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Int. J. Pure Appl. Sci. Technol., 6(1) (2011), pp. 54-61
International Journal of Pure and Applied Sciences and Technology
ISSN 2229 - 6107
Available online at www.ijopaasat.in
Research Paper
Weighted Average Rating (WAR) Method for Solving
Group Decision Making Problem Using an Intuitionistic
Trapezoidal Fuzzy Hybrid Aggregation (ITFHA)
Operator
A. Nagoor Gani 1,*, N. Sritharan 2 and C. Arun Kumar 3
1,2,3
PG & Research Department of Mathematics, Jamal Mohamed College, Trichy, Tamilnadu, India
* Corresponding author, e-mail: ([email protected])
(Received: 22-7-11; Accepted: 30-8-11)
Abstract: Intuitionistic Fuzzy numbers each of which is characterized by the
degree of membership and the degree of non-membership of an element are a very
useful means to depict the decision information in the process of decision making.
The aim of this article is to investigate the approach to multiple attribute group
decision making with intuitionistic trapezoidal fuzzy numbers, some operational
laws of intuitionistic trapezoidal fuzzy numbers are applied. We investigate the
group decision making problems in which all the information provided by the
decision makers is expressed as decision matrices where each of the elements are
characterized by intuitionistic trapezoidal fuzzy numbers and the information about
attribute weights are known. We first use the intuitionistic trapezoidal fuzzy hybrid
aggregation (ITFHA) operator to aggregate all individual fuzzy decision matrices
provided by the decision makers into the collective intuitionistic fuzzy decision
matrix. Furthermore, we utilize weighted average rating method and score function
to give an approach to ranking the given alternatives and selecting the most
desirable one. Finally we give an illustrative example.
Keywords: Intuitionistic Fuzzy Set, Multiple Attribute Group Decision
Making(MAGDM), Intuitionistic Trapezoidal Fuzzy Hybrid Aggregation Operator,
Weighted Average Rating, Score Function.
Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61.
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1. Introduction:
Atanassov[1] introduced the concept of intuitionistic fuzzy set (IFS) characterized by a
membership function and a non-membership function, which is a generalization of the
concept of fuzzy set[2] whose basic component is only a membership function. Li[5]
investigated MADM with intuitionistic fuzzy information and Lin[6] presented a new
method for handling multiple attribute fuzzy decision making problems, where the
characteristics of the alternatives are represented by intuitionistic fuzzy sets. Furthermore,
the proposed method allows the decision maker to assign the degree of membership and
the degree of non-membership of the attribute to the fuzzy concept ‘importance’. Wang[8]
gave the definition of intuitionistic trapezoidal fuzzy number and interval valued
intuitionistic trapezoidal fuzzy number. Wang and Zhang[9] gave the definition of
expected values of intuitionistic trapezoidal fuzzy number and proposed the programming
method of multi-criteria decision making based on intuitionistic trapezoidal fuzzy number
incomplete certain information.
In this paper, we focus our attention on the issue of multi attribute decision making under
intuitionistic fuzzy environment where all the information provided by the decision
makers is characterized by intuitionistic trapezoidal fuzzy numbers, and the information
about the attribute weights are known. We first use the intuitionistic trapezoidal fuzzy
hybrid aggregation (ITFHA) [4] operator to aggregate all individual fuzzy decision
matrices provided by the decision makers into the collective intuitionistic fuzzy decision
matrix. Next we calculate the weighted average rating [7] by using the aggregated matrix
and the given criteria weights. Finally we find the best alternative by using the score
function.
This Paper is organized as follows: In section 2, we give a review of basic concepts and
operator related with intuitionistic trapezoidal fuzzy numbers. Section 3, presents an
algorithm for weighted average rating method for solving Group decision making problem
using an Intuitionistic trapezoidal fuzzy Hybrid aggregation operator. Section 4, provides
a practical example to illustrate the developed approach and finally, we conclude the
paper in Section 5.
2. Basic Concepts:
2.1. Definition: Let a set X={x1, x2, x3, …, xn} be fixed, an IFS A in X is an object of the
following
form,
Where
the
functions
µ A : X → [0,1], x ∈ X , µ A ( x) ∈ [0,1] and υ A : X → [0,1], x ∈ X ,υ A ∈ [0,1] with the
condition
, the numbers
denote the
to the set
degree of membership and the degree of non-membership of the element
A respectively.
2.2. Definition: Let
its membership function as
be an intuitionistic trapezoidal fuzzy number,
Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61.
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Its non-membership function is
Where
and
.
2.3. Definition: Let
two intuitionistic trapezoidal fuzzy number and λ ≥ 0, then
1.
2.
3.
4.
be
2.4. Definition: An intuitionistic trapezoidal fuzzy hybrid aggregation (ITFHA) operator
of dimension n is a mapping
, that has an associated vector
such that wj> 0 and
Furthermore,
Where
is the jth largest of the weighted intuitionistic trapezoidal fuzzy numbers
be the weight vector of
, and n is the balancing coefficient. Where
is a permutation of (1,2,…,n), such that
for all
j=2,…,n.
2.5. Definition: We defined a method to compare two intuitionistic trapezoidal fuzzy
numbers which is based on the score function and the accuracy function. Let
and
be two intuitionistic fuzzy values,
and
be the scores of and , respectively, and let,
and
be the accuracy degrees of and , respectively, then
(i)
if S( ) < S( ), then
is smaller than , denoted by < ;
(ii)
if S( ) = S( ), then,
(a)
if H( ) = H( ), then
and
are the same, denoted by = ;
(b)
if H( ) < H( ), then is smaller than , denoted by < .
3. Weighted Average Rating (WAR) Algorithm:
Step 1: Form a intuitionistic fuzzy decision matrix
Step 2: Utilize the ITFHA operator
of k decision makers.
Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61.
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to derive the
of
collective overall preference intuitionistic trapezoidal fuzzy values
the alternative Ai, where V=(v1, v2, …, vn) be the weighting vector of decision makers
with vk in [0,1],
is the associated weighting vector of
the ITFHA operator, with
Step 3: Calculate the weighted aggregated decision matrix , using the multiplication
formula,
where
is the intuitionistic
trapezoidal fuzzy number.
Step 4: calculate the weighted average rating by using the aggregated matrix and the
given criteria weights with help of the formula
Step 5: To find the best alternative by using the score function.
5. Numerical Example:
Let us suppose there is a risk investment company which wants to invest a sum of money
in the best option. There is a panel with four possible alternatives to invest the money.
The risk investment company must take a decision according to the following four
attributes.
1.
G1 is the risk analysis,
2.
G2 is the growth analysis,
3.
G3 is the social political impact analysis,
4.
G4 is the Environmental impact analysis.
The four possible alternatives Ai (i=1,2,3,4) are to be evaluated using the intuitionistic
trapezoidal fuzzy numbers by the three decision makers with their weighting vector
v=(0.35,0.40,0.25)T under the above four attributes weighting vector w=(0.2,0.1,0.3,0.4)T
and construct the decision matrices
as follows,
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By step 2 using the ITFHA Operator to aggregate all the three decision matrices into
single collective decision matrix with Intuitionistic trapezoidal fuzzy ratings.
Consider,
Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61.
The Aggregated Matrix is,
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Int. J. Pure Appl. Sci. Technol., 6(1) (2011), 54-61.
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Next we calculate the weighted aggregated decision matrix by using step 3, we get
Calculate the Weighted Average Rating for each alternative by using step 4, we get
D(A1) = ([0.4233,0.5270,0.6623,0.7654];0.2810,0.5300)
D(A2) = ([1.2734,1.3860,1.5248,1.6730];0.5980,0.2880)
D(A3) = ([0.9573,1.1650,1.3870,1.5420];0.5060,0.2990)
D(A4) = ([1.0770,1.2310,1.3850,1.5370];0.3841,0.3520)
To find the ranking order of the alternatives, use the score function,
S(A1) = 0.2810 - 0.5300 = -0.249
S(A2) = 0.5980 – 0.2880 = 0.3100
S(A3) = 0.5060 – 0.2990 = 0.2070
S(A4) = 0.3841 – 0.3520 = 0.0321
S(A2) > S(A3) > S(A4) > S(A1)
i.e. A2 > A3 > A4 > A1
The Best option is A2.
4. Conclusions:
We have investigated the multiattribute decision making problems under intuitionistic
fuzzy environment and developed an approach to handling the situations where the
attribute values are characterized by intuitionistic trapezoidal fuzzy numbers and the
information about attribute weights are known. The approach first fuses all individual
intuitionistic fuzzy decision matrices into the collective intuitionistic fuzzy decision
matrix by using the intuitionistic trapezoidal fuzzy hybrid aggregation operator, then
based on the collective intuitionistic fuzzy decision matrix, we can utilize the weighted
average rating method and the score function to get the best alternative.
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