The Constitutive Error and Optimize Operator Research in the Grey ...

Proceedings of 2009 IEEE International Conference on Grey Systems and Intelligent Services, November 10-12, 2009, Nanjing, China

The Constitutive Error and Optimize Operator Research in the Grey Number’s addition and subtraction Inverse Operation Zhou Wei, Fang Zhigeng, and Chen Changjun 

Abstract—At first, this paper defined the grey number inverse operation, proposed a essential condition about the grey number’s addition and subtraction inverse operation, then discovered the constitutive error in process of the addition and subtraction inverse operation that is the generally result in the addition and subtraction inverse operation enlarges the length to

addition and subtraction or multiplication and division of the grey number. The fundamental reason is not reversible computing in the operation and inverse operation of the grey number. Just for this, a great deal of research couldn’t be further carried out in-depth. So that, the research about the inverse operation of the interval grey number has a certain theoretical value and practical significance. The problem that how to carry out inverse operation between some interval grey numbers has made the academic attention of many scholars, and this kind of research have done all the time. Such as, the professor Wang Qinyin put forward the concept of general-grey number and its law in computing[4-6]. So, this grey number could avoid the structural error in he inversion of the interval grey number, to achieve a regular expression against the request of each other, but the definition of general-grey number g [ x, a u , u b] is obvious different from the interval grey number in general, so the actual use would be is less effective and more different when use. Then, the professor Zhou Qiren and so on put forward the second interval grey number addition and subtraction operations, which could achieve the purpose of eliminating errors, but there are some deficiency in using this rule when calculate the interval grey number in general, even impossibility. In this paper, the result was found that the structural errors exist in the four basic inverse operation interval of the interval grey number. So the special structural errors were introduced to respectively restore the purpose of accuracy. The author considered only the grey addition and subtraction inverse operation for the limited in length of the paper. And, the grey multiplication and division inverse operation will be researched in next article.

b  a in the top and bottom sector comparing to the exact

solution. The author has introduced a kind Optimize Operator (b  a )  2(b  a ) r that is based on the standard grey number for eliminating this kind of constitutive error in the inverse operation, which was properly used in three forms of addition and subtraction inverse operation. At last, the author confirmed this method’s feasibility and usability through a classical example.

I. INTRODUCTION

T

HE Grey system theory is an emerging discipline, which was first introduced by the Chinese Professor Deng Julong scholars. This theory has been put forward since the 19th century 80's at home and abroad, has brought enormous impact and are widely used in the analysis of various industries [1-3]. However, the basic computing operation in the interval grey number is still not enough to improve operations, in particular, the inverse operation in the interval grey number is in serious shortage. As we all know, the inverse operation is the real medium: the addition and subtraction are inverse operation each other, multiplication and division are each other's inverse operation, which would be a simple and unchanged rule. But there are many errors in the operation of the interval grey number when using these rules. Even more, the Constitutive Error will emerge in the

Manuscript received April 9, 2009.This work was supported in part by the by National Natural Science Foundation of China though Grant NO.70473037, by the Research Fund for the Doctoral Program of National Ministry of Education though Grant NO.20020287001. At the same time the authors would like to acknowledge the partial support of the Science and Innovation Fund for Special Professor of NUAA via Grant 1009-260812, and Special thanks also goes to Graduate Student Innovation Fund of Jiangsu Province. Also acknowledgements are given to Science and Innovation Fund of NUAA and to Innovation Group of NUAA Zhou Wei is with School of Economics & Management, Nanjing

II. THE CONSTITUTIVE ERROR ANALYSIS IN THE GREY NUMBER’S ADDITION AND SUBTRACTION INVERSE OPERATION Definition 1:(the interval grey number): The number is called the interval grey number means only the border from top to bottom were known, but the specific value and its distribution are unknown. Then, the interval grey number could be represented by … [ a , a ] , a  a .

University of Aeronautics and Astronautics, Nanjing 210016, China (corresponding author to provide phone: 13675163825; e-mail: [email protected]). Fang Zhigeng is with School of Economics & Management,

Definition 2: (the inverse operation of the interval grey number): The inverse operation of the interval grey number means Some unknown numbers in an equation that there are some the interval numbers and the unknown numbers in the right or left side of the equation are resulted by it, and the equation is still established.

Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China (e-mail: [email protected]). Chen Changjun. Author is School of Economics & Management, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China, n (e-mail: [email protected]). 978-1-4244-4916-3/09/$25.00 ©2009 IEEE

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b  a in the results by the general grey addition and

So, there are six basic inverse operations in the inverse operation of the grey number as the following forms.

…1  x

…2 ΰ1α;  y2  …3 …4 

ΰ2α;

…5  y1

…6 ΰ3α; …7 u z

ΰ4α;

…8 

subtraction inverse operations when compared with the precise value. Proof: (a) In the grey addition inverse operations as: [a, b]  [ x, y ] [c, d ] (7)

…12 ΰ6α; The unknown numbers are separately x Ε y1 Ε y2 Ε

…9 y k1

…10 ΰ5α; k2 y …11

Thus a is lower then b , otherwise the numberwill be a white number and its count is all known. The x and y is unknown and necessary to be solved. The result that

z Ε k1 Ε k2 which are required to resulted by the inverse operation of the interval grey number, and the k1 u k2 z 0 .

[ x, y ] [c, d ]  [a, b] [c  b, d  a ] will be got by the general grey addition inverse operations. Then, we could use the result to the (7) formula and find that the (7) formula isn’t set up and the theorem 2.1 isn’t satisfied. So there will be some error in the calculation. But the precise value is

Only the first three the grey addition and subtraction inverse operations were researched in this paper for the limited space and time. Definition 3: (the length of the interval grey number): The length of the interval grey number means the length between upper and lower boundaries of the interval grey number, and expressed as … ( … So, …

[ x, y ] [ c  a , d  b ]

[ a , a ] ).

what is easy to get. So we can get it that the equidistant between upper and lower are enlarged b  a by comparing

(a  a ) and … t 0 . The grey number will

be a specific real number when …

the two results and the up value and down value are separately enlarged and shrunk (b  a ) what could see in the Fig. 1 as following.

0.

Theorem 1: The length of the both sides in the equation of the grey addition and subtraction inverse operations will be equal just as:

˄¦ …i ˅ULJKW ˄¦ …i ˅ OHIW .

Proof: There are two general addition and subtraction grey number operations as:

[a, b]  [c, d ] [a  c, b  d ] ΰ7α

Fig. 1:The Error Comparison in the grey addition inverse operations

[e, f ]  [ g , h] [e  h, f  g ] ΰ8α Then, the length˄¦ …i ˅ OHIW of the left side of the (7) formula and (8) formula will be separately

(b) There are two forms in the grey addition inverse operations as (2) formula and (3) formula. The (2) formula will be first discussed as following: [ x, y ]  [a, b] [c, d ] (8)

(d  c  b  a )

˄¦ … ˅

i ULJKW of the and ( f  e  h  g ) . And the length right side of the (7) formula and (8) formula will be separately (d  b  a  c) and ( f  g  e  h) .

¦…˅

So ˄

i OHIW

Thus a is lower then b , otherwise the number will be a white number and its count is all known. The x and unknown and necessary to be solved. The result that

˄¦ …i ˅ULJKW which means the

length of the both sides in the two equations of the grey addition and subtraction inverse operations are equivalent. Then the theorem was proved. Conversely, the basic equation will be not set up and the constitutive error will be exit in the grey addition and subtraction inverse operations if the conditions are not satisfied. The reason why the constitutive error will be exit in the grey addition and subtraction inverse operations is the theorem couldn’t be set up. Then, the following will be a detailed analysis. Theorem 2: There will be a structural error phenomenon that the equidistant between upper and lower are enlarged

y is

[ x, y ] [ c, d ]  [ a , b ] [ c  a , d  b ] will be got by the general grey subtraction inverse operations. Then, we could use the result to the (8) formula and find that the (8) formula isn’t set up and the theorem 2.1 isn’t satisfied. So there will be some error in the calculation. But the precise value is

[c  b, d  a ] what is easy to get. So we can get it that the equidistant between upper and lower are enlarged b  a by comparing the two results and the up value and down value are separately enlarged and shrunk (b  a ) what could see in 430

The (b  a ) in the operator to amend is the interval

the Fig. 2 as following.

distance [b  a ] , and the white way that the grey coefficient

r equivalent to the results of the general inverse operation of the grey factor. Proof: In the grey addition inverse operations as the (7) formula. The result that

Fig. 2: The Error Comparison in the grey subtraction inverse operations

[ x, y ] [c, d ]  [a, b] [c  b, d  a]

will be got by the general grey addition inverse operations. And we could know there are deviations in the result. So the operator to amend that the interval grey number additive

The (3) formula will be first discussed as following: [a, b]  [ x, y ] [c, d ] (9) The result that



inverse operation (b  a )  2(b  a ) r is put in and the deviations will be eliminated that would be proofed as following.

[ x, y ] [ a , b ]  [ c , d ] [ a  d , b  c ]

will be got by the general grey subtraction inverse operations. Then, we could use the result to the (9) formula and find that the (9) formula isn’t set up and the theorem 1 isn’t satisfied. So there will be some error in the calculation. But the precise value is

[ x, y ]   

[b  d , a  c] what is easy to get. So we can get it that the equidistant

[c, d ]  [a, b]  (b  a)  2(b  a)r [c  b, d  a ]  (b  a)  2(b  a)r (c  b)  (d  a  c  b)r  (b  a ) 2(b  a)r (c  a )  ( d  b  c  a ) r [ c  a , d  b ]

So the deviations were eliminated when the new result was put into the (7) formula, the results of inverse operation to satisfy the equation and the necessary conditions that the length on both sides of the equation are same in the grey additive inverse operation (Theorem 1) could also be meet. Therefore, the structural error in the general interval grey number inverse operation of addition is eliminated by the amending operation introduced. And the theorem3.1 was proofed. Theorem 4: The operator to amend in the interval grey number subtraction inverse operation is the (b  a )  2(b  a ) r , which could eliminate the structural error in the computing of the general standards for the number of interval grey. The (b  a ) in the operator to amend is the interval

between upper and lower are enlarged b  a by comparing the two results and the up value and down value are separately enlarged and shrunk (b  a ) . To sum up, there will be a structural error phenomenon that the equidistant between upper and lower are enlarged

b  a in the results by the general grey addition and subtraction inverse operations when compared with the precise value. Then the Theorem 2 was proofed. III. THE INTRODUCTION AND TO PROVE ABOUT THE AMENDMENTS OPERATORS IN THE GREY NUMBER’S ADDITION AND SUBTRACTION INVERSE OPERATION Definition 4:ΰthe standard interval grey numberα A

[a, b] could unified standard for the form a  (b  a ) r , and the r is a number between 0

distance [b  a ] , and the white way that the grey coefficient

and 1 and named the grey coefficient. Then, we could call the a  (b  a)r is a standard interval grey number. The nature and operation of the standard interval grey number could refer to the literature [8]-[10]. Definition 5: (the general standard interval grey number)The (b  a )  2(b  a )r form is called the general

Proof: In the grey subtraction inverse operations as the (8) formula. The result that

arbitrary interval grey number

r is the same to the above-mentioned one.

[ x, y ] [ a , b ]  [c , d ] [ a  c, b  d ] will be got by the general grey addition inverse operations. And we could know there are deviations in the result. So the operator to amend that the interval grey number subtraction inverse operation (b  a )  2(b  a ) r is put in and the deviations will be eliminated that would be proofed as following.

standard interval grey number if the grey coefficient r is a real number or could be white in a standard interval grey number. Theorem 3: The operator to amend the interval grey number additive inverse operation is  the (b  a )  2(b  a )r , which could eliminate the structural error in the computing of the general standards for the number of interval grey. 431

[ x, y ] [a, b]  [c, d ]  (b  a)  2(b  a)r  [a  c, b  d ]  (b  a)  2(b  a )r

number inverse operation of subtraction is eliminated by the amending operation introduced. The other grey subtraction inverse operations as the (9)  (a  c)  (b  d  a  c)r  (b  a )  2(b  a )r formula could be revised by introduced the interval grey number subtraction e inverse  (b  c)  (d  c  b  a)r operation (b  a )  2(b  a ) r .  [b  c, a  d ] In sum up, the theorem3.2 was proofed. So the deviations were eliminated when the new result was As a result, the structural error could be completely put into the (8) formula, the results of inverse operation to eliminated by introducing the amend operation in the interval satisfy the equation and the necessary conditions that the grey inverse operation of addition and subtraction. Then, length on both sides of the equation are same in the grey some conclusions are got as the following tabeOĉ. subtraction inverse operation (Theorem 1) could also be meet. Therefore, the structural error in the general interval grey TABLE I THE IMPROVED ALGORITHM TABLE OF THE GREY ADDITION AND SUBTRACTION INVERSE OPERATIONS

the grey inverse operation

[ a, b ]  [ x, y ]

[c, d ]

[ a, b ]  [ x, y ]

[c, d ]

[ x, y ]  [ a, b]

[c, d ]

range of the number the general standard interval grey the principle of inverse operation to improve number needed to be introduced [ a, b] [c, d ] the length b  a needed to reduced in the (b  a )  2(b  a ) r unlimited unlimited upper and lower range of the value the length b  a needed to reduced in the (b  a )  2(b  a ) r unlimited unlimited upper and lower range of the value the length b  a needed to reduced in the (b  a )  2(b  a ) r unlimited unlimited upper and lower range of the value

So, we can get the (3) just as IV.

2 x  3 y [11.1,18.4] AN EXAMPLE USED THE NEW METHOD

An example was given to show the effective and practical nature of the new method, and the old and new methods will all be used in this example. The detailed analysis is as following. Example 4.1: A company produced a number of products and plans to sale them by its two regions sale companies, and the total sales number must be [4.5, 7] millions TABLES

y [2.9.1,9.4] and x [4.9.1,9.9] by the (1) and (3) in the old grey inverse operation These results will be wrong for two obvious reasons. The one reason, the cost of any company is not negative number accordance with the reality of the existence. The other reason, the grey-scale range of the result is an unlimited zone by the grey-scale range formula. Then, there is not any basis and reason for decision-making. So, there are some errors or deviations in the old method. Even more, the errors or deviations will be enlarged for there are more grey inverse operations and equations to calculate. At the second, the new method was used. We can get [1.8, 2.2]  2 x  3 y [13.3, 20.2]  by the (2) formula. Then,

and the whole cost must be controlled in [13.3, 20.2] millions dollars for taking into account the productive capacity of the enterprise and the lowest opportunity cost of sales. Then we can get the fixed costs and sale cost of every produce in two sale companies are respectively [1,1.2] and [0.8,1] , and sale cost of every produce in two sale companies are respectively 2 and 3 by the old experience. The question: How should the corporate initial allocation of sales and get the effective cost control. This problem is a typical computing problem of interval grey number from the main subject content. Also, it’s a specific dual of the interval grey number. The grey operation and the equation are as following. We could the x and y respectively mean the sale number of two companies. So we can get the equation.

­ x  y [4.5, 7] ® ¯[1,1.2]  2 x  [0.8,1]  3 y [13.3, 20.2]

(3)

Then, we got

2 x  3 y [11.1,18.4]  0.4  0.8r  11.1  7.3r  0.4  0.8r   [11.5,18]

We can get the

(1) (2)

by (1) u 2 . So,

At the first, the old method was used.

432

[9,14]  y [11.5,18]

y [11.5,18]  [9.14]  5  10r  [2.5, 4]

Industrial Engineering.Gemini International Limited Dublin15, Ireland, 2003,pp.520Д525. [11] Wang Yezhen. The application of Grey system and fuzzy mathematics in environmental protection. Harbin: Harbin Industry University press. 2007:,pp.2-5.

At last, we can get

x

[4.5, 7]  [2.5, 4]  1.5  3r  [0.5, 4.5]  1.5  3r [2,3]

So, we got the result

x [2,3], y [2.5, 4] by the new method. These results are more reasonable. On one hand, the numbers are real and entirely possible for the situation for larger than 0. On the other hand, the structural errors are almost eliminated for the amended operation. So it will be a more credible and feasible basis and results for decision making. Therefore, enterprises can distribute the sales to two companies in accordance with the above result. The entire range of value is the probability of future sales volume, and the large and low values mean the big sales and the small sales within a certain credibility. V. CONCLUSION

The structural error in the general interval grey inverse operation was effectively eliminated by adding the amended operator, and some useful calculations were proved in the paper. At last, the author confirmed this method is feasible and usable through a classical example. But, the author didn’t research the grey multiplication and division inverse operation; however the research will appear in another paper.

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Deng Julong. The Basis of Grey Theory. Wuhan: Press of Huazhong University of Science Technology.2002,pp.1-3. [2] Liu Sifeng, Dang Yaoguo, Fang Zhigeng. Grey system theory and its application. Beijing: Science Press, 2004,pp.1-3 [3] Liu S F, Forrest J. The role and position of grey system theory in science development.The Journal of Grey System(UK),1997,9 (4),pp.351-356. [4] WANG Qingyin. The Math Grey. Wuhan: Huazhong University of Science and Technology Publishing House. 1996,pp.45-46 [5] WANG Qingyin, LU Ruihua. The Intrinsic Relation Between Universal Grey Number and Intervals Number Standard Expression as well as Four Arithmetic Operations. Mathematics In Practice and Theory.2005,6,pp.216-218. [6] WANG Qingyin. Foundations of Universal Grey Algebra. Journal of Huazhong University of Science and Technology. 1992,8,pp.152-153. [7] Zhou Zhiren; Zhou Junjinn; Zhang Zhihai; Reversed Operation of the Interval Number and Simple Interval Number Equation. Journal of Hebei institute of architectural science and technologe .1994,2,pp.194-196. [8] Fang Zhigeng , Liu Sifeng , Lu Fang. Study on Improvement of Token and Arithmetic ofInterval Grey Numbers and Its GM( 1 ,1) Model. Engineering Science,2005,7(2),pp.57Д60. [9] Fang Zhigeng , Liu Sifeng. The discussion about the grey game matrix mixed strategy solution based on the Interval number. Chinese Journal of Management Science,2004.10,pp.117-121 [10] Zhigeng Fang, Sifeng Liu. Grey matrix model based on pure strategy . Limerick , Ireland, Mohamed ID essouky, Cathal Heavey, eds. Proceedings of the 32nd International Conference on Computers &

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