10 D. Chakraborty, D. Guha / IJIM Vol. 2, No. 1 (2010) 9-20

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1 Aville online t Int. J. Industril Mthemtics Vol., No. (00) 9-0 Addition of Two Generlized Fuzzy Numers D. Chkrorty, D. Guh Deprtment of Mthemtics, IIT-Khrgpur Khrgpur-730, Indi Received 5 Mrch 00; revised 3 July 00; ccepted 6 July Astrct The ojective of this pper is to develop rithmetic opertions etween generlized trpezoidl (tringulr) fuzzy numers so tht the drwcks of the existing works re overcome. In this respect, the extension principle hs een used to clculte dierent rithmetic opertions. Keywords : Generlized fuzzy numer, Extension principle, Function principle { Introduction Present-dy science nd technology re fetured with complex process nd phenomen for which complete informtion is not lwys ville. For such situtions, mthemticl models hve to e set up using the ville dt which is only pproximtely known. To mke this possile Zdeh [5] introduced fuzzy set theory. In recent yers, this theory hs emerged s n interesting rnch of pure nd pplied sciences [, 8, 0]. In 985 nd 999, Chen [, 3] further developed the theory nd possile pplictions of generlized fuzzy numers. In the pper [, 3] dierent rithmetic opertions on generlized fuzzy numers were formulted y proposing the function principle. In 996, Chen et l. [] mentioned tht the function principle ws proposed in order to overcome the complictions rising due to the use of extension principle [4, 6]. It ws lso mentioned in [8] tht \the method known s the function principle is more useful thn the extension principle for the fuzzy numers with trpezoidl memership functions". There re few litertures [4{ 3, 5{7, 9, {4] involving generlized fuzzy numers theory nd pplictions sed on Chen's rithmetic opertions. Though the function principle ws used to develop rithmetic opertions on generlized fuzzy numers, in prctice it hs een relized tht there re some shortcomings of Chen's Corresponding uthor. Emil ddress: drdejnic@yhoo.co.in 9

2 0 D. Chkrorty, D. Guh / IJIM Vol., No. (00) 9-0 method. From mthemticl point of view it cn e sid tht, computing dierent rithmetic opertions using function principle is siclly point wise opertion (ddition, sutrction, multipliction nd division). Due to this reson, it hs een oserved tht rithmetic opertions of generlized trpezoidl (tringulr) fuzzy numers with function principle cuse the loss of informtion nd do not give exct results. This motivtes us to correct the results of rithmetic opertions of generlized trpezoidl (tringulr) fuzzy numers. The structure of this pper hs een orgnized s follows. In section, the sic rithmetic opertions etween generlized trpezoidl fuzzy numers hve een reviewed. This section lso give the exmples to show tht the existing opertions mke some pproximtion, therey cuses loss of informtion. In section 3 the extension principle hs een used to correct the result. Numericl exmples hve een given in section 4. The pper hs een concluded in Section 5. The fuzzy rithmetic opertions with function principle In this section the rithmetic opertions etween generlized trpezoidl fuzzy numers re reviewed from [, 3, 5]. Let us consider ~ A = ( ; ; c ; d ; w ) nd ~ A = ( ; ; c ; d ; w ) e two generlized trpezoidl fuzzy numers. (i) The ddition of ~ A nd ~ A : ~ A (+) ~ A = ( + ; + ; c + c ; d + d ; min(w ; w )). (ii) The sutrction of ~ A nd ~ A : ~ A ( ) ~ A = ( d ; c ; c ; d ; min(w ; w )). (iii) The multipliction of ~ A nd ~ A : ~ A () ~ A = ( ; ; c c ; d d ; min(w ; w )), if ; ; c ; d ; ; ; c ; d re ll positive rel numers. (iv) The division, ~ A is divided y ~ A : ~ A (=) ~ A = ( =d ; =c ; c = ; d = ; min(w ; w )), if ; ; c ; d ; ; ; c ; d re ll nonzero positive rel numers. It hs een found tht most of the work [4{3, 5{7, 9, {4] done using generlized fuzzy numers is sed on Chen's rithmetic opertions. Therefore, the importnce of these opertions nd the high numer of cittions of Chen's [, 3] work relted to this, wrrnts thorough study of their work. In course of this study some shortcomings of Chen's method hve een oserved which hve een illustrted with the help of following exmple: 0

3 D. Chkrorty, D. Guh / IJIM Vol., No. (00) 9-0. Exmple Let us consider two generlized tringulr fuzzy numers of the following form: ~ A = (0:7; 0:8; 0:9; 0:5) nd ~ A = (0:8; 0:9; :0; :0). After performing Chen's ddition opertion (dened ove) etween ~ A nd ~ A, the result hs een otined s ~ A (+) ~ A = ~ B = (:5; :7; :9; 0:5), which is generlized tringulr fuzzy numer. In our opinion this result does not give the exct vlue. The result fter performing Chen's ddition opertion hs een illustrted with the help of Figure. given elow. 0.5 A A B Fig.. Sum of two generlized tringulr fuzzy numers It my e oserved from the ove gure tht, min (; 0:5) = 0:5. If the oth fuzzy numers re tken to the sme level y "truncting the higher one" i.e. if we tke 0:5(since 0:5 < :0) cut of A ~ then A ~ is trnsformed into generlized trpezoidl (t) fuzzy numer. Therefore, it is necessry to conserve this tness into the resultnt generlized fuzzy numer. In this respect Chen's pproch is incomplete nd hence loses its signicnce. In view of this, there is need to develop rithmetic opertions for generlized fuzzy numers, which clcultes the result more ppropritely. Therefore, in the next sections the im is to develop the rithmetic opertions etween generlized fuzzy numers sed on the extension principle to improve the method. 3 Opertions on generlized fuzzy numers Let A ~ = ( ; ; c ; d ; w ) nd A ~ = ( ; ; c ; d ; w ) e two generlized t fuzzy numers with memership functions nd ~A ~ A respectively, which cn e written in the following form: 8 0 if < x >< w f (x) if x (x) = ~A w if x c w h (x) if c x d >: 0 if d < x

4 D. Chkrorty, D. Guh / IJIM Vol., No. (00) if < y >< w f (y) if y (y) = ~A w if y c w h (y) if c y d >: 0 if d < y Here, f (x) is strictly incresing from the intervl [ ; ] to the intervl [0; ] nd h (x) is strictly decresing from the intervl [c ; d ] to the intervl [0; ]. Similrly f (y) is strictly incresing from [ ; ]to the intervl [0; ] nd h (y) is strictly decresing from [c ; d ] to the intervl [0; ]. The memership function ~C of their composition A ~ () A ~, (where () is n extended inry opertion to comine two generlized fuzzy numers A ~ nd A ~ ) is dened using Zdeh's extension principle, [4, 6] s follows: ~C (z) = sup minf ~A (x); ~ (y)g A z=xy If A ~ () A ~ is denoted y C ~ from the ove denition it cn e sid tht C ~ will e generlized fuzzy numer with condence level w c = min(w ; w ): Dierent rithmetic opertions etween A ~ nd A ~ hve een estlished with the following theorems. 3. Sum of two generlized fuzzy numers Let ~ A nd ~ A e two generlized t fuzzy numers, s dened ove. If ~ C = ~ A (+) ~ A, then the following reltion cn e written Theorem 3.. Let ~C ~C (z) = sup minf ~A (x); ~ (y)g A z=x+y e the memership function of ~ A (+) ~ A ; then 8 >< 0 8z ( ; + ] [ [d + d ; ) ~C (z) = [0; w c ] 8z [ + ; f (w c ) + f (w c )] [ [h (w c ) + h (w c ); d + d ] >: w c 8z [f (w c ) + f (w c ); h (w c ) + h (w c )] where w c = min(w ; w ): Proof: For o ~C = A ~ (+) A ~ = n(z; ~C (z)) : z = x + y nd A ~ = (x; ~A (x)); A ~ = (y; (y)) ~A depending on the positions of x nd y the following three cses my rise: Cse : Let us consider x ( ; ] nd y ( ; ] ) x + y = z ( ; + ]. As ~A (x) = 0 8x ( ; ] nd ~A (y) = 0 8y ( ; ]; we need to prove ~C (z) = 0 8 z ( ; + ]:

5 D. Chkrorty, D. Guh / IJIM Vol., No. (00) To evlute ~C (z) we must consider every pir (p; r) such tht, z = p + r. Now for every pir (p; r) (for p + r = z) the following two possiilities hve een considered: (i) For p < x nd r > y, minf ~A (p); ~ (r)g = 0 s A ~ A (p) = 0 8p < x, (ii) For p > x nd r < y, minf ~A (p); ~ (r)g = 0 s A ~ A (r) = 0 8 r < y: From oth (i) nd (ii), sup minf ~A (p); ~ (r)g = 0 ) A C ~ (z) = 0: z=p+r Since this result holds for ny ritrry z, therefore, ~C (z) = 08z ( ; + ]. In similr mnner it cn e proved tht ~C (z) = 0 8 z [d + d ; ). Thus, ~C (z) = 0 8 z ( ; + ] [ [d + d ; ). Cse : Suppose min(w ; w ) = w For x [ ; f (w )] nd y [ ; ], i.e. x [ ; f (w )] nd y [ ; f (w )], ) x + y = z [ + ; f (w ) + f (w )]. We need to prove ~C (z) [0; w ] 8z [ + ; f (w ) + f (w )]. Now for z [ + ; f (w ) + f (w )), for every pir (p; r)(for p + r = z)the following two possiilities hve een considered: (i) For p < x nd r > y, () For p > nd r < f (w ) =, we hve 0 < minf ~A (p); ~ (r)g < w A. () For p > nd r > = f (w ), we hve minf ~A (p); ~ (r)g < w A. (c) For p < nd r >, we cn otin minf ~A (p); ~ A (r)g = 0. (d) For p < nd r <, we hve minf ~A (p); ~ A (r)g = 0. (ii) For p > x nd r < y, () For p < f (w ) nd r >, we hve minf ~A (p); ~ A (r)g = 0. () For p < f (w ) nd r <, we otin minf ~A (p); ~ A (r)g = 0. (c) For p > f (w ) nd r >, we hve minf ~A (p); ~ (r)g < w A. (d) For p > f (w ) nd r <, we hve minf ~A (p); ~ A (r)g = 0. Consequently for oth (i) nd (ii), 8z [ + ; f (w ) + f minf ~A (p); ~ (r)g < w A. sup z=p+r (w )) we hve, 0 And for z = f (w ) + f (w ), i.e. when x = f (w ) nd y = f (w ), the following holds: minf ~A (x); ~ (y)g = w A. As efore, if we consider the following possiilities: (iii) For p < x nd r > y, we get minf ~A (p); ~ A (r)g < w. (iv) For p > x nd r < y, we get minf ~A (p); ~ A (r)g < w. 3

6 4 D. Chkrorty, D. Guh / IJIM Vol., No. (00) 9-0 (v) For p = x nd r = y only, we get minf ~A (p); ~ A (r)g = w. Consequently for (iii), (iv) nd (v), we hve, sup minf ~A (p); ~ (r)g = w A. z=p+r Therefore, from the ove ve possiilities, we cn sy tht for ny ritrry z [ + ; f (w ) + f (w )], 0 sup z=p+r minf ~A (p); ~ A (r)g w ) 0 ~C (z) w Since this result holds for ny ritrry z [ + ; f (w ) + f (w )], so we cn sy tht this is true for ll z [ + ; f (w ) + f (w )]. Therefore, ~C (z) [0; w ] 8z [ + ; f (w ) + f (w )]. In similr mnner we cn prove tht ~C (z) [0; w ] 8 z [h (w ) + h (w ); d + d ]. Similr result cn e otined if we consider min(w ; w ) = w : Hence, in generl, it cn e written tht if min(w ; w ) = w c then ~C (z) [0; w c ] for ll vlues of z elongs to [ + ; f (w c ) + f (w c )] [ [h (w c ) + h (w c ); d + d ]. Cse 3: Let us consider x [f (w c ) + f (w c ); h z [f (w c ); h (w c ) + h (w c )] nd y [f (w c ); h (w c )] ) x + y = (w c )] Now (x) = ~A w c 8x [f (w c ); h (w c )] nd (y) = ~A w c 8y [f Therefore, it is ovious tht 8z [f Thus, the proof hs een completed. (w c ) + f (w c ); h (w c ); h (w c )]. (w c ) + h (w c )], ~C (z) = w c. Therefore, for ~ C = ~ A (+) ~ A, the memership function ~C (z) cn e written s follows: 8 0 if < z + >< w c f c (z) if + z f (w c ) + f ~C (z) = w c if f (w c ) + f (w c ) z h (w c ) + h w c h c (z) if h (w c ) + h (w c ) z d + d >: 0 if d + d < z (w c ) (w c ) Where f c (z) = sup minff (x); f (y)g nd h c (z) = sup minfh (x); h (y)g. z=x+y z=x+y 3.. Prticulr Cse: For generlized tringulr fuzzy numers Let us consider, two generlized tringulr fuzzy numers ~ A nd ~ A denoted s ~ A = ( ; ; c ; w ) nd ~ A = ( ; ; c ; w ). Theorem 3.. Addition of two tringulr fuzzy numers with dierent condence levels genertes trpezoidl fuzzy numer s follows: 4

7 D. Chkrorty, D. Guh / IJIM Vol., No. (00) ~A 3 = ~ A (+) ~ A = ( 3 ; 3 ; c 3 ; d 3 ; w 3 )where, 3 = + 3 = + + ( )w 3 =w + ( )w 3 =w c 3 = c + c (c )w 3 =w (c )w 3 =w d 3 = c + c nd w 3 = min(w ; w ); w 6= w. Proof: Generlized tringulr fuzzy numers ~ A nd ~ A hve the memership functions of the following form: 8 0 if < x >< (x) = w (x )=( ) if x ~A w (c x)=(c ) if x c >: 0 if c < x (3.7) 8 0 if < x >< (y) = w (y )=( ) if y ~A w (c y)=(c ) if y c >: 0 if c < y (3.8) It cn e sid tht for xed vlue of w [0; min(w ; w )], there exists (x; y) R such tht (x) = ~A ~ (y) = w = A ~ A (z) holds,where z = x + y. 3 For otining A3 ~, z needs to e found with respect to w. For this purpose the incresing prt of the memership functions of A ~ nd A ~ hs een considered. From (3.7), the incresing prt of ~A (x) gives the following reltion etween x nd w w (x )=( ) = w ) x = + ( ):w=w (3.9) And similrly from (3.8) we get the following w (y )=( ) = w ) y = + ( ):w=w (3.0) Therefore, to compute z, from (3.9) nd (3.0) we get w = z ( + ) ( )=w + ( )=w (3.) On the other hnd the incresing prt of the memership function of ~ A3 cn e written in the following form: 5

8 6 D. Chkrorty, D. Guh / IJIM Vol., No. (00) 9-0 ~A 3 (z) = w = w 3(z 3 )=( 3 3 ) (3.) Now eqution (3.) cn e expressed in the following form: w=w 3 = z ( + ) w 3 (( )=w + ( )=w ) (3.3) Compring (3.) nd (3.3), the following cn e written 3 = + nd 3 3 = ( )w 3 =w + ( )w 3 =w Therefore, 3 = + + ( )w 3 =w + ( )w 3 =w. In the sme wy considering the decresing prt of the memership functions of ~ A nd ~A the following cn e proved c 3 = c + c (c )w 3 =w (c )w 3 =w ; d 3 = c + c Thus, nlly the following cn e otined ~A 3 = ~ A (+) ~ A = ( 3 ; 3 ; c 3 ; d 3 ; w 3 )where, 3 = + 3 = + + ( )w 3 =w + ( )w 3 =w c 3 = c + c (c )w 3 =w (c )w 3 =w d 3 = c + c Thus, the required vlue of 3 ; 3 ; c 3 nd d 3 cn e otined. Hence, the proof hs een completed. 3.. More on the exmple of susection. As shown in the exmple of section., fter performing the ddition opertion etween ~A nd A ~, we otined A ~ (+) A ~ = B ~ = (:5; :7; :9; 0:5). But fter the discussion stted ove in Theorem (3.), it cn e sid tht the ddition of A ~ nd A ~ genertes generlized trpezoidl fuzzy numer nd tht should e A3 ~ = (:5; :65; :75; :9; 0:5) which gives etter result. The dierence etween the two opertions hs een shown in Figure 3.. From the gure it is now cler tht Chen's ddition cuses the loss of informtion. 6

9 D. Chkrorty, D. Guh / IJIM Vol., No. (00) A Conservtion of fltness 0.5 A B A Fig.. Comprison etween the two dditions Theorem 3.3. Addition of two generlized trpezoidl fuzzy numers ~ A = ( ; ; c ; d ; w ) nd ~ A = ( ; ; c ; d ; w ), with dierent condence levels genertes trpezoidl fuzzy numer s follows: ~A 4 = ~ A (+) ~ A = ( 4 ; 4 ; c 4 ; d 4 ; w 4 )where, 4 = + 4 = + + ( )w 4 =w + ( )w 4 =w c 4 = d + d (d c )w 4 =w (d c )w 4 =w d 4 = d + d nd w 4 = min(w ; w ); w 6= w. Proof: The proof is similr to Theorem Sutrction of two generlized tringulr fuzzy numers Let us consider two generlized tringulr fuzzy numers ~ A = ( ; ; c ; w ) nd ~ A = ( ; ; c ; w ). In order to compute the sutrction opertion etween ~ A nd ~ A, the vlue of ~ A ( ) ~ A cn e dened s ~ A ( ) ~ A = ~ A (+)( ~ A ). Now the ddition opertion on ~ A nd ( ) ~ A s discussed in the section 3.., cn e esily performed. Hence, the following cn e written ~A 5 = ~ A ( ) ~ A = ( 5 ; 5 ; c 5 ; d 5 ; w 5 )where, 5 = 5 = c + ( )w 5 =w + (c )w 5 =w c 5 = c (c )w 5 =w ( )w 5 =w d 5 = c nd w 5 = min(w ; w ); w 6= w 4 Numericl illustrtion Exmple 4.. Let ~ A = (; ; 4; 5; 0:5) nd ~ A = (5; 6; 8; 9; ) e two generlized trpezoidl fuzzy numers. Find ~ A (+) ~ A. Result: With the help of Theorem 3.3, the following result hs een otined ~ A (+) ~ A = (6; 6:5; :5; 4; 0:5). 7

10 8 D. Chkrorty, D. Guh / IJIM Vol., No. (00) 9-0 Exmple 4.. Let ~ A = (0:4; 0:6; 0:7; 0:8) nd ~ A = ( 0:; 0:; 0:4; ) e two generlized tringulr fuzzy numers. Find ~ A (+) ~ A. Result: With the help of Theorem 3., the following result hs een otined ~ A (+) ~ A = (0:3; 0:74; 0:84; :; 0:8). Exmple 4.3. Let ~ A = (0:5; 0:6; 0:7; 0:5) nd ~ A = (0:3; 0:4; 0:5; ) e two generlized tringulr fuzzy numers. Find ~ A ( ) ~ A. Result: Utilizing the process s stted in section 3., the result hs een otined s: ~A ( ) A ~ = (0; 0:5; 0:5; 0:4; 0:5). 5 Conclusion In this pper the shortcomings of the existing opertions on generlized fuzzy numers hve een discussed. In order to overcome this prolem the extension principle hs een used to derive dierent rithmetic opertions (ddition, sutrction). In future, the result of production nd division of generlized fuzzy numers my lso e nlyzed. Acknowledgement The uthors re grteful to the Editor in Chief nd nonymous referees for their vlule comments nd suggestions. The rst uthor grtefully cknowledges the nncil support provided y the Deprtment of Science nd Technology, New Delhi (ref: SR/S4/M:497/07).The second uthor would lso like to cknowledge the nncil support provided y the Council of Scientic nd Industril Reserch, Indi (Awrd No. 9/8(7)/08-EMR-I). References [] M. Brkhordri Ahmdi, N. A. Kini, Solving two-dimensionl fuzzy prtil dierentil eqution y the lternting direction implicit method, Interntionl Journl of Industril Mthemtics, (009) 05-0 [] S.H. Chen, Opertions on fuzzy numers with function principl, Tmkng Journl of Mngement Sciences. 6 (985) 3-5. [3] S.H. Chen, Rnking generlized fuzzy numer with grded men integrtion, in Proceedings of the 8 th interntionl fuzzy systems ssocition world congress. vol., Tipei, Tiwn, Repulic of Chin, (999) [4] S. J. Chen, S. M. Chen, Fuzzy risk nlysis sed on similrity mesures of generlized fuzzy numers, IEEE Trnsctions on Fuzzy Systems. (003) [5] J.H. Chen, S.M. Chen, A new method for rnking generlized fuzzy numers for hndling fuzzy risk nlysis prolems, in Proceeding of the 9th joint conference on informtion sciences, Kohsiung, Tiwn, Repulic of Chin (006) [6] S.J. Chen, S.M. Chen, Fuzzy risk nlysis sed on the rnking of generlized trpezoidl fuzzy numers, Applied Intelligence. 6 (007) -. 8

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12 0 D. Chkrorty, D. Guh / IJIM Vol., No. (00) 9-0 [] J.N. Sheen, Generlized fuzzy numers comprison y geometric moments, in proceedings of the 5th WSEAS interntionl conference on instrumenttion, mesurement, circuits nd systems, Hngzhou, Chin, April 6-8, (006) [] S. H. Wei nd S. M. Chen, A new pproch for fuzzy risk nlysis sed on similrity mesures of generlized fuzzy numers, Expert Systems with Applictions. 36 (009) [3] Z. Xu, S. Shng, W. Qin, nd W. Shu, A method for fuzzy risk nlysis sed on the new similrity of trpezoidl fuzzy numers, Expert Systems with Applictions. 37 (00) [4] D. Yong, S. Wenkng, D. Feng, L. Qi, A new similrity mesure of generlized fuzzy numers nd its ppliction to pttern recognition, Pttern Recognition Letters. 5 (004) [5] L.A. Zdeh, Fuzzy sets, Informtion nd Control. 8 (965) [6] H.J. Zimmermnn, Fuzzy Set Theory, Kluwer Acdemic, Boston, 99. 0