THE TECHNIQUE OF FINDING A BEST
OPTION TO MULTIPLE ATTRIBUTE
DECISION MAKING PROBLEMS BY
GREY SYSTEM THEORY USING FUZZY
CONCEPT
R.BIRLA
Assistant Professor of Mathematics, Shanmuga Industries Arts and Science College,
Tiruvannamalai, Abstract
This is one of best way to find an
alternative in the situation of multiple attribute
decision making. This paper presents how to find a
way the required alternative using fuzzy sets and
Grey system with interval numbers.
Keywords: Grey system theory, multiple attribute decision making, Fuzzy sets, AHP, TOPSIS.
I.INTRODUCTION
In the field of decision science, everyone do the research for minimization of cost through system engineering, management science, operations research and so on. Deng (1982) developed the Grey system theory and introduced (Deng, 1989) grey decision-making systems. The grey system has been applied in many areas as follows, Grey Relational space, Grey Generating space, Grey Forecasting, Grey Decision Making, Grey Control, Grey Mathematics, and Grey Theory Grey System theory
One of the main reason is the software researchers tend to be methodologically accepting, when they are grey mathematically efficient and sure of being grey methodologically innovative and some of the element techniques and concepts of Grey System theory have been implemented and investigated by a variety of areas.
Grey system theory using fuzzy concept Let
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,
,
,
/
{
]
,
[
a
b
x
a
x
b
a
b
a
b
R
A
, we call A is an interval number.
If
0
a
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, we call interval number A a positive interval number.Let
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[
],
,
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{[
a
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2b
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nX
be an n dimension interval number column vector. Definition: 1
If
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[
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,
[
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,
{[
a
1b
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2b
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nX
is an arbitrary interval number column vector, the norm of X is defined by
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2 2 1
1
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Definition: 2
If A= [a1, b1] and B= [a2, b2] are two arbitrary
interval number, the distance from A to B is defined by
}
,
{
2 1 2
1
a
b
b
a
Max
B
A
Definition: 3
If A= [a1, b1] is an arbitrary interval number, if k is
an arbitrary real number, the product of A and k is defined by
Definition: 4
If A= [a1, b1] and B= [a2, b2] are two arbitrary
interval number, the product of A and B is defined by
When b2>0: [a1, b1] *[a2, b2]=[a1a2, b1b2]
When b2<0: [a1, b1] *[a2, b2]=[b1a2, a1b2]
When b2=0: [a1, b1] *[a2, b2]={a1a2 }
Analytic Hierarchy Process (AHP)
The pair-wise comparison method and the hierarchical model were introduced in 1980 by T.L.Saaty in the context of the Analytical Hierarchy Process (AHP). It is most widely used MCDM approaches. AHP is an approach to decision making that involves structuring multiple choice criteria into a hierarchy, assessing the relative importance of these criteria, comparing alternatives for each criterion and determining an overall ranking of the alternatives. Some of its applications include technology choice, vendor selection of a telecommunications system, project selection, and budget allocation. The steps for implementing the AHP process are illustrated as follows: Define the Objectives, Identify the Criteria/Attributes, and Choose the Alternatives, Establish the Hierarchy, Design Questionnaire and survey, Construct the Pair wise Comparison matrices using Satty’s 9-point scale.
Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)
It is one of the most classical MCDM methods, was first introduced by Hwang and Yoon , it is based on the idea that the chosen alternative should have the shortest distance from the Positive Ideal Solution (PIS) and on the other side the farthest distance of the Negative Ideal Solution (NIS). The Positive Ideal Solution maximizes the benefit criteria and minimizes the cost criteria, whereas the Negative Ideal Solution maximizes the cost criteria and minimizes the benefit criteria [16, 17]. In the process of TOPSIS, the performance ratings and
the weights of the criteria are given as exact values. The steps of TOPSIS model are as follows: Find the normalized decision matrix, Find the weighted normalized decision matrix, Find out the Positive Ideal Solution and Negative Ideal Solution, Find the separation measures for each alternative from the positive and negative ideal solution. Find the relative closeness to the ideal solution for each alternative. Rank the preference order
Multiple attribute decision making:
Suppose that Multiple attribute decision making problem with interval number has m feasible alternatives X1, X2, …..,Xn, n indexes, the
weight value wj of index Gj is uncertain, but we
know that
w
j
[
c
j,
d
j]
.Here,0
c
j
d
j
1
j=1, 2… n, w1+w2+….+ wn=1 the index value of jth index Gj of feasible alternative Xi is an interval number [aij, bij], i=1, 2, …. , m and j=1, 2, …., n.whencj = dj, j=1, 2, …., n the multiple attribute decision making problem with interval numbers is an interval valued multiple decision making problem with crisp weights. When aij = bij, i=1, 2, …. , m and j=1, 2, …., n., the alternative scorescriteria are crisp.
Step of the method
Step 1: Construct decision matrix M with index number of interval numbers
If the index value of jth index Gj of
feasible alternative Xi is an interval number [aij,
bij], i=1, 2, …. , m and j=1, 2, …., n, decision
matrix M with index number of interval of interval numbers is defined as follow,
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1 1 12
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n n
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a
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a
b
Step 2: Transform contrary index into positive index
The index is called a positive index if a greater index value is better. The index is called a contrary index if a smaller index value is better. We may transform contrary index onto positive index if jth index Gj is contrary index
[cij, dij] = [-bij, -aij] i=1, 2, ….m
Without loss of generality, in the following, we supposed that the entireindexes are positive indexes’.
Step 3: Standardize decision matrix M with index number of interval numbers to gain standardizing decision matrix R = [uij, vij]
If we mark the column vectors of decision matrix M with interval valued indexes with M1,
M2… Mn, the element of standardizing decision
matrix R = [uij, vij] is defined as the following
j ij j ij ij ijM
b
M
a
v
u
,
]
,
[
i=1, 2…, m and j=1,2, …., n.,
NB: TOPSIS uses the root mean square to evaluate distance. Grey theory uses a different norm, based on minimization of maximum distance.
Step 4: Find interval number weighted matrix C = ([xij, yij])m x n
The formula for find the interval number weighted matrix C = ([xij, yij])m x n is
[xij, yij] = [cj, dj]*[uij, vij] , i=1, 2, …. , m and j=1,
2, …., n.,
Step 5: Determine number sequence
The vector for the reference number is determined as the set of optimal weighted interval values associates with each of the n attributes.
[
0(
1
),
0(
1
)],
[
0(
2
),
0(
2
)],...
,
[
0(
),
0(
)]
0
n
n
Z
is called a reference number sequence, if
ij m i ij
m
i
x
and
j
Max
y
Max
j
1 0
10
(
)
(
)
, j = 1, 2, …. n.
Step 6: Find the connection between the sequences composed of weight interval number standardizing index value of every alternative and reference number sequence.
The connection coefficient
i(
k
)
, between the sequences composed of weight interval number standardizing index value of every alternative
[
i1,
i1],
[
i2,
i2],...,
[
in,
in]
i
x
y
x
y
x
y
Z
andreference number sequence
[
0(
1
),
0(
1
)],
[
0(
2
),
0(
2
)],...
,
[
0(
),
0(
)]
0
n
n
Z
is calculated. The formula of
i(
k
)
is]
[
)]
(
),
(
[
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[
)]
(
),
(
[
]
[
)]
(
),
(
[
]
[
)]
(
),
(
[
)
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, 0 0 , 0 0 , 0 0 , 0 0 ik ik k i ik ik ik ik k i ik ik k i iy
x
k
k
Max
Max
y
x
k
k
y
x
k
k
Max
Max
y
x
k
k
Min
Min
k
Where
(
0
,
)
After calculating
i(
k
)
, the connection between ith alternative and referenced number sequence will be calculated according to the following formulam
k
n
r
n k ii
.
(
)
,
1
1
,
2
,....
1
1
Step 7: Calculate optimal alternative
The feasible alternative Xt is optimal by
Grey theory, if
r
t
Max
1imr
iII.REFERENCE
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