Nonparametric CUSUM Charts for Process Variability

Transcription

1 Journal of Academia and Indusrial Research (JAIR) Volume 3, Issue June 4 53 REEARCH ARTICLE IN: Nonparameric CUUM Chars for Process Variabiliy D.M. Zombade and V.B. Ghue * Dep. of aisics, Walchand College of Ars and cience, olapur, (M), India Dep. of aisics, olapur Universiy, olapur-4355, (M), India vbghue_sas@rediffmail.com*, digambarzombade@gmail.com; *, Absrac In his sudy, wo nonparameric CUUM ype conrol chars are developed for monioring he process variabiliy. The proposed chars are based on nonparameric wo sample ess for esing equaliy of variance. The average run lengh (ARL) performance of he proposed CUUM-ype chars is evaluaed hrough a simulaion sudy and compared wih he corresponding hewhar-ype nonparameric chars. The sudy indicaes ha he proposed CUUM-ype nonparameric conrol chars are beer in deecing small shifs in process variabiliy, while for larger shifs hewhar-ype conrol chars has beer performance. Keywords: Conrol char, average run lengh, process variabiliy, CUUM-ype chars, nonparameric ess. Inroducion Conrol chars are saisical process conrol ools ha are widely used for monioring mean and variabiliy of a process. Mos of he conrol chars ha have been developed in lieraure are designed and evaluaed under he assumpion ha he underlying disribuion of he qualiy characerisic is normal. In real applicaions, here are many siuaions in which he process daa come from a non-normal disribuion which need o be moniored by appropriae conrol chars. To monior such ype of daa, developmen of conrol chars ha do no depend on a paricular disribuional assumpion is desirable. Nonparameric conrol chars can serve his purpose. The main advanage of a nonparameric conrol char is ha i does no assume any probabiliy disribuion for he characerisic of ineres. A formal definiion of nonparameric or disribuion-free conrol char is given in erms of is in-conrol run lengh disribuion. If he in-conrol run lengh disribuion is same for every coninuous disribuion hen he char is called disribuion-free. In lieraure, several nonparameric conrol chars are proposed for monioring locaion of a univariae process. ome of hese are based on signs and/or rank saisics by assuming a known in-conrol arge value for process locaion. Amin e al. (995) developed hewhar and cumulaive sum (CUUM) conrol chars based on sign es saisic. Bakir and Reynolds (979) developed a nonparameric CUUM o monior a process cener based on wih-in group signed-ranks. Amin and earcy (99) used wih-in group signed ranks o develop exponenially weighed moving average (EWMA) conrol char. Bakir (4) developed a disribuion-free hewhar conrol char for monioring process cener based on he signed-ranks of grouped observaions. Bakir (6) proposed hewhar, CUUM and EWMA conrol chars based on signed-rank-like saisics of grouped daa for monioring a process cener when in-conrol arge cener was no specified and sudied he robusness of he chars agains ouliers. Chakrabori e al. () presened an exensive review of he lieraure on univariae nonparameric conrol chars. Ghue and hirke () developed nonparameric conrol char based on bivariae signed-rank es o monior he changes in he locaion of a bivariae process. Ghue (3) developed disribuion-free conrol char based on Hodges sign es o monior he locaion of a bivariae process. There exis only few aricles on nonparameric chars for monioring process variabiliy. Lehmann (975) suggesed using non-parameric ess for he equaliy of wo variances for use as conrol saisics in nonparameric conrol chars for variabiliy. Conrol chars using ess saisics for comparing wo variances would require obaining an iniial sample (of size m) when he process is considered o be in-conrol. Then a each sample ime i, a sample of size n is obained from he process and he pooled sample of size (m + n) is obained. The observaions in he pooled sample are hen ranked from smalles o larges and some saisic based on he ranks of he observaions is calculaed. Das and Bhaacharya (8) proposed a nonparameric conrol char for monioring process variabiliy based on Conver s squared rank es for variance. Das (8) developed wo nonparameric conrol chars for monioring process variabiliy based on wo nonparameric ess. Murakani and Masuki () proposed a nonparameric conrol char for dispersion based on he rank sum saisic. ince few works are repored in he lieraure on nonparameric conrol chars for monioring process variabiliy, he purpose of his sudy is o develop CUUM-ype nonparameric conrol chars for monioring process variabiliy for he case ha he locaion parameer is under conrol. Youh Educaion and Research Trus (YERT) jairjp.com Zombade & Ghue, 4

2 Journal of Academia and Indusrial Research (JAIR) Volume 3, Issue June 4 54 The proposed nonparameric conrol chars are based on wo sample nonparameric ess proposed by Mood (954) and ukhame (956). These are mos powerful es saisics for deecing scale shifs. The performance of he proposed chars is assessed for boh he in-conrol sae and ou-of-conrol sae under differen underlying disribuions. Maerials and mehods hewhar-ype nonparameric conrol char based on ukhame es: uppose we wan o compare wo independen random samples X (X,X,...,Xm ) and Y (Y,Y ) which are drawn from absolue coninuous disribuions and differ only in he scale parameers. Le and X Y be he arbirary measures of dispersion of X and Y respecively hen problem of esing of hypohesis is H : agains X Y H : X. Y The ukhame es saisic for esing null hypohesis is defined as: m n T D(X i,y j), () m n where i j D (X, Y) if eiher X Y or Y X oherwise We rejec hypohesis if T is oo large or oo small. The mean and variance of he saisic T is given by: E (T ) and 4 For a large sample, (m n 7) Var ( T ) 48m n T E (T) Z () Va (T) has a sandard normal disribuion and he es is performed on he basis of abulaed values of he sandard normal disribuion. We consider Z as he conrol char saisic for he hewhar ype nonparameric conrol char for monioring process variabiliy and he char is referred as NP- char. We consider X (X,X,...,X m ) as reference sample of size m from an in-conrol process and ha Y (Y,Y ) be an arbirary es sample of size n. The sample saisics Z compued from independen observaions from he process are ploed agains an upper conrol limi UCL = 3 and LCL = -3. The process is considered ou-of-conrol when a ploed poin lies above UCL or below LCL. hewhar-ype nonparameric conrol char based on Mood es: uppose X (X,X,...,Xm ) and Y (Y,Y ) are wo independen random samples drawn from absolue coninuous disribuions and differ only in he scale parameers. We wish o es H : agains X Y H : X. Le Y R R... R m be he combined samples ranks of he X-values in increasing order of magniude. The Mood es saisic for esing null hypohesis is defined as: m M R i The mean and variance of he saisic M is given as: m(n ) E (M) i N and, Var (M) where N m n m n (N )(N 8 4) For N greaer han or equal o 3, we may consider he normalized random variable W M E (M) W (4) Var(M) and perform he es on he basis of abulaed values of he sandard normal disribuion. We consider W as he conrol char saisic for he hewhar-ype nonparameric conrol char for monioring process variabiliy and he char is referred as NP-M char. We consider X (X,X,...,X m) as reference sample of size m from an in-conrol process and ha Y (Y,Y ) be an arbirary es sample of size n. The sample saisics W compued from independen observaions from he process are ploed agains an upper conrol limi UCL = 3 and LCL = -3. The process is considered ou-of-conrol when a ploed poin lies above UCL or below LCL. CUUM-ype nonparameric conrol chars: I is well known ha he hewhar-ype conrol chars are relaively inefficien in deecing small shifs of he process parameers. Memory based conrol chars such as cumulaive sum (CUUM), exponenially weighed moving average (EWMA) are developed as alernaive o he hewhar chars for he deecion of small process shifs in he process parameers. They use he addiional informaion from recen hisory of process hence are more effecive han a hewhar conrol char in deecing small process shifs. Therefore, in his sudy, wo nonparameric CUUM procedures are developed o monior process variabiliy. The proposed CUUM procedures are based on wo sample nonparameric ess proposed by Mood (954) and ukhame (956). (3) Youh Educaion and Research Trus (YERT) jairjp.com Zombade & Ghue, 4

3 Journal of Academia and Indusrial Research (JAIR) Volume 3, Issue June 4 55 To develop nonparameric CUUM char o monior a process variabiliy, nonparameric es saisic T is obained from wo independen random samples X (X,X,...,X m ) and Y (Y,Y ) Then one. sided nonparameric CUUM char saisic based on nonparameric saisic T is given by: Max (, T k), (5) Where and k >. The CUUM scheme signals a firs for which h, where h > and k > are parameers of he procedure. The parameer k is he reference value and h is he decision inerval for he CUUM. The one sided nonparameric CUUM char saisic based on ukhame saisic Z is given by: Max (, Z k), (6) and char based on his saisic is referred as NPCM- char. The one sided nonparameric CUUM char saisic based on Mood saisic W is given by: Max(, W k), (7) and char based on his saisic is referred as NPCM-M char. Resuls and discussion Performance of he proposed conrol chars: To examine he abiliy of proposed NPCM- and NPCM-M chars o deec variabiliy shif in a process, we consider underlying process disribuions as normal, double exponenial and uniform wih mean zero and variance one. The uniform disribuion is considered as process disribuion o see he effec of a ligh ailed disribuion and double exponenial disribuion is considered o see he effec of heavy ailed disribuion on he performance of proposed nonparameric conrol chars. Consider a process where qualiy characerisic of ineres X is disribued wih mean μ and sandard deviaion σ. Le μ and σ be he in-conrol values of μ and σ respecively. When a shif in process sandard deviaion occurs, we have change from he in-conrol value σ o he ou-of-conrol value σ δσ ( δ ). Therefore, when conrol char for variabiliy is employed, he process shifs are measured hrough. When δ, he process is considered o be in-conrol. For δ an increase in occurs and for, decrease in occurs. Compuer programs wrien in C language are used o sudy he performance of he proposed conrol chars. In he (upper) one-sided, CUUM procedure he reference value k is aken as. Using his value of k, he value of h should hen be chosen o achieve he desired in-conrol ARL. The in-conrol and ou-of-conrol ARL values of he proposed conrol chars are compued using simulaions for sample size of n = 5 and. Table and provide he ARL values of he hewhar-ype and CUUM-ype nonparameric conrol chars based on ukhame es saisic when he underlying process daa acually follows normal, double exponenial and uniform disribuions wih sample sizes n =5 and respecively. Examinaions of Table and lead o he following findings: In-conrol ARL values of he proposed NP- and NPCM- conrol chars for differen process disribuions are approximaely same. For small shifs, ou-of-conrol ARL values of NPCM- char are smaller han ha of he NP- char. Therefore, NPCM- char is more efficien han NP- char for deecing small shifs in process when underlying process disribuion is normal, ligh ailed uniform and heavy ailed double exponenial. For uniformly disribued daa, boh NPCM- and NP- chars perform beer han normally and doubly exponenial daa. Table. ARL values of NP- and NPCM- chars when n = 5. hif Normal Double exponenial Uniform NP- NPCM- NP- NPCM- NP- NPCM Youh Educaion and Research Trus (YERT) jairjp.com Zombade & Ghue, 4

4 Journal of Academia and Indusrial Research (JAIR) Volume 3, Issue June 4 56 Table. ARL values of NP- and NPCM- chars when n =. hif Normal Double exponenial Uniform NP- NPCM- NP- NPCM- NP- NPCM Table 3. ARL values of NP-M and NPCM-M chars when n = 5. hif Normal Double exponenial Uniform NP-M NPCM-M NP-M NPCM-M NP-M NPCM-M Table 4. ARL values of NP-M and NPCM-M chars when n =. hif Normal Double exponenial Uniform NP-M NPCM-M NP-M NPCM-M NP-M NPCM-M Table 3 and 4 provide he ARL values of he hewhar-ype and CUUM-ype nonparameric conrol chars based on Mood es saisic when he underlying process daa acually follows normal, double exponenial and uniform disribuions wih sample sizes n =5 and respecively. Examinaions of Table 3 and 4 lead o he following findings: In-conrol ARL values of he proposed NP-M and NPCM-M conrol chars for differen process disribuions are approximaely same. For small shifs, ou-of-conrol ARL values of NPCM-M char are smaller han ha of he NP-M char. Therefore, NPCM-M char is more efficien han NP-M char for deecing small shifs in process when underlying process disribuion is normal, ligh ailed uniform and heavy ailed double exponenial. For uniformly disribued daa, boh NPCM-M and NP-M chars perform beer han normally and doubly exponenial daa. Conclusion In his sudy, wo nonparameric CUUM-ype conrol chars are developed for monioring process variabiliy. The performance of he proposed conrol chars is sudied by simulaion and compared wih corresponding hewhar-ype conrol chars under normal, ligh ailed and heavy ailed disribuions. Our simulaion sudy indicaes ha he NPCM-M and NPCM-M conrol chars are more efficien han NP- and NP-M conrol chars for deecing small shifs in process variabiliy for differen process disribuions. Boh NP-M and NP- conrol chars perform beer when underlying process disribuion is ligh ailed. Acknowledgemens Auhors would like o hank he Edior for commens and suggesions. The firs auhor would like o acknowledge he suppor by Universiy Grans Commission, New Delhi for award of Teacher Fellowship under Faculy Improvemen Program during he XII h plan. Youh Educaion and Research Trus (YERT) jairjp.com Zombade & Ghue, 4

5 Journal of Academia and Indusrial Research (JAIR) Volume 3, Issue June 4 57 References. Amin, R. and earcy, A. 99. A nonparameric exponenially weighed moving average conrol scheme. Commun. a. imulaion Compu. : Amin, R., Reynolds, M. and Bakir, Nonparameric qualiy conrol chars based on he sign saisic. Commun. a. Theory Meh. 4: Bakir,. 4. A disribuion-free hewhar qualiy conrol char based on signed-ranks. Qualiy Engg. 6: Bakir,. 6. Disribuion-free qualiy conrol chars based on signed-rank-like saisics. Commun. a. Theory Meh. 35: Bakir,. and Reynolds, M A non-parameric procedure for process conrol based on wihin-group ranking. Technomerics. : Chakrabori,., Vanderlaan, P. and Bakir,.. Nonparameric conrol chars: An overview and some resuls. J. Qual. Technol. 3: Chakrabori,., Vanderlaan, P. and Vandewiel, M. 4. A class of disribuion-free conrol chars. J. Royal a. oc. eries C: Appl. a. 53(3): Das, N. 8. Non-parameric conrol char for conrolling variabiliy based on rank es. Econ. Qual. Conrol. : Das, N. and Bhaacharya, A. 8. A new nonparameric conrol char for conrolling variabiliy. Qual. Technol. Quan. Managemen. 5(4): Ghue, V. 3. Disribuion-free conrol char for bivariae process. J. Acad. Indus. Res. (): Ghue, V. and hirke, D.. A nonparameric signed-rank conrol char for bivariae process locaion. Qual. Technol. Quan. Managemen. 9(4): Lehmann, E Nonparameric aisical Mehods Based on Ranks. Holden-Day, an Fransisco, California. 3. Mood, A On he asympoic efficiency of cerain nonparameric wo-sample ess. Ann. Mah. a. 5: Murakami, H. and Masuki, T.. A nonparameric conrol char based on he Mood saisic for dispersion. In. J. Advan. Manuf. Techn. 49: ukhame, B On cerain wo-sample nonparameric ess for variances. Ann. Mah. a. 8(): Youh Educaion and Research Trus (YERT) jairjp.com Zombade & Ghue, 4