A New Approach For the Ranking of Fuzzy Sets With Different Heights

Transcription

1 New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts Pushpnder Sngh School of Mathematcs Computer pplcatons Thapar Unversty, Patala-7 00 Inda STCT ankng of fuzzy sets plays an mportant role n decson makng, optmzaton, forecastng, etc Fuzzy sets must be ranked before an acton s taken by a decson maker Fuzzy sets wth dfferent heghts are a generalzaton of the ordnary fuzzy sets In ths paper, wth the help of several counterexamples, t s proved that the rankng method proposed by ee Chen (Expert Systems wth pplcatons 3, 008, s ncorrect The man am of ths paper s to propose a new approach for the rankng of fuzzy sets wth dfferent heghts The man advantage of the proposed approach s that wth t the correct orderng of fuzzy sets wth dfferent heghts, also the results of the proposed rankng method the exstng rankng method, can be compared Keywords: sets, fuzzy sets wth dfferent heghts, rankng functon Introducton Fuzzy set theory [] s a powerful tool to deal wth real-lfe stuatons eal numbers can be lnearly ordered by or ; however, ths type of nequalty does not exst n fuzzy numbers Snce fuzzy numbers are represented by possblty dstrbuton, they can overlap wth each other t s dffcult to determne clearly whether one fuzzy number s larger or smaller than the other n effcent approach for orderng the fuzzy numbers s by usng a rankng functon : F(, where F( s a set of fuzzy numbers defned on the real lne whch maps each fuzzy number nto the real lne, where a natural order exsts Thus, specfc rankng of fuzzy numbers s an mportant procedure for decson makng n a fuzzy envronment, generally, has become one of the man problems n fuzzy set theory The method for rankng was frst proposed by Jan [] Yager [3] proposed four ndces that may be employed for the purpose of orderng fuzzy quanttes n [0, ] Campos Gonzalez [] proposed a subjectve approach for rankng of fuzzy numbers ou Wang [7] developed a rankng method based on the ntegral value ndex Cheng [0] presented a method for rankng fuzzy numbers by usng the dstance method Kwang ee [5] consdered the overall possblty dstrbutons of fuzzy numbers n ther evaluatons proposed a rankng method Modarres Sad-Nezhad [8] proposed a rankng method based on the preference functon whch measures the fuzzy numbers pont by pont, at each pont, the most preferred number s dentfed Chu Tsao [] proposed a method for rankng fuzzy numbers wth the area between the centrod pont the orgnal pont Deng u [] presented a centrod ndex method for rankng fuzzy numbers Chen Chen [5] presented a method for rankng generalzed trapezodal fuzzy numbers Wang ee [] used the centrod concept n developng ther rankng ndex Chen Tang [8] proposed a method for rankng nonnormal p-norm trapezodal fuzzy numbers ee Chen [6] presented a new method for rankng fuzzy sets used the proposed fuzzy rankng method to present a new fuzzy rsk analyss algorthm to deal wth fuzzy rsk analyss problems Chen Wang [9] studed the fuzzy rsk analyss based on the rankng of fuzzy numbers bbasby Hajjar [] ntroduced a new approach for rankng trapezodal fuzzy numbers based on the left rght spreads at some levels of trapezodal fuzzy numbers Chen Chen [6] presented a method for fuzzy rsk analyss based on of rankng generalzed fuzzy numbers wth dfferent Journal of ppled esearch Technology 9

2 New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts, Pushpnder Sngh / 9 99 heghts dfferent spreads aml Mohamad [9] presented the comprehensve survey of dfferent methods for the rankng of fuzzy numbers sady [] ndcated the shortcomngs of Wang method proposed a revsed method n whch shortcomngs for rankng fuzzy numbers are removed ssady [3] suggested an nterestng approach to crsp functon approxmaton of fuzzy numbers defne the epslon neghborhood of the fuzzy number The method leads to the crsp functon whch s the best one related to a certan measure of dstance between the fuzzy number a crsp functon of the set support functon Chen et al [7] presented a new method for fuzzy rsk analyss based on the proposed new fuzzy rankng method for rankng generalzed fuzzy numbers wth dfferent left heghts rght heghts lso, they proposed a new method for fuzzy rsk analyss based on the proposed fuzzy rankng method Ezzat et al [3] modfed the method of bbasby Hajjar [] In ths paper, wth the help of counterexamples, t s shown that the rankng approach proposed by ee Chen [6] s ncorrect new approach s proposed for the rankng of fuzzy sets wth dfferent heghts esdes, the results of the proposed approach the exstng rankng approach are compared Ths paper s organzed as follows: In Secton, some basc defntons arthmetc operatons between fuzzy sets wth dfferent heghts are presented In Secton 3, ee Chen s [6] rankng approach s revewed Shortcomngs of ee Chen s approach [6] are ponted out n Secton In Secton 5, a new approach s proposed for the rankng of fuzzy sets wth dfferent heghts esults of the proposed approach the exstng rankng approach are compared n Secton 6 The fnal secton s for conclusons Prelmnares element of the unversal set X fall wthn a specfed range e : X [0,] The assgned value ndcates the membershp grade of the element n the set The functon s called the membershp functon, the set {( x, ( x; x X} defned by, ( x for each x X, s called a fuzzy set Defnton00[6] fuzzy set ( a, b, c, d ; H, H, n s sad to be a fuzzy set wth dfferent heght f ts membershp functon s gven by 0, ( x 0, H H H x a (, b a c x ( c b x d (, c d H x b (, b c b x a a x b x c c x d d x where H H denotes the left rght heght of fuzzy sets wth dfferent heghts, respectvely IH IH In ths secton some basc defntons arthmetc operatons are presented asc defntons Defnton [] The characterstc functon of a crsp set X assgns a value, ether 0 or, to each member n X Ths functon can be generalzed to a functon such that the value assgned to the 0 a b c d Fgure Fuzzy set wth dfferent heght 9 Vol 0, December 0

3 New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts, Pushpnder Sngh / 9 99 rthmetc Operatons et ( a, b, c, d ; H, H ( a, b, c, d ; H, H be two fuzzy sets wth dfferent heghts [6] then mn ( H mn ( ( a a, b b, c c, d,, mn(, H ( a H, d, b H H, mn( H H c, c ( a, b, c, d; ( d, c, b, a;, H H d (I b, d a ; H,, H H 3 ee Chen s rankng approach p (II ; 0 < 0 (III et,,, n be a set of trapezodal fuzzy numbers, where ( a, a, a, a ;,, n 3 H H The proposed method for rankng fuzzy numbers s now presented as follows: Step Fnd the values of s, H, s T s of each fuzzy number H, M, M,, where H denotes the rght heght of the fuzzy number H denotes the left heght of the fuzzy number ( a3 a M denotes the average of the rght elements a 3 a ( a a M denotes the average of the left elements a a S ( aj aj denotes the j3 j3 stard devaton of the rght elements a 3 a S ( aj aj denotes the j j stard devaton of the rght elements a a T S ( aj aj denotes the stard j j devaton of the elements a, a, a3, a n Step Choose two proper values for, where denotes the expert s degree of confdence, [0,], denotes the ndex of optmsm of the decson maker [0,] In general, let be proper values for rankng fuzzy numbers Step 3 Calculate the rankng values ank of each fuzzy number ank ( [ ( M H ( ( ( ] ( 3 as follows: s ( ] ( [ ( ( H s ( T ( M s ( ( ] The larger the rankng value ank, the better the rankng of the fuzzy number ( Journal of ppled esearch Technology 93

4 New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts, Pushpnder Sngh / 9 99 Shortcomngs of ee Chen s rankng approach Wang Keere [] proposed the followng reasonable propertes for the valdaton of any rankng functon: If are normal fuzzy sets then ( C ( C ( C ( C (I (II : ( C : ( C (III where, C s normal fuzzy set For the fuzzy sets wth dfferent heghts, the same property can be wrtten as If ( a, b, c, d ; H, H ( a, b, c, d ; H, H are two fuzzy sets wth dfferent heghts then ( C ( C (I ( C ( C (II : ( C : ( C (III where, C ( a3, b3, c3, d3; 3 H, 3 H s a type-ii fuzzy set ( 3H 3H H H H H, ( mn(,, mn(, There may exst several fuzzy sets wth dfferent heghts for whch the exstng rankng functons [6] do not satsfy the reasonable property C C, e, accordng to exstng rankng approaches C C whch s a contradcton accordng to Wang Keere [] Example0 et (,,3,;0 6,0, (0,3,,5;0,0 C (,3,,5;0,0 be three fuzzy sets wth dfferent heghts Then, accordng to exstng rankng approaches [6], but C C e,, whch s a contradcton Example0 et (,5,6,7;0 6,0, (3,,5,6;0 8,06 C (,3,7,8;0 6,0 be three fuzzy sets wth dfferent heghts Then, accordng to exstng rankng approaches [6], but C C e,, whch s a contradcton 5 new approach for the rankng of fuzzy sets wth dfferent heghts In ths secton, a new approach s proposed for the rankng of fuzzy sets wth dfferent heghts To overcome the shortcomngs dscussed n Secton, the followng defntons are proposed Defnton 3 For any fuzzy set wth dfferent heght ( a, b, c, d ;, H H, the expectaton value of centrod s defned as follows: M d a d a xf f ( x ( x ( Defnton For any fuzzy set wth dfferent heght ( a, b, c, d; H, H, the transfer coeffcent of,,,, n, s gven by M M mn ( M max M mn where M max( M, M,, M max n M mn( M, M,, M mn n 9 Vol 0, December 0

5 New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts, Pushpnder Sngh / 9 99 Defnton 5 et ( a, b, c, d; H, H be a fuzzy set wth dfferent heght, a mn( a, a,, a mn n d max max( d, d,, dn The areas ( ( s s of the left rght sde of fuzzy set wth dfferent heght s ( s ( are defned as follows: H 0 H 0 ( b a y ( a a ( d max mn dy (3 ( c d y d dy ( From Equatons, 3, the proposed rankng ndex of s defned as follows:,,,, n s ( s( s ( ( (5 Defnton 6 For any two fuzzy sets wth dfferent heghts, j, based on (5, ther order s defned by f only f j s( s( (I f only f j s( s( : f only f j s ( : s( (II (III 5 lgorthm et ( a, b, c, d ; H, H ( a, b, c, d ; H, H be fuzzy sets wth dfferent heghts n F ( Use the followng step to compare : * * Step Transform, nto as follows: * ( a, b, c, d; H, H * ( a, b, c, d ; H, H where, ( mn(,, mn(, ( H H H H H H Step Usng Equaton, fnd the expectaton * * values of centrod, M M of, respectvely Fgure Fuzzy set left rght area Step 3 Usng Equaton, fnd the transfer * * coeffcents, of, respectvely * Step Obtan s * ( s ( by usng (3 ( Journal of ppled esearch Technology 95

6 New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts, Pushpnder Sngh / 9 99 * * Step 5 Obtan the values of s ( s ( by usng (5 Step 6 The fuzzy sets wth dfferent heghts, * ( can be compared as follows: * * f only f s( s( (I * * f only f s( s( (II : * * f only f s ( : s( 6 Comparatve study (III In ths secton, dfferent generalzed fuzzy sets fuzzy sets wth dfferent heghts are taken to compare the results of the proposed rankng method the exstng rankng method Set et (0,0,06,08;035 (0,0,03,0;07 be two generalzed fuzzy sets Use the followng steps to compare : Step Transform, * nto *, where * (0,0,06,08;035 * (0,0,03,0;03 5 Step Usng Equaton, M 3766 M 59 * * Step 3 Usng Equaton, * 0 * * Step Usng Equatons 3, ( 007 * s * ( 0035 ( 075 * s ( 0075 * s, s, * Step 5 Usng Equaton 5, s ( 0 * s( 0075 * * Snce s( s(, so Set et (0,0,0,05; (0,03,0 3,05; be two generalzed fuzzy sets Use the followng steps to compare : Step Transform, nto *, where * (0,0,0,05; * (0,03,03,05; Step Usng Equaton, M 0877 M 005 * * * Step 3 Usng Equaton, 0 * * * Step Usng Equatons 3, s ( 05, * s * ( 0 s * ( 0, s ( 0 * Step 5 Usng Equaton 5, s ( 05 * s ( 0 * * Snce s( s(, so Set 3 et (,,3,;0 6,0 (0,3,,5;0,0 be two fuzzy sets wth dfferent heghts Use the followng steps to compare : Step Transform, * nto *, where * (,,3,;0,0 * (0,3,,5;0,0 Step Usng Equaton, M 78 M 50 * * 96 Vol 0, December 0

7 New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts, Pushpnder Sngh / 9 99 Step 3 Usng Equaton, 0 * * * Step Usng Equatons 3, s ( 06, * s * ( 03 s * ( 06, s ( 0 * Step 5 Usng Equaton 5, s ( 0 * s ( 06 * * Snce s( s(, so Set et (,5,6,7;06,0 (3,,5,6;0 8,06 be two fuzzy sets wth dfferent heghts Use the followng steps to compare : Step Transform, nto, where * (,5,6,7;0 6,0 * (3,,5,6;06,0 Step Usng Equaton, M 383 M 387 * * * * Fgure 3 Generalzed fuzzy sets fuzzy sets wtth dferent heghts Journal of ppled esearch Technology 97

8 New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts, Pushpnder Sngh / 9 99 Step 3 Usng Equaton, 0 * * * Step Usng Equatons 3, s ( 09, * s * ( 0 s * ( 09, s ( 06 * Step 5 Usng Equaton 5, s ( 0 * s ( 09 * * Snce s( s(, so Methods Set Set Set 3 Set Cheng [0] Chu Tsao [] Chen Chen [5] bbasby Hajjar [] Chen Chen [6] ou Wang [7] ee Chen [6] ommelfang er [0] Proposed approach : N N : N N N N N : N N N N : N N N N N Table Comparson of the proposed rankng approach wth the exstng rankng approach Note:- where N denotes the not applcable 7 Conclusons In ths paper, wth the help of counterexamples, t s proved that the rankng method proposed by ee Chen [6] s ncorrect new approach for the rankng of fuzzy sets wth dfferent heghts, as well as the results of the proposed rankng method the exstng rankng method are compared cknowledgements The authors would lke to thank the Edtor-n-Chef the anonymous referees for the varous suggestons whch have led to an mprovement n both the qualty clarty of the paper eferences [] S bbasby T Hajjar, new approach for rankng of trapezodal fuzzy numbers, Computers Mathematcs wth pplcatons, 57, 3-9, 009 [] ssady, The revsed method of rankng fuzzy number based on devaton degree, Expert Systems wth pplcatons, 37, [3] ssady, evson of dstance mnmzaton method for rankng of fuzzy numbers, ppled Mathematcal Modellng, 35, , 0 [] M Campos Gonzalez, subjectve approach for rankng fuzzy numbers, Fuzzy Sets Systems, 9, 5-53, 989 [5] S J Chen S M Chen, Fuzzy rsk analyss based on the rankng of generalzed trapezodal fuzzy numbers, ppled Intellgence, 6, -, 007 [6] S M Chen J H Chen, Fuzzy rsk analyss based on rankng generalzed fuzzy numbers wth dfferent heghts dfferent spreads, Expert Systems wth pplcatons, 36, , 009 [7] S M Chen et al, Fuzzy rsk analyss based on rankng generalzed fuzzy numbers wth dfferent left heghts rght heghts, Expert Systems wth pplcatons, 39, , 0 [8] CC Chen HC Tang, ankng nonnormal p- norm trapezodal fuzzy numbers wth ntegral value, Computers Mathematcs wth pplcatons, 56, 30-36, Vol 0, December 0

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