CHPTER FUZZY NUMBER ND FUZZY RITHMETIC 1 Introdction Fzzy rithmetic or rithmetic of fzzy nmbers is generlistion of intervl rithmetic, where rther thn considering intervls t one constnt level only, severl levels re considered in [0, 1] This is becse, the definition of fzzy set which llows degree of membership for n element of the niversl set It plys n importnt role in mny pplictions, sy, fzzy control, decision mking, pproximte resoning, optimiztion nd sttistics with imprecise probbilities Some pproprite reference for this chpter re Dbois nd Prde [14] nd [15], Kfmnn nd Gpt [1] nd [] nd Moore [39] Fzzy rithmetic 1 Intervl rithmetic The fndmentls of fzzy rithmetic is nothing bt intervl rithmetic In given closed intervl R, how to dd, sbtrct, mltiply nd divide These intervl in R is lso clled n intervl of confidence s its limits the ncertinty of dt to n intervl definitions Let = [1, ] nd B= [b1, b] be two closed intervl in R then we hve following Definition: (ddition nd Sbtrction) x 1 1 If [, ], y [b, b ], then x y [1 b1, b] nd x y [1 b1, b] 3
therefore +B = [1, ] + [b1, b] = [1+b1, +b] B = [1, ] - [b1, b] = [1-b, -b1] Exmple: 3 [, 5] + [1, 3] = [3, 8] [0, 1] [ 6, 5] = [ 5, 7] Definition 4 (Imge of n intervl) x 1 then its imge x, If [, ] by is defined s 1 Therefore the imge of is denoted,, 1 1 Definition: 5 (Mltipliction) The mltipliction of two closed intervls = [1, ] nd B = [b1, b] of R denoted by B is defined s B = [min (1b1, 1b, b1, b), mx (1b1, 1b, b1, b)] Exmple: 6, 05,, 05, mx, 05,, 05 [ 1, 1] [, 05] min [,] 4
Definition: 7 (Sclr Mltipliction nd Inverse) Let, be closed intervl in R nd kr identifying the sclr k s the 1 closed intervl [k, k], the sclr mltipliction k is defined s k = [k, k] [1, ] = [k1, k] 1 1 1 for [1, ] R if x [1, ] nd if 0[ 1, ] then x 1 Therefore the inverse of is denoted by 1 nd it is defined s 1 1 1 1 [1, ], provided 0 [1, ] 1 Definition: 8 (Division) The division of two closed intervls = [1, ] nd B = [b1, b] of R denoted by /B is defined s the mltipliction [1, ] nd 1 b 1 b 1 provided 0[b1, b] Therefore /B = [1, ] / [b1, b] 1 1 = [1, ], b b 1 = 1 1 1 1 min,,,,mx,,, b b1 b b1 b b1 b b1 Exmple: 9 [4, 10] / [1, ] = [, 10] 5
rithmetic opertions on closed intervls stisfy some sefl properties, to overview, let = [1, ], B = [b1, b], C = [c1, c], 0 = [0, 0] nd 1= [1, 1], sing these symbols, the properties re formlted s follows: 1 +B = B+ nd B = B (Commttive) (+B)+C = + (B+C) nd (BC) = ( B) C (ssocitive) 3 =0+ = +0 nd =1 = 1 (Identity) 4 (B+C) B+ C (Sbdistribtivity) 5 If b c) 0, forevery b B, c C, then (B C) B C(Distribtive Frthermore, if = [, ] then (B+C) = B + C 6 0 in nd 1 in / 7 If E nd B F then B E F, B E F, B E F, / B E/ F (Inclsion monotonicity) s n exmple we prove only the less obvios properties of Sbdistribtivity First we hve (B+C) = {(b+c) /, b B, c C} = { b+ c /, bb, cc } { b ' c/ ', b B, c C }= B + C Hence (B+C) B+ C b1 1 ssme now withot loss of generlity, tht 0, c 0 Then we hve to consider the following three cses: 1 If 1 0then (B+C) = [1 (b1+c1), (b+c)] = [1 b1+ b] + [1 c1, c] 6
= B + C If 1 < 0 nd 0 then 0, ()=[, 1] nd () (B+C) = ( ) B + ( ) C Hence (B+C) = B+ C 3 If 1 < 0 nd > 0 then (B+C) = [1 (b+c), (b+c)] = [1 b, b] + [1 c, c] = B + C To show tht distribtivity does not hold in generl, Let = [0, 1], B = [1, ], C = [, 1], then B = [0, ], C = [, 0], B+C = [ 1, 1] nd (B+C) = [ 1, 1] [, ] = B + C Definition 10 (Mx nd min opertions) Let = [1, ] nd B = [b1, b] be two closed intervls in R then the Mx nd min opertions on nd B re defined s B [1, ] [b1, b ] [1 b1, b ] B [1, ] [b1, b ] [1 b1, b ] Note: 11 It cn be verified tht ddition nd mltipliction opertions on closed intervls s defined bove re commttive nd ssocitive bt sbtrction nd division re neither commttive nor ssocitive + ' = [1, ] + [, 1] [0, 0] = 0 7
In cse = [1, ] R nd 0 [1, ], then -1 = -1 [1, 1] 1 Note: 1 In the next section, fzzy nmbers nd fzzy rithmetic re introdced It is more significnt in rithmetic of cts which re closed nd bonded intervls of type,, (0,1], when fzzy set in R is fzzy nmber Therefore ll ides presented in this section cn be borrowed for intervls of type,, (0,1] rithmetic of fzzy nmbers cn be developed nd meningfl 3 Fzzy nmber nd their representtion There re mny in rel life sittion, in which, the res like decision mking nd optimiztion, where rther thn deling with crisp rel nmbers nd crisp intervls one hs to del with pproximtion of nmbers which re close to given rel nmber The prpose of this section is to nderstnd, how fzzy sttement cn be conceptlized by certin pproprite fzzy sets in R to be termed s fzzy nmbers Let s consider fzzy sttement the nmbers tht re closed to given nmber r since r in R is close to itself, ny fzzy set in R which represents the fzzy sttement shold hve property tht (r) 1 Which imply mst be norml fzzy set lso, jst prescribing n intervl rond r is not enogh The intervl shold be considered t vrying levels (0,1] to hve proper grdtion of cts in nd it mst be closed intervl of type, for(0,1] Frther ] tht spport of is bonded (0, 1 mst be of finite length nd for tht one needs 8
Definition 31 (Fzzy nmber) fzzy set in R is clled fzzy nmber if it stisfies the following conditions: 1 is norml fzzy set is closed intervl for every (0,1] 3 Spport of is bonded Theorem 3 Let be fzzy set in R then is fzzy nmber if nd only if there exists closed intervl (which my be singleton) [, b] sch tht 1 if x [,b] (x) (x) if x (,) r(x) if x (b, ) Where i) : (-, ) [0, 1] is monotonic incresing, continos from the right sch tht (x) = 0 for x in ii) r :(b, ) 0, 1, k, k1 < is monotonic decresing, continos form the left sch tht r(x) = 0 for x in (k, ), k > b Note: 33 In cse the membership fnction of the fzzy set in R tkes the form 1 (x) 0 if x if x 9
It becomes the chrcteristic fnction of the singleton set {} nd therefore it represents the rel nmber The following figre exhibits both continos fnction (x) nd r(x) re hving continity in the membership Note: 34 Fzzy set in X is convex set if nd only if ll its α-cts α re convex crisp sets for [0,1] Frther when the membership fnction of the convex fzzy set in R is pper semi-continos, then ll these α-cts α for (0,1] re closed intervls Since the bsic reqirement to define fzzy nmber is tht ll it (0,1] re closed nd bonded intervls The following is n lternte definition of fzzy nmber Definition: 35 fzzy sbset of the rel line R with membership fnction : R [0, 1] is clled fzzy nmber if 1 is norml there exists n element x0 sch tht (x0) = 1 is convex, (x1 + (1-)x) (x1) Λ (x) (x1, x) R nd [0, 1] 30
3 is pper semi-continos 4 Spport of is bonded, where Spp = {x R: (x) 0} 4 rithmetic of fzzy nmbers rithmetic of fzzy nmbers re defined by the following methods: 1 Intervl rithmetic on α-cts of given fzzy nmbers Mthemticl pproch which decomposes fzzy set in terms of specil fzzy set, (0, 1] 3 Zdeh s extension principle 41 pproch on α-cts Let nd B be ny two fzzy nmbers nd [, ] nd B [b, b ] be cts where (0, 1] of nd B respectively Let * denote ny rithmetic opertions, on fzzy nmbers, then we hve following definitions: Definition 4 (Opertions on two fzzy nmbers) Let nd B be ny two fzzy sets Let nd B be fzzy nmbers, then * opertion on fzzy nmbers gives fzzy nmbers in R B ( B) nd B B, (0, 1] Here B be fzzy nmber bt not generl fzzy set The sets B,, B, re ll closed intervls for (0, 1] lso for given in (0, 1], the closed intervl ( B) cn be compted by pplying the intervl rithmetic on the closed intervls respect to the opertions * nd B with 31
L R In prticlr, B [ b, b ] L R B [ b, b ] Frther, fzzy nmber in R, k > 0 cn gin be defined s erlier in the context of intervl rithmetic k k [k,k ] 5 pproch sing Zdeh s extension principle Let nd B two fzzy nmbers nd * be ny of the rithmetic opertions described bove Using extension principle, fzzy nmber *B is defined s sp min μ B (z) = zxy μ (x), μ B (y) z in R In prticlr we hve Z) sp minμ (x), μ (y) B( B zxy B B ( z) sp min μ (x), μ B(y) zxy ( z) sp min μ (x), μ B (y) zxy Z) spmin μ (x), μ (y) ( B x B z y 51 Specil types of fzzy nmbers nd their rithmetic Some specil types of fzzy nmbers nd fzzy rithmetic re discssed here, which will be sed extensively in lter Chpters on fzzy mthemticl progrmming nd fzzy gmes 3
Definition 5 (Tringlr fzzy nmber (TFN)) fzzy nmber is clled tringlr fzzy nmber (TFN) if its membership fnction is defined by μ 0 x (x) x x, x x x The TFN is denoted by triplet (3-tples) =,, nd hs the shpe of tringle shown below is the closed intervl Frther the cts of the TFN,, [, ] = ( )α, ( )α (0, 1] Next =,, nd B=b, b, two TFN then sing one cn compte *B where opertion * my be +, -, *, /,, b In this context B b, b, b,,,, 33
k= k, k, k, k > 0, B = b, b, b re TFN bt B, 1, / B, B, B need not be TFN Exmple: 53 [1, ] Let = (-3,, 4), B= (-1, 0, 5) be two TFN s then sing the formle for the ddition nd sbtrction of TFN s +B = (-3,, 4) + (-1, 0, 5) = (-4,, 9) -B = (-3,, 4) - (-1, 0, 5) = (-8,, 5) Using the cts α nd B α for the given fzzy nmber nd B we hve [, ] = ( ) α, ( ) α (0,1], = ( 3) 3, () 4, (0,1] 5α 3, α 4, (0,1 = ] nd B [b,b ] = ] (b b )α b, (b b) α b (0,1 = α 1, 5α 5, (0,1] B 5 3, 4 1, 5 5, (0,1 Therefore ] = 6 4, 7 9 [c,c ] (sy), (0,1] To find the membership fnction of B(x), we hve to find the rnge x in R where α level sets re vlid Ths c L, 34
6 4 x 7 9 if if x 4, from c 6 9 x, from c 7 L R frther α becomes, for x = the membership fnction becomes 0 x 4 B (x) 6 9 x 7 if if if x 4 4 x x 9 or x 9 Which is tringlr fzzy nmber (TFN) (-4,, 9) s shown in the below given digrm The following exmple shows B need not be TFN in generl: Exmple: 54 [1, ] Let = (, 3, 5) nd B = (1, 4, 8) be two TFN in R s noted erlier one hs to compte B by the first principle sing α-cts nd B where, 5 nd B 3 1, 4 8 35
B 3 1, 5 4 8 3 7, 8 36 8 c, c For,, B,40nd for, B 1,1 0 0 0 1 1 1 Therefore membership fnction B tkes 1 for x = 1 nd 0 for x < nd lso for x > 40 lso in between nd 1 nd lso between 1 nd 40, the segments of the membership fnctions re not stright lines bt it is prbol For this it is fond tht the rnge x in R, in which α level set re vlid This cn be ccomplished from L c by solving x 3 7 3 7 x 0 7 51x 6 Here only + sign will be tken becse t x=1, α = 1 nd α cnnot be negtive Similrly one hs to solve 8 36 8 x 0 9 1 x 5 Therefore membership fnction B 0 7 51x 6 9 1 x 5 if x or x 40 if x 1 if 1 x 40 Which is not the membership of TFN nd it cn be seen from the digrm 36
For more detils in tringlr pproximtions of fzzy one cn refer Kffmnn nd Gpt [0, 1] nd Dbois nd Prde [14, 15] Definition 55 (Trpezoidl fzzy nmber TrFN) fzzy nmber is clled trpezoidl fzzy nmber (TrFN) if its membership fnction μ is given by μ 0 x (x) 1 x if if if x x x, x if x The TrFN is denoted by qdrplet (4-tples) =,,, nd hs the shpe of trpezoid 37
The TrFN is denoted by qdrplet =,,, nd hs the shpe of trpezoid Frther α-ct of TrFN =,,, is the closed intervl α =[ α L, α R ] = )α, ( )α where [0,1] ( Next =,,, nd B= b,b,b, two TFN s then sing one cn compte *B b where opertion * my be +, -, *, /,, In this context it cn be verified B b, b, b, b, B b, b, b, b nd k = k, k, k, k k > 0 re TrFN bt -1, B, / B, B nd B need not be TrFN 38
Definition 56 (Left Right fzzy nmber) fzzy nmber is clled n L-R fzzy nmber if its membership fnction μ : X [0, 1] hs the following form: x L α 1 μ (x) x b R β 0 if α x, α 0 if x b if b x b β, β 0 otherwise The L-R fzzy nmber s described bove nd it is denoted by (,b,, ) LR Here L nd R re clled strting s the left nd right reference fnctions, nd b re respectively clled strting nd end points of flt intervl, is clled the left spred nd is clled the right spred The generl shpe of n L-R fzzy nmber will s follows: Let ( 1,b,α,β) nd B(,b,γ, δ) 1 LR LR be two L-R fzzy nmbers then it cn be verified tht 39
B (, b b, α γ, β δ) 1 1 LR B ( b, b, α δ,β γ) 1 1 LR Frther, similr to TFN nd TrFN, -1, B, / B re not L-R fzzy nmbers in generl nd it needs certin L-R pproximtions if they re to be sed s pproximte L-R fzzy nmbers 6 Rnking of fzzy nmbers Rnking of fzzy nmber plys importnt role in fzzy mthemticl progrmming nd fzzy gmes Let N(R) be the set of ll fzzy nmbers in R nd, B in N(R) Let F: N(R) R, clled rnking fnction or rnking index is defined nd F () F (B) is treted s eqivlent to B Since F () = F (B), in generl, it does not men =B This rnking is to be nderstood in the sense of eqivlence clsses only Yger [59] proposed the following indices h () 1 xμ μ (x)dx (x)dx where nd re the lower nd pper limits of the spport of The vle h1 () represents the centroid of the fzzy nmber in N(R) If in prticlr =,, is TFN, then nd re the lower nd pper limits of the spport of Here is the modl vle verified tht In this cse by ctl sbstittion of the membership fnction of the TFN, it cn be 40
h1() 3 Therefore for given two TFN s =,,, B =, b, b b B with respect to h1 if nd only if,, b, b, b mx h () = m, d, where α mx is the height of 1, α = 0 α ct, where α in (0, 1] nd m ct For TFN =,, α mx =1 nd α =, L R m, L R, L R is n is the men vle of elements of tht n α- = ), ( ) ( α, α = ( ) h () 4 In this view of the bove, we conclde tht given two TFN =,, nd B = b b, b, B with respect to h if nd only if,, b b b Chpter 5 The pplictions of the bove fzzy nmbers shll be discssed in Chpter 3 nd 41