Grey relationanal analysis1

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.

1

P.E.S. College Of Engineering Aurangabad


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

event) is shown in Table 4.1.4.

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Event

Comparative

series

100

M Shot

Put

High

Jump

400

M

110 M

High

Hurdles

Discus

Throw

Pole

Vault

Javelin

Throw

1500

M

X1 0.833 1.000 0.919 0.646 0.679 0.654 1.000 0.912 0.758

X2 0.866 0.693 1.000 0.904 0.893 0.737 0.683 0.583 0.667

X3 0.500 0.500 0.558 1.000 1.000 0.522 0.500 0.879 0.500

X4 0.722 0.871 1.000 0.585 0.500 1.000 0.556 1.000 1.000

X5 1.000 0.574 0.500 0.500 0.543 0.500 0.510 0.500 0.508

Table 4.1.4: The Grey relational coefficient

4.1.5. Determination of the relational grade for each contestant

The relation grade for each comparative series is determined by averaging the

Grey relation coefficient of each individual contest event. The Grey relation grade can

be expressed by:

Where n = 10

Substituting the coefficient of Grey relation into above equation, we can get each

contestant’s Grey relation grade, which are

Γ01= 0.832; Γ02= 0.769; Γ03= 0.646;

Γ04= 0.823; Γ05= 0.596.

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This Grey relation grade is the overall performance for the decathlon.

4.1.6. Obtaining the ranking

Since we get Γ01 >Γ04 >Γ02 >Γ03 >

Γ05, the ranking order for these five contestants is A, D, B, C and E. Although Grey

relational analysis does agree well with the traditional method, it possesses an

overwhelming advantage to solve the problems that traditional method could not

overcome such as tie score dispute.

4.2. Applying Grey Relational Analysis to the Vendor Evaluation Model

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This section will concentrate on an example to illustrate how to apply the Grey

multiple attributes to the vendor evaluation. Since variations of managerial conditions

among vendors must be accounted, and the requirements for their manufacturing

process are quite different, it is not our intention to develop an evaluation model with

considering all variations as mentioned above. Therefore, this paper will only provide

a feasibility study on the new model for vendor evaluation.

4.2.1. Implementation with evaluation factors and measure parameters

Since the evaluation factors are much dependent on the enterprise environment,

the top management of the enterprise may invite the members of the department of

purchasing, production control, and quality control to meet together, and decide the

appropriate evaluation factors and measure parameters for vendor evaluation.

Traditionally, quality, price, delivery date, quantity, and services are chosen to be

typical evaluation factors. The measure parameters for these five evaluation factors are

shown in Table 4.2.0.

Evaluation

factors

Quality Price Delivery date Quantity Services

Measure

parameters

Defects Quotation Delay rate Shortage

rate

Score

Quantify

criteria

rejects/(gross

no of

batches)

Unit price

no of delays/gross

no of ships

no of

short./gross

no of ships

Five

classification

levels

Table 4.2.0: Evaluation factors and measure parameters for vendor evaluation

Regarding the services of evaluation factors, the corresponding measure

parameter can be determined by

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1. Operated as better methods for defects elimination, and active involvement

with vendors and customers;

2. Delivery speed;

3 service condition. A qualitative ranking will be given by a review committee,

then convert these qualitative ranking into a rating scale of 1 to 5 (larger-the-better). A

more subjective overall fuzzy evaluation method may be used to get fuzzy number.

4.2.2. The corresponding weighting value for evaluation factors

Once the evaluation factor has been determined, we are in a position to find the

corresponding weighting value for each individual evaluation factor. The weighting

value determination can be done by Delphi Method or Eigenvector [14]. Table4. 2.1

show the corresponding weighting value for each evaluation factor.

4.2.3. Implementation with evaluation matrix

It is assumed that five vendors are able to supply certain raw materials. The

delivery record is rearranged by purchasing staff as shown in Table4.2.2. Using the

data from Table 4.2.2 an evaluation matrix can be formed. It is noted that evaluation

factor is indicated in attribute column, each vendor is comparative series.

4.2.4. Data rationalizing

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The expected goal can be rationalized according to each attribute. A group of

assumptions are made for the following:

1. Quality: the quantifying value for reject (smaller-the-better) is 0;

2. Price: the quantifying value for unit price (smaller-the-better) is 0;

3. Delivery date: the quantifying value for delay (smaller-the-better) is 0;

4. Quantities: the quantifying value for shortage (smaller-the-better) is 0;

5. Services: the quantifying value for the service score (larger-the-better) is 5.

The measure value of each attribute is further standardized based on above-

mentioned expected goal [9]. The matrix for comparative series is obtained as shown

in Table 4.2.3.

4.2.5. Establishing standard series

According to our expected goal for each evaluation factor, an ideal standard

series (X0 = 1) is established in the last line in Table 4.

Evaluation

factors

Quality Price Delivery

date

Quantity Services

Measure

parameter

0.30 0.20 0.15 0.15 0.20

Table4. 2.1: The corresponding weighting value for each evaluation factor

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Measure

Vendors

Quality

(Defects)

Price (Unit

price)

Delivery

date (Delay

rate)

Quantity

(Shortage

rate)

Services

(Score)

A 0.15 12 0.15 0.05 2

B 0.22 10 0.25 0.08 4

C 0.15 8 0.15 0.05 5

D 0.08 13 0.30 0.15 4

E 0.12 9 0.05 0.20 3

Table 4.2.2: Measurement value for each evaluation attributes

(Delivery data for a period of two years)

Item

Comparative series

Quality Price Delivery date Quantity Services

X1 0.500 0.200 0.600 1.000 0.000

X2 0.000 0.600 0.200 0.800 0.667

X3 0.500 1.000 0.600 1.000 1.000

X4 1.000 0.000 0.000 0.383 0.667

X5 0.714 0.800 1.000 0.000 0.333

Standard series (X0) 1 1 1 1 1

Table 4.2.3: Data rationalizing

4.2.6 Determination of Grey relational coefficient for each evaluation factor

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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 candidate vendor is shown in Table

4.2.4.

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Item

Comparative

series Quality Price

Delivery

date Quantity Quantity

X1

0.667 0.555 0.714 1.000 1.000

X2 0.500

0.714 0.555 0.833 0.833

X3 0.667 1.000 0.714 1.000 1.000

X4 1.000

0.500 0.500 0.618 0.618

X5 0.777 0.833 1.000 0.500 0.500

Table4.2.4: The Grey relational coefficient

4.2.7. Determination of the relational grade for each candidate vendor

Using the corresponding weighting value for each evaluation factor (see

Table4.2.2), we can calculate the relational grade of each candidate vendor by:

Where is the corresponding weighting value for each evaluation factor. Substituting

the relational coefficient from Table 5 into above equation 6, the Grey relational

coefficient can be obtained

Γ01 = 0.602; Γ02 = 0.601; Γ03 = 0.791;

Γ04 = 0.618; Γ05 = 0.659

This value of Grey relation is the overall performance that the enterprise requires.

4.2.8. Obtaining the ranking

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Because of Γ03 >Γ05 >Γ04 >Γ01 > Γ02, the ranking order for all candidate vendors

is: (1) C; (2) E; (3) D; (4) A; (5) B. It is noted that the ranking order will change while

we change the weighting value for each evaluation factor. In other words, the owner of

an enterprise may select a suitable vendor based on his own requirements.

5. APPLICATION

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Grey relational analysis in Grey theory had been largely applied to project

selection, prediction analysis, and performance evaluation. Hence grey relational

analysis is used in following application:

1. A Novel Prediction for Stock Index.

2. A Novel Prediction for Stock Index.

3. Airline Network Optimization.

4. In gas breakdown and var compensator finding.

5. Artificial Neural Network (ANN) to measure the impacts of key elements

on the forecasting performance of real estate investment trust.

6. Grey relational analysis has mainly applied to decision-making in

economics, medicine, computer science, social science, geometry,

chemistry, Management.

7. Empirical modeling of EDM parameters.

8. Supplier selection.

9. Sheet metal forming for multi-response quality characteristics.

10. For ranking material options.

11. Fault type identification on power transmission line.

6. ADVANTAGES

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1. The major advantage of Grey theory is that it can handle both incomplete

information and unclear problems very precisely.

2. It serves as an analysis tool especially in cases when there is no enough

data.

3. Grey multiple attributes decision method is very accurate.

4. It can overcome the uncertainty arising from the measured parameters of

each attribute.

5. For new vendor evaluation, it is very convenient to perform overall

measurement based on each enterprise’s requirements. The overall

performance can determine the order for selecting the suitable vendors.

6. Grey relational analysis requires less data and can analyze many factors

that can overcome the disadvantages of statistics method.

7. DISADVANTAGES

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1. It must have plenty of data.

2. Data distribution must be typical.

3. A few factors are allowed and can be expressed functionally.

8. CONCLUSION

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Based on our study in this paper, the Grey relational analysis can be applied

project selection, prediction analysis, and performance evaluation. Through

quantitative analysis of Grey relation, it provides more accurate and subjective data.

Grey theory handles both incomplete information and unclear problems very precisely.

9. REFERENCES

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www.google.com

http://search.yahoo.com/

http://en.wikipedia.org/wiki/Special:Search

http://www.journal.au.edu/ijcim/2003/sep03/

ijcimv11n3_art5.pdf

http://knol.google.com

http://www.ijcim.th.org/past_editions/2003V11N3/

ijcimv11n3_art4.pdf

http://www.m-hikari.com/imf-password2007/13-16-2007/

chienhowuIMF13-16-2007.pdf

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http://www.m-hikari.com/imf-password2007/13-16-2007/chienhowuIMF13-16-2007.pdf

http://www.m-hikari.com/imf-password2007/13-16-2007/chienhowuIMF13-16-2007.pdf

http://www.ijcim.th.org/past_editions/2003V11N3/ijcimv11n3_art4.pdf

http://www.ijcim.th.org/past_editions/2003V11N3/ijcimv11n3_art4.pdf

http://knol.google.com/

http://www.journal.au.edu/ijcim/2003/sep03/ijcimv11n3_art5.pdf

http://www.journal.au.edu/ijcim/2003/sep03/ijcimv11n3_art5.pdf

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Grey relationanal analysis1