Centre for Computational Intelligence, De Montfort University, ... Department of Mathematics, Slippery Rock University, Slippery Rock,. Pennsylvania, USA.
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General grey numbers and their operations
General grey numbers
Sifeng Liu and Zhigeng Fang Institute for Grey Systems Studies, Nanjing University of Aeronautics and Astronautics, Nanjing, People’s Republic of China
341
Yingjie Yang Centre for Computational Intelligence, De Montfort University, Leicester, UK, and
Jeffrey Forrest Department of Mathematics, Slippery Rock University, Slippery Rock, Pennsylvania, USA Abstract Purpose – The purpose of this paper is to advance new rules about operations of grey numbers. Design/methodology/approach – The paper first puts forward the definitions of basic element of grey number and general grey number. The operation axiom, operation rules of general grey numbers and a new algebraic system for general grey numbers are built based on the “kernel” and the degree of greyness of grey numbers. Findings – Up to now, the operation of general grey numbers has been transformed to operation of real numbers; thus, the difficult problem for set up operation of general grey numbers has been solved to a certain degree. Practical implications – The method exposed in the paper can be used to integrate information from a different source. The operation of general grey numbers could be extended to the case of grey algebraic equation, grey differential equation and grey matrix which includes general grey numbers. The operation system of general grey numbers also opened a new passageway for research on grey input-output and grey programming, etc. Originality/value – The new conception of a basic element of grey number and general grey number was given for the first time in this paper. The novel operation rules of general grey numbers were also constructed. Keywords Basic element, General grey numbers, Operations, Kernel, Greyness, Number theory, Mathematical programming Paper type Research paper
The relevant research done in this paper is supported by the joint research project of both the Natural Science Foundation of China (NSFC, No. 71111130211)) and the Royal Society (RS) of UK, the Natural Science Foundation of China (No. 90924022, 70971064 and 70901041), the Social Science Foundation of the China(No. 10zd&014, No. 08AJY024), the Soft Science Foundation of China(2008GXS5D115), the Foundation for Doctoral Programs (200802870020) and the Foundation for Humanities and Social Sciences of the Chinese National Ministry of Education (No. 08JA630039). At the same time, the authors would like to acknowledge the partial support of the Science Foundation for the Excellent and Creative Group in Science and Technology of Jiangsu Province (No. Y0553-091), the Foundation for Key Research Base of Philosophy and Social Science in Colleges and Universities of Jiangsu Province, and the Foundation for National Outstanding Teaching Group of China (No. 10td128).
Grey Systems: Theory and Application Vol. 2 No. 3, 2012 pp. 341-349 q Emerald Group Publishing Limited 2043-9377 DOI 10.1108/20439371211273230
GS 2,3
342
1. Introduction Grey system theory has emerged as an effective model for systems with partial information known and partial information unknown (Deng, 1982; Lin et al., 2004; Liu and Lin, 2006, 2011). As a basis of grey system theory, the studies on grey number, operations of grey numbers and grey algebraic systems have drawn the attention of scholars for a long time, but up to now we have not had a satisfied result. The conception of interval numbers could date back to Young (1931). Moore and many other scholars did some further studies on interval numbers (Moore, 1979; Ishihuchi and Tanaka, 1990). However, the operation of interval numbers is the most disputed. In the 1980s, we put forward the concept of the mean whitenization number of grey numbers at first (Liu, 1989) and based on the concept we tried to construct a new algebraic system of grey numbers. According to the standard definition of the degree of greyness of grey numbers (Liu, 1996, 2006; Yang, 2007, 2011), it is possible for us to deal with the grey intervals after the operation of grey numbers with the help of the conception of the degree of greyness. In this paper, a definition for the grey “kernel” has been put forward first. The axioms for the operation of grey numbers and a grey algebraic system is built based on the grey “kernel” and the degree of greyness of grey numbers. The properties of the operation are studied and up to now, the operation of grey numbers has been transformed to the operation of real numbers. So, the difficult problem of setting up the operation of grey numbers and the grey algebraic system has been solved to a certain degree. The paper is structured as follows. In the next section a brief overview of the basic concepts relevant to grey numbers is provided. The general grey number and its reduced form and the propagation of the associated “kernel” and degrees of greyness of general grey number are defined in Section 3. Then the operations of general grey numbers are derived and their applications are shown by an example. Finally, we draw out some concluding remarks in Section 5. 2. Basic concepts As a different model for uncertainty representation, grey systems were proposed by Deng (1982). In grey systems, the information is classified into three categories: white with completely certain information, grey with insufficient information and black with totally unknown information. Interval grey numbers are the basic concepts in grey systems. Definition 1 (interval grey number). The following: ^ [ ½a; a ; a , a
ð1Þ
is called an interval grey number, where a and a are the upper and lower limits of the information separately (Liu, 2010; Liu and Lin, 2011). The arithmetic of the interval grey number is very similar to interval values (Alefeld and Herzberger, 1983). An interval grey number has only one number which belongs to interval ½a; a ; a , a . It is much different from an interval: Definition 2 (the “kernel” of grey number (Liu, 2010)) . (1) Suppose an interval grey number ^ [ ½a; a ; a , a , in the case of a lack of the ^ ¼ ð1=2Þða þ a Þ is distributing information of the values of grey number ^, then ^ called the “kernel” of grey number ^.
(2) If a grey number ^ is a discrete one and ai [ ½a; a Pði ¼ 1; 2; . . . ; nÞ are all the ^ ¼ ð1=nÞ n ai is called the “kernel” possible values for grey number ^, then ^ i¼1 of grey number ^. (3) Suppose that grey number ^ [ ½a; a ; a , a is a random grey number which ^ ¼ Eð^Þ is called the contains distribution information of the values. Then ^ “kernel” of grey number ^. Definition 3 (white number). If a ¼ a ¼ a in equation (1), then ^ ¼ a is called a white number. The “kernel” of a white number is itself. ^ the “kernel” of a grey number ^ as the representation of grey numbers ^, has great ^, significance which cannot be exchangeable in the course of transforming the operation of grey numbers to the operation of real numbers. In fact, the “kernel” of grey number ^, as a real number, can be completely operated by the operation of real numbers, such as plus, minus, multiplication, division, power, extract and so on. It is reasonable to take the operation results of the “kernels” as the “kernel” of operation results of grey numbers. Definition 4 (the degree of greyness of grey number (Liu, 2011)). Suppose that the background which makes grey number ^ come into being is V and m is the measure of V, then: mð^Þ ð2Þ g8ð^Þ ¼ mðVÞ is called the degree of greyness of grey number ^ (denoted as g8 for short). According to the properties of measure and ^ , V, the definition of the degree of greyness of grey number given in definition 4 satisfies normality, that is: 0 # g8ð^Þ # 1
ð3Þ
The degree of greyness of the grey number reflects the uncertainty degree of the thing described by the grey number. The white number is a completely assured number and its degree of greyness is 0. The black number is a completely unknown number and its degree of greyness is 1. For most interval grey numbers, their degrees of greyness g8 are between 0 and 1. The nearer g8 to 0, the smaller the uncertainty of the values of grey number is. Contrarily, the nearer g8 to 1, the bigger the uncertainty of the values of grey number is. 3. General grey number Definition 5 (basic element of grey number). The grey interval and white number are called by a joint name of basic element of the grey number. Based on the work on grey numbers (Liu, 2010; Yang, 2007; Yang and Liu, 2011), we give our definition of general grey number as follows. Definition 6 (general grey number). Let g ^ [ R be an unknown real number within a union set of closed or open grey intervals: n
g ^ [ < ½ai ; a i
ð4Þ
i¼1
i ¼ 1, 2, . . . ,n, n is an integer and 0 , n , 1, ai ; a i [ R and a i21 # ai # a i # a iþ1 , for any grey interval ^ i [ ½a i ; a i ,
