Application of FEWMA Control Chart for Monitoring Yarn Process in the Textile Industry

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1 Adves i Fuzzy Mthemtis. ISSN Volume 1 Numer 3 (017) pp eserh Idi Pulitios Applitio of F Cotrol Chrt for Moitorig Yr Proess i the Textile Idustry S. Suulkshmi 1 A. Khimohidee d. Ssikumr 3 1 Deprtmet of Sttistis Dr. MG Jki College of Arts d Siee for Wome Chei 8 Tmil Ndu Idi. Deprtmet of Sttistis Periyr E.V.. College Tiruhirpplli Tmil Ndu Idi. 3 Deprtmet of Sttistis Momim Sudrr Uiversity Tiruelveli Tmil Ndu Idi. Astrt A proess tht uses sttistil tehiques to oserve d otrol produt qulity is lled Sttistil Qulity Cotrol (SQC) where otrol hrts re test tools ommoly used for moitorig the mufturig proess. Sttistil Proess Cotrol (SPC) is proess to get etter the qulity of produts d lesse rework d srp so tht the qulity d produtivity prospet e met. Cotrol hrtig is the most importtprt ofspc. Oe of themost importt otrol hrts is Expoetilly Weighted Movig Averge () to detet the smll shifts. But uertity i hum ogitive proesses mkes the trditiol otrol hrts ot e pproprite sie they give oly prtiulr iformtio. Thus Fuzzy Expoetilly Weighted Movig Averge (F) otrol hrt whih is ustomed to use whe the sttistil dt uder oer re usure or hzy.i this pper F otrol hrt is pplied tomoitor the yr proess i the textile idustry. Keywords: Sttistil Qulity Cotrol Sttistil Proess Cotrol Expoetilly Weighted Movig Averge Cotrol Chrt Fuzzy Trsformtio Tehique.

2 748 S. Suulkshmi A. Khimohidee d. Ssikumr 1. INTODUCTION Sttistil Qulity Cotrol (SQC) d developmet is rh of idustril sttistis whih iludes primrily the res of epte smplig sttistil proess moitorig d otrol (SPC) desig of experimets d pility lysis. Most SQC reserh hs foussed o preisio.the mufturigsetor plys very importt role i promotig the eoomi developmet d pushig the developmet forwrd. So the mufturig setor for mkig the right deisio eeds to hve sietifi wy for iresig the qulifitio of the produtio proesses. Oe wy to get this is y pplyig Qulity Cotrol.SPC is employed to moitor produtio proesses over time to detet hges i proess performe. The si fudmetls of SPC d otrol hrtig were proposed y Wlter A Shewhrt i the 190 s d 1930 s. Util the mid to lte 1970 s there were my importt dves ut reltively few idividuls odutig reserh i the re ompred to other res of pplied sttistis. eserh tivity hs gretly iresed sie out 1980 owrds. Muh of the irese i iterest ws due to the qulity revolutio whih ws used y iresigly ompetitive itertiol mrket ple. Improvemets i qulity were required for survivl i my idustries. SPC methods re developed d pplied lrgely i the disrete prts idustries.motgomery provides exellet disussio out SPC proedure i the mufturig idustry.a otrol hrt is vlule sttistil tool tht ids prtitioers i sttistilly otrollig d moitorig oe or more vriles whe the qulity of the produt or the qulity of the proess is hrterised y erti vlues of these vriles. I geerl otrol hrt is very esy to e implemeted i y type of proess. Thus otrol hrts reextesively used i mufturig re owdys preservig the qulity of the proess or the fil produt. A otrol hrt is hrt thtshows whether smple of dt flls iside the ommo or orml rge of vritio.a otrol hrt hs upper d lower otrol limits tht divide ommo fromssigle uses of vritio. The geerl rge of vritio is defied y the utilize ofotrol hrt limits. A proess is out of otrolwhe plot of dt revelstht oe or more smples fll outside the otrol limits.i this pper we usedf otrol hrt for moitorig the yr proess. The utilizesll previous oservtios ut the weight lose to dt is expoetillywig s the oservtios get older d older. By hgig the prmeterof the sttisti the `memory' of the otrol hrt e ilied.. EVIEW OF LITEATUE Sttistil theory eg to e futio effetively to qulity otrol i the yer 190 d i194 Shewhrt prepred the first sketh of fresh otrol hrt. His ttempt ws lter developed y Demig d the efore time work of Shewhrt Demig Dodge d omig ostitutes gret del of wht tody omprises the theory of

3 Applitio of F Cotrol Chrt for Moitorig Yr Proess SPC. I 1931 Wlter A. Shewhrt prit ook out the fudmetl oept of sttistil otrol d sketh out five eoomi dvtgesessile through sttistil otrol of worth muftured produt.the mi impetus for implemetig SPC i mufturig proesses is eed for higher d stle qulity. Grvi (1987) defied qulity i vrious dimesios: level of performe reliility durility servieility estheti fetures pereived qulity d oforme to stdrds. Motgomery (009) eluidted importt moder defiitio tht Qulityimprovemet is the redutio of vriility i proess d produts. I this view SPC is see s mehism for otrollig vriles.wg d z (1990) exemplify two pprohes for ostrutig vrile otrol hrts sed o liguisti dt. oerts (1959) itrodued the otrol hrt sed o the. Most urret referees ilude Huter (1986) Crowder (1987) d Lus d Sui (1990).The survivlof fuzzy uertity i mufturig system is idisputletruth. I glol mrket ompies must del with high rte of hges i usiess eviromet. The prmeters vriles d restritios of the produtio system re turllyvgueess. Suh hoestidetifitioertily roughtfuzzy mthemtis iitited y Zdeh(1965)defied fuzzy set s lss of ojets with grdes memership.z d Wg (1990) proposed pproh sed o the fuzzy set theory y ssigig fuzzy set to eh liguisti term. El-Shl d Morris (00) desried reserh to mke use of of fuzzy logi to lter SPC rules with the pl of sikig the ohort of flse lrms. 3. FUZZY TANSFOMATION TECHNIQUES With referee to Wg d z (1990) fuzzy trsformtio tehiques re wor to overt the fuzzy umers ito ruhy vlues. The four fuzzy mesures of etrl tedey fuzzy mode α- level fuzzy midrge fuzzy medi d fuzzy verge whih fmous i desriptive sttistis re give elow: The fuzzymode(fmode) :The memership futio of the vlue of the se vrile of fuzzy set equls 1 i the fuzzy mode. Tht is fmode = {x μf(x) =1} x F. Also Fuzzy mode is uique if the memership futio is uimodl. The α-level fuzzy midrge( f α mr ): F the verge of the ed poits of α-ut is α o fuzzy suset of the se vrile x otiig ll the vlues with memership futio vlues greter th or equl to α. Thus F α ={x μ F x α }. If d α α re ed poits of α ut F α suh tht α =Mi { F α } d α =Mx { F α } the f α = 1 mr ( + ) α α

4 750 S. Suulkshmi A. Khimohidee d. Ssikumr The fuzzy medi(fmed):fuzzy medi is the poit whih divides the urve of the fuzzy set i to two equl regios uder the memership futio stisfyig the followig equtios: fmed μ xdx = F μ x dx = 1 F fmed μ F vrile of the fuzzy set F suh tht is less th. x dx Where d re the poits i the se The fuzzy verge(fvg ) : Aordig to Zdeh the fuzzy verge is fvg=av(x;f)= 1 xμ α0 1 α0 μ F F x dx x dx It should e oted tht there is o hypothetil sis followig y oe prtiulrly or the hoiemog them. I geerl whe the memership futio is olier the first two methods re esier to lulte th the lst two. The fuzzy mode my go hed to ised results whe the memership futio is tremedously symmetril. The fuzzy midrge is strethier euse oe hoose differet levels of memership (α) of iterest. If the re uder the memership futio is mesured to e pproprite mesure of fuzziess the fuzzy medi is pproprite. Figure 1: Trigulr fuzzy umers 4. FUZZY EPONENTIALLY WEIGHTED MOVING AVEAGE CONTOL CHAT The proess ws moitored y the otrol hrts. Fuzzy otrol hrts re ommoly used whe dt re overed y miguity d vgueess for providig litheess o otrol limits to put off flse lrms. The is hose to pereive smll shifts i the proess. The trditiol otrol hrt ws itrodued y oerts d Huter detils of theses hrts re s elow:

5 Applitio of F Cotrol Chrt for Moitorig Yr Proess Zt= λ + t 1 Zt-1where Ztis the tth expoetilly weighted movig verge deotes the t th smple verge 0< λ 1 is ostt is the overll me m is thesmpleumer d t=1... m.z0= If i re idepedet rdom vriles with vrie σ / (σ is the popultio stdrd devitio d kow) the the vrie of Zt isσzt = [ t 1 1 ] where is the smple size.as t ireses σzt ireses to limitig vlue:σz=. If the smple umer t is modertely lrge the trditiol otrol hrt is give s follows: UCL = +3 CL= LCL= -3 the trditiol otrol hrt is give s follows:. For smll t t UCL = +3 [1 1 t ] CL= LCL= -3 [1 1 t ] If σ is estimted from smple is used for ostrutig trditio otrol hrt like followig: UCL = +A CL= LCL= - A where is the verge of the i s while i is rgefor eh smple. 4.1 F Cotrol Chrts Whe (σ σ σ) re Kow First lulte the rithmeti mes of the lest possile vlues the most possile vlues d the lrgest possile vlues of elogtio of otto i yr proess respetively.(1 1 1) d( ) re otied s follows:firstly (j j j ) vlues re lulted where j=mx{j}-mi{j}

6 75 S. Suulkshmi A. Khimohidee d. Ssikumr j =mx{j}-mi{j} j=mx{j}-mi{j}d mx{ij} is the mximum of fuzzy umers i the smple d mi{ij} is the miimum of fuzzy umers i the smple. Tle 1: Fuzzy Numer epresettio of the smples SAMPLE x x = = = = = = = = = =9.79 =9.75 = =1.41

7 Applitio of F Cotrol Chrt for Moitorig Yr Proess = = = =9.74 C 5 =.1 5 =.13 =9.78 =9.80 =9.81 =.13 Whe detetig the smll shift i proess with fuzzy oservtiosthe F otrol hrt should e used to evlute the proess. Fuzzy oservtios (111) re olleted from proess whe the fuzzy oservtios re represeted y trigulr memership futio with smple size. ( t t )represets the fuzzy t verge of r th smple(tle ). Where z 0= (z0 z0 z0) = ( )While kowig the fuzzy stdrd devitios the fuzzy verges fuzzy stdrd devitios d λ re used to ostrut the fuzzy otrol hrt.if the smple umer t is modertely lrge the fuzzy otrol hrt is give s follows: U C L= ( ) + 3 (σσ σ) = C L= ( ) L C L= ( ) _ 3 (σσ σ) = _ 3 _ 3 _ 3

8 754 S. Suulkshmi A. Khimohidee d. Ssikumr If the smple umer t is smll followig equtios re otied: U C L= ( [ t 1 1 ] + 3 ) + 3 (σσ σ) [ t 1 1 ] C L= ( + 3 ) t 1 1 ]= + 3 [ [ 1 1 ] t L C L= ( t 1 1 ] _ 3 ) _ 3 (σσ σ) [ t 1 1 ] _ 3 t 1 1 ]= _ 3 [ [ 1 1 ] t [ Tle : Fuzzy verges d fuzzy expoetilly weighted movig verges T M ( 1 t 1 ) 1 ) ( ( )... ( m m ) m Z 1 =λ( 1 Z t 1 )+(1-λ) ( 1 Z =λ( )+(1-λ) Z 1 Z 3 =λ( Z m =λ( m m m )(1-λ) Z )+(1-λ) m 1 ) Z 4. α-uts F otrol hrts whe (σ σ σ ) re kow A α-uts is restrited fuzzy set whih iludes the elemets whose memership degrees re greter th equl to α.after pplyig the α-uts o mes d stdrd devitios the α-uts fuzzy overll mes d α-uts fuzzy stdrd devitio re lulted s follows respetively: = + α ( - ) = - α ( - ) = - α ( - ) d = + α( )

9 Applitio of F Cotrol Chrt for Moitorig Yr Proess If the smple umer t is modertely lrge the α-uts fuzzy otrol hrt is give s follows: + 3 CL = ( LCL = ( UCL = ( + 3 ) )- 3-3 )+ 3 ( ) ( ) = - 3 = If the smple umer t is smll the α-uts F otrol hrt is otied y the followig equtios: UCL =( )+ 3 t [1 ( )] t [1 ( )] 1 CL = ( LCL = ( = - 3 ) )- 3 t t [1 ( )] 1 ( [1 ( )] 1 = + 3 ) t 1 t [1 ( )] + 3 ( [1 ( )] 1-3 [1 ( )] 1 ) t t 1 [1 ( )] - 3

10 756 S. Suulkshmi A. Khimohidee d. Ssikumr 4.3 α-level fuzzy medi for α-uts F otrol hrt for (σ σ σ ) re kow The α-level fuzzy medi trsformtio tehiques is pplied o α-uts F otrol hrts for otiig the risp vlues of otrol limits. The α-level fuzzy medi for α-uts F otrol hrt is otied for the smple umer t is modertely lrge d t is smll s follows respetively; UCL = med CL + 1 med ( ) CL = 1 med 3 ( ) LCL = med CL - 1 med ( UCL = med CL + 1 med ( ) [1 ( )] 1 ) t CL = 1 med 3 ( ) LCL = med CL - 1 med ( [1 ( )] 1 While evlutig ) t the smple with F otrol hrt we lulte the α-level fuzzy medi vlue. S = med j 1 ( ) 3 j j j F otrol hrt for ukow stdrd devitios re lulted for yr proess dt s follows: U C L= + A +A + A + A = *.1 0. = A + A = * = = * =10. 0.

11 Applitio of F Cotrol Chrt for Moitorig Yr Proess U C L=( ) CL =( ) = ( ) CL = ( ) L C L = - A - A - A - A = * = A - A = * = = *.1 0. = L C L = ( ) where λ=0.due to geerl pproh i produtio proess. d re lulted y usig the followig equtios where α=0.65. Beuse there re quite lot of pplitios i literture i whih α-uts is preferred 0.65 for the mufturig proess. = = = = + α( - α( +α ( -α( ) = ( ) = 9.79 ) = ( ) = 9.80 ) = (.13.10) =.1 ) = (.13.13) =.13 The limits of α-uts F otrol hrt re give s follows for yr proess: UCL = + A +A + A + A = *.1 0. = A = * =10.1

12 758 S. Suulkshmi A. Khimohidee d. Ssikumr + A = * = 10.1 UCL = ( ) CL = ( CL = ( ) ) =( ) LCL = - A - A -A = * A = A = * = A = * = 9.39 LCL = ( ) Fuzzy medi trsformtio tehique is itegrted to the α-level fuzzy medi for α-utf otrol hrt s follows: UCL = med CL + 1 med 3 A( + + ) = *0.577( ) = 10.1 CL = 1 med 3 ( ) = 1 3 ( )=9.80 LCL = med CL - 1 med ( ) = *0.577( ) 0. = For eh smple α-level fuzzy medi vlue ( S ) is lulted. α uts for med j verges for the 5 smples re lulted s follows:

13 Applitio of F Cotrol Chrt for Moitorig Yr Proess = = ( - 1 ) = ( ) = ( - )= ( ) = = ( - 5 ) ( )= = = = ( - 1 ) = ( ) = ( - ) = ( )= ( - 5 )= ( ) = After lultig fuzzy umer with α-uts for verges for the first smple α-level fuzzy medi vlue is otied. Sie this vlue is etwee otrol limits the first smple is i otrol. Also the oditio of proess otrol for eh smple is defied y usig the followig: LCL S UCL i otrol for Proess otrol = out ofotrol otherwise give i tle. d med med j med S = med 1 1 ( ) 3 = 1 ( ) = ( ) = 1 ( ) = S = med 3... S = med 5 1 ( ) 3 = 1 ( )=9.73 3

14 760 S. Suulkshmi A. Khimohidee d. Ssikumr Tle 3: Cotrol limits of F α-levelfuzzy medi vlue d the proess oditios. smple S 9.39<S α <10.1 med j I otrol 9.75 I otrol I otrol I otrol I otrol I otrol I otrol I otrol Out of otrol I otrol I otrol I otrol I otrol I otrol I otrol I otrol I otrol I otrol I otrol Out of otrol I otrol I otrol I otrol I otrol I otrol

15 Applitio of F Cotrol Chrt for Moitorig Yr Proess As see i Tle 3 the yr proess i textile idustry is out of otrol due to the ieth d twetieth smple. Eve though twety three smples revel uder otrol proess two smples idites ssigle uses. So the produtio proess is out of otrol. The ssigle uses for this shift should e serhed d fter removig this use the proess ru gi. 5. CONCLUSION I vrious rel-world exertios dt re hzy d vgue; the hypothesis of ruhy profiles for proesses is ot sesile. Fuzzy set theory is ompetet tool to tkle this limittio. I this pper ew system of idetifyig smll hges of proess profile hs ee pled i whih profile prmeters were ssumed hzy d vgue. For this purpose we hve used F otrol hrts d disussed the ompetee of yr proess i textile idustry. The results hve show tht this method ws highly ple i disoverig ssigle uses i profiles. EFENCES [1] Cheg Chi-Bi. 005 Fuzzy proess otrol: ostrutio of otrol hrts with fuzzy otrol hrts with fuzzy umers Jourl of fuzzy sets d systems 154. [] Crowder S. V A simple method for studyig ru-legth distriutios of expoetilly weighted movig verge hrtstehometris 9(4) [3] Ertugrul Irf d Ayt Esr 006 Costrutio of qulity otrol hrts y usig proility d fuzzy pprohes d pplitio i Textile Compy Jourlof Itelliget Mufturig 0. [4] Grvi D. A Competig i the Eight Dimesio of Qulity. Hrvrd Busiess eview 97(6) 101. [5] Huter S. J The expoetilly weighted movig verge Jourl of Qulity Tehology 18(4) [6] Lus J. M. d Sui M.S expoetilly weighted movig verge otrol shemes: Properties d ehemets" Tehometris 3(1) 1-9. [7] Motgomery D.C. 009 Itrodutio to Sttistil Qulity Cotrol 6 th ed. Joh Wiley & sos Jourl of fuzzy sets d systems 154. [8] oerts S. W Cotrol hrt tests sed o geometri movig verges Tehometris1(3)

16 76 S. Suulkshmi A. Khimohidee d. Ssikumr [9] Wg J.H d z T 1990 O the ostrutio of otrol hrts usig liguisti vriles Itertiol Jourl of produtio d eserh 8(3)

SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is

$SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is$ SPH3UW Uit 7.5 Sell s Lw Pge 1 of 7 Notes Physis Tool ox Refrtio is the hge i diretio of wve due to hge i its speed. This is most ommoly see whe wve psses from oe medium to other. Idex of refrtio lso lled