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Ranking of Intuitionistic Fuzzy Number by Centroid Point

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  Ranking of Intuitionistic Fuzzy Number by Centroid Point Satyajit Das and Debashree Guha Department of Mathematics, Indian Institute of Technology, Patna, India Email: {satyajit, debashree}@iitp.ac.in Abstract   —  The notion of Intuitionistic Fuzzy Numbers (IFNs) has been improved in many decision making problems. Ranking of IFNs is one of the techniques that conceptualize IFNs to illustrate order or preference in decision making. Ranking of IFNs plays a very important role in multi-criteria decision making, optimization and in many different fields but ranking of IFNs is not a very easy process. As far as our knowledge is concerned, the number of existing methods for ranking of IFNs in the literature is very few. In this paper a new method has been proposed for ranking of IFNs by determining centroid point of IFNs. Examples have been given to compare the proposed method with the existing ranking results. The results show that the new method can overcome the drawbacks of the existing methods. Index Terms   —  intuitionistic fuzzy number, ranking, centroid point. I.   I  NTRODUCTION   The idea of intuitionistic fuzzy set (IFS) introduced by Atanassov [1] is the generalization of Zadeh’s [2 ] fuzzy set. An IFS is characterized by membership degree as well as non-membership degree. Since its introduction, the IFS theory has been studied and applied in different areas including decision making. Now in modeling a decision problem, ranking is a very important issue. In this regard, many authors have paid considerable attention to investigate the ranking methods of  IFSs. In 1994 Chen and Tan [3] defined a score function of intuitionistic fuzzy values (IFVs) for ranking IFVs. Li and Rao [4] defined different types of score function to compare IFVs. Some time two IFVs may have same score value. In this situation ranking is not possible by using score function. To overcome this case, Hong and Choi [5] defined a new function known as accuracy function of IFVs. Xu and Yager [6] used both the score function and accuracy function for ranking IFVs. However, it has been observed that the research concentrated on finite universe of discourse only. In view of this, recently the research on the concept of intuitionistic fuzzy numbers (IFNs), with the universe of discourse as the real line, has received attention and definitions of IFNs [7]-[9] have been proposed. Further, several ranking methods have also been proposed to solve the ranking problems of IFNs. Chen and Hwang [10] Manuscript received April 15, 2013; revised May 7, 2013. introduced a crisp score function to rank IFNs. In 2008  Nayagam et al  . [11] introduced a new score function for ranking triangular intuitionistic fuzzy numbers (TIFNs) and further they modified it in [12]. Jianqiang and Zhong [13] used both the score function and accuracy function to ranking TrIFNs. In case of inter-valued intuitionistic fuzzy numbers (IVIFNs) Lee [14] proposed a novel method for ranking of IVIFNs by utilizing score function and deviation function. In 2008 Xu and Yager [15]  proposed a new method for ranking IFNs by determining the distance from the IFNs to the positive and negative ideal points. By calculating normalized Hamming distance from IFNs to positive and negative ideal solution a ranking method has been given by Wu and Cao [16] and they applied it in multi attribute group decision making problem. A method for comparing IFNs based on metrics in the space of IFNs was proposed by Grzegorzewski Wei and Tang [17] proposed a possibility degree method for ranking IFNs. A new ranking method was developed by Li [18] on the basis of the concept of a ratio of the value index and ambiguity index of IFNs. Rezvani [19] also proposed a ranking process of TrIFNs  by determining value and ambiguity of TrIFNs. However, after analyzing the aforementioned ranking  procedures it has been observed that, for some cases, they fail to calculate the ranking results correctly. Furthermore, many of them produce different ranking outcomes for the same problem. Under these circumstances, the decision maker may not be able to carry out the comparison and recognition properly. This creates problem in practical applications. In order to overcome these problems of the existing methods, a new method for ranking IFNs has  been proposed in this paper which is based on centroid  point of IFNs. This paper has been organized as follows: In section-II some basic concepts of IFNs have been reviewed. Section-III represents the centroid formula for trapezoidal intuitionistic fuzzy numbers (TrIFNs). This section also describes the proposed ranking process of normal TrIFNs. A set of examples have also been provided in section-IV, to compare the proposed ranking method with the existing   methods . Some   conclusions   have    been   made   in section -V. II.   P RELIMINARIES  This section describes basic definition and some arithmetic operations related to IFN. 107 ournal of Industrial and Intelligent Information Vol. 1, No. 2, June 2013 ©2013 Engineering and Technology Publishing doi: 10.12720/jiii.1.2.107-110   Definition 1 : [20] Let  A is a TrIFN and its membership and non-membership functions are defined as follows: ( ), ;( ), ;( )( ), ;( )0 , .  A  x aw a x bb aw b x c xd xw c x d d c x a or x d             …. (1) ( ) ( '), ' ;( '), ;( )( ) ( ' ), ';( ' )0 , ' '.  A b x x a ua x bb au b x c x x c d x uc x d d c x a or x d                 ….  (2) where 0 1; 0 1; 1; , , , , ', ' w u w u a b c d a d R         For sake of simplicity, throughout this paper we have considered ' a a   and '. d d   Symbolically, then TrIFN has been represented as ([ , , , ]; , ).  A a b c d w u  In  particular, if b c   then TrIFN transform to TIFN.  Definition 2 : [16] Let 1 1 1 1 1 1 ([ , , , ]; , )  A a b c d w u   and 2 2 2 2 2 2 ([ , , , ]; , )  B a b c d w u   be two TrIFNs and 0      be a scalar, then 1 1 2 2 1 2 1 2 1 2 1 2 1 2 ) ([ , , , ]; , ) i A B a a b b c c d d w w ww u u         1 2 1 2 1 2 1 2 1 2 1 2 1 2 ( ) ([ , , , ]; , ) ii A B a a bb c c d d ww u u u u     1 1 1 1  1 1 ( ) ([ , , , ,];1 , ) (1 ) iii A a b c d   w u                1 1 1 1 1 1 ( ) ([ , , , ]; , ) (1 ) iv  a b c d w u A              III.    N EW R  ANKING M ETHOD  Let ([ , , , ]; , )  A a b c d w u   be a TrIFN, which has been shown in Fig-1. In order to find out the centroid of TrIFN, the area under the membership and non membership function has been considered together. First of all the whole TrIFN has been split into five rectangles: ARUP, REBU, EFCB, FSVC and SDQV where coordinates of the corner points of rectangles have been given below: : ( ,0), : ( , ), : ( , ), : ( ,0),  A a B b w C c w D d         : ( ,0), : ( ,0), : ( ,1), : ( ,1),: ( )/( 1),0 ,: ( )/( 1),0 ,: ( )/( 1), /( 1) ,: ( )/( 1), /( 1) .  E b F c P a Q d  R aw au b w uS dw du c w uU aw au b w u w w uV dw du c w u w w u                    Now, the centroid point has been determined by using the formulae ( )( )  xf x dx X   f x dx     and ( )( )  yg y dyY  g y dy     where the specific region bounded by continuous function ( )   f x  and ( )  g y  respectively. The required centroid point ( , )  A A  X Y  of TrIFN  A has been given below: 12  A  x x  X    , where 11 1 aw au bb cw u L Law au ba bw u  x xg dx xf dx xwdx            11  R dw du cd w u R dw du ccw u  xf dx xg dx           (3) and 1111 2  R aw au bbw u L Law au baw ucbdw du cd w u R dw du ccw u  x g dx f dxwdx  f dx g dx                 (4)  1,2  A  y y Y     where, 01 110 011 110 01 1 ( )[ ][ ]  L w R Lww u R Rww uww u Lww u  y y h h dy yd dy yh dy yk dy yh dy yk dy aydy                  (5) and 01 110 011 110 01 2 ( )[ ][ ]  L w R Lww u R Rww uww u Lww u  y h h dyd dy h dy k dyh dy k dy ady                  (6) where :[ , ] [0, ]  L   f a b w  and :[ , ] [0, ]  R   f c d w   are the left and right part of the membership function of TrIFN .  A   Figure 1. Trapezoidal intuitionistic fuzzy number :[ , ] [0, ]  L  g a b u  and :[ , ] [0, ]  R  g c d u   are the left and right part of the non-membership function of TrIFN  A which have been shown in Fig. 1.  :[0, ] [ , ]  L h w a b  and :[0, ] [ , ]  R h w c d    are the inverse functions of  L   f   and  R  f    respectively; :[0, ] [ , ]  L k u a b   and :[0, ] [ , ]  R k u c d    are the inverse functions of  L  g  and  R  g   respectively which have been shown in Fig. 2. In case of TrIFN, functions 108 ournal of Industrial and Intelligent Information Vol. 1, No. 2, June 2013 ©2013 Engineering and Technology Publishing  ( ), ( ), ( )  L R L   f x f x g x and  ( )  R  g x and their inverse functions ( ), ( ), ( )  L R L h y h y k y and  ( )  R k y can be analytically expressed as follows:  ( )( ) , ;( )( )( ) , ;( )  L R w x a  f x a x bb aw x d   f x c x d c d         ( ) ( )( ) , ;( )( ) ( )( ) , ;( )  L R  x b u a x g x a x ba b x c u d x g x c x d d c            ( )( ) , 0 ;( )( ) , 0 ;( ) ( )( ) , 1;1( ) ( )( ) , 1.1  L R L R b a yh y a y wwd c yh y d y wwa b y b auk y u yud c y c duk y u yu                 In particular if 1 w   and 0 u   then 2 2 2 2 (3 3 ),2(3 3 )7( ) 5( )18( ) 6( )  A A a b c d a b c d d a c bd a c b  X Y            (7) It is known that  X   A  denotes the representative location of IFN  A on the real line and Y   A  presents the average height of the IFN. In order to rank IFNs, the importance of the degree of representative location is higher than the average height. Therefore, the ranking may be done in the following way [21]: For any two different IFNs  A and  B , we have (a)If   A B  X X   , then  A B  ; (b)If   A B  X X   , then  A B  ; (c)If   A B  X X   , then if   A B Y Y   , then  A B  ; else if   A B Y Y   , then  A B  ; else  A B Y Y   , then  A B  . We rank A and B based on their  X  ’s values if they are different. If their  X  ’s values are equal then the attention has been given to the Y  ’s values . Figure 2. Inverse function of TrIFN IV.   C OMPARISON WITH THE E XISTING M ETHOD  In this section some examples of IFNs have been  presented (see Table I) to compare the proposed ranking  process with the existing methods [16], [18], [20], [21]. A comparison between the results of the proposed process and the result of the existing methods has been illustrated in Table I. TABLE I. A   C OMPARISON OF THE P ROPOSED R  ANKING P ROCESS WITH THE E XISTIN M ETHOD   The expressions of existing ranking process Examples The proposed method Wu and Cao [18] 1 1 2 21 1 2 21 1 2 21 1 2 2 ( , )1[ (1 ) (1 ).18(1 ) (1 ).1(1 ) (1 ).1(1 ) (1 ).1 ] d A r abbb                                         Where  [( , , , ); , ]  A a b c d       and [(1,1,1,1);1,0] r      If ( , ) ( , ) i j d A r d A r      then i j  A A    Example-1 ([0.57,0.73,0.83];0.73,0.20),([0.58,0.74,0.819];0.72,0.20).  A B  ( , ) 0.45( , ) 0.45 d A r d B r     A B    0.69730.36100.69570.3600  A A B B  X Y  X Y      A B   Jianqiang and Zhong [20] 1( ) [( ) (1 )]8( ) ( ) ( )( ) ( ) ( )  I A a b c d S A I A H A I A                     Where  [( , , , ); , ]  A a b c d         If ( ) ( ) i j S A S A  then i j  A A  ; If ( ) ( ) i j S A S A  then i j  A A   if ( ) ( ) i j  H A H A    Example-2 ([0.56,0.74,0.80,0.90];0.50,0.50),([0.50,0.70,0.85,0.95];0.50,0.50).  A B  ( ) 0( ) 0( ) 0.3750( ) 0.3750 S AS B H A H B     A B    0.72360.37150.71460.3542  A A B B  X Y  X Y      A B    Rezvani [21] ( 2 2 )( )6 a b c d V A       Where  [( , , , ); , ]  A a b c d        If ( ) ( ) i j V A V A   then i j  A A    Example-3 ([0.55,0.60,0.70,0.75];1,0)([0.45,0.65,0.70,0.75];1,0).  A B  ( ) 0( ) 0 V AV B     A B    0.65000.45240.60390.4123  A A B B  X Y  X Y      A B    Li [16] ( , )( , )1 ( , ) V a R a A a       Where 1 2 3 [( , , ); , ] a w u a a a    ( , ) ( ) ( ( ) ( )) V a a a a V V V            ( , ) ( ) ( ( ) ( ))  A a a a a  A A A           1 1 2 3 ( )( )6 4 a  w a a aV        1 2 31 )( )( )6 (1 4 a u a a aV          1 3 1 ( )( )3 a  w a a A      3 11 )( )( )3 (1 a u a a A        Example-4 ([ 6,1,2];0.6,0.5)([ 6,1,2];0.7,0.4).  A B       A B    2.31430.38992.28880.3797  A A B B  X Y  X Y        B A    109 ournal of Industrial and Intelligent Information Vol. 1, No. 2, June 2013 ©2013 Engineering and Technology Publishing  From Table I we can see some drawbacks of the existing methods and some advantages of the proposed method, which have been elaborate below: 1)   In Example 1, two different TIFNs have been considered. By Wu and Cao’s [18] method these two different TIFNs are not comparable. But by the proposed method the ranking result is .  A B   2)   From Example 2, it is observe that for two different TrIFNs, the ranking indices [20] give the same value and thus they are not comparable. However, by utilizing the proposed ranking method we may get the ranking result as .  A B   3)   Similarly, in Example 3, Rezvani’s  [21] approach the ranking result is same for two different TrIFNs. But by the proposed method ranking result is .  A B   4)   In Example 4, by the ratio ranking method [16] it is clear that the given two numbers (see Table I) are not comparable because their ratio ranking result is ( , ) ( , ) 0  R a R b      , although they have different membership and non-membership values. But by utilizing proposed method we can compare these two TIFNs and ranking result is .  B A   Therefore, from Table I it is clear that in all the above cases the proposed method finds the ranking result correctly and overcomes the drawbacks of the existing methods. V.   C ONCLUSION  In this paper, a new method for ranking IFNs has been introduced by utilizing centroid point of IFNs. For this  purpose, the centroid point of IFNs has also been computed. Examples have been given to compare the  proposed ranking method with the existing methods. This ranking approach may be applicable to multi-criteria decision making problem, which will be topic of our future research work. A CKNOWLEGEMENT  The authors would like to express their great thanks for the anonymous referees for their careful reading and valuable comments. R  EFERENCES   [1]   K. Atanassov, “ Intuitionistic fuzzy sets, ”    Fuzzy Sets and Systems , vol. 20, no 1 pp. 87-96, 1986. [2]   L. A. Zadeh, “ Fuzzy sets, ”    Information Control  , vol. 8, no 3,  pp. 338-353, 1965. [3]   S. M. Chen and J. M. Tan, “ Handing multi-criteria fuzzy decision-making problems based on vague set theory, ”    Fuzzy Sets and Systems , vol. 67, no 2, pp. 163  –  172, 1994. [4]   F. Li and Y. Rao, “ Weighted methods of multi-criteria fuzzy decision making based on vague sets, ”   Computer Science , vol. 28,  pp. 60  –  65, 2001. [5]   D. H. Hong and C. H. Choi, “ Multicriteria fuzzy decision making  problems based on vague set theory, ”    Fuzzy Sets and Systems , vol. 114, no 1, pp. 103  –  113, 2000. [6]   Z. S Xu and R. R Yager, “ Some geometric operators based on intuitionistic fuzzy sets, ”    International Journal of General Systems , vol. 35, no 4, pp. 417  –  433, 2006. [7]   P. Grzegorzewski, “ Distances and orderings in a family of intuitionistic fuzzy numbers, ”  in  Proc. Third Conference on Fuzzy  Logic and Technology, 2003, pp. 223-227.   [8]   P. Burillo, H. Bustince, and V. Mohendano, “Some definitions of Intuitionistic fuzzy numbers. first properties, ”  in  Proc. 1st Workshop on Fuzzy Based Expert Systems,  Sofia, 1994, pp53-55. [9]   R. Parvathi and C. Mal athi, “Arithmatic operations on symmetric trapezoidal intuitionistic fuzzy numbers, ”    International Journal of Soft Computing and Engineering  , vol. 2, no 2, pp. 2231-2307, May 2012. [10]   S. J. Chen and C. L. Hwang,  Fuzzy Multiple Attribute Decision  Making  , New York: Springer-Verlag, 1992. [11]   V. L. G. Nayagam, G. Venkateshwari, and G. Sivaraman, “ Ranking of intuitionistic fuzzy numbers, ”  in  Proc. IEEE  International Conference on Fuzzy Systems, Hong Kong, 2008, pp. 1971-1974. [12]   V. L. G. Nayagam, G.Venkateshwari, and G. Sivaraman, “ Modified ranking of intuitionistic fuzzy numbers, ”    NIFS  , vol. 17, no.1, pp. 5  –  22, 2011. [13]   W. Jianqiang and Z. Zhong, “ Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems, ”    Journal of Systems  Engineering and Electronics , vol. 20, no 2, pp. 321  –  326, 2009. [14]   W. Lee, “A novel method for ranking interval-valued intuitionistic fuzzy numbers and its application to decision making, ”  in  Proc.  International Conference on Intelligent Human-machine Systems and Cybernetics,  2009, pp. 282-285. [15]   Z. S Xu and R. R Yager, “ Dynamic intuitionistic fuzzy multiple attribute decision making, ”    International Journal of Approximate  Reasoning  , vol. 48, pp. 246  –  262, 2008. [16]   J. Wu and Q. Cao, “ Same families of geometric aggregation operators with intuitionistic trapezoidal fuzzy numbers, ”    Applied  Mathematical Modeling  , vol. 37, pp. 318-327, 2013. [17]   C. Wei and X. Tang, “ Possibility degree method for ranking intuitionist fuzzy numbers, ”  in  Proc. IEEE/WIC/ACM     International Conference on Web Intelligence and Intelligent  Agent Technology , 2010, pp. 142-145. [18]   D. F. Li, “ A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems, ” Computers and  Mathematics with Applications , vol. 60, no. 6, pp. 1557-1570, Sep 2010. [19]   S. Rezvani, “Ranking method of trapezoidal intuitionistic fuzzy numbers, ”    Annals of Fuzzy Mathematics and Informatics , 2012. [20]   J. Q. Wang and Z. Zhang, “ Multi-criteria decision making method with incomplete certain information based on Intuitionistic fuzzy number, ”   Control and Decision , vol. 24, no 4, pp. 226-230, 2009. [21]   Y. J. Wang and H. S. Lee, “ The revised method of ranking fuzzy numbers with an area between the centroid and srcinal points, ”   Computers and Mathematics with applications , vol. 55, no 9, pp. 2033-2042, 2008. Satyajit Das  received his M.Sc. degree in mathematics with ComputerApplications from National Institute of Technology, Durgapur, India in 2012. Presently, he is Research Fellow in the department of mathemati-cs in Indian Institute of Technology, Patna, India. His current research area: Fuzzy Logic, Intuitionistic Fuzzy Set theory. Debashree Guha  received her B.Sc. and M.Sc. degree in mathematics from Jadavpur University, Calcutta, India in 2003 and 2005, and resepectively. She received her Ph.D degree in Mathematics from Indian Institute of Technology, Kharagpur, India in 2011. Presently, she is the assistant professor of department of mathematics in Indian Institute of Technology, Patna, India. Her current research interest includes: multi-attribute decision making, fuzzy mathematical programming, aggregation operators and fuzzy logic. 110 ournal of Industrial and Intelligent Information Vol. 1, No. 2, June 2013 ©2013 Engineering and Technology Publishing
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