,7,9) (7.8,9.2,9.8)

A3 (7,9,10) (7,9,10) (5,7,9) (9,10,10) (7,9,10) (7,8.8,9.8)

P3 A1 (5,7,9) (3,5,7) (7,9,10) (9,10,10) (3,5,7) (5.4,7.2,8.6)

A2 (5,7,9) (9,10,10) (7,9,10) (9,10,10) (7,9,10) (7.4,9,9.8)

A3 (3,5,7) (9,10,10) (9,10,10) (5,7,9) (5,7,9) (6.2,7.8,9)

We get

P1 P2 P3

A1 (5.4,7.2,8.6) (5.8,7.8,9.2) (5.4,7.2,8.6)

A2 (8.6, 9.8, 10) (7.8,9.2,9.8) (7.4,9,9.8)

A3 (7,8.8,9.8) (7,8.8,9.8) (6.2,7.8,9)

STEP 4

Aggregate the fuzzy numbers column wise based on decision makers P1, P2 ,…..

Total evaluation

A1 (5.53,7.4,8.8)

A2 (7.93,9.33,9.86)

A3 (6.73, 8.46, 9.53)

43

STEP 5

Find the deviation in triangular fuzzy number by using the formula

Df =( ) + −+ .

For A1 ( l=5.53,m=7.4,u=8.8) Df=3.489

For A2 ( l=7.93,m=9.33,u=9.86) Df=2.058

For A3 ( l=6.73, m=8.46, u=9,53) Df=2.977

STEP 6

The fuzzy number A2 with minimum deviation Df=2.058 comes first in ranking order

[ascending]. The final ranking order is A2, A3, A1.

Also we can use the following aggregation (geometric mean) formula.

Lag =Mag -∏

,

Mag = [∏ ] &

Uag = Mag + ∏

for all fuzzy numbers (li,mi,ui).

44

EXAMPLE

Suppose 3 expert decision makers want to select a most suitable computer programmer

from 3 alternatives based on 5 criteria which are attitude, communication skills,

hardworking, general knowledge, and programming knowledge.

STEP 1

Collect the evaluation of alternatives by expert decision makers with respect to all criteria

in terms of linguistic variables and we can form a decision matrix[6].

C1 C2 C3 C4 C5

A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3 A1 A2 A3

P1

MG G VG G VG MG F VG G VG VG VG F VG G

P2

G G G MG VG G G VG MG G VG VG F MG G

P3

MG MG F F VG VG G G VG VG VG MG F G MG

STEP 2

Replace each linguistic variable by corresponding fuzzy number.

FUZZY DECISION MATRIX

C1 C1 C3 C4 C5

P1 A1 (5,7,9) (7,9,10) (3,5,7) (9,10,10) (3,5,7)

A2 (7,9,10) (9,10,10) (9,10,10) (9,10,10) (9,10,10)

45

A3 9,10,10 (5,7,9) (7,9,10) (7,9,10) (7,9,10)

P2 A1 (7,9,10) (5,7,9) (7,9,10) (7,9,10) (3,5,7)

A2 (7,9,10) (9,10,10) (9,10,10) (9,10,10) (5,7,9)

A3 (7,9,10) (7,9,10) (5,7,9) (9,10,10) (7,9,10)

P3 A1 (5,7,9) (3,5,7) (7,9,10) (9,10,10) (3,5,7)

A2 (5,7,9) (9,10,10) (7,9,10) (9,10,10) (7,9,10)

A3 (3,5,7) (9,10,10) (9,10,10) (5,7,9) (5,7,9)

STEP 3

Aggregate the fuzzy numbers in column wise based on criteria C1, C2 ,……..

C1 C2 C3 C4 C5 Aggregation

P1 A1 (5,7,9) (7,9,10) (3,5,7) (9,10,10) (3,5,7) (6.57,6.91,10.12)

A2 (7,9,10) (9,10,10) (9,10,10) (9,10,10) (9,10,10) (8.82, 9.79, 10)

A3 9,10,10 (5,7,9) (7,9,10) (7,9,10) (7,9,10) (6.71, 8.74, 9.66)

P2 A1 (7,9,10) (5,7,9) (7,9,10) (7,9,10) (3,5,7) (6.39, 7.6, 9.24)

A2 (7,9,10) (9,10,10) (9,10,10) (9,10,10) (5,7,9) 8.69, 9.12,9.71)

46

A3 (7,9,10) (7,9,10) (5,7,9) (9,10,10) (7,9,10) (6.71,8.74,9.66)

P3 A1 (5,7,9) (3,5,7) (7,9,10) (9,10,10) (3,5,7) (3.97,6.91,8.05)

A2 (5,7,9) (9,10,10) (7,9,10) (9,10,10) (7,9,10) (6.71, 8.93, 9.55)

A3 (3,5,7) (9,10,10) (9,10,10) (5,7,9) (5,7,9) (5.27, 7.55, 9.02)

Then we get

STEP 4

Aggregate the fuzzy numbers column wise based on decision makers P1, P2 ,…..

Total evaluation

A1 (6.07, 7.02, 8.93)

A2 (8.46, 9.07, 9.73)

P1 P2 P3

A1 (6.57,6.91,10.12) (6.39, 7.6, 9.24) (3.97,6.91,8.05)

A2 (8.82, 9.79, 10) 8.69, 9.12,9.71) (6.71, 8.93, 9.55)

A3 (6.71, 8.74, 9.66) (6.71,8.74,9.66) (5.27, 7.55, 9.02)

47

A3 (6.73, 8.15, 9.40)

STEP 5

Find the deviation in triangular fuzzy number by using the formula

Df =( ) + −+

For A1 ( l=6.07,m=7.02,u=8.93) Df=2.96

For A2 ( l=8.46,m=9.07,u=9.73) Df=1.32

For A3 ( l=6.73, m=8.15, u=9.40) Df=2.82

STEP 6

The fuzzy number A2 with minimum deviation Df=1.32 comes first in ranking order

[ascending]. The final ranking order is A2, A3, A1.

Method A1 A2 A3 Ranking Order

Aggregation Mean using Arithmetic mean

3.409 2.508 2.977 A2 > A3 > A1

Aggregation Mean using Geometric mean

2.96 1.32 2.82 A2 > A3 > A1

48

4.2 CONCLUSION & FUTURE PLANS

The important fields Engineering, Management and Social science domain involve

various Decision Making problems. These problems often involve data that are

imprecise, uncertain and vague in nature. A number of solutions have been proposed for

such problems using probability theory, Rough set theory, Approximate Reasoning

theory etc. But, inorder to deal with vagueness and ambiguity of human preferences ,

fuzzy set theory is needed.

In Multi Criteria Group Decision Making with linguistic variables, the decision matrix

may have vague information, limited attention and different information processing

capabilities. This research work proposes a user friendly fuzzy approach with linguistic

variables. Linguistic variable fits to the habits of the human being and makes the

communication among individuals more conveniently. A numerical example has been

worked out. The results are very similar to other methods. The approach is

computationally simple and facilitating its implementation in a computer-based system.

The proposed method can be converted to computer program in future to make the

method more accessible.

49

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