CHAPTER - IV STANDARDIZED CUSUM MEDIAN CONTROL CHART

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1 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess CHAPTER - IV STANDARDIZED CUSUM MEDIAN CONTROL CHART. Itroducto: I motorg e process mea, e Mea ( X ) cotrol charts, ad cumulatve sum (CUSUM) - X chart have bee vestgated extesvely by Page 954]. The X chart for motorg e process mea may sometmes lead to hgh level false alarms. But e sample meda X ~ s a robust estmator of locato for samples. Recetly, X ~ cotrol charts such as EWMA - X ~ chart by Castaglola ], Shewhart - X ~ chart by Motgomery 5] ad GWMA - X ~ charts by Yag ad Sheu 6] were developed successo to study e process shft usg meda. Subsequetly e CUSUM cotrol charts for meda have bee studed by may auors such as Ga 99], Woodall et al. 993], Lg Yag et al. ], etc. The CUSUM cotrol charts have bee appled to detect eve a small shft e process quckly whe e process has o outler.. Proposed meodology: Varadharaja ad Paduraga ] have establshed at e stadardzed cusum cotrol charts are hghly sestve detectg eve a small shft e process mea. I s chapter, a stadardzed CUSUM Meda cotrol chart s developed for motorg e process mea w FSS ad VSS. The performace of s chart s compared w e covetoal charts. 3. The CUSUM cotrol charts: Let X ad X ~ be e sample average ad sample meda of e subgroup respectvely, where subgroup s X s N (, ). For a sample of subgroups, e sample mea for e X j X, j / Ad e sample meda s 9

2 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess ~ X X X, ], ] ; X f s odd, ] ; f s eve where X, j] s e j order statstc for e sample. Based o e above statstcs, CUSUM meda cotrol charts are developed. 3.. The CUSUM cotrol charts Motgomery 5] has proposed a algormc (Tabular) CUSUM for motorg process mea. He has suggested oe sded upper ad lower cusums respectvely as C Max, X ( K) C ] () C Max,( K) X C ] () Where e startg values are about halfway betwee e target value C = detfyg e process shft quckly. That s, K. C =, K s e referece value ad s ofte chose o ad e out of cotrol value at we are terested If e shft s expressed stadard devato uts as ( / ), e K s oe half e magtude of e shft or ( or K ( ) e, K ( ) k x,where x ad k ad also deotes e stadard devato of e sample mea, deote e magtude of e process mea shft (multple of ). Let H be e decso terval, defed as H h x h 3

3 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess If eer or exceeds H, e process s cosdered to be out of cotrol. Motgomery has recommeded a geeral procedure for selectg H ad K. Accordg to at procedure, H h ad K k, where s e stadard devato of e sample varable used formg cusum. He has recommeded h=4 or 5 ad k=/ for a cusum to have good ARL propertes agast a shft of about e process mea. The same values for H ad K are take s chapter to develop a ew cocept. 3.. The tabular CUSUM - cotrol charts Based upo CUSUM - X cotrol charts developed by Motgomery 5], Lg Yag, et al. ] have proposed e CUSUM meda cotrol chart. Accordg to em, e upper ad lower CUSUMs are defed as respectvely. ~ Max, X ( K) ]. (3) ~ Max,( K) ]. (4) X W startg values of = =. The referece value K ca be represeted as K. If e shft s expressed stadard devato uts as ( ), e K d x Where d ~, (. Let M be e decso terval, defed as follows ) M m ~ m ~ x, ~, - s e stadard devato of sample meda for N (, ) For a specfc value of d, choose m to acheve e desred cotrol ARL. If eer or exceeds M, e process s cosdered to be out of cotrol. Reasoably oe ca take e value for M as fve tmes e process stadard devato. 3

4 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess 4. The Stadardzed CUSUM - cotrol chart: May researchers of e CUSUM prefer to stadardze e varable X, before performg e calculatos sce e stadardzed CUSUM performs better to cotrol e process varablty, reduce e specto ad testg costs. Accordg to Motgomery, Lg Yag, et al. ], e referece value K ad e decso terval M are to be chose as, ad K k ~ x, M m ~ x where ~ x s e stadard devato of e sample varable used formg cusum. Lg Yag, et al. ] gve m=4 or 5 ad k=/ wll geerally provde a cusum at has good ARL propertes agast a shft of about e process mea. The varable follows Z ~ ( X ) ; for =,, 3, m S Where X ~ s e sample meda of e subgroup X ca be stadardzed as Where S ( x ~ x ) The e stadardzed CUSUM calculated usg Max, Z K ] to be out of cotrol. Ad Max, K ] Z Startg w = =. If eer or exceeds M, e process s cosdered 5. Applcato Numercal llustrato: I order to establsh e proposed stadardzed CUSUM meda cotrol charts, umercal examples are cosdered. The FSS ad VSS are take respectvely to hghlght e usage. 6. CUSUM Meda cotrol chart w FSS: To check e performace of stadardzed CUSUM - cotrol chart, cosder a smulato producto process. Let e producto process follows ormal dstrbuto w mea 3

5 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess ad stadard devato 3. To apply e CUSUM - chart, let us smulate ormal varable w µ = ad σ = 3 for e frst samples of sze 5 each ad µ = 5 ad σ = 3 from sample owards. The smulated ormal observatos, sample medas ad stadard devatos are gve e followg table. Table Calculato of medas ad Stadard devatos S. No Sample observatos Meda SD:S Average Covetoal Meda cotrol chart: The covetoal Shewhart cotrol lmts for chart w subgroup sze = 5 are LCL = = UCL = = 5.98 Takg =.536, L = 3 for =5 accordg to Castaglola 998] The above table dcates a sgal at e 3 sample sce ts mea value exceeds e UCL. 33

6 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess 6.. Stadardzed CUSUM Meda cotrol chart: To compare s procedure w Stadardzed CUSUM - X ~, proceed as follows The CUSUM values are calculated as detaled below, The tal values are take as. The stadardzed ormal values are Z ~ ( X ) = S =.87 The Max, Z.5 ] Max,.87.5 ] ad Max,.5 Z ] Max,.5.87 ].373 Smlarly e oer values are computed ad gve Table. Table Computato of stadardzed CUSUMs S. No Z N H N L

7 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess The above table dcates at e mea value has shfted at e sample sce s greater a 5. Thus e stadardzed CUSUM - cotrol chart gves e exact tme at whch a shft e mea has occurred. oly at e I e above case t s foud from e Shewhart cotrol chart, e sgal was obtaed 3 sample. But as per e stadardzed CUSUM - cotrol specfcato e sgal was obtaed at e stadardzed CUSUM - sample, whch was e tme e shft has occurred. As far as e cotrol chart, oly 55 sample observatos were eeded to get a alarm. 7. CUSUM Meda cotrol chart w VSS: Cosder a producto process of forged psto rgs w outer dameter (OD) as mm. The qualty characterstcs of e psto rg OD ca be checked. I s case, e cotrol producto process s modeled as a ormal process whose mea s equal to w stadard devato 5mm. Sample observatos are smulated takg = ad for e frst sample (subgroups) ad = ad from sample (subgroups) owards. The sample szes are take as 9 or 4. The smulated sample observatos ad subgroup averages are show Table 3 35

8 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess Table 3 Calculato of medas ad Stadard devatos S. No Sample Varables Meda SD:S Average Covetoal Meda cotrol chart w VSS: The Shewhart cotrol lmts for X ~ charts w varable sample sze are LCL = L ~, = UCL = L ~, = 5.58 for sample sze = 9 ad UCL = ; LCL = for sample sze = 4. The above table dcates at e process s out of cotrol at e 5 sample, sce e sample mea exceeds e UCL for = 9. But a shft has occurred from e sample owards. I s case, sample observatos have bee take to produce e sgal e process. 36

9 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess 7.. Stadardzed Meda cotrol chart w VSS: By usg e subgroup average X ~ stadardzed CUSUM - X ~ for each sample s calculated. The tal values are take as: Z ~ ( X ).89 - =. 399 S.87 Max, Z.5 ] Max, ] ad Max,.5 Z ] Max, ].998 ad so o. The computatos are gve e Table 4 below Table 4 Computato of Stadardzed CUSUMs S. No Z N H N L

10 A Study o Process Varablty usg CUSUM ad Fuzzy Cotrol Charts Ph.D Thess The stadardzed CUSUM - X ~ from e above Table 4 dcates at e process s out of cotrol at sample 3 sce s greater a 5. But, e covetoal Shewhart cotrol chart has sgaled at e 5 subgroup for whch e total sample specto was. The stadardzed CUSUM - X ~ cotrol chart system w tal values takes 87 sample observatos to sgal w varable sample sze (VSS). Thus, e stadardzed CUSUM - X ~ cotrol chart system s more ecoomcal a e covetoal Shewhart - cotrol chart w FSS ad VSS. 8. Cocluso: The Stadardzed CUSUM - X ~ cotrol chart s used to motor e process mea. It s easly detfy eve a small shft e process mea. Ths meod s compared w e covetoal Shewhart cotrol chart for Fxed Sample Sze (FSS) ad Varable Sample Sze (VSS). The stadardzed CUSUM - X ~ cotrol chart s foud to be more ecoomcal a e covetoal Shewhart - X ~ cotrol chart for FSS ad VSS. The stadardzed CUSUM meda cotrol chart takes lesser umber of observatos to sgal. I e above dscussed two cases, e stadardzed CUSUM w FSS foud to be more ecoomcal, because t takes oly 55 observatos to detfy e sgal e process where as t was 87 e case of stadardzed CUSUM w VSS. The ARL for CUSUM Meda cotrol chart s e best sutable meodology a e covetoal Meda chart for FSS ad VSS. Ths meod ca be exteded to Markova evromet, by selectg sample szes as doe by Paduraga ]. 38