FUZZY MULTINOMIAL CONTROL CHART WITH VARIABLE SAMPLE SIZE

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1 A. Paduaga et al. / Iteatoal Joual of Egeeg Scece ad Techology (IJEST) FUZZY MUTINOMIA CONTRO CHART WITH VARIABE SAMPE SIZE A. PANDURANGAN Pofesso ad Head Depatmet of Compute Applcatos Vallamma Egeeg College, Kattaulathu, Chea , Ida. e-mal d : apga6@gmal.com R. VARADHARAJAN Assstat Pofesso Statstcs Depatmet of Maagemet Studes VES Uvesty, Pallavaam, Chea 600 7, Ida e-mal d: vaa@gmal.com Abstact: The cotol chat techque s beg wdely used dustes to moto a pocess fo qualty mpovemet. Oe of the basc chats fo attbutes s the p-chat. Fo a p chat each tem s classfed as ethe ocofomg o cofomg to the specfed qualty chaactestc. I Some cases, a tem may be classfed moe tha two categoes such as bad, medum, good, ad excellet. Based o ths cocept, Amzadeh et al. [] have developed Fuzzy Multomal chat (FM-chat) wth the fxed sample sze (FSS). I ths pape a Fuzzy Multomal pocess wth Vaable Sample Sze (VSS) s poposed ad the cotol lmts fo the FM chat have bee obtaed usg multomal dstbuto. The poposed method s compaed wth the covetoal p chat. It s see that FM chat wth VSS pefoms bette tha the covetoal chat. Key wods: Multomal dstbuto; FM-chat; p chat; Vaable sample sze; gustc vaable; Membeshp fucto; Fuzzy statstcs.. Itoducto: Statstcal Pocess Cotol (SPC) s used to moto the pocess stablty whch esues the pedctablty of the pocess. The powe of cotol chats les the ablty to detect pocess shft ad to detfy abomal codto the pocess. I 94, Walte Shewhat desged the fst cotol chat as follows: et w be a sample statstc that measues some qualty chaactestc of teest ad suppose that the mea of w s µ w ad the stadad devato of w s σ w. The the cete le (C), the uppe cotol lmt (UC) ad the lowe cotol lmt (C) ae defed as UC µ w + d σ w C µ w C µ w d σ w whee d s the dstace of the cotol lmts fom the cete le, expessed stadad devato uts. A sgle measuable qualty chaactestc such as dmeso, weght o volume s called a vaable. I such cases, cotol chats fo vaables ae used. These clude the -chat fo cotollg the pocess aveage ad the R -chat (o S -chat) fo cotollg the pocess vaablty. If the qualty-elated chaactestcs such as chaactestcs fo appeaace, softess, colo, taste, etc., attbutes cotol chats such as p-chat, c-chat ae used to moto the poducto pocess. Some tmes the poduct uts ae classfed as ethe "cofomg" o" ocofomg", depedg upo whethe o ot they meet some specfcatos. The p -chat s used to moto the pocess based upo the facto of ocofomg uts. ISSN : Vol. 3 No. 9 Septembe

2 A. Paduaga et al. / Iteatoal Joual of Egeeg Scece ad Techology (IJEST). Fuzzy logc ad gustc vaables: The cocept of fuzzy logc plays a fudametal ole fomulatg quattatve fuzzy vaables. These ae vaables whose states ae fuzzy membes. The membes epeset lgustc cocepts, such as vey small, small, medum ad so o, as tepeted a patcula cotext. The esultg costucts ae usually called lgustc vaables. The lgustc tems ae commoly used dusty to expess popetes o chaactestcs of a patcula poduct. The cofomty to specfcatos o a qualty stadad s evaluated oto a two-state scale, fo example, acceptable o uacceptable, good o bad, ad so o. I some stuatos the bay classfcato mght ot be sutable, whee poduct qualty ca assume moe temedate states. The assgmet of weghts, to eflect the degee of sevety of poduct ocofomty has bee adopted may ccumstaces. Whe the poducts ae classfed to mutually exclusve lgustc categoes, fuzzy cotol chats ae used. Dffeet pocedues ae poposed to costuct these chats. Raz ad Wag [7,0] developed fuzzy cotol chats fo lgustc data whch ae maly based o membeshp ad pobablstc appoaches. I ths pape, a fuzzy multomal cotol chat (FM chat) fo lgustc vaables wth vaable sample sze s poposed. The FM chat deals wth a lgustc vaable whch s classfed to moe tha two categoes. The FM chat wth VSS foud to be moe effectve tha p chat fo studyg the shft pocess mea. 3. Methodology: Based upo Fuzzy set theoy, a lgustc vaable s chaactezed by the set of mutually l l... l that eflects the degee of membeshp exclusve membes{, l }. We attach a weght m to each tem the set. The t ca be wtte by a fuzzy set as (). {( l, m ),( l, m ),...( l, m )} To moto the out of cotol sgal the poducto pocess tae depedet samples of dffeet szes. The sze of the sample to be daw each tme s decded by choosg a membe adomly fom {,,... s }. 4. Fuzzy Multomal Cotol Chat: I ths secto a ew appoach fo costucto of Fuzzy multomal cotol chat based o vaable sample sze s poposed. The statstcal pcples udelyg the fuzzy multomal cotol chat (FM - chat) wth vaable sample sze ae based o the multomal dstbuto. As defed (), s a lgustc vaable whch ca tae mutually exclusve membes{ l, l... l }. Assume that the poducto pocess s opeatg a stable mae ad p s the pobablty that a tem s l,,. ad successve tems poduced ae depedet. Suppose that a adom sample of sze uts of the poduct s selected ad let,,, be the umbe of tems of the poduct that ae l,,. The {,... } has a multomal dstbuto wth paametes ad p, p... p. It s ow that each,,, magally has a bomal dstbuto wth the mea ad vaace p p ), p (,, espectvely. The weghted aveage of the lgustc vaable wth sample sze s defed by m m, {,,... s } () The cotol lmts fo FM chat ae UC E [] + d va( ) ; C E [] ; ad C E [] d va( ), whee d s the dstace of the cotol lmts fom the cete le. The pocedue fo computg E [] ad va( ) fo each sample s gve the followg theoem. ISSN : Vol. 3 No. 9 Septembe

3 A. Paduaga et al. / Iteatoal Joual of Egeeg Scece ad Techology (IJEST) 5. Theoem: be a lgustc vaable such that et {( l, m ),( l, m ),...( l, m )} tem s l,,,. If sample of sze, the () E [] p m,, s the umbe of uts of the poduct that ae () va( ) m p ( p ) mmp < whee,,... s ae pe-detemed sample szes. Poof: I a sample of uts, p p ),,, ad ( Cov (, ) () The mea s: E [] p ), {... } p s the pobablty that a,, s has a bomal dstbuto wth the mea p p, f ad the m E p m m E( ) m, {... },, s (3) m () The vaace s: va( ) va va( m ), [ va( m x + m x m )] x p l,, a p ad vaace m va( ) + cov(, ) mm < m va( ) + cov(, ) mm < m p ( p ) + mm( p p ) < m p ( p ) mmpp), whee {,,... s }.... (4) < The esults fo E [] ad va( ) become same as that of the esults due to Amzadeh et al. [], whe the sample sze s fxed. That s, whe fo all, the mea ad vaace deved (3) ad (4) fo a lgustc vaable wth vaable sample sze wll be equal to the mea ad vaace of the lgustc vaable fo fxed sample sze. ISSN : Vol. 3 No. 9 Septembe

4 A. Paduaga et al. / Iteatoal Joual of Egeeg Scece ad Techology (IJEST) 6. Choce of Sample sze: May authos, fo example [], [8] have ecommeded vaable sample szes (VSS) fo the costucto of cotol chats fo vaables as well as attbutes. ate, the Maov depedet sample sze (MDSS) was poposed by Svasamy et al [9], ad Paduaga [6] fo the advatage of ecoomc samplg specto. Now, to costuct FM cotol chat the sample sze fo each daw ca be adomly chose fom the pe detemed set{,,... s }. The advatage of tag vaable sample sze les ASN ad cosequetly the costs of samplg specto. 7. Numecal Example: O a poducto le, a vsual cotol of the alumum de-cast of a lghtg compoet mght have the followg assessmet possbltes. "eect" f the alumum de-cast does ot wo;. "poo qualty" f the alumum de-cast wos but has some defects; 3. "medum qualty" f the alumum de-cast wos ad has o defects, but t has some aesthetc flaws; 4. "good qualty" f the alumum de-cast wos ad has o defects, but has few aesthetc flaws; 5. "excellet qualty" f the alumum de-cast wos ad has ethe defects o aesthetc flaws of ay d. To moto the qualty of ths poduct, 5 samples of dffeet szes ae selected. The degees of membeshp fo the above assessmet ae tae as, 0.75, 0.5, 0.5 ad 0 espectvely. The data wth pˆ ae gve Table. ad o. The value of ca be calculated to the followg ways m m, {... },, s {( ) + (0 0.75) + ( 0.5) + (54 0.5) + ( 0) } 00 (8 ) + (7 0.75) + (9 0.5) + (48 0.5) + (8 0) 80 (6 ) + ( 0.75) + ( 0.5) + (43 0.5) + (8 0) 80 { } { } Ad so o, ad the value of pˆ, the cotol lmts fo p chats ca be calculated as follows D p ˆ, {,,... s } D D 8 D3 6 p ˆ 0.0, p ˆ 0. 00, ˆ p ; ad so 3 ISSN : Vol. 3 No. 9 Septembe

5 A. Paduaga et al. / Iteatoal Joual of Egeeg Scece ad Techology (IJEST) Sample No Table Sample Poo Medum Good Excellet Reect pˆ sze Qualty(PQ) Qualty(MQ) Qualty(GQ) Qualty(EQ) The cotol lmts fo p chat s obtaed as follows p s UC D p d p( p) ( 0.096) 00 Fo Sample : C p 0.096; p( p) 0.096( 0.096) C p d , 00 p( p) Fo Sample : UC p + d ad so o ( 0.096) 80 C p 0.096; p( p) 0.096( 0.096) C p d , 80 ISSN : Vol. 3 No. 9 Septembe

6 A. Paduaga et al. / Iteatoal Joual of Egeeg Scece ad Techology (IJEST) Fg. : The p chat fo the 5 samples. I Fgue, the out of cotol sgal s see coespodg to th sample, fo whch the sample sze s 0. Fom the p chat, out of 450 sample obsevatos, 070 sample obsevatos wee eeded to get the sgal. The coespodg cete le ad cotol lmts ae as ude C 0.096, UC 0.80 ad C 0.0 To costuct the FM chat, the UC ad C values ae computed fo each sample as follows Fo Sample : UC E ] + d C C va( ) [ p m + d m p ( p ) mmpp) < E[ ] p m ] d va( ) E[ p m - d m p ( p ) mmpp) < Fo Sample : UC E[ ] + d va( ) ISSN : Vol. 3 No. 9 Septembe

7 A. Paduaga et al. / Iteatoal Joual of Egeeg Scece ad Techology (IJEST) C C p m + d m p ( p ) mmpp) < E[ ] p m E[ ] d va( ) p m - d m p ( p ) mmpp) < , ad so o. FM chat fo the 5 samples s gve Fgue. Fg. : The FM chat fo the 5 samples. Fgue shows that the pocess s out of cotol at samples 8 ad, the espectve sample szes ae 00 ad 0 ad the coespodg cete les ad cotol lmts ae () C , UC ad C 0.89 fo sample sze 00. ( P R 0.0, P PQ 0., P MQ 0.8, P GQ 0.45, P EQ 0.05) () C , UC ad C fo sample sze 0. ( P R 0.88, P PQ 0.8, P MQ 0.09, P GQ 0.473, P EQ ) Fom fgue, t s see that the FM chat gves the fst sgal coespodg to 8 th sample. Wheeas, p chat the fst sgal fo the exstece of assgable causes s see oly at the th sample. That s, the case of FM chat, oly 780 samples ae spected to get the fst out of cotol sgal. But, 070 samples ae to ISSN : Vol. 3 No. 9 Septembe

8 A. Paduaga et al. / Iteatoal Joual of Egeeg Scece ad Techology (IJEST) be spected to get the alam wth the help of a p chat. Thus, the FM s moe ecoomcal ad moe sestve detfyg ay shft the specfed qualty level. Cocluso: FM chat has bee poposed fo lgustc data set. To daw the chat, samples of vayg szes ae chose adomly fom a pe detemed set. The FM-chat has bee compaed wth the covetoal p chat wth VSS. It s show that the FM chat s moe ecoomcal ad moe sestve gvg the alam fo shft the specfed qualty level. Ths wo ca be exteded fo Maov depedet sample szes. REFERENCES [] Amzadeh, V.; M. Mashch; M.A. Yaghoob. (008). Costucto of Cotol Chats Usg Fuzzy Multomal Qualty. Joual of Mathematcs ad Statstcs 4 (): pp.6-3. [] Costa A.F.B, (994). -chat wth vaable sample sze. Joual of Qualty Techology, 6, pp [3] Facesch, F. ad D. Romao, (999). Cotol chat fo lgustc vaables: A method based o the use of lgustc quatfes. Iteatoal Joual of Poducto Reseach. 37: pp [4] Gulbay, M., C. Kahama ad D. Rua, (004). α - cut fuzzy cotol chats fo lgustc data. Iteatoal Joual of Itellget Systems. 9: pp [5] Kaagawa, A., F. Tama ad H. Ohta, (993).Cotol chats fo pocess aveage ad vaablty based o lgustc data. Iteatoal Joual of Poducto Reseach. 3:pp [6] Paduaga A., (00). Some Applcatos of Maov Depedet Samplg Scheme SQC, Upublshed Ph.D Thess, Bhaathdasa Uvesty, Tuchappall, [7] Raz, T. ad J.H. Wag, (990). Pobablstc ad membeshp appoaches the costucto of cotol chats fo lgustc data. Poducto Plag ad Cotol. J. Math. & Stat. 4 (): 6-3, [8] Sawlapua, U, Reyolds, M.R.J, Aold, J.C (990). Vaable Sample Sze -chats. Peseted at the wte cofeece of the Ameca Statstcal Assocato, Olado, F. [9] Svasamy, R, Sathaumaa, A ad Subamaa, C (000). Cotol chats fo Maov Depedet Sample Sze. Qualty Egeeg, (4), pp [0] Wag, J.H. ad T. Raz, (990). O the costucto of cotol chats usg lgustc data. Iteatoal Joual of Poducto Reseach. 8: ISSN : Vol. 3 No. 9 Septembe 0 699