Grey relational analysis 1. INTRODUCTION Grey analysis uses a specific concept of information. It defines situations with no information as black, and those with perfect information as white. However, neither of these idealized situations ever occurs in real world problems. In fact, situations between these extremes are described as being grey, hazy or fuzzy. Therefore, a grey system means that a system in which part of information is known and part of information is unknown. With this definition, information quantity and quality form a continuum from a total lack of information to complete information – from black through grey to white. Since uncertainty always exists, one is always somewhere in the middle, somewhere between the extremes, somewhere in the grey area. Grey analysis then comes to a clear set of statements about system solutions. At one extreme, no solution can be defined for a system with no information. At the other extreme, a system with perfect information has a unique solution. In the middle, grey systems will give a variety of available solutions. Grey analysis does not attempt to find the best solution, but does provide techniques for determining a good solution, an appropriate solution for real world problems. 1 P.E.S. College Of Engineering Aurangabad
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Grey relational analysis
1. INTRODUCTION
Grey analysis uses a specific concept of information. It defines situations
with no information as black, and those with perfect information as white. However,
neither of these idealized situations ever occurs in real world problems. In fact,
situations between these extremes are described as being grey, hazy or fuzzy.
Therefore, a grey system means that a system in which part of information is known
and part of information is unknown. With this definition, information quantity and
quality form a continuum from a total lack of information to complete information –
from black through grey to white. Since uncertainty always exists, one is always
somewhere in the middle, somewhere between the extremes, somewhere in the grey
area.
Grey analysis then comes to a clear set of statements about system solutions.
At one extreme, no solution can be defined for a system with no information. At the
other extreme, a system with perfect information has a unique solution. In the middle,
grey systems will give a variety of available solutions. Grey analysis does not attempt
to find the best solution, but does provide techniques for determining a good solution,
an appropriate solution for real world problems.
The proposition of Grey theory occurring in the 1990 to 1999 time period
resulted in the uses of Grey theory to each field, and the development is still going on.
The major advantage of Grey theory is that it can handle both incomplete information
and unclear problems very precisely. It serves as an analysis tool especially in cases
when there is no enough data. It was recognized that the Grey relational analysis in
Grey theory had been largely applied to project selection, prediction analysis,
performance evaluation, and factor effect evaluation due to the Grey relational analysis
software development. Recently, this technique has also applied to the field of sport
and physical education.
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2. Grey Theory
The black box is used to indicate a system lacking interior information.
Nowadays, the black is represented, as lack of information, but the white is full of
information. Thus, the information that is either incomplete or undetermined is called
Grey.
A system having incomplete information is called Grey system. The
Grey number in Grey system represents a number with less complete information. The
Grey element represents an element with incomplete information. The Grey relation is
the relation with incomplete information. Those three terms are the typical symbols
and features for Grey system and Grey phenomenon. There are several aspects for the
theory of Grey system:
1. Grey generation: This is data processing to supplement information. It is aimed
to process those complicate and tedious data to gain a clear rule, which is the
whitening of a sequence of numbers.
2. Grey modeling: This is done by step 1 to establish a set of Grey variation
equations and Grey differential equations, which is the whitening of the model.
3. Grey prediction: By using the Grey model to conduct a qualitative prediction,
this is called the whitening of development.
4. Grey decision: A decision is made under imperfect countermeasure and unclear
situation, which is called the whitening of status.
5. Grey relational analysis: Quantify all influences of various factors and their
relation, which is called the whitening of factor relation.
6. Grey control: Work on the data of system behavior and look for any rules of
behavior development to predict future’s behavior, the prediction value can be fed
back into the system in order to control the system.
The Grey relational analysis uses information from the Grey system to
dynamically compare each factor quantitatively. This approach is based on the level of
similarity and variability among all factors to establish their relation. The relational
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analysis suggests how to make prediction and decision, and generate reports that make
suggestions. This analytical model magnifies and clarifies the Grey relation among all
factors. It also provides data to support quantification and comparison analysis. In
other words, the Grey relational analysis is a method to analyze the relational grade for
discrete sequences. This is unlike the traditional statistics analysis handling the relation
between variables.
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3. GREY RELATIONAL ANALYSIS
The validity of traditional statistical analysis techniques is based on
assumptions such as the distribution of population and variances of samples.
Nevertheless sample size will also affect the reliability and precision of the results
produced by traditional statistical analysis techniques. J. Deng argued that many
decision situations in real life do not conform to those assumptions, and may not be
financially or pragmatically justified for the required sample size. Making decisions
under uncertainty and with insufficient or limited data available for analysis is actually
a norm for managers in either public or private sectors. To address this problem, J.
Deng developed the grey system theory, which has been widely adopted for data
analysis in various fields.
The grey relational analysis introduced in the following is a method in grey
system theory for analyzing discrete data series. A procedure for the grey relational
analysis consists of the following steps.
1. Generate reference data series x0.
X0 = (d01, d02, ..., d0m)
Where m is the number of respondents. In general, the x0 reference data
Series consists of m values representing the most favoured responses.
2. Generate comparison data series xi.
Xi = (di1, di2, ..., dim)
where i = 1, ..., k. k is the number of scale items. So there will be k comparison
data series and each comparison data series contains m values.
3. Compute the difference data series Δi.
Δi=(|d01 – di1| , |d02 – di2| , ..., |d0m – dim|)
4. Find the global maximum value Δmax and minimum value Δmin in the
difference data series.
Δmax =∀imax (max Δi)
Δmin =∀imin (min Δi)
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5. Transform each data point in each difference data series to grey relational
coefficient. Let γi(j) represents the grey relational coefficient of the jth
data point in the ith difference data series, then
γi(j) = (Δmin + ςΔmax)/( Δi(j) + ςΔmax)
Where Δi(j) is the jth value in Δi difference data series. ς is a value
Between 0 and 1. The coefficient ς is used to compensate the effect of
Δmax should Δmax be an extreme value in the data series. In general
The value of ς can be set to 0.5.
6. Compute grey relational grade for each difference data series. Let Γi
Represent the grey relational grade for the ith scale item and assume that
Data points in the series are of the same weights 1, then
The magnitude of Γi reflects the overall degree of standardized deviance
of the ith original data series from the reference data series. In general,
a scale item with a high value of Γ indicates that the respondents, as a
Whole, have a high degree of favoured consensus on the particular item.
7. Sort Γ values into either descending or ascending order to facilitate the
Managerial interpretation of the results.
This is brief procedure for the grey relational analysis. Now we will discuss in details
the Grey theory and method as follow:
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3.1. Influence space, measurement space, and Grey relational space
Let P(X) represent the factor set of a specific topics, Q is the influence
relation, then {P(X); Q} is influence space. It must have the following properties:
1. Existence of key factors: for example, the key factors of basketball player are
height, weight, and rebound.
2. Numbers of factors are limited and countable: for example each of the height,
weight, and rebound are countable.
3. Factor in dependability: each factor must be independent.
4. Factor expandability: For example, besides the height, weight, and rebound, the
free throw attempt can be added as a factor.
The series formed by P(X) is:
xi(o) (k) = ( x1
(o) (1),…, xi(o) (k))?X;
Where i = 0,…,m. k=1,…,n.?N
If the following conditions are satisfied:
1. No dimension: the numeric value for all factors must be no dimension.
2. Scaling: the factor value for various series must be at the same level.
3. Polarization: if the factor value in the series is described as the same direction, the
series is comparable. Then the measurement space is expressed as {P(X); x i*(k)}, the
Grey relational space formed by the satisfaction of both factor space and comparability
is termed by {P(X); Γ}.
3.2. Generation of Grey relation
Under the principle of series comparability, to achieve the purpose of Grey
relational analysis, we must perform data processing. This processing is called
generation of Grey relation or standard processing. The expected goal for each factor
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is determined by Wu based on the principles of data processing. They are described in
the following:
1. If the expectancy is larger-the-better (e.g., the benefit), then it can be expressed
by
Xij = (Xij – (Xij)min)/((Xij)max – ( Xij)min) (1)
2. If the expectancy is smaller-the-better (e.g., the cost and defects), then it can
be expressed by
Xij = ((Xij)max – Xij)/((Xij)max – ( Xij)min) (2)
3. If the expectancy is nominal-the-best (e.g., the age), and when the targeted
value is X0 : (Xij)max ? X0 ? ( Xij)min ,then it can be expressed by
Xij = (Xij – X0 )/ ((Xij)max – X0) (3)
3.3. The Grey relational grade
The measurement formula for quantification in Grey relational space is called
the Grey relational grade. When we are determining Grey relation and taking only one
series, X0(X) , as a referenced series, it is called the grade of local Grey relation. If
anyone of the series, Xi(X) , is referenced series, it is called the grade of global Grey
relation. Additionally, the Grey relational coefficient must first be determined before
we obtain the Grey relational grade.
In the Grey relational space, {P(X); Γ}, there is a series
xi= ( xi( 1 ), xi( 2 ), ?, ?? ,xi( k )) ?X
Where i = 0,…,m. k=1,…,n.?N
If the grade of local Grey relation is brought to define the Grey relational coefficient,
γ ( xi( k ),xj( k ) ) it can be expressed as following:
γ ( xi( k ),xj( k ) ) = (Δmin +Δmax)/( Δ0i(k) + Δmax) (4)
Where i = 0,…,m. k=1,…,n. j?i; x0 is a referenced series, xi is a specific
comparative series;
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Representing the k’s absolute value of the difference of X0 and Xi ;
After obtaining the Grey relational coefficient, we normally take the average of the
Grey relational coefficient as the Grey relational grade:
(5)
However, since in real application the effect of each factor on the system is not exactly
same, Eq. (5) can be modified as:
(6)
Where represents the normalized weighting value of a factor and
When equating both Equations (5) and (6).
3.4. The Grey relational series
The Grey relational grade represents the correlation between two series. It is
not important in a decision-making. Rather, the ranking order of the relational grade is
the most important information. Therefore, m’s comparative series with its
corresponding Grey relational grade is rearranged according to the order of their
magnitudes. A Grey relational series is defined as following:
In the Grey relational space, {P(X); Γ}, referenced series, x0, and comparative
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series, xi and xJ:
If g(x0 , xi) ? g( x0 , xj), the situation indicating the relational grade of xi vs. x0 is
greater than that of xj vs. x0, or represented by Γ0i >Γ0j. This is the relational series for
xiand xj.
3.5. Decision for Grey multiple attributes.
To solve problems, if many ways or feasible methods exist, we normally
make a complete evaluation on those resolutions. Then decision is made based on the
evaluation results. It is noted that the multiple attributes decision-making is defined
when more than one-evaluation factors are considered. Hence, the application of Grey
relational analysis to multiple purposes and attributes is called as Grey multiple
attributes decision.
Moreover, this method regards each comparative series as a feasible
solution, and the numeric score for each evaluation factor becomes the numeric value
for each comparative value. The relational grade between comparative series and
standard series is then determined. Finally, the decision can be made based on the
ranking of each feasible solution.
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4. ILLUSTRATION OF NUMERICAL RESULTS
In the above section we have discussed, what grey relational analysis is
and detail procedure of grey relational analysis. To understand the application of grey
relational analysis and where it should be used, we will discuss two numerical or
application of grey relational analysis.
4.1 Applying Grey Relational Analysis to the Decathlon Evaluation Model
For showing the significance of Grey relational analysis in sport score
evaluation and resolving the problems of scoring dispute, the method for decathlon
ranking using Grey relational analysis is discussed herein. We assume there are five
contestants attending decathlon competition. The score is shown in Table 2. By
utilizing traditional method, the results in Table 4.1.0 showing the ranking order are A,
D, B, C and E.
Pole Vault Javelin Throw
Meter Points Meter Points
3.45469 101.26 1,375
3.44467 101.20 1,374
3.43464 101.14 1,373
3.42462 101.08 1,372
3.41459 101.02 1,371
3.40457 100.96 1,370
Table 4.1.0: Typical score table for Pole Vault and Javelin Throw
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Event
Contest
ant
100 M
(Secs)
(M)
Long
Jump
(M)
Shot
Put
(M)
High
Jump
(M)
400
M
(Sec
s)
110 M
high
Hurdles
(Secs)
Discu
s
Thro-
w
(M)
Pole
Vaul
t
(M)
Javelin
Throw
(M)
1500
M
(Secs
A 11.13 7.34 14.5
0 2.25
48.4
5 14.56 43.67 5.70 60.50
256.6
5
B 11.10 6.95 13.7
5 2.27
47.2
0 14.18 45.50 5.12 55.25
265.5
5
C 11.65 6.50 12.8
0 2.08
46.9
0 14.05 39.55 4.45 60.15
290.1
5
D 11.25 7.43 14.2
5 2.27
48.9
0 15.13 49.28 4.70 61.32
240.9
5
E 11.00 7.23 13.1
5 2.03
49.7
3 14.96 38.66 4.50 52.82
288.5
7
Table 4.1.1: Statistical data for the decathlon competition
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Even
-t
Cont
estan
t
100
M
Lon
-g
Jum
p
Sho
-t
Put
Hig-
h
Jum
p
400
M
110
M
High
Hurdl
es
Dis
cus
Thr
ow
Pole
Vaul
t
Javel
in
Thro
w
150
0 M
Total
score
s
R
a
n
k
A 832 896 759 104
1 887 903 740 1090 746 839 8,733 1
B 838 802 713 106
1 948 951 777 947 667 778 8,482 3
C 721 697 655 878 963 968 655 746 740 619 7,642 4
D 806 918 744 106
1 866 834 855 819 758 885 8,546 2
E 861 869 676 831 829 854 638 760 630 631 7,599 5
Table 4.1.2: Event score obtained by traditional method
Moreover, if five contestants attending decathlon competition, the following procedure
indicates how the top of three can be determined.
4.1.1. Implementation with evaluation matrix
Generate an evaluation matrix by arranging game event as attribute column,
and contestants as comparative sequence. In the analysis, the record for the five
contestants is used to form evaluation matrix as shown in Table 4.1.1.
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4.1.2. Data rationalizing
Expected goal can be rationalized according to each attribute. A group of assumptions
are made for the following:
1. 100 meters, 400 meters, 1500 meters, and 110 meters high hurdles: In
this category, it is obviously that the expectancy is shorter-the-better for the
time, which can be determined by:
In which Xij represents the score both at attribute i and comparative series j.
2. Long Jump, Shot Put, and High Jump: The expectancy in this category is
longer-the-better for the distance, which can be determined by
Based on the expectancy of each individual event, the scoring points for each attribute
are normalized to obtain the matrix table of comparative series as shown in Table 4.4.
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4.1.3. Establishing standard series
In accordance with our expected goal for each individual contest event, an ideal
standard series (X0 = 1) is established in the last line in Table 4.1.3.
Event
Comparati
ve
Series
100
M
Lon
g
Jum
p
Shot
Put
Hig
h
Jum
p
400
M
110 M
High
Hurdl
es
Discu
s
Thro
w
Pole
Vaul
t
Javeli
n
Thro
w
150
0 M
X1
0.80
0
0.90
3
1.00
0
0.91
2
0.45
2 0.528 0.472
1.00
0 0.904
0.68
1
X2
0.84
6
0.48
4
0.55
9
1.00
0
0.89
4 0.880 0.644
0.53
6 0.286
0.50
0
X3
0.00
0
0.00
0
0.00
0
0.20
8
1.00
0 1.000 0.084
0.00
0 0.862
0.00
0
X4
0.61
5
1.00
0
0.85
3
1.00
0
0.29
3 0.000 1.000
0.20
0 1.000
1.00
0
X5
1.00
0
0.78
5
0.25
8
0.00
0
0.00
0 0.157 0.000
0.04
0 0.000
0.03
2
Standard
series (X0) 1 1 1 1 1 1 1 1 1 1
Table 4.1.3: Data rationalizing
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4.1.4. Determination of Grey relational coefficient for each contestant
1. Calculate the maximum and minimum difference by:
The resulting maximum difference is one;
The resulting minimum difference is zero.
2. Calculate the Grey relational coefficient by:
By substituting the value of maximum and minimum difference into above
equations, the Grey relational coefficient for each contestant (individual contest