Run Length Distribution of Synthetic Double Sampling Chart

Transcription

1 Iteratioal Joural of Applied Egieerig Research ISSN Volume, Number 4 (7) pp Research Idia Publicatios Ru egth Distributio of Sythetic Double Samplig Chart You Huay Woo ASASIpitar, PERMATApitar Natioal Gifted Cetre, Uiversiti Kebagsaa Malaysia, 46 Bagi, Selagor, Malaysia Orcid: Abstract The average ru legth (AR) has bee used as a commo characteristic to exae the performace of a cotrol chart The computatio of AR requires quality practitioers to detere the shift size i advace This requiremet is restricted as the quality practitioers may ot kow the shift size i advace I light of this, expected average ru legth (EAR) is proposed to evaluate the performace of a sythetic type chart whe the shift size is radom I this paper, sythetic double samplig (SDS) chart was ivestigated based o AR ad EAR Results demostrate that EAR ca be employed as a alterative performace measure whe the process shift size is ukow i advace Keywords: Quality; cotrol chart; expected average ru legth INTRODUCTION Quality is a crucial elemet to be cosidered i the world of productio ad maufacturig This is because quality products or services will icrease customers satisfactio towards these products or services I view of this, improvig quality is a key factor for a successful busiess Statistical Process Cotrol (SPC) is a collectio of statistical tools, with the objective of reducig variability i the process Hece, this will idirectly produce high quality products or services Cotrol chart is oe of the SPC tools that is widely adopted i the maufacturig ad service idustries [] I 94, Dr Walter A Shewhart developed the first cotrol chart ad amed it as the Shewhart chart The Shewhart chart is commoly used i detectig large process mea shifts However, oe major limitatio of the Shewhart chart is that it is isesitive i detectig small ad moderate mea shifts This has motivated may researchers to search for ways to improve the sesitivity of the cotrol chart The sythetic type chart is a alterative method proposed to ehace the performace of the Shewhart chart It is well kow that the sythetic type chart is a powerful statistical process cotrol tool i moitorig the process mea There are two types of sythetic chart, amely, the sythetic chart ad sythetic double samplig (SDS) chart I this paper, the SDS chart is cosidered due to its advatages i speedig up detectio ability i compariso to the sythetic chart ad double samplig (DS) chart [] The performace of a cotrol chart is importat i order to select the appropriate cotrol chart to be used i a process Oe commo performace measure is average ru legth (AR) The AR refers to the average umber of samples plotted o a cotrol chart before it sigals a out-of-cotrol [] By usig AR as the performace measure, quality practitioers eed to specify the magitude of shift Nevertheless, i practice, practitioers may ot kow the shift size i advace I view of this, it is crucial to exae the expected average ru legth (EAR) as a alterative performace measure, where the computatio of EAR does ot require the practitioers to detere the process shift size Istead, the EAR computes the expected value of the AR over the distributio fuctio of the shift size [4] Other studies evaluatig the performace of the cotrol chart whe the process shift size is ukow ca be foud i [5] ad [6], to ame a few I this regard, the performace of the SDS chart was ivestigated usig AR ad EAR The paper is structured as follows: Sectio briefly itroduces the SDS chart, ad this is followed by reviewig the ru legth properties of the SDS chart Performace aalysis usig AR ad EAR is preseted i Sectio 4 Fially, a coclusio is draw OVERVIEW OF THE SDS CHART The SDS chart was itroduced by [] It is a itegratio of the double samplig (DS) sub-chart ad a coforg ru legth (CR) sub-chart I 974, Croasdale [7] itroduced the first DS cotrol chart Daudi et al [8] ad Daudi [9] developed a modified DS chart Iriato ad Shiozaki [] demostrated that the performace of the ewly developed DS chart ([8]; [9]) is superior to the origial DS chart ([7]) Sice the, the DS cotrol chart has bee extesively studied by several researchers icludig amog other, [] ad [] A graphical view of the DS sub-chart is preseted i Figure Here, the regios i Figure are defied as: I [, ], I [, ) (, ], I (, ) (, ) ad I [, ] Meawhile, the CR sub-chart has oly oe 4 468

2 Iteratioal Joural of Applied Egieerig Research ISSN Volume, Number 4 (7) pp Research Idia Publicatios lower limit, ie A CR value is the umber of coforg uits betwee two ocoforg uits (iclusive the edig ocoforg uit), i Y, i / (b) If,i is i I, the i th samplig time is ocoforg, or (c) If,i is i I, take the secod sample with, ad (i) calculate the sample mea, Y Y, ad, i j j (ii) i i combied samples, Y Note that Y Y Y If i is i I 4, the i, i, i i th samplig time is coforg Otherwise, the i th samplig time is ocoforg Note that if,i is i I or (,i is i I ad i ot i I 4 ), it merely idicates the existece of a ocoforg sample The CR sub-chart is required to detere whether or ot the sample is out-of-cotrol If CR >, the process is icotrol Otherwise, the process is declared as out-of-cotrol Thus, appropriate actio(s) are required to eliate the assigable cause(s) i Yi / (a) First sample THE RUN ENGTH PROPERTIES OF THE SDS CHART Assume that the process is idepedetly ad idetically distributed (iid) havig a ormal distributio with kow icotrol mea,, ad i-cotrol stadard deviatio, et P be the probability of a ocoforg samplig time ad deoted as follows []: (b) Combied samples Figure : DS sub-chart The operatio of the SDS chart is elaborated as follows Take the first sample of size from the moitorig process The, compute the sample mea ad stadardised statistic, ad i i Y Y, i j j respectively Usig Figure,, Y,, (a) if,i is i I, the i th samplig time is coforg, or with P Pr I a ad, i P Pa P () b P Pr I ad I b i 4, i c rc z c rc z ( z) dz * zi Here, ( ) ad ( ) are the cumulative distributio fuctio (cdf) ad probability desity fuctio (pdf) of the stadard ormal distributio, respectively I additio, * I,,, ad r ad c r () () 469

3 Iteratioal Joural of Applied Egieerig Research ISSN Volume, Number 4 (7) pp Research Idia Publicatios Fially, the AR is computed as: AR P, P where is the lower cotrol limit for the CR sub-chart Computatio of the AR requires the practitioers to detere the process shift size i advace I practical applicatio, practitioers may ot kow the process shift size i advace as they do ot have historical kowledge of the process I view of this, the EAR criterio is used as the performace measure of the SDS chart ad is defied as: EAR f AR d (4) (5) Note that f is the probability desity fuctio of the magitude of the shift i a process, ie PERFORMANCE ANAYSIS OF THE SDS CHART I geeral, performace of the cotrol chart is measured usig average ru legth (AR) There are two commo ARs used, amely, the i-cotrol AR AR ad out-of cotrol AR AR Table displays the AR for differet combiatios of sample size, = {, 5} ad shift size, = {, 5, 9,, 5, 9} I Table, the optimal chartig parameters,,,,, are displayed i colums 8 Results i Table were obtaied based o these parameters Note that the optimal chartig parameters,,,,, will give a iteded AR 74 whe = For example, whe = ad = 5, the optimal chartig parameters are,,,,, = (, 6, 8, 584, 867, 8) ad the correspodig AR is 4, ad yet achieved AR 74 I reality, the shift size,, may ot be kow i advace because quality practitioers do ot have ay historical kowledge o the process or without ay experiece i moitorig the process to detere the shift size Furthermore, if a practitioer deteres a particular shift size ad employs the correspodig optimal chartig parameter, the performace of the SDS chart will be sigificatly deteriorated if a differet shift size is occurred i the process Therefore, it is crucial to measure the process usig the performace measure that takes ito accout a radom shifts, ie EAR Here, two EARs are usually of iterest, amely the icotrol EAR, EAR ad out-of-cotrol EAR, EAR Note that the EAR is set at 74 For compariso purposes, Table summarises the EAR for the same combiatios The shift iterval, ie,, is show i colums Table : Optimal chartig parameters,,,,, ad the correspodig ARs for = {, 5} whe AR = 74 AR

4 Iteratioal Joural of Applied Egieerig Research ISSN Volume, Number 4 (7) pp Research Idia Publicatios I order to iclude the exact shifts size cosidered i Table,, = (, ) ad, = (, ) were cosidered For illustratio, the shift iterval, = (, ) ad, = (, ) iclude the shifts, = {, 5, 9} ad = {, 5, 9}, respectively For example, whe = 5, = ad =, the optimal chartig parameters,,,,, = (4, 9, 58,, 6, 9) yields the lowest EAR = 8 By cosiderig = 5 ie, for the same value, the AR = 4 is obtaied usig the optimal chartig parameters,,,,, = (4, 4, 5, 544, 9647, ) from Table This fidig shows that the AR value obtaied usig the optimal chartig parameters,,,,,, by imisig EAR i Table is almost similar with the AR value obtaied usig the optimal chartig parameters,,,,,, AR i Table, as log as imisig by, Table : Optimal chartig parameters,,,,, ad the correspodig EARs for = {, 5} whe EAR = 74 EAR CONCUSION I this paper, it is clearly show that EAR ca be used i place of AR for the SDS chart whe the process shift size is ukow I the productio ad maufacturig idustries, it is a usual situatio where quality practitioers do ot have historical kowledge o which process shift size to be implemeted Hece, quality practitioers ca employ the proposed optimal chartig parameters based o imisig EAR This study is based o the i-cotrol process parameters, ie mea, ad stadard deviatio, are kow I reality, the process parameters are usually ukow Hece, this study ca be exteded to cosider the SDS chart with estimated process parameters based o imisig EAR REFERENCES [] Haq, A, Brow, J, ad Moltchaova, E, 5, New sythetic cotrol charts for moitorig process mea ad process dispersio, Qual Reliab Eg It, (8), pp 5-5 [] Khoo, M B C, ee, H C, Wu,, Che, C H, ad Castagliola P,, A sythetic double samplig cotrol chart for the process mea, IIE Tras, 4(), pp -8 [] Faraz, A, Heuchee, C, ad Saiga, E, 7, The p chart with guarateed i-cotrol average ru legths, Qual Reliab Eg It, (5), pp [4] Wu,, Shamsuzzama, M, ad Pa, E S, 4, Optimizatio desig of cotrol charts based o Taguchi s loss fuctio ad radom process shifts, It J Prod Res, 4(), pp 79-9 [5] Castagliola, P, Celao, G, ad Psarakis, S,, Moitorig the coefficiet of variatio usig EWMA charts, J Qual Techol, 4(), pp [6] You, H W, Khoo, M B C, Castagliola, P, ad Qu,, 6, Optimal expoetially weighted movig average charts with estimated parameters based o media ru legth ad expected media ru legth, It J Prod Res, 54(7), pp [7] Croasdale, R, 974, Cotrol charts for a doublesamplig scheme based o average productio ru legths, It J Prod Res, (5), pp [8] Daudi, J J, Duby, C, ad Trécourt, P, 99, Plas de cotrôle double optimaux (maȋtrise des procédés et cotrôle de receptio), Revue De Statistique Appliquée, 8, pp [9] Daudi, J J, 99, Double samplig X chart, J Qual Techol, 4(), pp [] Iriato, D ad Shiozaki, N, 998, A optimal double samplig cotrol chart, It J of Id Eg Theory, Appl Pract, 5, pp 6-4 [] You, H W, Khoo, M B C, ee, M H, ad Castagliola, P, 5, Sythetic double samplig X chart with estimated process parameters, Qual Techol Quat Maag, (4), pp

5 Iteratioal Joural of Applied Egieerig Research ISSN Volume, Number 4 (7) pp Research Idia Publicatios [] Castagliola, P, Oprime, P C, ad Khoo, M B C, 7, The double samplig s chart with estimated process variace, Commu Stat Theory Methods, 46(7), pp