Ming Ha Lee. School of Engineering, Computing and Science Swinburne University of Technology (Sarawak Campus), Malaysia.

Transcription

1 VRIBLE SMPLING RTE HOTELLING S T 2 CONTROL CHRT WITH RUNS RULES Ming H Lee School of Engineering, Computing nd Science Swinburne University of Technology (Srwk Cmpus), Mlysi mhlee@swinburne.edu.my BSTRCT This study investigtes the properties of the vrible smpling rte Hotelling s T 2 chrt with. The verge time to signl checking of the control chrt is derived using Mrkov chin pproch. Numericl comprisons show tht the vrible smpling rte Hotelling s T 2 chrt with performs better thn the vrible smpling rte Hotelling s T 2 chrt without for smll nd moderte process men shifts. OPSOMMING Dié studie ondersoek die eienskppe vn Hotelling se T 2 -grfiek met vernderlike proefnemingstempo en lopiereëls. Die gemiddelde tyd wt ndui wnneer om die kontrolekrt te ondersoek, word fgelei deur vn n Mrkovkettingbendering gebruik te mk. Numeriese vergelykings wys dt Hotelling se T 2 -grfiek met vernderlike proefnemingstempo en lopiereëls beter presteer s Hotelling se T 2 -grfiek met vernderlike proefnemingstempo sonder lopiereëls vir klein en mtige prosesvernderings. South fricn Journl of Industril Engineering My 2012 Vol 23(1): pp

2 1. INTRODUCTION Runs rules re used to increse the performnce of control chrts in detecting smll shifts. The Western Electricl Compny [6] suggested set of to be used with the X control chrt. prisi et l. [1] investigted the performnce of Hotelling s chi-squre control chrt with supplementry. They concluded tht the verge run length of the chrt with supplementry decreses considerbly when there is shift in the process. Koutrs et l. [4] introduced simple modifiction of the chi-squre control chrt, which mkes use of the notion of runs to improve the sensitivity of the chrt in the cse of smll nd moderte process men vector shifts. stndrd control chrt uses fixed smpling rte (FSR) in which the smple size nd the smpling intervl re fixed. In the literture, severl pproches such s vrible smpling rte (VSR) hve been suggested to improve the performnce of the control chrt, in which the smple size nd the smpling intervl re llowed to vry ccording to the smple sttistic. The VSR procedure is to use smll smple size nd long smpling intervl if there is no indiction of chnges in the process men. However, it uses lrge smple size nd short smpling intervl if there is n indiction tht the process men my hve shifted from the trget vlue. The VSR Hotelling s T 2 chrt ws developed by prisi & Hro [2]. They showed tht there is significnt improvement in the verge time to signl vlue for the Hotelling s T 2 chrt with the VSR feture. Cui & Reynolds [3] considered the X chrt with, in which the smpling intervl is llowed to vry depending on wht is observed from the current smple nd pst smples. Comprisons with the usul fixed smpling intervl chrt with show tht the vrible smpling intervl X chrt with is more efficient. This study considers the Hotelling s T 2 chrt with in which the smple size nd smpling intervl re llowed to vry ccording to current nd pst smple sttistics. This chrt is clled vrible smpling rte (VSR) Hotelling s T 2 chrt with. 2. VRIBLE SMPLING RTE HOTELLING S T 2 CHRT WITH RUNS RULES The performnce of the control chrt cn be determined by the verge number of smples to signl (NSS), defined s the expected number of smples from the strt of the process until the control chrt signls. nother performnce mesure of the control chrt is the verge number of observtions to signl (NOS), defined s the expected number of items from the strt of the process until the control chrt signls. When the smpling intervl cn be vried, the performnce of the control chrt cn be determined by the verge time to signl (TS). The TS is defined s the expected length of time from the strt of the process until the control chrt signls. This zero-stte TS is n pproprite mesure if it is ssumed tht the process strts to shift t the strt of the process. In prctice, however, in mny cses the process my strt in control, nd shift t some rndom time in the future where the time of shift my fll between two smples. In this sitution, it is more relistic to consider the stedy stte TS (STS), which llows for the shift to occur between two smples (Reynolds et l. [5]). Therefore in this study the performnce of the out-of-control condition for the VSR Hotelling s T 2 chrt with is evluted bsed on the STS insted of the zero-stte TS. The VSR Hotelling s T 2 chrt with consists of three regions: the sfety region (below the wrning limit WL), the wrning region (between the wrning limit WL nd the control limit CL), nd the out-of-control region (bove the control limit CL). Let (n 1, h 1 ) be pir of lrge smple size nd short smpling intervl, nd let (n 2, h 2 ) be pir of smll smple size nd long smpling intervl, such tht n 2 < n < n 1 nd h 1 < h < h 2, where n denotes the in-control verge smple size nd h denotes the in-control verge smpling intervl. Let Y S nd Y W denote the number of consecutive smpling points flling in the 123

3 sfety region nd the wrning region respectively. Then the pir of smple size nd smpling intervl for the next smpling point will be () (n 2, h 2 ) if Y S s 1 ; (b) (n 1, h 1 ) if (i) s 1 < Y S s 2 or (ii) Y W w, where s 1, s 2 nd w re positive integers with s 1 < s 2. Consider p correlted qulity chrcteristics s being mesured simultneously, nd the p qulity chrcteristics follow multivrite norml distribution with men vector nd covrince mtrix. Let X t represent the t th smple verge vector for the p qulity chrcteristics with smple of size n i where i = 1, 2. For monitoring process men, the 2 0chrt sttistic Tt ni( Xt 0) ( Xt 10) is plotted on the Hotelling s T 2 chrt with control limit CL = 2 p, where 0 nd 0 re the in-control process men vector nd the incontrol covrince mtrix respectively, nd 2 is the upper percentge point of the p, chi-squre distribution with p degrees of freedom. Let p 1 = Pr( w 2( p) CL)/ Pr( 2( p) CL ) be the proportion of smpling point flling in the wrning region, given tht this smpling point does not fll in the out-of-control region, nd let p 2 = Pr( 2( p) w)/ Pr( 2( p) CL ) be the proportion of smpling point flling in the sfety region, given tht this smpling point does not fll in the out-of-control region, where 2( p ) denotes the centrl chi-squred rndom vrible with p degrees of freedom. Then p 1 n 1 + p 2 n 2 = n (1) nd p 1 n 1 + p 2 n 2 = h. (2) It is noted tht p 1 n 1 + p 2 n 2 = 1. The wrning limit WL cn be determined from Eqution (1): WL = F -1 n n1 Pr( 2( p) CL) (3) n n 2 1 where F -1 is the cumultive distribution function of the stndrd norml distribution. From Eqution (2) we cn obtin h 2 = ( h p 1 h 1 )/p 2 by fixing p 1, p 2, h nd h 1. The performnce of the Hotelling s T 2 chrt depends solely on the distnce of the out-ofcontrol men vector μ 1 from the in-control men vector μ 0. The mgnitude of shift is expressed by 1 ( μ μ ) Σ ( μ μ ) (4) When the process is out-of-control, the smple sttistic t is distributed s non-centrl chi-squre distribution with p degrees of freedom nd non-centrlity prmeter c = n i 2. The VSR Hotelling s T 2 chrt with is considered to be out-of-control if: (i) s consecutive smpling points fll in the sfety region ( T 2 t w); or (ii) w + 1 consecutive smpling points fll in the wrning region (w < T 2 t CL); or (iii) one smpling point flls in the out-of-control region ( T 2 t > CL). Mrkov chin pproch is used to evlute the properties of the VSR Hotelling s T 2 chrt with. Consider the Mrkov chin with the following sttes: Stte 1: one smpling point flls in the wrning region Stte 2: two consecutive smpling points fll in the wrning region Stte 3: three consecutive smpling points fll in the wrning region Stte w: w consecutive smpling points fll in the wrning region Stte w + 1: one smpling point flls in the sfety region Stte w + 2: two consecutive smpling points fll in the sfety region Stte w + 3: three consecutive smpling points fll in the sfety region Stte w + s 2 : s 2 consecutive smpling points fll in the sfety region Stte w + s 2 + 1: bsorbing stte (out-of-control) T 2 124

4 Let p S nd p w denote the probbility for smpling point flling in the sfety region nd the wrning region respectively. Then p S = Pr( 2 (p,c) WL) (5) nd p W = Pr(w < 2 (p,c) WL) (6) where 2 (p,c) denotes the non-centrl chi-squred rndom vrible with p degrees of freedom nd non-centrlity prmeter c = n i 2 (n i for p S is given s n 2, wheres n i for p W is given s n 1 ). Suppose we consider the with w = 3, s 1 = 4 nd s 2 = 7. Then the sttes for the Mrkov chin re defined s follows: Stte 1: one smpling point flls in the wrning region Stte 2: two consecutive smpling points fll in the wrning region Stte 3: three consecutive smpling points fll in the wrning region Stte 4: one smpling point flls in the sfety region Stte 5: two consecutive smpling points fll in the sfety region Stte 6: three consecutive smpling points fll in the sfety region Stte 7: four consecutive smpling points fll in the sfety region Stte 8: five consecutive smpling points fll in the sfety region Stte 9: six consecutive smpling points fll in the sfety region Stte 10: seven consecutive smpling points fll in the sfety region Stte 11: bsorbing stte (out-of-control) nd the trnsitionl mtrix of the trnsient sttes for the Mrkov chin is given s pw pw pw pw pw pw P 5 pw pw 6 pw pw 7 pw pw 8 pw pw 9 pw pw 10 pw pw where the lst row nd column represent the bsorbing stte. Let s denote the initil probbility vector where ll components equl zero, except where the (w + 1) th component equls one. Let h denote the vector of smpling intervl with the (w + 1)th, (w + 2)th, (w + 3)th, nd (w + s 2 s 1 + 1)th elements equl h 2 nd h 1 otherwise. Let n denote the vector of smple size with the (w + 1)th, (w + 2)th, (w + 3)th, nd (w + s 2 s 1 + 1)th elements equl n 2 nd n 1 otherwise. When the process is in-control, the zerostte NOS, the zero-stte NSS, nd the zero-stte TS respectively re clculted s: in-control NOS = s(i Q 0 ) -1 1 (7) in-control NSS = s(i Q 0 ) -1 n (8) nd in-control TS = s(i Q 0 ) -1 h, (9) where Q 0 is the (w + s 2 ) (w + s 2 ) sub-mtrix of the trnsient sttes for the in-control process ( = 0), obtined by deleting the lst row nd column of mtrix P; I is the identity mtrix; nd 1 is column vector of ones. The in-control verge smple size n nd the incontrol verge smpling intervl h cn be expressed s: 125

5 n nd h in - control NOS = in - control NSS (10) in - control TS = in - control NSS. (11) Let b denote the vector of modified stedy-stte probbility with elements b i = i h i /h where i = 1, 2,, (w + s 2 ), nd is the in-control stedy-stte probbility vector, which cn be obtined by solving the following equtions: Q 01 = nd 1 = 1 (12) where Q 01 is obtined by dividing ech element of Q 0 by the sum of its row. Therefore the out-of-control STS cn be obtined by: STS = [(I Q ) -1 h 0.5h (13) where Q is the (w + s 2 ) (w + s 2 ) sub-mtrix of the trnsient sttes for the out-of-control process ( 0), obtined by deleting the lst row nd column of mtrix P. 3. PERFORMNCE OF VRIBLE SMPLING RTE HOTELLING S T 2 CHRT WITH RUNS RULES Tbles 1, 2, nd 3 give the vlues of the in-control TS nd the out-of-control STS of the VSR Hotelling s T 2 chrt with for different vlues of w, s 1, s 2, respectively. From Tbles 1 nd 3, the flse lrm rte decreses nd the out-of-control STS increses when the vlues of w nd s 2 increse; wheres from Tble 2, the flse lrm rte increses nd the out-of-control STS decreses when the vlue of s 1 increses. Chrt (w, s 1, s 2) = (6, 4, 8) (w, s 1, s 2) = (8, 4, 8) (w, s 1, s 2) = (10, 4, 8) Tble 1: The vlues of in-control TS nd out-of-control STS of VSR Hotelling s T 2 chrt with runs rules for p = 2, (n 1, h 1) = (5, 0.1), (n 2, h 2) = (1, 1.863), CL = , WL = nd different vlues of w = 6, 8, 10 ( n = 3 nd h = 1). Chrt (w, s 1, s 2) = (8, 6, 12) (w, s 1, s 2) = (8, 8, 12) (w, s 1, s 2) = (8, 10, 12) Tble 2: The vlues of in-control TS nd out-of-control STS of VSR Hotelling s T 2 chrt with runs rules for p = 2, (n 1, h 1) = (5, 0.1), (n 2, h 2) = (1, 1.863), CL = , WL = 1.386, different vlues of s 1 = 6, 8, 10 ( n = 3 nd h = 1). 126

6 Chrt (w, s 1, s 2) = (7, 5, 8) (w, s 1, s 2) = (7, 5, 10) (w, s 1, s 2) = (7, 5, 12) Tble 3: The vlues of in-control TS nd out-of-control STS of VSR Hotelling s T 2 chrt with runs rules for p = 2, (n 1, h 1) = (5, 0.1), (n 2, h 2) = (1, 1.863), CL = , WL = 1.386, different vlues of s 2 = 8, 10, 12 ( n = 3 nd h = 1). Tbles 4 to 7 give the vlues of the in-control TS nd the out-of-control STS of the VSR Hotelling s T 2 chrt with nd without for p = 2, four qulity chrcteristics, incontrol verge smple size of n = 3 nd 5, nd in-control verge smpling intervl of h = 1.0. The tbles lso give the in-control TS nd the out-of-control STS of the FSR Hotelling s T 2 chrt with nd without. The control limit is CL = p 2,0.004 for the VSR nd FSR Hotelling T 2 2 chrts with, where p, is the upper percentge point of the chi-squre distribution with p degrees of freedom. The wrning limit is WL = p 2,0.5 for the VSR Hotelling s T 2 chrt with. The in-control verge smple size n nd incontrol verge smpling intervl h of the VSR control chrt re mtched with the fixed smple size n 0 nd the fixed smpling intervl h 0 of the FSR control chrt respectively. This mens tht the vlues of (n 1, h 1 ) nd (n 2, h 2 ) for the VSR control chrt re selected such tht the VSR control chrt hs the sme in-control performnce s the FSR control chrt in the sense tht the flse lrm rte or the in-control TS of the VSR control chrt is the sme s the FSR control chrt. Comprisons of the VSR Hotelling s T 2 chrt with nd without in Tbles 4 to 7 show tht the improve the performnce of the VSR Hotelling s T 2 chrt for smll nd moderte process men shifts. The tbles lso show tht the VSR Hotelling s T 2 chrt with performs better thn the FSR Hotelling s T 2 chrt with nd without for wide rnge of process men shifts, except for lrge shifts. Suppose tht it is desirble to improve the performnce of the VSR Hotelling s T 2 chrt by dding with (w, s 1, s 2 ) = (9, 4, 10); (n 1, h 1 ) = (4, 0.1); (n 2, h 2 ) = (2, 1.885); CL = ; nd WL = to monitor process with p = 4 qulity chrcteristics (in-control TS = ). Then the pir of smple size nd smpling intervl for the next smpling point will be (n 2, h 2 ) = (2, 1.885) if the number of consecutive points flling in the sfety region ( T 2 t 3.357) is less thn or equl to s 1 = 4; wheres the pir of smple size nd smpling intervl for the next smpling point will be (n 1, h 1 ) = (4, 0.1) if the number of consecutive points in the sfety region ( T 2 t 3.357) is more thn s 1 = 4 nd less thn or equl to s 2 = 10; nd if the number of consecutive points in the wrning region (3.357 < Tt ) is less thn or equl to w = 9. The process is considered to be out-of-control if (s 2 + 1) = 11 consecutive smpling points fll in the sfety region ( T 2 t 3.357), or (w + 1) = 10 consecutive smpling points fll in the wrning region (3.357 < T 2 t ). The results in Tble 6 show tht this VSR Hotelling s T 2 chrt with performs better thn the corresponding VSR Hotelling s T 2 chrt without for smll nd moderte process men shifts, nd this control chrt lso performs better thn the corresponding FSR Hotelling s T 2 chrt with nd without. 127

7 Chrt VSR chrt with runs rules VSR chrt without FSR chrt with FSR chrt without (w, s 1, s 2) = (9, 4, 10) (n 1, h 1) = (4, 0.1) (n 2, h 2) = (2, 1.885) (n 1, h 1) = (4, 0.1) (n 2, h 2) = (2, 1.9) CL = n 0 = 3 CL = n 0 = 3 CL = CL = WL = WL = WL = Tble 4: The vlues of in-control TS nd out-of-control STS of VSR nd FSR Hotelling s T 2 chrts with nd without for p = 2, n = 3 nd h = 1. Chrt VSR chrt with runs rules VSR chrt with-out FSR chrt with FSR chrt without (w, s 1, s 2) = (9, 4, 10) (n 1, h 1) = (7, 0.1) (n 2, h 2) = (3, 1.883) (n 1, h 1) = (7, 0.1) (n 2, h 2) = (3, 1.9) CL = n 0 = 5 CL = n 0 = 5 CL = CL = WL = WL = WL = Tble 5: The vlues of in-control TS nd out-of-control STS of VSR nd FSR Hotelling s T 2 chrts with nd without for p = 2, n = 5 nd h = 1. Chrt VSR chrt with runs rules VSR chrt without FSR chrt with FSR chrt without (w, s 1, s 2) = (9, 4, 10) (n 1, h 1) = (4, 0.1) (n 2, h 2) = (2, 1.885) (n 1, h 1) = (4, 0.1) (n 2, h 2) = (2, 1.9) CL = n 0 = 3 CL = n 0 = 3 CL = CL = WL = WL = WL = Tble 6: The vlues of in-control TS nd out-of-control STS of VSR nd FSR Hotelling s T 2 chrts 128 with nd without for p = 4, n = 3 nd h = 1.

8 Chrt VSR chrt with runs rules VSR chrt without FSR chrt with FSR chrt without (w, s 1, s 2) = (9, 4, 10) (n 1, h 1) = (7, 0.1) (n 2, h 2) = (3, 1.883) (n 1, h 1) = (7, 0.1) (n 2, h 2) = (3, 1.883) CL = n 0 = 5 CL = n 0 = 5 CL = CL = WL = WL = WL = Tble 7: The vlues of in-control TS nd out-of-control STS of VSR nd FSR Hotelling s T 2 chrts with nd without for p = 4, n = 5 nd h = CONCLUSIONS The ppliction of to the VSR Hotelling s T 2 chrt shows tht the out-of-control STS chieved improvements for smll nd moderte process men shifts. Thus dding runs rules improves the performnce of the VSR Hotelling s T 2 chrt. The Mrkov chin pproch is used to clculte the in-control TS nd the out-of-control STS of the control chrts. From the numericl results, the VSR Hotelling s T 2 chrt with is lso superior to the FSR Hotelling s T 2 chrt with nd without in detecting process men shifts. REFERENCES [1] prisi, F., Chmp, C.W. & Grci-Diz, J.C performnce nlysis of Hotelling s 2 control chrt with supplementry, Qulity Engineering, 16, pp [2] prisi, F. & Hro, C.L comprison of T 2 control chrt with vrible smpling schemes s opposed to MEWM chrt, Interntionl Journl of Production Reserch, 41, pp [3] Cui, R.Q. & Reynolds, M.R., Jr X -chrts with nd vrible smpling intervls, Communictions in Sttistics Simultion nd Computtion, 17, pp [4] Koutrs, M.V., Bersimis, S. & ntzoulkos D.L Improving the performnce of the chisqure control chrt vi, Methodology nd Computing in pplied Probbility, 8, pp [5] Reynolds, M.R., Jr, min, R.W., rnold, J.C. & Nchls, J X chrts with vrible smpling intervls, Technometrics, 30, pp [6] Western Electricl Compny Sttisticl qulity control hndbook, mericn Telephone nd Telegrph Compny. 129