On Efficient Monitoring of Process Dispersion using Interquartile Range

Transcription

1 Open Journal of Appled Scences Supplement01 world Congress on Engneerng and Technolog On Effcent Montorng of Process Dsperson usng Interquartle Range Shabbr Ahmad 1, Zhengan Ln 1, Saddam Akber Abbas, Muhammad Raz 3,4 1 Department of Mathematcs, Insttute of Statstcs, Zhejang Unverst, 31007, Hangzhou, Chna Department of Statstcs, Unverst of Auckland, New Zealand 3 Department of Statstcs, Quad--Azam Unverst, Islamabad Pakstan 4 Department of Mathematcs and Statstcs, Kng Fahd Unverst of Petroleum and Mnerals, Dhahran, 3161, Saud Araba Emal: 1 shabbrahmad786@ahoo.com Abstract: The presence of dsperson/varablt n an process s understood and ts careful montorng ma furnsh the performance of an process. The nterquartle range IQR) s one of the dsperson measures based on lower and upper quartles. For effcent montorng of process dsperson, we have proposed aular nformaton based Shewhart-tpe IQR control charts namel IQR r and IQR p charts) based on rato and product estmators of lower and upper quartles under bvarate normall dstrbuted process. We have developed the control structures of proposed charts and compared ther performances wth the usual IQR chart n terms of detecton ablt of shft n process dsperson. For the sad purpose power curves are constructed to demonstrate the performance of the three IQR charts under dscusson n ths artcle. We have also provded an llustratve eample to justf theor and fnall closed wth concludng remarks. Kewords:Aular Informaton, Bvarate Normal Dstrbuton, Control Carts, Interquartle Range, Lower and Upper Quartles, Power Curves. 1. Introducton Statstcal Process Control SPC) s a collecton of fundamental tools whch are used to montor process behavor. In the earl 190s Walter A. Shewhart developed control chartng as a useful tool of Statstcal Process Control SPC) to montor process parameters such as locaton, dsperson etc. The estence of varablt s unavodable n an process and ts careful montorng s necessar to mprove the performance of an process. The varablt n a process can be classfed n two parts namel natural and unnatural. Natural/normal varaton has a consstent pattern whle unnatural/unusual varaton has an unpredctable behavor over the tme. The presence of natural varaton n a process ensures that the process s ncontrol state, otherwse out-of-control. Control charts assst dfferentatng between natural and unnatural varatons and hence declarng the process to be n-control or out-ofcontrol. To montor process varablt [1] proposed usual range and standard devaton charts namel R and S charts). The effcenc of R chart s affected wth the ncrement n sample sze where as the performance of S chart becomes poor due to estence of outlers n data cf. []). Later on dfferent estmators of nterquartle range IQR) have been used to establsh desgn structures of dsperson charts such as: [3] and [4] have used nterquartle range b restrctng the poston of lower and upper quartles as nteger, whch become cause of some uneven patterns n desgn structure of control chart. Rocke [5] proposed IQR based R q chart whch out performs the R chart for detectng shfts n process dsperson n outler scenaro. To avod some rregulartes of R q chart, [] proposed a new method of usual IQR chart based on the defnton of [6]. Abbas & Mller [7] compared the performances of dfferent dsperson charts under normall and non-normall dstrbuted envronments and concluded that for small sample sze the IQR chart ehbts reasonable performance whle the performances of R and S charts are sgnfcantl nfluenced for hghl skewed process envronments. Much of the work related to dsperson control charts ma be seen n the bblographes of the above authors. In ths artcle we have proposed IQR control charts namel IQR r and IQR p charts to montor the process dsperson n Shewhart setup. These charts are based on rato and product estmators of lower and upper quartles of stud varable Y usng one aular varable X under bvarate normall dstrbuted process. The rest of the artcle s organzed as: Secton II provdes the desgn structure of IQR charts based on dfferent quantle estmators consdered here. In Secton III the performance of IQR charts are nvestgated under the assumpton of normalt. An llustratve eample s provded n Secton IV to justf our proposal and fnall the stud s concluded wth some recommendaton n Secton V.. Quantle Estmators and IQR Chartng Structures Let the qualt characterstc of nterest s Y e.g. nner dameter of shaft) and X be an aular characterstc e.g. Coprght 01 ScRes. 39

2 outsde dameter) assocated wth Y. Let Q ) & Q ) be the -quantle of Y & X respectvel and f Q ) ) & f Q ) ) be the values of denst functon at Q ) & Q ) respectvel whch can also be obtaned b the kernel method or the k th nearest neghbor cf. [8]). Also be the Cramer s coeffcent defned as:, ) ) 1 )) P where les between 0 and 1 11 dependng upon the choce of quantle; P, ) P X Q ) & Y Q ) ) cf. [9]). 11 Let & 1,,, n) be a sample of sze n to get estmated values of quantle of Y & X as Q ) & Q ) respectvel. We consder three estmators of Q ), one usual and two based on an aular characterstc X usng rato and product patterns) defned as: Usual : Q ) Qˆ ) u Rato : Q ) Q ) Q ) Q ) r. Product : Q ) Q ) Q ) Q ) p 1) It s to be mentoned that we are takng Q ) to be a known quantle of aular characterstc X. For the case of unknown Q ) we ma estmate t b applng two phase samplng procedure cf. [10] and [11]). The propertes of the estmators, gven n 1), can be easl obtaned up to frst order degree of appromaton, followng [10] and [1]. In our stud we have consdered normall dstrbuted process envronment under bvarate setup Y, X) wth denst functon gven as: ep[ 0.5 w / 1 )] <,, 0 f,, for 1 <, 1 1 ) where, w ) ) ) ) ), & are means of Y & X respectvel, & are varances of Y & X respectvel, s covarance between Y & X, and ) be the correlaton coeffcent between Y & X. The bvarate normal denst plot s gven as: Based on the estmators, gven n 1) for 0.5 & 0.75, we defne nterquartle range statstc as: IQR = Q 0.75) Q 0.5) u, r & p). 3) We ma defne the control chartng structures based on IQR u, r & p) to montor the dsperson parameter of qualt characterstc Y. The probablt lmts of IQR based chartng structures can be descrbed as: LPL IQRl wth F IQR IQRl l Prob Lmts: CL = IQRc UPL IQRu wth F IQR IQRu1u 4) where CL, LPL and UPL refer to the Central Lmt, Lower Probablt Lmt and Upper Probablt Lmt respectvel of IQR charts and s a prespecfed false alarm rate whch s equall dvded on both l u tals of the probablt dstrbuton of IQR to defne the probablt lmts. It s to be mentoned that we ma also defne K-sgma lmts of the structures based on IQR u, r & p) followng [13]. It s to be noted that a varet of senstzng rules are avalable n qualt control lterature whch are used to dfferentate between n-control and out-of-control states of process cf. [14], [15], [16]). In our stud we focus on frst senstzng rule to decde about process status for the control structures defned n 4). The frst rule s defned as: Smulate bvarate random samples, ) of sze n from the probablt model ) and compute the sample statstcs IQR s for each sample. Plot the values of IQR s aganst the control lmts defned n 4) or the appropratel defned K- sgma lmts. B frst senstzng rule, an value of IQR fallng outsde the control lmts ndcates an out-of control sgnal for the dsperson parameter of qualt characterstc Y. 3. Power Stud of IQR Charts To quantf the effcenc of a desgn structure, the dscrmnator power s ver famous performance measure n control chartng setups. In ths secton we have evaluated the effcenc of IQR charts under nvestgaton n terms of detecton ablt for shfts n process dsperson parameter and created power curves followng [17], [7] and [18]. The 0 n-control value of s consdered as whle the out-ofcontrol value s consdered as 1 whch can be defned n 1 0 terms of and as, where s amount of shft n process dsperson. It s generall desred that for ncontrol state of process the false alarm rate should be low/close to pre-fed value of whle for out-of-control 40 Coprght 01 ScRes.

3 state the power of charts should be hgh to detect the shfts n process parameters. In order to nvestgate the performance of the IQR u, r & p) charts n terms of sgnalng probablt, the power epresson can be defned as: 1 0 Power Pr[ IQR LPL or IQR UPL) ] 5) A Monte Carlo smulaton stud wth 100,000 replcatons s conducted under probablt model ) for dfferent parameter values and dfferent choces of n, and. B varng the values of from we have evaluated 5) and provded the resultng power curves of IQR charts n Fgures 1-3 for = 5, = 1, 0.50, 0.70 & 0.90, n 10 and In Fgures 1-3, s plotted on horzontal as and power values are plotted on vertcal as. The smbols IQR u, IQR r & IQR p refer to the power curves of IQR u, IQR r & IQR p charts respectvel. The power curve analss reveals the followng ponts for the chartng structures under dssecton. The usual IQR u chart performs better than rato and product tpe IQR r and IQR p charts for low correlatons cf. Fgure 1), whle for moderate and hgh correlatons IQR r chart outperforms the IQR u and IQR p charts cf. Fgures & 3). The performance of IQR r chart keeps mprovng wth the ncrease n the amount of correlaton between Y and X, whch s not the case wth the IQR p chart. The most nferor performance s ehbted b IQR p chart. The reason behnd ths nferort ma be due to the fact that IQR p chart s based on product estmator of quantle and accordng to [10], the product estmator s less effcent than usual estmator for 0. Fgure 1. Power Curves of IQR u, IQR r and IQR p charts under bvarate normal dstrbuton for n=10, =0.50 & =0.00 Fgure. Power Curves of IQR u, IQR r and IQR p charts under bvarate normal dstrbuton for n=10, =0.70 & =0.00 The performance of all the charts has a drect relatonshp wth the values of and n as epected. Fgure 3. Power Curves of IQR u, IQR r and IQR p charts under bvarate normal dstrbuton for n=10, =0.90 & = Illustratve Eample In order to justf our fndngs of power stud n Secton III, an eample s provded to compare the Coprght 01 ScRes. 41

4 performances of usual IQR u chart and an aular nformaton based IQR r chart. In real lfe eamples the varables Y and X ma refer as ) Y: the tensle strength n ps) and X: the outsde dameter n mm) to montor producton of steel wre; ) Y: producton of pharmaceutcal products n unts) and X: the temperate n C) n montorng of pharmaceutcal products etc. For the sad purpose we have smulated 30 samples each of sze n=10 from probablt model ) wth = 5, = 1 and The frst m 0 0 samples are generated from n-control state.e. 1 whereas the remanng m1 0 observatons are generated from an outof-control state wth.5 and computed the values of the chartng statstcs IQR u and IQR r. The resultng values are demonstrated n the form of control chart b plottng sample number on horzontal as and values of IQR u and IQR r on vertcal as n Fgure 4. The sold lnes refer to the control lmts and values of IQR r chart whle the dotted lnes refer to the control lmts and values of IQR u chart. Fgure 4. Control Chart Plots of IQR u and IQR r under bvarate normal dstrbuton for n=10, =0.90, =.5 & =0.000 It s obvous from Fgure 4 that after 0 th sample the IQR u chart has detected 6 out-of-control ponts whle the IQR r chart has ndcated 9 out-of-control sgnals. It mean that IQR r chart has gven 3 more out of control sgnals as compare to IQR u chart whch s n accordance wth the fndng of power stud n Secton III. 5. Summar, Concluson and Recommendatons For an mporved montorng of process dsperson, we have nvestgated Shewhart-tpe nterquratle range charts namel IQR r and IQR p charts. The desgn structures of these charts are based on rato and product estmators of lower and upper quartles of qualt charcterstc Y wth the asstance of an aular charcatersc X. For comparson purposes we have also ncluded the usual nterquratle range chart namel IQR u chart. We have observed that the detecton ablt of IQR r chart s posstvel releated wth the correlaton between Y & X. For low correlatons the IQR r chart offers lower detecton ablt than the usual IQR u chart but wth the ncreament n, IQR r chart outperforms IQR u and IQR p charts. The most nferor performance s ebted b IQR p chart for an posstvl correlated process envronmet because the product estmator s more effcent than usual estmator n case of negatve correlaton between Y and X. The scope of the stud ma be etended for dfferent contamnated process scenaros under Shewhart, EWMA and CUSUM setups. Moreover, the multvarate versons of these control charts ma be another drecton to be eplored. 6. Acknowledgement Ths work was supported b the Natonal Natural Scence Foundaton of Chna No ) and the Specalzed Research Fund for the Doctor Program of Hgher Educaton No ). REFERENCES [1] W.A. Shewhart, "Economc control of qualt manufactured product. D. Van Nostrand", reprnted b the Amercan Socet for Qualt Control n 1980, Mlwauker, WI), New York, [] M. Raz, "A Dsperson Control Chart", Communcatons n Statstcs - Smulaton and Computaton, 376), , 008. [3] D.M. Rocke, "Robust Control Charts", Technometrcs, 31), , [4] L.G. Tatum, "Robust Estmaton of the Process Standard Devaton for Control Charts", Technometrcs, 39), , [5] D.M. Rocke, " X Q and R Q Charts: Robust Control Charts", Journal of the Roal Statstcal Socet. Seres D The Statstcan), 411), , 199. [6] H.A. Davd, "Earl sample measures of varablt", Statstcal Scence, 134), , [7] S.A. Abbas, and A. Mller, "On proper choce of varablt control chart for normal and non-normal processes", Qualt and Relablt Engneerng Internatonal, 83), 79-96, 01. [8] B.W. Slverman, "Denst Estmaton for Statstcs and Data Analss", Chapman & Hall, London, [9] M.D. MartÃnez - Mranda, M. Rueda, and A. Arcos, "Lookng for optmal aular varables n sample surve quantle estmaton", Statstcs, 413), 41-5, 007. [10] P.V. Sukhatme, and B.V. Sukhatme, "Samplng Theor of Surves wth Applcaton", Iowa Statstcal Unverst Press, New York, [11] M.M. Rueda, A. Arcos, J.F. Muñoz, and S. Sngh, "Quantle estmaton n two-phase samplng", Computatonal Statstcs & Data Analss, 515), , Coprght 01 ScRes.

5 [1] M.M. Rueda, A. Arcos, and M.D. Martínez, "Dfference Etmators of Quantles n Fnte Populatons", TEST, 1), , 003. [13] D.C. Montgomer, "Introducton to Statstcal Qualt Control", 6, Wle, New York, 009. [14] R.J.M.M. Does, and B.F. Schrever, "Varables control chart lmts and tests for specal causes", Statstca Neerlandca, 464), 9-45, 199. [15] M.B.C. Khoo, "Desgns of runs rules schemes", Qualt Engneerng, 17, 7-43, 004. [16] M.V. Koutras, S. Bersms, and P.E. Maravelaks, "Statstcal process control usng shewhart control chart wth supplementar Runs Rules", Methodolog and Computng n Appled Probablt, 9, 07-4, 007. [17] M. Raz, and A. Saghr, "A mean devaton-based approach to montor process varablt", Journal of Statstcal Computaton and Smulaton, 79, , 009. [18] A. Saghr, Z. Ln, S.A. Abbas, and S. Ahmad, "The Use of Probablt Lmts of COM Posson Charts and ther Applcatons", Qualt and Relablt Engneerng Internatonal, 1-16, 01. Coprght 01 ScRes. 43