Numerical Comparisons of Quality Control Charts for Variables

Transcription

1 Global Vrtual Coferece Aprl, Nuercal Coparsos of Qualty Cotrol Charts for Varables J.F. Muñoz-Rosas, M.N. Pérez-Aróstegu Uversty of Graada Facultad de Cecas Ecoócas y Epresarales Graada, pa jfuoz@ugr.es; perez@ugr.es Abstract Cotrol charts for varables ca be obtaed by usg both saple varaces ad saple rages. Cotrol charts ca be affected by varous paraeters, such as the saple sze ad the uber of saples used to obta the cotrol lts. Whe calculatg a cotrol chart, the ga of usg the saple varaces stead the saple rages s ot clear for a specfc stuato. Assug dfferet scearos ad Mote Carlo sulatos, cotrol chart for varables based upo saples varaces are eprcally copared to cotrol chart based upo saples rages. Keywords- Mote Carlo sulatos; qualty cotrol; oral dstrbuto; varace; rage. I. INTROUCTION Research o qualty volves a rage of cocers about deftos, practces ad such specfc echass as statstcal qualty cotrol (QC). These techques are used to cotrol the qualty of the product by aalyzg oe or ore product characterstcs. The ost wdely used techques QC are cotrol charts. Cotrol chart s a tool used to detere f a busess or aufacturg process s a state of statstcal cotrol. [3] developed ths techque ad provded a fraework for decdg whether the varato the result s due to assgable causes. He created a atheatcal odel that assues the Cetral Lt Theore, so t s usual to adopt ea ad rage type charts. Recet research shows that the X cotrol chart s very sple to uderstad, pleet ad desg, ad ay be ore sutable ay QC applcatos, whch both the ea ad varace of a varable eed to be otored (see [4]). A dspesable assupto for the correct developet of cotrol charts s that the process paraeters are assued kow but, practce, the process paraeters are rarely kow. May studes do ot cosder the effects of -cotrol paraeter estato o cotrol chart propertes ad recoeded alteratve estators for ea ad stadard devato. I ths sese, [2] ad [4] cosder several robust locato estators for X cotrol chart. Cotrol charts ca be affected by varous paraeters, such as the saple sze ad the uber of saples used to obta the cotrol lts. Whe calculatg a cotrol chart, the ga of usg the saple varaces stead the saple rages s ot clear for a specfc stuato. Assug dfferet scearos ad Mote Carlo sulatos, cotrol chart for varables based upo saples varaces are eprcally copared to cotrol E. Álvarez-Verdejo Uversdad Poltécca de Cartagea Cetro Uverstaro de la efesa. Acadea Geeral de Are a Javer (Murca), pa Ecar.alvarez@cud.upct.es chart based upo saples rages. The paper proceeds as follows: the et secto suarzes the lterature revew about cotrol charts for varables ad detfes gaps the prevous research. Followg ths, we develop several uercal coparsos order to copare uercally cotrol chart based upo saples varaces ad saple rages. The paper cocludes wth the a cotrbutos. II. CONTROL CHART FOR VARIABLE Cotrol chart s the ost cooly used tool for otorg the producto process. Ths tool s sply a graph that shows whether a saple of observatos falls wth the oral rage of varato, whch s defed by usg the called cotrol lts. Thus, a process s out of cotrol whe a plot of a set of observatos reveals that oe or ore saples fall outsde the cotrol lts. There ets alteratve tests to aalyze out-of-cotrol stuatos. For eaple, a process s sad to be out of cotrol whe at least eght successve saples fall o the sae sde of the ceter le. However, ths study s based upo the dea of coparg two cotrol charts by aalyzg the percetage of saples whch fall outsde the cotrol lts, hece the study of alteratve tests to aalyze whether a process s out of cotrol s a topc whch s beyod the scope of ths paper. The dfferet characterstcs that ca be easured by cotrol chart ca be dvded to varables ad attrbutes. X-bar chart, R chart ad chart are soe eaples of cotrol chart for varables. We assue X-bar charts ths paper. Let N(, ) be the varable of terest, where s the true process ea ad s the true process stadard devato. We assue that the stadards ad are ukow, hece they eed to be estated by usg saple forato. Assug ths scearo, cotrol lts are obtaed by selectg prevous saples wth sze each oe. It s coo to assue that each saple wth sze s selected fro a lot. We assue that each lot has sze N. The average of the observatos the -th saple, wth, s gve by be defed by,,, ad the ceter le of the X-bar chart ca CL, where s the grad st Global Vrtual Coferece Busess Maageet

2 average. Cotrol lts ca be obtaed by usg or R j j a( ) ( R 2, where / 2 ad R ) are, respectvely, the stadard devato ad the rage for the -th saple. Cotrol lts based o the stadard devato () are gve by LCL A 3 ad UCL A 3 LCL R A 2 R UCL R A 2 R, where A2 ad 3 lts based o the rage (R) are gve by ad GLOBAL VIRTUAL, whereas cotrol A are costats based o. Tables for A2 ad A3 costats ad for varous values of ca be see by Motgoery (2009) Apped A VI. The ga of usg the X-bar chart based o coparso to the X-bar chart based o R ay deped o the paraeters, N, or, whch are prevously defed. III. NUMERICAL COMPARION Assug dfferet scearos, Mote Carlo sulatos are ow carred out to copare the ga of usg X-bar charts based o stead usg X-bar chart based o R. For ths reaso, we geerated observatos fro a Noral dstrbuto wth ea 0 ad stadard devato 0.5,, 2. At the frst terato ru, cotrol lts were obtaed by selectg saples wth sze. I practce, t s coo to cosder that saples wth sze are selected fro a lot wth N observatos, hece ths stuato s also cosdered our eprcal coparsos. We cosdered the values 5, 20, 50, N 25,00,000 ad values of betwee 3 ad 25. Whe cotrol lts are obtaed, 000 saples wth sze are selected fro lots wth sze N, ad t s studed whether such saples falls outsde the cotrol lts. At the secod terato ru, ew saples wth sze are obtaed order to calculate the ew cotrol lts ad saples are selected to aalyze f they are out of cotrol. Ths process was repeated 000 tes. The crteru to copare the dfferet cotrol charts s the eprcal probablty (EP) of the varous saples of fallg sde the cotrol lts,.e., value for cotrol chart based o 3 sga crteru. Note also that EP s related to the eprcal Type I error (ETI), whch s defed as the eprcal probablty of saples falls outsde the cotrol lts eve though o specal cause are operatg. I partcular, EP=-ETI. Note that the Type I error s fed at for the 3 sga crteru. Fgure gves the values of EP for the varous cotrol charts wth 0. 5 ad uder dfferet scearos. We observe that the value of N does ot a pact o the perforace of the varous cotrol charts. Ths was epected because saples are selected fro the sae dstrbuto. We also observe that cotrol charts geerally have a slar perforace whe s saller tha 5. Whe 5 ad 5, cotrol charts obta values of EP saller tha the crtcal value , whch dcates that the eprcal Type I error s larger tha the fed Type I error. However, the cotrol chart based o s better tha the cotrol chart based o R ths stuato. Whe 20 ad 5, the cotrol chart for has a very good perforace, wth values of EP close to 99.73%. Whe 50 ad 5, the values of EP for the cotrol chart based o are slghtly larger tha 99.73%, whereas the values of EP for the cotrol chart based o R are slghtly saller tha 99.73%. Results derved fro the sulato study are very slar whe the true process stadard devato takes other values (Fgures 2 ad 3), hece the value of does ot a relevat pact o the perforace of the varous cotrol charts. IV. CONCLUION Global Vrtual Coferece Aprl, Whe stadards ad are ukow, cotrol lts for X-bar charts ca be obtaed by usg the saple varaces or the saple rages. The perforace of such cotrol chart ay deped o varous paraeters, such as, or. Assug Mote Carlo sulatos ad dfferet scearos, cotrol charts based o ad R have bee copared, ad soe relevat coclusos have bee obtaed. Frst, eprcal results dcate that values of saller tha 5 ca gve a larger Type I error for both cotrol charts. ecod, results dcate that a value of close to 20 ca be a good choce, specally for the cotrol chart based o, whch acheves values of EP close to the requred 99.73%. Thrd, we observed that the cotrol chart based o geerally has a better perforace that the cotrol chart based o R. Fally, we also cocluded that the values of N ad does ot a relevat pact o the perforace of the varous X-bar charts. EP, EP j Pj j where P f the saple the -th terato ru ad the j j-th saple s sde the cotrol lts ad P 0 otherwse. Note that EP should be close to , whch s the crtcal j P j st Global Vrtual Coferece Busess Maageet

3 Global Vrtual Coferece Aprl, Fgure. Values of EP for X-bar charts based o ad R. The true process stadard devato s st Global Vrtual Coferece Busess Maageet

4 Global Vrtual Coferece Aprl, Fgure 2. Values of EP for X-bar charts based o ad R. The true process stadard devato s. st Global Vrtual Coferece Busess Maageet

5 Global Vrtual Coferece Aprl, Fgure 3. Values of EP for X-bar charts based o ad R. The true process stadard devato s 2. REFERENCE [].C. Motgoery, tatstcal qualty cotrol. A oder troducto. 6th ed. New York, Wley, [2] M. choohove, H. Nazr, M. Hafz, R. oes, Robust Locato Estators for the X Cotrol Chart, J. Qual. Tech. Vol. 43, No. 4, pp , 20. [3] W.A. hewhart, W.A, Ecooc cotrol of qualty of aufactured product. Mlwaukee, AQC Qualty Press, 93. [4] M. Yag, Z. Wu, K. Lee ad M. Khoo, The X cotrol chart for otorg process shfts ea ad varace. Iter. J. Prod. Resear. Vol 50, No 3, pp , 202. [5] Y. Zhag ad P. Castaglola, Ru rules X charts whe process paraeters are ukow, Iter. J. Relab. Qual. af. Eg. Vol. 7, No. 4, pp , 200. st Global Vrtual Coferece Busess Maageet