Estimating Time of a Simple Step Change in Nonconforming Items in High-Yield Processes

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Internatonal Journal of Industral Engneerng & Producton Management (22) March 22, Volume 22, Number 4 pp. 39-33 http://ijiepm.ust.ac.

Unversty of Scence and Technology, Kamran Peynabar, Ph.

1 Internatonal Journal of Industral Engneerng & Producton Management (22) March 22, Volume 22, Number 4 pp Estmatng Tme of a Smple Step Change n Nonconformng Items n Hgh-Yeld Processes Rassoul Noorossana*, Kamran Peynabar & Mona Moradmanesh Rassoul Noorossana, Industral Engneerng epartment, Iran Unversty of Scence and Technology, Kamran Peynabar, Ph.. Student of Industral and Operatons Engneerng, Unversty of Mchgan, Ann Arbor, MI , Mona Moradmanesh, Industral Engneerng epartment, Islamc Azad Unversty South Tehran Branch, Tehran, Iran Keywords Statstcal Process Control (SPC), Change Pont, Smple Step Change, Bult-n Estmator, Cumulatve Sum control chart (), Exponental Weghted Movng Average control chart (), Average Run Length (ARL), Confdence Set ABSTRACT Knowng about the real tme of a change n the parameter(s) of a statstcal process would enable users to dentfy root causes more quckly and precsely. ue to the senstvty and mportance of reachng zero defects n hgh qualty processes, to be aware of the change tme would be so precous. In ths paper, we consder the performance of the Maxmum Lkelhood Estmator n comparson wth bult-n change pont estmators of Cumulatve Sum () and Exponental Weghted Movng Average () control charts. Usng lkelhood functon approach, we also proposed a confdence set for the change pont. The results of a Monte-Carlo smulaton have shown the prorty of the MLE n dentfyng the real tme of the change. 22 IUST Publcaton, IJIEPM. Vol. 22, No. 4, All Rghts Reserved * Correspondng author. Rassoul Noorossana Emal: rassoul@ust.ac.r

2 ISSN: * () () MLE MLE * rassoul@ust.ac.r 2 3 Bult-n Confdence Set m.moradmanesh@gmal.com

3 L(.) X MLE p MLE (-) (-) T,2,...,, 2,...,T () (ˆ) V 2 3 V-mask Tabular Observaton (MLE)

4 ,ˆ ARL (C) Z X Z X (Z ) LCL E Z L Var Z, UCL E Z L Var Z Var Z E Z L H max, H k H X L mn, L k L X X.25 L hl H hh H u, u hh Q L v, hl v kh, kl (SPRT) ARL ARL ˆ max : H, ˆ max : L. L, k kl kh Z loge loge 2 L 2 UCL CL LCL kber c, hber h, uber w c c. 2 L 2 k H c, hh ch c, u w 3 Robustness 2 Corrected ffuson Sequental Probablty Rato Tests

5 N ˆ ˆ seˆ seˆ se T ˆ max : Z ˆ max : Z T f Z UCL, f Z LCL.,ˆ T τ MLE T ˆ ˆ ˆ ˆ ˆ ARL τ (C) ˆ ˆ

6 Type u kh hh.5 Lower-sde Upper-sde Lower-sde Upper-sde Lower-sde Upper-sde E T ˆ E T ARL seˆ se ˆ E T (PPM) E T se ˆ seˆ ˆ ˆ seˆ seˆ MLE

7 ˆ ˆ ˆ ˆ (PPM) ˆ ˆ ˆ E T seˆ seˆ MLE ARL E T ARL

8 E T.5,. ARL MLE ˆ MLE L ˆ ˆ ˆ MLE PPM PPM ˆ ˆ E T seˆ MLE.2 se ˆ ˆ ˆ ˆ.5,.5 E T (PPM) E T 5 se ˆ seˆ

9 .5,. E T E T se ˆ seˆ (PPM) ˆ (PPM)

10 efectve Percent (PPM) 35.8 Coverage Probablty Value Average Cardnalty, 5, CS t : loge L t loge L ˆ loge L ˆ.353,.5,.75,2.,2.25,2.5,2.75, efectve Percent (PPM) Coverage Probablty Value Average Cardnalty, 5, Confdence Set

11 efectve Percent (PPM) 35 Coverage Probablty Value Average Cardnalty, 5PPM,,.5.5, efectve Percent (PPM) [] Noorossana, R., Abd, S., Saghae, A., Paynabar, K., Identfyng the Tme of a Change n Hgh Yeld Processes, Proceedng of 4th Internatonal Industral Engneerng Conference, Tehran, Iran, 5. Coverage Probablty.353,.5,.75,2.,2.25,2.5,2.75, [2] Noorossana, R., Saghae, A., Peynabar, K., Abd, S., Identfyng the Perod of a Step Change n HghYeld Processes, Qualty and Relablty Engneerng Internatonal, 9; 25: pp [3] Pgnatello, J., Samuel, R., Estmaton of the Change Pont of a Normal Mean n SPC Applcatons Value Average Cardnalty, 5,,.5

12 Qualty Technology and Quanttatve Management, 6; 3(3): pp [4] Perry, M.B., Pgnatello, J.J., JR., Smpson, R., Estmatng the Change Pont of a Posson Rate Parameter for SPC Applcatons, (under revew). [5] Perry M.B., Pgnatello, J.J., JR., Estmaton of the Change Pont of the Process Fracton Nonconformng n SPC Applcatons, Internatonal Journal of Relablty, Qualty and Safety Engneerng, 5; 2(2): pp [6] Xe, M., Goh, T.N., Kuralman, V., Statstcal Models and Control Charts for Hgh-Qualty Processes, Kluwer Academc Publshers, Norwell, 2. [7] Brook,., Evans,.A., An approach to the probablty dstrbuton of run length. Bometrka, 99; 59: [8] Reynolds, M.R., Jr. Stoumbos, Z.G.. A Chart for Montorng a Proporton when Inspectng Contnuously. Journal of Qualty Technology. 999; 3: 87 8 [9] Chang, T.C., Gan F.F., Chartng Technques for Montorng a Random Shock Process, Qualty and Relablty Engneerng Internatonal, 999; 5: pp [] Page, E.S. Cumulatve Sum Control Charts. Technometrcs 96; 3(): -9. [] Roberts, S.V., Control Chart tests Based on Geometrc Movng Averages, Technometrcs 959; : pp [2] Quesenberry, C.P., Geometrc Q Charts for Hgh Qualty Process Journal of Qualty Technology, 995; 27(4): pp [3] Borror, C.M., Montgomery,.C., Runger, G.C., Robustness of the Control Cchart to NonNormalty. Journal of Qualty Technology, ;3(3): pp [4] Nshna, K., A Comparson of Control Charts from the Vewpont of Change Pont Estmaton, Qualty and Relablty Engneerng Internatonal, 992; 8: pp [5] Box, G.E.P., Cox,.R., An Analyss of Transformatons, Journal of the Royal Statstcal Socety. Seres B (Methodologgal), 964; 26(2): pp [6] Segmond,., Boundary Crossng Probabltes and Statstcal Applcatons, The Analyss of Statstcs, 986; 4(2): pp

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