FUZZY ARITHMETIC WITHOUT USING THE METHOD OF - CUTS

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1 International Journal o Latest Trens in Computin (E-ISSN: ) 73 Volume, Issue, December 00 FUZZ ARITHMETIC WITHOUT USING THE METHOD OF - CUTS Supahi Mahanta, Rituparna Chutia, Hemanta K Baruah 3 (Corresponin author) Research Scholar, Department o Statistics, Gauhati University, supahi_mahanta@reimail.com Research Scholar, Department o Mathematics, Gauhati University, Rituparnachutia7@reimail.com 3 Proessor o Statistics, Gauhati University, Guwahati, hemanta_bh@yahoo.com, hemanta@auhati.ac.in Abstract: In this article, an alternative metho to evaluate the arithmetic operations on uzzy number has been evelope, on the assumption that the Dubois- Prae let an riht reerence unctions o a uzzy number are istribution unction an complementary istribution unction respectively. Usin the metho, the arithmetic operations o uzzy numbers can be one in a very simple way. This alternative metho has been emonstrate with the help o numerical eamples. Key wors an phrases: Fuzzy membership unction, Dubois-Prae reerence unction, istribution unction, Set superimposition, Glivenko-Cantelli theorem.. INTRODUCTION The stanar metho o -cuts to the membership o uzzy number oes not always yiel results. For eample, the metho o -cuts ails to in the uzzy membership unction (m) o even the simple unction when is uzzy. Inee, or in particular, Chou (009) has orware a metho o inin the m or a trianular uzzy number. We shall in this article, put orwar an alternative metho or ealin with the arithmetic o uzzy numbers which are not necessarily trianular. Dubois an Prae (see e.. Kaumann an Gupta (984)) have eine a uzzy number a, c with membership unction L,, L a b R b c (.) 0, otherwise bein continuous non-ecreasin unction in the interval [a, b], an R bein a continuous nonincreasin unction in the interval [ c], with L a Rc 0 an L b Rb Prae name. Dubois an L as let reerence unction an R as riht reerence unction o the concerne uzzy number. A continuous non-ecreasin unction o this type is also calle a istribution unction with reerence to a Lebesue-Stieltjes measure (e Barra (987). pp- 56). In this article, we are oin to emonstrate the easiness o applyin our metho in evaluatin the arithmetic o uzzy numbers i start rom the simple assumption that the Dubois-Prae let reerence unction is a istribution unction, an similarly the Dubois- Prae riht reerence unction is a complementary istribution unction. Accorinly, the unctions L an R woul have to be associate with ensities L an [c] respectively (Baruah (00 a, b)). R in [a,b] an. SUPERIMPOSITION OF SETS

2 International Journal o Latest Trens in Computin (E-ISSN: ) 74 Volume, Issue, December 00 The superimposition o sets eine by Baruah (999), an later use successully in reconizin perioic patterns (Mahanta et al. (008)), the operation o set superimposition is eine as ollows: i the set A is superimpose over the set B, we et A(S) B= (A-B) (A B) () (B-A) (.) where S represents the operation o superimposition, an (A B) () represents the elements o (A B) occurrin twice. It can be seen that or two intervals [a, b ] an [a, b ] superimpose ives [a, b ] (S) [a, b ] = [a (), a ()] [a (), b ()] () [b (), b ()] where a () = min (a, a ), a () = ma (a, a ), b () = min (b, b ), an b () = ma (b, b ). Inee, in the same way i [a, b ] (/) an [a, b ] (/) represent two uniormly uzzy intervals both with membership value equal to hal everywhere, superimposition o [a, b ] (/) an [a, b ] (/) woul ive rise to [a, b ] (/) (S) [a, b ] (/) = [a (), a ()] (/) [a (), b ()] () [b (), b ()] (/). (.) So or n uzzy intervals [a, b ] (/n), [a, b ] (/n) [a n, b n ] (/n) all with membership value equal to /n everywhere, [a, b ] (/n) (S) [a, b ] (/n) (S) (S) [a n, b n ] (/n) = [a (), a ()] (/n) [a (), a (3)] (/n) [a (n-), a (n)] ((n-) /n) [a (n), b ()] () [b (), b ()] ((n-) /n) [b (n- ), b (n-)] (/n) [b (n-), b (n)] (/n), (.3) where, or eample, [b (), b () ] ((n-) /n) represents the uniormly uzzy interval [b (), b () ] with membership ((n-) /n) in the entire interval, a (), a (),, a (n) bein values o a, a,, a n arrane in increasin orer o manitue, an b (), b (),, b (n) bein values o b, b,, b n arrane in increasin orer o manitue. We now eine a ranom vector = (,,, n ) as a amily o k, k =,,, n, with every k inucin a sub-σ iel so that is measurable. Let (, n ) be a particular realization o, an let (k) realize the value (k) where (), (),, (n) are orere values o,,, n in increasin orer o manitue. Further let the sub-σ iels inuce by k be inepenent an ientical. Deine now Ф n () = 0, i < (), = (r-)/n, i (r-) (r), r=, 3,, n, =, i (n) ; (.4) Ф n () here is an empirical istribution unction o a theoretical istribution unction Ф(). As there is a one to one corresponence between a Lebesue-Stieltjes measure an the istribution unction, we woul have Π (a, b) = Ф (b) Ф (a) (.5) whereπ is a measure in (Ω, A, Π), A bein the σ- iel common to every k. Now the Glivenko-Cantelli theorem (see e.. Loeve (977), pp-0) states that Ф n () converes to Ф() uniormly in. This means, sup Ф n () - Ф() 0 (.6)

3 International Journal o Latest Trens in Computin (E-ISSN: ) 75 Volume, Issue, December 00 Observe that (r-)/n in (.4), or (r-) (r), are membership values o [a (r -), a (r)] ((r -) /n) an [b (n r + ), b (n - r)] ((r -) /n) in (.3), or r =, 3,, n. Inee this act oun rom superimposition o uniormly uzzy sets has le us to look into the possibility that there coul possibly be a link between istribution unctions an uzzy membership. In the sections 3, 4, 5 an 6 we are oin to iscuss the arithmetic o uzzy numbers. 3. ADDITION OF FUZZ NUMBERS Consier a, c an p, r two trianular uzzy numbers. Suppose Z a p b c r be, be the uzzy number o. Let the m o an be an y as mentione below an,, L a b R b c (3.) 0, otherwise y, y, L a y b y R b y c (3.) 0, otherwise where L an y an Ran Ry respectively. We assume that L an y istribution unction an Ran y L are the let reerence unctions are the riht reerence unctions L are R are complementary istribution unction. Accorinly, there woul eist some ensity unctions or the istribution unctions L an R,. Say, L a b an R, b c Rwith y y We start with equatin L with L y R. An so, we obtain y, an an respectively. Let z y, so we have an, so that z z an z z, say. Replacin by z an by Now let, an in z in, we obtain z z say. The istribution unction or by a p z z z m z z z, z m z, an, woul now be iven z m z z a p b q an the complementary istribution unction woul be iven by z m z z, b q c r bq We claim that this istribution unction an the complementary istribution unction constitute the uzzy membership unction o as, zm zz, a p b q a p zm zz, b q c r bq 0 4. SUBTRACTION OF FUZZ NUMBERS

4 International Journal o Latest Trens in Computin (E-ISSN: ) 76 Volume, Issue, December 00 Let a, c an p, r be two uzzy numbers with uzzy membership unction as in (3.) an (3.). Suppose Z. Then the uzzy membership unction o by Z. number o Z woul be iven Suppose r p reerence unctions, be the uzzy. We assume that the Dubois-Prae L y an y R as istribution an complementary istribution unction respectively. Accorinly, there woul eist some ensity unctions or the istribution unctions Say, y, y y Ly p y q L y an Ry an y Ry, q y r Let y t. y t y so that mt, say. Replacin t in yan y y t an y t y woul be iven by y r y q 0 t mt t, t mt, we obtain t,, say. Then the m o r y q q y p Then we can easily in the m o by aition o uzzy numbers an as escribe in the earlier section. 5. MULTIPLICATION OF FUZZ NUMBERS Let a, c, a, c 0 an p, r, p, r 0 be two trianular uzzy numbers with uzzy membership unction as in (3.) an (3.). Suppose Z. a. p, b. c. r be the uzzy number o.. Lan L y are the let reerence unctions an Ran R y are the riht reerence unctions respectively. We assume that L an L y are istribution unction an Ran R y are complementary istribution unction. Accorinly, there woul eist some ensity unctions or the istribution unctions L an R,. Say, L a b an R, b c an an We aain start with equatin L with L y, R with R y. An so, we obtain y y respectively. Let z. y, so we have. an., so that z z an z z, say. Replacin by z in z in, we obtain z z say. an by Now let, an The istribution unction or by z z z m z z z z m z, an., woul now be iven

5 International Journal o Latest Trens in Computin (E-ISSN: ) 77 Volume, Issue, December 00 z z m z, ap bq ap an the complementary istribution unction woul be iven by z z m z, bq cr bq We are claimin that this istribution unction an the complementary istribution unction constitute the uzzy membership unction o zm ap bq 0 z z m z,. as, z ap bq z, bq cr 6. DIVISION OF FUZZ NUMBERS Let a, c, a, c 0 an p, r, p, r 0 be two trianular uzzy numbers with uzzy membership unction as in (3.) an (3.). Suppose Z membership unction o by Z... Then the uzzy Z woul be iven At irst, we have to in the m o. Suppose,, r q p o unctions be the uzzy number. We assume that the Dubois-Prae reerence L y an y R as istribution an complementary istribution unction respectively. Accorinly, there woul eist some ensity unctions or the istribution unctions y, y y Lyan Ry Ly p y q an y Ry, q y r Replacin Let.Say, y t y so that mt t t y t in y y y t an y t woul be iven by y y r y q 0 t mt t mt an t, t,, say., we obtain, say. Then the m o q r y q y p Net, we can easily in the m o multiplication o uzzy numbers an as escribe in the earlier section. by In the net section we are oin to cite some numerical eamples or the above iscusse methos. 7. NUMERICAL EAMPLES Eample : Let,,4 an 3,5,6 trianular uzzy numbers with m be two, 4, 4 (7.) 0

6 International Journal o Latest Trens in Computin (E-ISSN: ) 78 Volume, Issue, December 00 An y y 3, 3 y 5 6 y, 5 y 6 0 (7.) Here 4,7,0. Equatin the istribution unction an complementary istribution unction, we obtain y 8 an y. Let z y, so we shall have z 3 an so that z z, z an z respectively. Replacin by ensity unctions z 8, 3 z an z an in the respectively, we have z an z z Now m z z an z z Then the m o Eample :. 3 m. z 3 woul be iven by, 4, , Let,,4 an 3,5,6 be two trianular uzzy numbers with membership unctions as in (7.) an (7.). Suppose, Z or Z ( ). Now, 6, 5, 3 be the uzzy number o ( ). Let t y so that y t, which implies t m. Then the ensity unction yan y woul be, say, y 3 y t, 3 y 5 an y y 6 y t, 5 y 6. y Then the m o woul be iven by y 6 y, 6 y y, 5 y Then by aition o uzzy numbers,,4 6, 5, 3 the m o Eample 3: an is iven by, 5, 5 3, Let,,4 an 3,5,6 be two trianular uzzy numbers with membership unctions as in (7.) an (7.). Suppose,. 3,0,4 uzzy number o be the.. Equatin the istribution unction an complementary istribution unction, we obtain y an 8 y.

7 International Journal o Latest Trens in Computin (E-ISSN: ) 79 Volume, Issue, December 00 Let z. y, so we shall have z. an 8 z., so that z an z 4 6 z 8z 4. Replacin by z an z in the ensity unctions an respectively, we have z. an z z Now z z an z z Then the m o. Eample 4: m. woul be iven by. z m 8 5, , Let,,4 an 3,5,6 be two trianular uzzy numbers with membership unctions as in (7.) an (7.). Suppose, Z or Then the m o 6,5,3 is iven as Z.. y 6 y, y y, y Then by multiplication o uzzy numbers,,4 an 6,5,3 the uzzy membership unction o woul be iven by, Eample 5: Let,4,5 an 4,6,5 6, , be a trianular uzzy number which is a non-trianular uzzy number with membership unctions respectively as,, 4 5, an y We can in the m o y, 4 y 6 5 y, 6 y 5 0 which is iven by,

8 International Journal o Latest Trens in Computin (E-ISSN: ) 80 Volume, Issue, December 00 4, , All our emonstrations above can be veriie to be true, usin the metho o -cuts. 9. CONCLUSION The stanar metho o -cuts to the membership o a uzzy number oes not always yiel results. We have emonstrate that an assumption that the Dubois-Prae let reerence unction is a istribution unction an that the riht reerence unction is a complementary istribution unction leas to a very simple way o ealin with uzzy arithmetic. Further, this alternative metho can be utilize in the cases where the metho o -cuts ails, e.. in inin the [3]. Baruah, Hemanta K.; (00 b), The Mathematics o Fuzziness: Myths an Realities, Lambert Acaemic Publishin, Saarbrucken, Germany. [4]. Chou, Chien-Chan; (009), The Square Roots o Trianular Fuzzy Number, ICIC Epress Letters. Vol. 3, Nos., pp [5]. e Barra, G.; (987), Measure Theory an Interation, Wiley Eastern Limite, New Delhi. [6]. Kaumann A., an M. M. Gupta; (984), Introuction to Fuzzy Arithmetic, Theory an Applications, Van Nostran Reinhol Co. Inc., Wokinham, Berkshire. [7]. Loeve M., (977), Probability Theory, Vol.I, Spriner Verla, New ork. [8]. Mahanta, Anjana K., Fokrul A. Mazarbhuiya an Hemanta K. Baruah; (008), Finin Calenar Base Perioic Patterns, Pattern Reconition Letters, 9 (9), m o. 0. ACKNOWLEDGEMENT This work was une by a BRNS Research Project, Department o Atomic Enery, Government o Inia. REFERENCES []. Baruah, Hemanta K.; (999), Set Superimposition an Its Applications to the Theory o Fuzzy Sets, Journal o the Assam Science Society, Vol. 40, Nos. &, 5-3. []. Baruah, Hemanta K.; (00 a), Construction o The Membership Function o a Fuzzy Number, ICIC Epress Letters (Accepte or Publication: to appear in February, 0).