8th Internatonal Conference on Informaton Fuson Washngton DC - July 6-9 5 evsed Method for ankng Generalzed Fuzzy Numbers Yu uo a Wen Jang b DeYun Zhou c XYun Qn d Jun Zhan e abcde School of Electroncs and Informaton Northwestern Polytechncal Unversty X an 77 Chna. jangwen@nwpu.edu.cn bstract number of fuzzy number rankng methods have been proposed by the researchers n recent years. However most of them have ehbted several shortcomngs assocated wth non-dscrmnatve and counter-ntutve problems especally when the fuzzy numbers are symmetrc fuzzy numbers or crsp numbers. In ths paper we propose a new method for rankng generalzed fuzzy numbers where the weght of centrod ponts degrees of fuzzness and the spreads of fuzzy numbers are taken nto consderaton whch can overcome the drawbacks of etng methods and s effcent for evaluatng symmetrc fuzzy numbers and crsp numbers. t last several numercal eamples are provded to llustrate the superorty of the proposed method. Keywords generalzed fuzzy numbers the degree of fuzzness centrod pont spread weght. Introducton In recent years a lot of achevements about the theory and method of rankng fuzzy numbers are obtaned. Snce Jan [] proposed the frst fuzzy number rankng method varous methods have been devsed for rankng fuzzy numbers [ 9]. Yager (978 [] proposed the centrod nde rankng method wth a weghtng functon. Cheng (998 [] presented an approach for rankng fuzzy numbers by usng the dstance method where the dstance represents the orgnal pont to the centrod pont. Chu and Tsao ( [] proposed an approach for rankng fuzzy numbers wth the area between the centrod pont and orgnal pont. Chen and Chen (7 [6] presented a method for rankng generalzed fuzzy numbers based on the centrod ponts and the standard devatons of generalzed fuzzy numbers. Wang et al. (9 [8] presented a method for rankng fuzzy numbers by combnng the transfer coeffcent and devaton degree of a fuzzy number. Chen and Chen (9 [9] presented a method for rankng generalzed fuzzy numbers by consderng the defuzzfed values the heghts and the spreads of the generalzed fuzzy numbers. Nejad and Mashnch ( [] proposed a method for rankng fuzzy numbers based on the areas of the sdes of the fuzzy numbers. Chen and Sanguansat ( [] proposed a method by consderng the areas on the postve sde the areas on the negatve sde and the heghts of the generalzed fuzzy numbers to evaluate the rankng scores of the generalzed fuzzy numbers. Chen ( [] presented a method for rankng generalzed fuzzy numbers wth dfferent left heghts and rght heghts. Emrah et al. ( [5] presented a new method for rankng generalzed trapezodal fuzzy numbers based on the ncenter and nradus of a trangle whch rank crsp numbers and fuzzy numbers wth the same centrod pont. Madhur et al. ( [7] proposed a new method for rankng generalzed trapezodal fuzzy numbers based on the Crcumcentre ponts. Bakar and Gegoy ( [8] proposed a novel method for rankng fuzzy numbers whch ntegrated the centrod pont and the spread approaches and overcomes the lmtatons and weaknesses of some ested methods. Wang (5 [9] frst proposed a fuzzy preference relaton wth membershp functon representng preference degree to compare two fuzzy numbers. Then a relatve preference relaton was constructed on the fuzzy preference relaton to rank a set of fuzzy numbers. The rankng methods stated above are commonly used approaches whch are hghly cted and have wdely appled. However they stll have some drawbacks such as nondscrmnatve and counter-ntutve problems. Therefore n ths paper we present a new method for rankng generalzed fuzzy numbers based on centrod ponts degrees of fuzzness and spreads of fuzzy numbers whch can overcome the drawbacks of estng methods. Meanwhle the proposed method consders that the weghts of all factors whch nfluence the rankng result should be dfferent. Fnally we make a comparson of the calculaton results wth the estng methods to llustrate the superorty of the proposed method. The rest of ths paper s organzed as follows. In Secton we brefly revew the basc concepts of generalzed fuzzy numbers. In Secton we present a new method for rankng generalzed fuzzy numbers based on centrod ponts degrees of fuzzness and spreads of fuzzy numbers. In Secton we make a comparson of the rankng results of the proposed fuzzy rankng method wth the estng methods. In Secton 5 the conclusons are dscussed. Prelmnares In ths secton we brefly revew some basc concepts of generalzed fuzzy numbers. 978--99657--7 5 ISIF
. Generalzed fuzzy numbers Chen ( [] proposed the concept of generalzed fuzzy numbers wth the dfferent left heght and rght heght. et be a generalzed fuzzy number wth the dfferent left heght and rght heght ( a a a a ; where a a a a are real values s called the left heght of the generalzed fuzzy number s called the rght heght of the generalzed fuzzy number [] and []. If a a a a then s called a standardzed generalzed fuzzy number. If then becomes a normal trapezodal fuzzy number. If a a s called a generalzed trangular fuzzy number. If a a a a s called a crsp number. The membershp functon f ( of a generalzed fuzzy number and the membershp functon f ( F of a fuzzy number F are shown n Fgure. f F f a a f T f a a Fgure. Membershp functons of generalzed fuzzy numbers and F. The degree of fuzzness Deuca and Termn proposed Shannon entropy as the measure of degree of fuzzness n 97 []. et be a fuzzy set n the unverse of U and all the fuzzy sets on U are denoted by F ( U. The membershp functon of s denoted by ( the degree of fuzzness of s denoted by d(. For F( U the degree of fuzzness d( or d( B satsfes the followng propertes: (If F( U d(. (If ( d(. (If ( B( or ( B( d( d( B. GuoChun Tang [] proposed a revsed degree of fuzzness whch called area degree of fuzzness n 999. Smultaneously he also presented the Mnkowsk degree of fuzzness on the contnuous fuzzy set. Now we brefly revew the area degree of fuzzness and Mnkowsk degree of fuzzness on the contnuous fuzzy set. If U [ ] fuzzy number ( a a a a; w U then the area degree of fuzzness of s defned as follows: d ( [ ( d ( ( ] a d ( ( ( where d ( s the area degree of fuzzness of. The Mnkowsk degree of fuzzness of s defned as follow: p p d( [ ( ( d]. a ( If p the Mnkowsk degree of fuzzness of becomes area degree of fuzzness. new method for rankng general - zed fuzzy numbers In ths secton we proposed a new method for rankng generalzed fuzzy numbers based on centrod ponts degrees of fuzzness and spreads of fuzzy numbers. The proposed method ndcates that the value of the centrod pont on the horzontal as( s the most mportant nde for rankng generalzed fuzzy numbers whereas the value of the centrod pont on the vertcal as( y the spread and the degree of fuzzness are the ad ndees. Therefore all the ndees have dfferent weght on the process of rankng generalzed fuzzy numbers.. The proposed method ssume that there are n generalzed fuzzy numbers n to be ranked where ( a a a a ; a a a a ; [] [] denotes the left heght of fuzzy number denotes the rght heght of fuzzy number and n.the proposed method s shown as follows: Step : ccordng to the method proposed by Wang et al. (6 [] calculate the each centrod pont ( y of generalzed fuzzy numbers shown as follows. Case : If ( a a a a ; and then the centrod pont s shown as follows: y a a a ( f ( d a d a f d a a a a ( f d ( a d a f d a a ( a a a a a a a ( a a ( a a ( yg yg dy a a ( g g dy a a a a ( ( ( ( (
where n g and g are the nverse functons of and f respectvely. f Case : If ( a a a a ; and then the centrod pont s shown as follows: y a a a a a a a a a a a a ( f d d ( f d ( f d d ( f d T ( yg ( yg dy yg yg dy T ( g g dy ( g g dy T g where n g g and are the nverse functons of f f and T f respectvely. Case : If then the centrod pont s shown as follows: y a a a a a a a a a a a a ( f d d ( f d ( f d d ( f d T ( yg ( yg dy yg yg dy T ( g g dy ( g g dy T g where n g g and are the nverse functons of f f and T f respectvely. Step : Calculate the Mnkowsk degree of fuzzness of each generalzed fuzzy numbers when p accordng to the method proposed by Guo-Chun Tang shown as follows: d( [ f ( f ( d] a a [ ( f ( d a a a a T a a f ( d f ( d] where ( a a a a ; [ ] and. Step : Calculate the spread of each generalzed fuzzy number shown as follows: (5 (6 (7 (8 (9 ( a j ( j STD a a a a where nand. Step : Calculate the score ( of each generalzed fuzzy number shown as follows: ( [ ( ] y score( STD d( ( where and they were gven by the decson-maker. ( s shown as follows: []; ( [. If ( a a a a ; s a crsp number the centrod pont s shown as follows: a ( y. ( The Mnkowsk degree of fuzzness of crsp number s shown as follows: d(. ( The spread of crsp number s shown as follows: STD. (5 The score of crsp number s shown as follows: ( a [ a ( a ] score (6 where n (. The propertes of proposed fuzzy score functon In the followng we present four propertes of the proposed fuzzy score functon. Property (Normalzaton property: If be a generalzed fuzzy number where ( a a a a; a a a a and [] then score( [ ] Proof: If ( a a a a; where a a a a and [] then based on (-( we can know [] d ( []. So ( y [] STD [] ( y STD d( then ( ] y []. STD d( Therefore we can get ( [ ( ] y score( [ ]. STD d( Property (symmetrc property : If ( a a a a; and ( a a a a; be generalzed fuzzy numbers where a a a a and [] then Score( Score(. 5
Proof: If ( a a a a ; and ( a a a a; where a a a a and [] then based on (-(8 we can see that y y. Based on (9 and ( we can see that STD STD d( d (. Based on ( we can see that ( ( y y STD d( STD d( ( ( Therefore we can get Score( Score( Property. If ( a a a a; a a where a then Score( a Proof: If ( a a a a; a a where a then based on ( ( (and(5 we can see that: a y a STD d(. So ( a( ( y score( STD d( ( a( a a If a ( a then ( a( a a a a score( a If a ( a then ( a( a a score( [( a ( a] a a a Therefore f ( a a a a; a a where a then score( a. Property: ( ; then Score(. Proof: If ( ; based on ( ( (and(5 we can see that: y STD d( then ( a( ( y score( STD d( (. comparson of the proposed rankng method wth estng methods. The eght sets of generalzed fuzzy numbers shown n the paper of Deng and u (5 are classcal eamples for the fuzzy number and usually are made as the benchmarks by the other papers such as the paper of Chen et al. ( and Chen & Chen (9. So we wll use them to compare the proposed method wth the estng methods n ths paper. Besdes the other four sets of generalzed fuzzy numbers are added to the paper to supplement the stuaton whch has not been consdered by Deng and u (5. Therefore the sets of generalzed fuzzy numbers are shown n Fgure. can comprehensvely represent the generalzed fuzzy numbers n dfferent stuatons and the rankng results of them can be etend to the other fuzzy numbers n the uncertan stuaton. Though the weght of factors s gven by the decsonmaker they should satsfy... and are suggested because the rankng result concdes wth the analytcal geometry and ntuton of human bengs better than the others at ths tme. The centrod pont on X-as s the most mportant factor so ts weght s always. The results shown n Table are obtaned when.. and. From Table we can see that (For the fuzzy numbers and B shown n Set of Fgure. the estng rankng methods and the proposed method get the same rankng order whch concdes wth the ntuton of human bengs. (For the fuzzy numbers and B shown n Set of Fgure. Bakar and Gegoy s method ( Madhur et al. s method ( and the proposed method get a reasonable rankng order.e. B whch concdes wth the ntuton of human bengs due to the fact that the centrod pont of s the larger than the centrod pont of B on the Y-as and the degree of fuzzness of B s larger than.yager s method (978 Cheng s method (998 Chu and Tsao s method ( Chen s method (9 Chen and Sanguansat s method ( Chen et al. s method ( Emrah s method ( and Wang s method (5 get an unreasonable rankng order.e. B or B. ( For the fuzzy numbers and B shown n Set of Fgure. Yager s method (978 Cheng s method (998 Chu and Tsao s method ( Chen and Sanguansat s method ( Chen et al. s method ( and Wang s method (5 get an unreasonable rankng order.e. B whereas Chen and Chen s method (9 Emrah s method ( Bakar and Gegoy s method ( Madhur et al. s method ( and the proposed method get the same rankng order.e. B whch concdes wth the ntuton of human bengs. 6
(For the fuzzy numbers and B shown n Set of Fgure. Yager s method (978 get an ncorrect rankng order.e. B and Wang s method (5 can not calculate the generalzed fuzzy numbers whereas Cheng s method (998 Chu and Tsao s method ( Chen s method (9 Chen and Sanguansat s method ( Chen et al. s method ( Emrah s method ( Bakar and Gegoy s method ( Madhur et al. s method ( and the proposed method get the same rankng order.e. B whch concdes wth the ntuton of human bengs. (5For the fuzzy numbers and B shown n Set 5 of Fgure. Yager s method (978 Cheng s method (998 Chu and Tsao s method ( and Madhur et al. s method ( cannot calculate the crsp number whereas Chen and Chen s method (9 Chen and Sanguansat s method ( Chen et al. s method ( Emrah s method ( Bakar and Gegoy s method ( Wang s method (5 and the proposed method get the same rankng order.e. B whch concdes wth the ntuton of human bengs. (6For the fuzzy numbers and B shown n Set 6 of Fgure. Yager s method (978 Chu and Tsao s method ( Chen and Chen s method (9 Chen and Sanguansat s method ( Chen et al. s method ( Emrah s method ( Bakar and Gegoy s method ( Wang s method (5 and the proposed method get the same rankng order.e. B whch concdes wth the ntuton of human bengs due to the fact that the centrod pont of B on the X-as s larger than the centrod pont of on the X-as whereas Cheng s method (998 and Madhur et al. s method ( get an ncorrect rankng order.e. B. (7 For the fuzzy numbers and B shown n Set 7 of Fgure. all the rankng methods ecept Madhur et al. s method ( get the same rankng order.e. B whch concdes wth the ntuton of human bengs. (8For the fuzzy numbers B and C shown n Set 8 of Fgure Cheng s method (998 Chu and Tsao s method ( Chen and Chen s method (9 Chen and Sanguansat s method ( Chen et al. s method ( Bakar and Gegoy s method ( Madhur et al. s method ( Wang s method (5 and the proposed method get the same rankng order.e. C B whch concdes wth the ntuton of human bengs. However Yager s method (978 and Emrah s method ( get the ncorrect rankng order.e. B C. (9For the fuzzy numbers and B shown n Set 9 of Fgure Cheng s method (998 and Wang s method (5 cannot calculate the generalzed fuzzy numbers and B when they themselves are symmetrcal about the Y- as. Yager s method (978 Chu and Tsao s method ( Chen and Chen s method (9 Chen and Sanguansat s method ( Chen et al. s method ( and Bakar and Gegoy s method ( get the ncorrect rankng order.e. B whereas Emrah s method ( Madhur et al. s method ( and the proposed method get the reasonable rankng order.e. B whch concdes wth the ntuton of human bengs due to the fact that the centrod pont of B on Y-as s larger than the centrod pont of on Y-as. (For the fuzzy numbers and B shown n Set of Fgure Cheng s method (998 cannot calculate the generalzed fuzzy numbers and B when they themselves are symmetrcal about the Y-as. Yager s method (978 Chu and Tsao s method ( Chen and Chen s method (9 Chen and Sanguansat s method ( Chen et al. s method ( Emrah s method ( Bakar and Gegoy s method ( Madhur et al. s method ( and Wang s method (5 get the ncorrect rankng order.e. B or B whereas the proposed method get the rankng order.e. B whch s reasonable due to the fact that the centrod pont of on Y-as s larger than the centrod pont of B on Y-as and the degrees of fuzzness of and B are dentcal. (For the fuzzy numbers B and C shown n Set of Fgure Cheng s method (998 and Wang s method (5 cannot calculate the generalzed fuzzy numbers B and C when they are symmetrcal about the Y-as. Yager s method (978 Chu and Tsao s method ( Chen and Chen s method (9 Chen and Sanguansat s method ( and Chen et al. s method ( Emrah s method ( Bakar and Gegoy s method ( and Madhur et al. s method ( get the ncorrect rankng order.e. B C or B C whereas the proposed method get the reasonable rankng order.e. B C whch concdes wth the ntuton of human bengs due to the fact that the centrod pont of on Y-as s B C and the degrees of fuzzness of B and C s B C (For the fuzzy numbers B and C shown n Set of Fgure only Chen et al. s method ( Bakar and Gegoy s method ( and the proposed method can calculate the generalzed fuzzy numbers wth dfferent left heghts and rght heghts correctly and they get the same rankng order.e. B C whch concdes wth the ntuton of human bengs. In summary from Fgure and Table we can see that the proposed method gves the reasonable rankng order and overcomes the drawbacks of the estng rankng methods. Especally the proposed method has advantage when the generalzed fuzzy numbers are symmetrcal about the Y- as. 7
Table comparson of the rankng results of the proposed method wth estng methods shown n Fgure. Methods Set Set Set Set B B B B Yager s method (978 Cheng s method(998 8 7 8 8 8 8 6 8 Chu and Tsao s method (. Chen and Chen s method(9 79 98 7 79 79 7.6 79 Chen and Sanguansat s method( Chen et al. s method( 5 5 5 5 5 6 5 Emrah et al. s method( 87 788 87 87 66 87 Bakar and Gegoy s method(.867.96.867.867.9.75.867 Madhur et al. s method( 77 885 77 77 87 9 77 Wang s method(5 5 5 N N The proposed method 67 7 67 67 9 8 67 Methods Set5 Set6 Set7 Set8 B B B B C Yager s method (978 N - 5 Cheng s method(998 N 8 8 67 8 57 6 Chu and Tsao s method ( N - 7 9 8 Chen and Chen s method(9 7. -79 79 8 5 79 96 Chen and Sanguansat s method(. - 75 5 5 5 5 Chen et al. s method( 5. -5 5 77 667 57 Emrah et al. s method(. - 87 68 87 6 97 Bakar and Gegoy s method(.96 -.867.867 8 7 6 5 Madhur et al. s method( 885 N 77 77 79 5 68 Wang s method(5.... 85 65 65 9 75 The proposed method 7. -67 67 677 95 559 77 89 Methods Set9 Set Set Set B B B C B C Yager s method (978....... N N Cheng s method(998 N N N N N N N 8 N N Chu and Tsao s method (....... N N Chen and Chen s method(9....... 7 N N Chen and Sanguansat s method(....... N N Chen et al. s method(....... 87 Emrah et al. s method( -.568 -.5 -.568 -.78 -.568 -. -.7 6 N N Bakar and Gegoy s method(........96 5.99 Madhur et al. s method( 6 5 6 86 6 85 8 885 N N Wang s method(5 N N N N N The proposed method.8.69.8.6.8..5 7 5 89 Note: N denotes cannot be calculated; gray background denotes unreasonable results. 8
... B... B... B... (. ; B ( ; Set... (.. ; B (. ; Set... (. ; B (. ; Set... B... B... B... (. ; B (. ; Set... (.. ; B ( ; Set5.... (- - - -.; B (. ; Set6...... (.; B (. ; Set7 B... B C... ( ; B (. ; C (. ; Set8....... (- -.. ; B (- -.. ; Set9 B....... (- -.. ; B (-. -...; Set 5 Conclusons B....... (- -.. ; B (- ; C (- ; Set In the data fuson processes or systems no matter whatever the form of data s t can be converted to the fuzzy number. Fuzzy number has become an ecellent math tool for the problem of data fuson. Especally nearly all the problems of decson-makng can be solved by rankng fuzzy numbers. So rankng fuzzy numbers properly s a key and necessary step n the fuson processes or systems. In ths paper we present a new method for rankng generalzed fuzzy numbers based on centrod ponts degrees of fuzzness and spreads of fuzzy numbers. The centrod pont reflects the ntegratve character of the generalzed fuzzy number. The degree of fuzzness and spread of fuzzy number reflect how fuzzy a generalzed fuzzy number s. B C Fgure. Twelve sets of generalzed fuzzy numbers.... B C... (.. ; B (.. ; C (.. ; Set The larger the values of the centrod pont on X-as and Y- as are the better the rankng order. The larger the degree of fuzzness and spread are the worse the rankng order. lthough the centrod pont degrees of fuzzness and spreads are all consdered by the proposed method they should have dfferent status when rankng generalzed fuzzy numbers. s the man factor the weght of the centrod pont should be larger than the degrees of fuzzness and spreads. Ths concdes wth the analytcal geometry and ntuton of human bengs. Comparng wth the estng methods the proposed method has ntroduced both degree of fuzzness and spread of fuzzy numbers whch can reflect how fuzzy a generalzed fuzzy number s and should be consdered carefully n decson makng. The proposed method has consders dfferent factors of generalzed fuzzy numbers should have 9
dfferent weght whch can overcome some drawbacks of the estng methods and has obvous advantage when the generalzed fuzzy numbers such as Set9 Set and set n Fgure are symmetrcal about the Y-as. 6 cknowledge Ths work was supported n part by a grant from Natonal Natural Scence Foundaton of Chna (No. 6 and Foundaton for Fundament esearch of Northwestern Polytechncal Unversty Grant No.JC5. eferences []. Jan (976. Decson makng n the presence of fuzzy varables IEEE. Trans. Syst. Man Cybern 6(976 pp.698 7. [].. Yager (978. ankng fuzzy subsets over the unt nterval Decson and Control ncludng the 7th Symposum on daptve Processes 978 IEEE Conference pp.5 7. [] C.H. Cheng. (998. new approach for rankng fuzzy numbers by dstance method Fuzzy Sets and Systems 95( pp.7 7. [] T.C. Chu C.T. Tsao. ( ankng fuzzy numbers wth an area between the centrod pont and orgnal pont Comput. Math. ppl ( pp. 7. [5] Y. Deng Q. u (5. TOPSIS-based centrodnde rankng method of fuzzy numbers and t s applcaton n decson-makng Cybernetcs and Systems 6(7 pp.58 595. [6] S.J. Chen S.M. Chen (7. Fuzzy rsk analyss based on the rankng of generalzed trapezodal fuzzy numbers ppled Intellgence 6( pp.. [7] B. sady. Zendehnam. (7. ankng fuzzy numbers by dstance mnmzaton ppl. Math. Modell. (7 pp. 589 598. [8] Z.X. Wang Y.J. u Z.P. Fan B. Feng ankng fuzzy number based on devaton degree Inf. Sc. 79 (9 pp. 7 77. [9] S.M. Chen J.H. Chen (9. Fuzzy rsk analyss based on rankng generalzed fuzzy numbers wth dfferent heghts and dfferent spreads. Epert Systems wth pplcatons 6( pp.68 68. [] Y.M. Wang (9. Centrod defuzzfcaton and the mamzng set and mnmzng set rankng based on alpha level sets. Computers & Industral Engneerng 57 pp.8 6. [].M. Nejad M. Mashnch. (. ankng fuzzy numbers based on the areas on the left and the rght sdes of fuzzy number Comput. Math. ppl. 6 ( pp.. [] S.M. Chen K. Sanguansat. (. nalyzng fuzzy rsk based on a new fuzzy rankng method between generalzed fuzzy numbers Epert Systems wth pplcatons 8 ( pp.6 7. [] S.Y. Chou.Q. Dat V.F. Yu. (. revsed method for rankng fuzzy numbers usng mamzng set and mnmzng set Computers & Industral Engneerng 6 ( pp.-8. [] S.M. Chen. Munf G.S. Chen H.C. u B.C. Kuo. (. Fuzzy rsk analyss based on rankng generalzed fuzzy numbers wth dfferent left heghts and rght heghts Epert Systems wth pplcatons 9 ( pp.6 6. [5] Emrahkyar H. kyar S.. D uzce. (. Fuzzy rsk analyss based on a geometrc rankng method for generalzed trapezodal fuzzy numbers Journal of Intellgent & Fuzzy Systems 5 ( pp. 9 7. [6] V.F. Yu H. Th X. Ch. (. ankng generalzed fuzzy numbers n fuzzy decson makng based on the left and rght transfer coeffcents and areas ppled Mathematcal Modellng 7 ( pp.86 87 [7] K.U. Madhur S.S. Babu N.. Shankar (. Fuzzy rsk analyss based on the novel fuzzy rankng wth new arthmetc operatons of lngustc fuzzy numbers Journal of Intellgent & Fuzzy Systems 6 ( 9 [8].S.. Bakar. Gegov (. ankng of fuzzy numbers based on centrod pont and spread Journal of Intellgent & Fuzzy Systems 7( 79 86. [9] Y.J. Wang (5 ankng trangle and trapezodal fuzzy numbers based on the relatve preference relaton ppl. Math 5 9 (5 586-599. []. Deluca S. Termn. (97 defnton of nonprobalstc entropy n the settng of fuzzy sets theory. Informaton and Control 97 ( pp.. [] G.C. Tang (999.The rea Degree of Fuzzness n Fuzzy Schedulng Journal of Shangha Second Polytechnc Unversty 999 ( pp.9 56. [] Y.M. Wang J.B. Yang D.. Xu K. S. Chn (6 On the centrods of fuzzy numbers. Fuzzy Sets and Systems 6(57 pp.99-9