Evaluation of Fuzzy Quantities by Distance Method and its Application in Environmental Maps

Transcription

1 Joural of pplied Sciece ad griculture, 8(3): 94-99, 23 ISSN Evaluatio of Fuzzy Quatities by Distace Method ad its pplicatio i Evirometal Maps Saeifard ad L Talebi Departmet of pplied Mathematics, Urmia Brach, Islamic zad Uiversity, Urmia, Ira bstract: The preset paper aimed at desigig a ew method to rak fuzzy umbers by distace method The purpose of the preset study was to give a ew method to value various fuzzy umbers effectively ad at the same time overcome the shortcomigs of the previous studies The proposed method is based o the cetral iterval ad the maximum crisp value of fuzzy umbersthe results of the aalysis idicated that the followig method effectively rak various fuzzy umbers ad their images The fidigs are discussed as far as a ew method is cocered to be far simpler tha the other approaches Fially, this method compared the preset proposed defiitio with some other kow oes Key words: Fuzzy umbers, akig, Defuzzificatio, Distace, Evirometal maps INTODUCTION akig fuzzy umbers is oe of the importat issues i the realm of decisio makig process Most of the problems that exist i ature are fuzzy, tha probabilistic or determiistic Problems cotaiig fuzzy risk aalysis, fuzzy optimizatio, etc, are to be used i fuzzy theory, therefore, at oe of the stages fuzzy umbers must be raked before a actio is take by a decisio maker Sice fuzzy umbers are represeted by possibility distributio they ofte overlap with each other ad i this case discrimiatig them is a complex task tha discrimiatig real umbers where a atural order exist betwee them mog the existig rakig methods oe of the efficiet approaches to order the fuzzy umbers is defuzzificatioto this ed, we defie a rakig fuctio from the set of all fuzzy umbers F ( ) to the set of all real umbers, which maps each fuzzy umber ito the real lie, where a atural order exist Usually by reducig the whole of the aalysis to a sigle umber, much of the iformatio is lost ad most of the rakig methods cosider oly oe poit of view with regard to comparig fuzzy quatities Hece, a attempt is to miimize this loss Due to the importace of fuzzy umbers, may researchers have proposed the related methods or applicatios to rak fuzzy umbers For example; Jai(976), Dubios ad Prade(983) itroduced the relevat cocepts of fuzzy umberselated to the rakig methods Bortola ad Degai(985) compared ad reviewed the existig approaches, poitig out some illogical coditios ad coclusios that arise out of the research Furthermore, Yao ad Wu (2) proposed a ew approach for rakig fuzzy umbers based o the fuzzy simulatio aalysis method i which a combiatio method icludig computer ad math applicatio was developed (Lee et al, 998) Saeifard ad llahviraloo(27) cosidered a fuzzy origi for fuzzy umbers ad the based o the distace of fuzzy umbers with respect to this origi, rak thembbasbady adsaeifard(2)suggested a defuzzificatio method based o the cetral iterval of fuzzy umberssady ad Zedeham(27)poited out a defuzzificatio method usig miimizer of the distace betwee the two fuzzy umbers O the other had, aother commoly used techique is the cetroid-based fuzzy umber rakig approach Cheg(998)argued that iterms of a cetroid idex rakig approach, the distace of the cetroid poit of each fuzzy umber ad origial poit are calculated to lead to the improvemet ofyager's approach Lee ad Li (998) coducted the compariso of fuzzy umbers based o the uiform ad proportioal probability distributios i which the mea ad stadard deviatio values are used to rak fuzzy umbers I order to overcome the above metioed limitatios, Chu ad Tsao(22) proposed a approach to rak fuzzy umbers with the area betwee the cetroid ad origial poits Havig reviewed the previous methods, this paper proposes a coceptual procedure ad method to use the cocepts of cetral iterval, maximum crisp value ad Euclidea distace i order to fid the order of fuzzy umbers The advatage of this method is that ca distiguish the alteratives clearly I this paper,defuzzificatio ca be used as a crisp approximatio of a fuzzy umber Thus, the mai purpose of the preset study is to obtai a crisp approximatio with respect to a fuzzy quatity, ad defie a method for rakig of fuzzy umbers Therefore, by meas of this defuzzificatio, the preset study aimsat presetig a ovel method for rakig of fuzzy umbers The reset of this paper is orgaized as follows:i Sectio 2, some defiitio ad fudametal results o fuzzy umbers are recalled I Sectio 3, a ew approach is proposed for rakig fuzzy umbers Discussio Correspodig uthor: Saeifard, Departmet of pplied Mathematics, Urmia Brach, Islamic zad Uiversity, Urmia, Ira srsaeeifard@yahoocom 94

2 J ppl Sci & gric, 8(3): 94-99, 23 ad compariso of this work ad other method are carried out i Sectio 4 The paper eds with coclusio i Sectio 5 Fially the refereces are listed at the ed 2 Basic Defiitios ad Notatios: Here, we review some basic otios of fuzzy umbers (take from (bbasbady et al, 2)) These otios are expressed as follows Defiitio 2 fuzzy umber is a mappig µ ( x ) : [,] with the followig properties: µ is a upper semi- cotiuous fuctio o 2 µ ( x ) = outside of some iterval [ a, b2] 3 There are real umbers a, a 2, b adb 2 ad such that a a2 b b2 ad 3 µ ( x ) is a mootoic icreasig fuctio o [ a, a 2], 32 µ ( x ) is a mootoic decreasig fuctio o [ b, b 2], 33 µ ( x ) = for all x i [ a2, b ] The set of all fuzzy umbers is deoted by F Defiitio22The membership fuctio µ of a fuzzy umber is expressed by (2) as: L f ( x ) whe x [ a, b ), w whe x [ b, c], µ ( x ) = (2) f ( x) wh e x ( c, d], otherwise Where < w is acostat The left fuctio f L ( x) :[ ab, ] [, w] ad the right fuctio f ( x) :[ c, d] [, w ] are two strictly mootoicad cotiuous mappig from to closed iterval[, w ] Therefore, the left iversed fuctio g L ( x) :[, w] [ ab, ] ad the right iversed fuctio g ( x) :[, w] [ cd, ] also exist Based o the basic theories of fuzzy umbers, is a ormal fuzzy umber if w =, whereas is a o-ormal fuzzy umber if < w < Notice that (2) is a L fuzzy umber fuzzy L umber with shape fuctio f ( x) ad f ( x) is defied by: L x a f ( x ) =, (22) b a ad d x f ( x ) = d c, (23) respectively, where >, will be deoted by = abcd,,, If be o-ormal fuzzy umber it will be deoted by = abcdw,,, ; If =, we simply wright = abcd,,,, which is kow as a ormal trapezoidal fuzzy umber ad if b = c, it is reduced to a triagular fuzzy umber ad represeted by = abd,,, therefore, triagular fuzzy umbers are special cases of trapezoidal fuzzy umbers Each fuzzy umber described by (2) has the followig α-level sets (α-cuts) α = [ aα, bα], aα, b, α α [,], L [ g ( α), g ( α )] for all α (,) 2 = [ bc, ] 3 = [ ad, ] If = abcd,,, the for all α [,], α = a+ α ( b a), d α ( d c) (24) 95

3 J ppl Sci & gric, 8(3): 94-99, 23 Defiitio 23ccordig to Dubios ad Prade(993) the iterval-valued probabilistic mea of a fuzzy umber with α-cuts = [ a, b ], α [,] is the iterval E( ) = [ E ( ), E ( )], where α α α E ( ) = a dα ad E ( ) = b dα (25) α α Carlsso ad Fuller (22) itroduced the iterval-valued possibilistic mea of a fuzzy umber as the iterval M ( ) = [ M ( ), M ( )], where M ( ) = 2 α aα dα ad M ( ) = 2 α bα dα (26) Defiitio 24(Bodaova, 25)Let be a fuzzy umber characterized by(2) Let ml ( ab, ) ad (, cd) be such that m ml b µ ( x ) dx = µ ( x ) dx, (27) a ad m ml d µ ( x ) dx = µ ( x ) dx, (28) c m espectivelythe M e( ) = [ ml, m] is called the media iterval(iterval-valued media) of b a Propositio 2(Bodaova, 25)Let = abcd,,, The M e( ) = [ ml, m],where ml = a+ ad m d c = d (29) I order to determie the cetral iterval C( ) = [ C ( ), C ( )] of a fuzzy umber, Bodaova (25) provides the followig cetral iterval formulae as: C( ) = E( ) M ( ) M ( ), (2) e Where E( ), M ( ) ad M e ( ) are the iterval-valued probabilistic mea ad the iterval-valued possibilistic mea ad the iterval-valued media of, respectively The cetral iterval C( ) has the lower boud C ( ) = max E{ ( ), M ( ), m ( )}, L (2) d the upper boud C ( ) = mi E{ ( ), M ( ), m ( )} (22) Defiitio 25(bbasbady et al, 2)Let be a arbitrary fuzzy umber, ad C( ) = [ C ( ), C ( )] be its cetral iterval (calculated by (2)) The measure of C( ) is as: M ( C( )) = sig ( C ( )) C ( ) C ( ), (23) I The ew measure of fuzzy umber is defied as follow: M ( ) = p( α) M ( C( )) dα (24) C I 96

4 J ppl Sci & gric, 8(3): 94-99, 23 Thefuctio P :[,] [, + ) deotes the distributio desity of the importace of the degrees of fuzziess, where P( α) dα = I particular cases, it may be assumed that k p( α) = ( k + ) α, k =,, Throughout this study the researchers assumed that k =, ie P ( α) = 2α (25) 3 New Method for akig Fuzzy Numbers: I this sectio, based o the distace method we preset a ew approach for rakig fuzzy umbers The methodot oly cosiders the cetral iterval of a fuzzy umber, but also takes ito accout the maximum crisp value of fuzzy umbers I order to rak fuzzy umbers, firstly,this study defies a maximum crisp value amax to be the bechmark ssume that there are fuzzy umbers, 2,,, where = ( a, a2, a3, a4), The maximum crisp value a max is the maximum value of the a, a 2, a 3, a 4, ad The proposed method for rakig fuzzy umbers, 2,, is ow preseted as follow: StepUse formula (2) to calculate the cetral iterval C( ) = [ C ( ), C ( )] of each fuzzy umbers,where Step2 Calculate the maximum crisp value a max of all fuzzy umbers, where Step3 Use the cetral iterval C( ) = [ C ( ), C ( )] to calculate the rakig value MS ( ) of the fuzzy umbers,where,as follow: MS ( ) = M ( ) a (36) C max From(36), oe may that MS ( ) could be cosidered as the Euclidea distace betwee the poit ( M C ( ),) ad the poit ( a max,) We ca see that the smaller the valueof MS ( ), the better the rakig of, where Sice the aim of the preset study is to approximate a fuzzy umber by a scalar value, the researchers have to use a operator MS : F (a space of all fuzzy umbers ito a family of real lie) MS isa crisp approximatio operator Sice the above metioeddefuzzificatio ca be used as a crisp approximatio of a fuzzy umber, it ca be cocluded that the resultat value is used to rak the fuzzy umber Thus, MS is used to rak fuzzy umbers Thesmaller MS, the larger fuzzy umber Let B, F be two arbitrary fuzzy umbers Defie the rakig of ad B o MS () as follow: MS ( ) > MS ( B ) if oly if B, 2 MS ( ) < MS ( B ) if oly if B, 3 MS ( ) = MS ( B ) if oly if ~ B The this article formulate the order ad as B if ad oly if B or ~ B, B if ad oly if B or ~ B The ew rakig idex ca sort may differet fuzzy umbers simultaeously I additio, the idex also satisfies the commo properties of rakig fuzzy umbers as follow: (a) Trasitivity of the order relatio, ie if B ad B C, the we should have C (b) Compatibility of additio, that is if there is B o { B, } the there is + C B + C o{ + C, B + C} emark 3If B the, B Hece, this article ca ifer rakig order of the images of thefuzzy umbers 4 Numerical Examples: I this Sectio, the curret study compares proposed method with some of other method Example 4Cosider the followig sets, see bbasbady et al(2) Set: = (4, 5,), B = (4,7,), C = (4,9,) Set2: = (3, 4, 7, 9), B = (3,7,9), C = (5, 7, 9) Set 3: = (3, 5, 7), B = (3,5,8,9), C = (3, 5, 9) Set4: = (, 4, 8, 9), B = (2, 5, 9), C = (, 6, 8) 97

5 J ppl Sci & gric, 8(3): 94-99, 23 To compare the proposed method with other methods, researchers refer the reader to Table (4) Note that, i Table (4) ad i set4, for Sig Distace with (P=) i (bbasbady et al, 26), sady&zedeham(27), Yao&Wu(2) ad Baldwi&Guild(979) methods, the rakig order for fuzzy umbers B ad C is B ~ C, which seems ureasoable regardig to figures The preset study utilizes the proposed method to rak fuzzy umbers i the above metioed example Table 4: Comparative esults of Example 4 uthors Fuzzy Set Set 2 Set 3 Set 4 umbers Proposed method B C esult B C B C C B B C bbasbady et al(2) B C esults B C B C C B B C Sig Distace method with p= (26) B C esults B C B C C B B C Sig Distace method with p=2 (26) B C B C C B B C B C B C C B B C B C B C C B B C esults B C sady ad Zedeham (27) esults B C bbasbady ad Haari(29) esults B C Choobieh ad Li (993) B C esults B C B C B C B C Baldwi ad Guild (979) B C esults B C B C B C B C Chu ad Tsao(22) B C esults B C B C C B C B Yao ad Wu (2) B C esults B C B C C B B C Cheg (998) CV uiform distributio B C esults C B B C B C C B Cheg (998) CV proportial distributio B C 3 25 B C B C C B esults C B Example42Cosider the fuzzy umbers = (,4,7,8), B = (2, 5, 9), C = (, 6, 8), (Figure 4) We rak these fuzzy umbers with our proposed method MS ( ) = 695, MS ( B ) = 646, MS ( C ) = 62 Therefore, the rakig order is B C The images of these fuzzy umbers are = ( 8, 7, 4,), B = ( 9, 5, 2), C = ( 8, 6, ) lso, by the providedmethod MS ( ) = 25, MS ( B ) = 254, MS ( C ) = 298 Therefore, we have C B Obviously, the proposed method ca effectively rak the images of fuzzy umbers 98

6 J ppl Sci & gric, 8(3): 94-99, 23 B C Fig 4: Fuzzy Numbers BCi,, Example 42 5 Coclusios: It seems that fuzzy umbers are coveiet for represetig imprecise umerical quatities i a vague eviromet, ad their compariso or rakig is very importat for applicatio purposessice fuzzy umbers do ot form a atural liear order,i this regard like real umbers, a key issue i operatioalizig fuzzy set theory is to show how to compare fuzzy umbersseveral aalytic methods have bee proposed to solve this problem, but i this paper, we suggest a ew approach to the problem of rakig usig distace method ad employed the cetral iterval ad the maximum crisp value i the proposed defiitio EFEENCES bbasbady, S, B sady, 26 akig of fuzzy umbers by sig distace, Iformatio Sciece, 76: bbasbady, S, T Haari, 29 ew approach for rakig of trapezoidal fuzzy umbers, Computer ad Mathematics with pplicatios, 57: bbasbady, S, Saeifard, 2 method for defuzzificatio based o cetral iterval ad its applicatio i decisio makig, Joural of merica Sciece, 7(6): sady, B, Zedeham, 27akig fuzzy umbers by distace miimizatio, ppl Math Model, 3: Baldwi, JF, NCF Guild, 979 Compariso of fuzzy umbers o the same decisio space, Fuzzy Sets Syst, 2: Bodaova, S, 25 Media value ad media iterval of a fuzzy umber, Iformatio Sciece, 72: Bortola, G, Degai, 985 review of some methods for rakig fuzzy umbers, Fuzzy Sets ad Systems, 5: -9 Carlsso, C, Fuller, 22Fuzzy reasoig i decisio makig ad optimizatio, Physica Verlag, Heidelberg Cheg, CH, 998 ew approach for rakig fuzzy umbers distace method, Fuzzy Sets ad Syst, 95(3): Choobieh, F, H Li, 993 idex for orderig fuzzy umbers, Fuzzy Sets ad Systems, 54: Chu, TC, CT Tsao, 22 akig fuzzy umbers with a area betwee the cetroid poit ad origial poit, Computers ad Mathematics with applicatio, 43: -7 Dubios, D, H Prade, 983 akig of fuzzy umbers i the settig of possibility theory, If Sci, 3: Jai,, 976 Decisio makig i the presece of fuzzy variables, IEEE Trasactios o Systems, Ma ad Cyberetics, 6: Lee, ES, J Li, 998 ew approach for rakig fuzzy umbers by distace method, Fuzzy Sets ad Systems, 95: Saeifard, S, T llahviraloo, F Hosseizadeh, N Mikaeilvad, 27 Euclidea rakig DMUs with fuzzy data i DE, ppl Math Sci, 6: Yager,, 98 O choosig betwee subsets, Kyberetes, 9: 5-54 Yager,, 98 procedure for orderig fuzzy subsets of the uit iterval, If Sci, 24:43-6 Yao, J, K Wu, 2 akig fuzzy umbers based o decompositio priciple ad siged distace, Fuzzy Sets ad Systems, 6: