Key words: Fuzzy linear programming, fuzzy triangular, fuzzy trapezoidal, fuzzy number, modified S-curve function, fuzzy simplex method.

Transcription

1 Irqi Journl of Sttistil Siene () The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis pp [-] Trnsformtion Liner Memership Funtion y Using the Modified S- Curve Shorish Omer dull Dr. dul Rhim K.Rhi strt: In this pper, we onentrte on two ses of fuzzy liner progrmming prolems (FLPP): LPP with oeffiient of oetive funtion nd oeffiient of ville resoures nd onstrint re trpezoidl nd tringulr of fuzzy numers respetively. Though y using - ut,fuzzy numers (fuzzy tringulr nd fuzzy trpezoidl ) n e trnsformed into intervl numers nd y tking one type of non-liner memership funtion this ment logisti funtion, we propose here the "modified S-urve funtion y simple method ". Key words: Fuzzy liner progrmming, fuzzy tringulr, fuzzy trpezoidl, fuzzy numer, modified S-urve funtion, fuzzy simple method. تحویل دالة الانتماء الخطیة باستخدام منحني S المعدلة م. شورش عمر عبدالله د.عبدالرحیم خلف راھی المستخلص في ھذا البحث نركز على إثنین من مشاكل البرمجة الخطیة الضبابیة :(FLPP) LPP بمعامل (دالة الھدف) ومعامل (موارد متاحة وقید) رباعي ومثلثیة من الا عداد الضبابیة على التوالي. مع ذلك با ستعمال - القطع (رباعیة ضبابیة ومثلثیة ضبابیة)بالاعداد ضبابیة تتحول ا ىل الفترة المغلقة الواحد من نوع دالة الانتماء غیر الخطیة یستخدم ھنا دالة اللوجستیكیة نستخدم في ھذ البحث "منحني."Simple المعدلة بالطریقة S Letuter\ Shool of Siene Edution \ University Grmin \rhistt@yhoo.om ssit Prof. \ College of dministrtion nd Eonomi \ University lmustinsriy \Mthshorishli@yhoo.om Reeived:/ / epted: / /

2 Trnsformtion Liner Memership Funtion Introdution: Fuzzy sets theory ws first introdued y "Prof. Lotfi Zdeh" in. Suessful pplitions of fuzzy sets theory on ontroller systems in 8 dedes led to the development of this theory in other fields suh s simultion, rtifiil intelligene, opertions reserh, mngement nd mny industril pplitions. In the rel world, mny pplied prolems re modeled s mthemtil progrmming nd it my e neessry to formulte these models with unertinty. Mny prolems of these kinds re liner progrmming with fuzzy prmeters. The first formultion of fuzzy liner progrmming is proposed y Zimmermnn (78). fter the pioneering works on fuzzy liner progrmming, severl kinds of Fuzzy Liner Progrmming Prolems (FLPP) hve ppered in the literture nd different methods hve een proposed to solve suh prolems. One onvenient pproh for solving the (FLPP) is sed on the onept of omprison of fuzzy numers y using the simple method. Usully in suh methods uthors define risp model whih is equivlent to the (FLPP) nd then use optiml solution of the model s the optiml solution of the (FLPP). We hve etended their results nd hve estlished the simple prolem of the liner progrmming with modified S-urve fuzzy vriles prolem nd hene dedue some simple results. These results will e useful for post optimlity nlysis. - Fuzzy Set: [],[3] Let X e universe set, whose generi elements re denoted,memership in suset of X is often viewed s hrteristi funtion m from X to {,} suh tht iff Î é m ( ) = ê êë iff Ï () If the set {, } is llowed to e the rel intervl [, ], is lled fuzzy set (Zdeh, []). m () is the grde of memership of in. The loser the vlue of m () is to, the more elong to. There is nother epression of fuzzy sets:- - is ompletely hrterized y the set of pirs = {(, m ()), ÎX}. - When X is finite set,..., }, fuzzy set on X is epressed s: { n n = m ( )/ m ( n )/ n = åm ( i )/ i.. () i= []

3 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis 3-When X is infinite, the following is written: = ò m ( )/ X.. (3) - Memership Funtion:[],[],[3] suset of universl set X is lled memership funtion if we n defined y the formlly: iff Î é m ( ) = ê ; We n sy tht the funtion m mps êë iff Ï the elements in the universl set X to the set {, }, : X {, }. 3- Support of Fuzzy Set:[3] Let e fuzzy set in X.Then the support of, denoted y S () is the risp set given y S( ) = { Î X : m ( ) > }.. () - Norml Fuzzy Set:[],[] The height of is defined s h() = sup m ( ) [3] m, if h() = then the fuzzy set is lled norml fuzzy set ; otherwise, it is lled su norml,if < h() <, then the su norml fuzzy set n e normlized, i.e.,it n e mde norml y redefining the memership funtion s m ( ) / h(), Î X. - Conve Set:[] ssuming universl set X is defined in n dimensionl Euliden Vetor speâ n. If ll the - ut sets re onve, the fuzzy set with these sets is onve, in other words, if the reltion: m ( ) ³ Min[ m ( ), m ( y)] n n Where ", y ÎÂ ndl + ( -l) yîâ ; l. Holds, the fuzzy set is onve. - ut - Fuzzy Numer:[],[],[] Let e fuzzy set in R, then is lled fuzzy numer, if stisfies the following onditions:- - is norml. - is onve. 3- m is upper semi ontinuous. - The support of is ounded ut:[],[7] Let e fuzzy set in X nd Î (, ], the - ut of the fuzzy set is

4 the risp set given y: { Î X m ( ) } = ³ 8- Liner Memership Funtion: Trnsformtion Liner Memership Funtion :. () Liner memership funtion represents the vlue of memership elements into fuzzy set the form of stright line, whih re importnt shpes of liner memership funtions: - Tringulr Funtion: [3],[] It is the funtion to user rumor; it hs three prmeters epressed s follows m : X [,] : T,, g nd n e ì mt (,,, g ) = í î - - g - g - if if if if < g > g. () Fig (-) Tringulr funtion - Trpezoidl funtion: [3],[],[] It is the funtion whih ontins four prmeters,, g, d nd n e epressed s follows: : X [,] m Tr []

5 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis m Tr ì if < - if (,,, g, d ) = í -. (7) if g d - if g d d -g î if > d Fig (-) Trpezoidl funtion - Non Liner Memership Funtion: There re mny types of non liner memership funtion in this tke one type is follows: - Logisti Funtions: [],[],[7] The logisti funtion for the non liner memership funtion nd (< < ) is fuzzy prmeter whih mesures the degree of vgueness. If indites risp, the fuzziness is very lrge if. This funtion n e epressed s: ì w m( ) = í + ue î < < < > (8) Where m () : is the degree of memer ship of,( < m () < ), : the upper nd the lower vlue of. : fuzzy prmeter tht determines the shpe of the memership funtion, ( < < ). u, w : onstnts. []

6 Trnsformtion Liner Memership Funtion Fig (Logisti funtion) There re mny types of logisti funtion in this tke one type is follows: i - Modified S Curve Funtion: This funtion is speil se of logisti funtion of the speified vlues (, w,u ) nd. m().. It n e defined: ì <. = w < < m ( ) = í - () [ ] - + u e. = î > Fig (Modified S- urve funtion) - Fuzzy Liner Progrmming Model: The formultion of FLP model hs the following form: i- ll oeffiients re fuzzy n M Z = s. t n å = å = i i, i =,..., m, ³ ; =,..., n Where Z : the vlue of the oetive funtion. : the vetor of deision vriles.,, : the fuzzy oeffiients of fuzzy LP model. i i ii- One oeffiient is fuzzy. () []

7 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis M s. t Z = n å = å = i n i, i =,..., m, iii- Doule oeffiients re fuzzy M s. t Z = n å = å = i n i, i =,..., m, ³ ³ ().. () - Fuzzy Simple pproh:[],[7] This method is used for solving fuzzy LPP for ny intervl s the oeffiient fuzzy model tkes vrious spets nd depends on lulting fuzzy memership funtion of oeffiient nd from lulting fuzzy oeffiient model on the sis of memership funtion for ll oeffiient from oeffiients model, it n e represented y the following steps: - The memership funtion is seleted ( liner or non- liner ) of fuzzy oeffiient progrmming models nd lulte the vlue of fuzzy oeffiient nd in tht method type from mny types of non liner logisti is used, modified S- urve memership funtion tht depends on two ftors, first is the epted level (degree ) Mu (. Mu. ), nd the other is the fuzzy ftor ( ),( ) nd the oeffiient of FLP is lulted model relted the oth Mu nd nd fuzzy ftor, where the memership funtion of the oeffiient of oetive funtion in FLPP s follows : - Memership m ì = í + î. w u e. - [ ] - < = = > < <. (3) Where m : memership of ;, : the upper nd the lower of fuzzy oetive funtion. nd to otin the vlue. [7]

8 e m = - [ - + ue ] = w - [ - [ u ] w m - ] - w [ ] = ln [ -] - u m Sine - w = + [ ]ln [ - ] u m is fuzzy - w = + [ ]ln [ - ] u m Trnsformtion Liner Memership Funtion. () Similrly the vlue - Fuzzy i Fig (Memership m of ) i fuzzy nd vlue i fuzzy n e otined. = i i - Fuzzy i i + [ - i w ]ln [ u m i -] i - i w = + [ ]ln [ - i i ] u m i [8].. ().... () The fuzzy LPP n e written y the oeffiient tht otined s ove. M Z = S.t = 3 - Consider the FLP = nd ³, Where is m n mtri nd is n m vetor.now, suppose tht rnk (, ) = rnk () =m. Prtition fter possily rerrnging the olumns of s [, N], m m is non zero squre mtri of rnk () = (m).the point T T T = (, ) where, = is lled si solution onsider the N FLP prolem s is defined: = - N

9 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis M S.t z = + N N + N.. (7) N =, N ³ Where : si mtri. N: non si mtri. : Component of si vriles. N : Component of non si vriles. + = N = - - N - N - - N So tht the oetive funtions s: z ( - + N - ) = - N - - = ( + z N ) + N (8) si fesile solution to the system, where = -, nd N =. Then, the orresponding fuzzy oetive vlues z - =, nd if ³ the si fesile solution of the system, z = nd then the ove FNLP prolem is rewritten in the following tleu formt: sis z N R.H.S z - N - N N The ove tleu gives us ll the required informtion to proeed with the simple method. The fuzzy ost row in the ove tleu is - = ( g - ) ¹, whih onsists of the g = z - for the non si i N N N vriles. ording to the optimlity ondition for these prolems one would e t the optiml solution if g ³, for ll ¹ i. On the other hnd, if g k <, for ll k ¹ i then r with k my e ehnged.then the vetor g k = - k is omputed. If g k,then k n e inresed indefinitely, nd then the optiml oetive is unounded. []

10 Trnsformtion Liner Memership Funtion Emple : [] M s. t Z = (,8,,), ³ + (,,, ) Where (, 8,, ) nd (,,, ) re fuzzy trpezoidl. Emple : [] M s. t Z = (,,) (,, ), ³ + + (, 3,) + (,.,) (,,8) (,,3) Where the oeffiient of ville resoures nd onstrint re fuzzy tringulr. - Formultion to Trnsform Fuzzy Intervl: [7] If M = ( m, g, ) is tringulr fuzzy numer then L R M = [ g ( - ) + m, m - ( -)] nd if M = ( m, m, g, ) is trpezoidl fuzzy L R numer then M = [ g ( -) + m, m - ( -)]. ssume tht the =.. The emples ove fter trnsformtion re s follows: Emple s the form M Z = [.,] + [.,.] s. t, ³ Emple s the form M Z = + s. t [3.,.] [3.,.8], ³ + [3.8,.] + [.8,.] [, 7.] [.,3.] 3- Solving prolem: y the ove formultion, we trnsformed the two emples into fuzzy intervl, y the following steps onstrut the mthemtil model of fuzzy liner progrmming prolem of emples y writing the progrm (MTL) []

11 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis to outome the deision vrile vlue nd the oetive funtion vlue, illustrted s follows: First Step: y Seleting the memership funtion of fuzzy oeffiient in the model through the ove emples, there in this study one type(modified S-urve memership funtion) from mny types of non liner logisti funtion e eplined in formul () sine this funtion hrterizes from fleiility in desriing unertin in fuzzy oeffiients of LPP through ontining fuzzy ftor, wheres seleting the memership funtion of (oeffiient of oetive funtion) y depending on the onstnt vlue of modified S-urve memership funtion of logisti funtion. u =. w = Seond Step: = 3.83 y using the progrm (MTL) with ( = 3.83), the vlue of fuzzy oeffiient ) is lulted y modified S-urve funtion. Third Step: ( The FLP model is written y formul () whih indites the fuzzy oeffiient vlue tht is otined y the ove step. Forth Step: The optiml oetive funtion vlue ( Z ) is lulted with respet to different epted degrees Mu, (. Mu.) y inresing (.) for fuzzy ftor of modified S- urve funtion ( = 3.83), nd the result otined is illustrted in tle () nd figure () : - Tle () (Optiml vlue with deision vrile of emple (oeffiient of oetive funtion is fuzzy)) []

12 Mu Z X Trnsformtion Liner Memership Funtion X Z Fig () ( Z with the epted degree Mu t(oeffiient of oetive funtion is fuzzy)) []

13 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis - The oetive funtion vlue Z degree (. dereses y inresing the epted Mu.83), nd the highest vlue of the oetive funtion ( Z =.87) t the higher level of epted degree (Mu =.) nd the lower vlue of the oetive funtion ( Z =.8) t the lower level of the epted degree (Mu=.). - The different volume etween one vlue nd nother of oetive funtion dereses t the epted degree (. Mu.) wheres ( DZ =.),(.-.) nd this inreses ( DZ =.7)t the epted degree (..8) nd this differene inreses t the epted degree ( Mu ³.8) wheres ( DZ =.373)t the epted degree (.-.). 3- With respet to the deision vrile vlues of these se, the oeffiient of oetive funtion suffered from fuzzy prolem, nd the deision vriles ( X, X ) hve the sme vlue t ll levels of the epted degree with inrese of the epted degree (. Mu.). - From figure () it n e onluded tht the optiml vlue ( Z ) dereses whenever the epted degree inreses. nd to study the effet of fuzzy ftor on optiml vlue Z, the optiml vlue Z is lulted of emple t different degree from fuzzy ftor ( ) y dding ( ) with different epted degree ontins the rnge (. Mu.) nd this ontins () volume y inresing (. ), the result otined is illustrted in tle () nd figure (),the following is onluded:- [3]

14 Trnsformtion Liner Memership Funtion See tle () Tle () with epted degree nd fuzzy ftor of different vlue t( oeffiient of oetive funtion is fuzzy )) ( Z fuzzy ftor of different vlue t( oeffiient of oetive funtion is fuzzy )) []

15 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis []

16 Trnsformtion Liner Memership Funtion Fig () Z with the epted degree nd fuzzy ftor of different vlue t(oeffiient ( of oetive funtion is fuzzy )) - The optiml vlue of oetive funtion ( Z =.87) t the lower level of the epted degree (Mu =.) nd ( Z =.837) t the higher level t the epted degree (Mu=.) t ll levels of fuzzy ftor studied ( ) y dding (). - The optiml vlue Z inreses y inresing fuzzy ftor ( ) nd this indites tht there is reltion etween Z, t ll degrees from the epted degree (. Mu.). 3- The optiml vlue Z t the epted degree (Mu=.) is very lose in ll levels of fuzzy ftor ( ), nd t the epted degree (Mu =.) the vlue of Z hs the sme vlue t ( 3 ) nd this dereses t ( ³ 3). - The optiml vlue Z dereses t fuzzy ftor ( = ) y inresing the epted degree (. Mu.) nd the different volume etween one vlue nd nother of Z t fuzzy ftor ( = ) dereses y inresing the epted degree (. [] Mu

17 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis.) wheres ( DZ =.377) t (..) nd this differene dereses t ( Mu ³.) wheres( DZ =.) t (..). The optiml vlue Z dereses t ( =) y inresing the epted degree(. Mu.), nd the different volume etween one vlue nd nother of oetive funtion t fuzzy ftor ( = ) dereses y inresing the epted degree (. Mu.) wheres ( DZ =.)t (..) nd this different inreses t (. Mu.) wheres( DZ = 7.7)t (..). nd with respet to fuzzy ftor ( ³ 3) whih hs the sme different of fuzzy ftor ( = 3). - From figure () it n e onluded tht there is reltion etween Z, nd Mu. In generl the optiml vlue Z inreses y inresing the epted degree.the deision mker n mke the higher vlue of Z ws e profit is (.87). - Coeffiient of ville Resoures nd Constrint re Fuzzy: In this se,the seletion nd lulting the memership funtion of fuzzy oeffiient in the model y the steps ( first nd seond )in ses (ove) would e,either the third step tht is written s the FLP model y depending on the formul () to outome with the set of deision vrile vlues nd oetive funtion vlues t the oeffiient of ville resoures nd onstrint of the model whih suffer from fuzzy prolem either oeffiient of oetive funtion is non fuzzy (norml ), wheres lulting the vlue Z nd deision vriles vlue to different epted degree (. Mu.)y inresing vlue (.) t fuzzy ftor =3.83, nd illustrted in tle (C) nd figure (C) the following n e onluded:- [7]

18 Trnsformtion Liner Memership Funtion Tle (C) (Oetive funtion vlue with emple t (oeffiient of ville resoures nd onstrint re fuzzy)) Mu Z X X Z [8]

19 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis Fig (C) ( Z with the epted degree Mu t ( Coeffiient of ville resoures nd onstrint re fuzzy )) - The vlue of oetive funtion ( Z ) inreses y inresing the epted degree, the smll vlue of oetive funtion ( Z = 3.7) t the lower level of the epted degree (Mu=.) nd this vlue inreses to reh t the higher vlue of oetive funtion ( Z =.8) t the higher level of the epted degree (Mu =.). - The different volume etween one vlue nd nother of oetive funtion dereses t the epted degree (. Mu.) wheres ( DZ =.38) t (.-.) nd this differene inreses t (Mu ³.) wheres ( DZ =.7) t (..). 3- With respet to this se,the oeffiient of onstrint nd ville resoures suffer from fuzzy prolem, oserve tht of ( X ) []

20 Trnsformtion Liner Memership Funtion dereses whenever inresing the epted degree (. Mu.) nd this inreses t the epted degree (. Mu.), with respet to the deision vrile ( X inresing the epted degree (. Mu.). ) inreses y - From figure (C) it n e onluded tht whenever the epted degree inreses the optiml vlue ( Z ) inreses. nd to study the effet of fuzzy ftor on optiml vlue of oetive funtion, the est vlue Z is lulted to emple t different degree from fuzzy ftor ( ) y dding () t different dgree the epted degree ontins rnge (. Mu.) nd ontins () volume y inresing (.) nd the result otined is illustrted in tle (D) nd figure (D) s follows:- See tle (D) Tle (D) with epted degree nd fuzzy ftor of different vlue t( oeffiient of ville resoure nd onstrint re fuzzy)) Z Mu ( []

21 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis []

22 Trnsformtion Liner Memership Funtion Fig (D) ( Z with the epted degree nd fuzzy ftor of different vlue t(oeffiient of ville resoure nd onstrint re fuzzy )) - The optiml vlue of oetive funtion ( Z = 3.7) t the lower level of the epted degree (Mu =.) nd ( Z =.) t the higher level of the epted degree (Mu =.), t ll levels of fuzzy ftor studied ( ) y dding (). - The optiml vlue Z dereses y inresing fuzzy ftor ( ) nd this indites tht there is refle reltion etween Z, t the epted degree (. Mu.). 3- The optiml vlue Z hs the sme vlue t fuzzy ftor ( ) t (Mu =.). The optiml vlue Z t (Mu =.) hve the sme vlue t fuzzy ftor ( 38 ) nd this vlue inreses t ( ³ ). - The vlue Z t fuzzy ftor ( = ) ( Z = 3.7) this vlue inreses till it rehes ( Z =.8),the different vlue etween one vlue nd nother dereses y inresing the epted degree(. Mu.)wheres ( DZ=.7)t the epted degree (.. ),this inreses it rehes ( DZ = []

23 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis.7) t the epted degree(..). If ( = ) the different vlue etween one vlue nd nother dereses y inresing t the epted degree (. Mu.) wheres ( DZ =.) nd this differene inreses y inresing t the epted degree (. Mu.) wheres ( DZ =.78) t (..). nd with respet to nother fuzzy ftor similrly to fuzzy ftor =. - From figure (D) it n e onluded tht there is reltion etween Z, nd Mu. The optiml vlue inreses whenever inresing the epted degree inresing is in fuzzy ftor. - Performne Mesure: In this pper study ( ) irumstnes of fuzzy tht suffered y emples is studied through pplying FLP where the fuzzy oeffiient is lulted y using modified S- urve non liner memership funtion whih is hrterized y eisting the fuzzy ftor( ), n intervl optiml vlue Z is otined with respet to, Mu of ll ses under study. To otin the optiml tht the deision mker depends on under fuzzy irumstne (unertin), omprison mong these ses were done y using the performne mesure M = m Z / DZ Where DZ = m Z - min Z m Z : higher vlue of Z. min Z : lest vlue of Z. Tle (E) (Performne mesure) Cse Fuzzy oeffiient m Z min Z DZ M = m / DZ Z i i, From tle (E) we onlude the following: [3]

24 Trnsformtion Liner Memership Funtion Comprison etween ses one nd two, then the se two is the est mong them nd the oeffiient of ville resoures nd onstrin re suffered from fuzzy prolem, this se hs one oeffiient from norml (non fuzzy).this mens tht ( ), nd the higher profit of this se is (.), n e mde y deision mker, where the performne mesure is (8.83).This is very useful for the deision mker to mke speifi deision. ) Conlusions: n industril pplition of FLP through the modified S- urve memership funtion hs een investigted y using set of dt olleted from two emples. The prolem of fuzzy hs een defined.two possile ses were identified depending upon sets of fuzzy nd non fuzzy oeffiients. Neessry equtions in eh se hve een formulted; profits nd stisftory level hve een omputed using FLPP. It n e onluded tht the model ontins doule oeffiient from fuzzy prolem is the est of the se hve one oeffiient suffered from fuzzy prolem. Referenes: [] llhvirnloo T., Lotfi F. H., Kisry M. Kh., Kini N.. nd lizdeh L., Solving Fully Fuzzy liner progrmming prolem y the Rnking funtion, pplied mthemtil Sienes ( 8 ),V., PP. 3. [] etor C. R. nd Chndr Suresh, Fuzzy Mthemtil progrmming nd Fuzzy mtri gmes, Springer Verlg erlin Heidelerg (). [3] Fuller R., Fuzzy Resoning nd fuzzy optimiztion, Turku Centre for Computer Siene (8). [] Klir George J. nd Yun o, Fuzzy sets nd Fuzzy logi theory nd pplitions, Prentie Hll, INC (). [] Nsseri S.H., rdil E., Yzdni. nd Zefrin R., Simple method for solving Liner progrmming prolems with Fuzzy Numers, proeedings of world demy of Siene ( ),V., PP [] Nsseri S.H., rdil E., Yzdni. nd Zefrin R., Simple method for solving Liner progrmming prolems with Fuzzy Numers, proeedings of world demy of Siene ( ),V., PP [7] Nehi H. M. nd lineghd M., Solving Intervl nd Fuzzy Multi oetive liner progrmming prolem y Neessrily Effiieny points, Interntionl mthemtil forum (8), V.3, PP.. []

25 The Fourth Sientifi Conferene of the College of Computer Siene & Mthemtis [8] Vsnt P.M., pplition of Fuzzy Liner progrmming in prodution plnning, Fuzzy optimiztion nd Deision Mking ( 3), V.3,PP.. [] Vsnt P. nd rsoum N. N., Fuzzy optimiztion of units produts in mi produt seletion prolem using fuzzy liner progrmming pproh, Springer siene( ), V., PP.. [] Zdeh L.., Fuzzy sets, Informtion nd Control (), V.8, PP []