Internatonal Journal of Industral Engneerng & Producton Management (22) March 22, Volume 22, Number 4 pp. 39-33 http://ijiepm.ust.ac.r/ 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 489-27, 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
http://ijiepm.ust.ac.r/ ISSN: 8-487 * () () MLE MLE * rassoul@ust.ac.r 2 3 Bult-n Confdence Set m.moradmanesh@gmal.com
L(.) X MLE p MLE (-) (-) T,2,...,, 2,...,T () (ˆ) V 2 3 V-mask Tabular Observaton (MLE)
,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
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
Type u kh hh.5 Lower-sde.3 2554.894 5.5 Upper-sde.7-6822 6823.756 5. Lower-sde.6 277.95 5. Upper-sde.3-549 5492.5 Lower-sde.3 2554.2523 5.9 5.5 Upper-sde.7-68 682.749 5 E T ˆ E T ARL seˆ se ˆ E T (PPM) E T se ˆ seˆ 3.54 99.83.5 96.82.26 5 5.54 8.47..7.2.28 97.33 98.9.26.28 25 3.8 2.4.38 99.82.32 35 2.48 6.25.58 4.45.48 39.25 6.66. 8.7.99 4 85.32 49.42.27 64.4.249 45 55 3.5 34.3 27.33 265.69.66.64 29.25 266.97.656.64 6 2.59 62.35.277 57.5.29 65 68.26 28.5.6 6.33.5 7 48.84 4.3..8.95 75 38.29 7.5.85 94.53.72 8 32. 3.94.7 9.97.62 85 9 27.86 24.86.77.4.6.52 9.55 89.8.58.56 95 22.62 2.5 99.52 99..47.43 89.2 89..55.55 ˆ ˆ seˆ seˆ MLE
ˆ ˆ ˆ ˆ (PPM) 45 55 5 25 35 4 6 65 7 75 8 85 9 95.459..326.94.224.8.46.67.87.49.46.29.9.3.6.5.5.6.2.5.25.27.38.4.53.5.7.6.9.68.4.73.24.76.37.82.74.48.555.47.44.344.293.265.88.86.8.4.49.43.7.5.5.7.35.44.64.84.95.25.28.6.64.9.99.24.23.233.26.247.288.263.88.626.685.575.54.497.398.4.267.286.59.66.76.69.27.24.25.28.55.7.98.27.42.86.9.233.237.269.283..322.32.362.333.393.35.88.7.77.667.63.599.48.5.332.367.24.28.99.94.36.33.34.39.74.92.28.66.84.238.242.294.298.334.35.367.395.385.439.399.475.44.97.746.826.726.698.672.546.576.388.434.242.264.2.5.44.4.42.49.92.4.55.2.22.284.288.345.35.388.47.42.454.438.52.45.54.465.939.779.866.768.75.725.62.637.436.49.277.34.4.36.52.49.5.58.9.33.8.232.255.323.329.389.395.434.454.466.56.484.555.497.594.57.977.874.953.878.893.866.779.85.6.68.47.454.224.27.9.85.9.3.82.22.289.365.389.48.48.555.559.6.624.628.68.639.725.644.762.652.987.924.977.928.948.925.87.897.723.788.59.558.293.282.23.5.25.4.244.292.372.466.487.59.588.666.668.75.73.726.783.733.822.735.85.739.992.953.986.956.97.954.923.94.8.856.6.635.354.335.53.43.57.77.297.352.443.545.566.673.67.745.747.779.85.794.85.799.883.797.95.8.995.97.99.972.98.97.952.965.856.9.667.695.47.382.8.68.86.29.345.46.54.6.632.738.735.84.88.833.86.843.897.846.922.844.939.847.996.982.993.982.986.982.969.979.896.93.723.745.456.424.27.92.24.238.388.45.556.662.69.788.787.848.855.872.9.88.93.882.948.879.96.882.997.989.994.989.989.988.979.987.925.952.769.785.498.462.23.24.238.265.428.49.64.76.738.828.83.883.892.93.928.97.952.99.964.97.972.99.998.993.996.993.992.993.985.992.945.967.86.87.538.498.254.235.262.289.465.527.646.744.779.86.866.99.99.925.948.928.966.929.975.929.98.929 ˆ ˆ ˆ E T seˆ seˆ MLE ARL E T ARL
E T.5,. ARL MLE ˆ MLE L.5.5 2.5427.5..995..5 2.5426...9885.5.5 2.5427.5..995 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 3.46 99.79.5 96.7.36 5.33.8.9 96.57.37 8..8.25 97.38.38 25 2.8 2.6.34 98.83.4 8.64 6.54.5.86.48 35 3.32 6.9.82.24.75 4 6.73 42.23.66 36.9.64 45 243.43 224.32.426 22.79.429 55 726.53 42.24.72 68.79.966 6 49.36 9.67.445 345..932 65 259.8 34.97.225 75.4.395 7 96.82 6.25.43 8.8.8 75 67.89 7.97.4.3. 8 52.8 3.99.82 95.6.74 85 42.8.53.69 93.67.67 9 36.64..6 92.73.64 95 32.28 99.3.54 92.9.63 29.7 98.6.49 9.74.63
.5,. E T E T se ˆ seˆ 4.74 99.74.6 89.46.7 5 7.46 99.87.22 9.8.72.4.4.29 9.5.74 25 6.47.83.39 92.72.77 24.59 5.33.56 95..83 35 38.83 4.64.9.72.97 4 69.63 4.35.73 7.52.54 45 259.8 228.95.448 94..43 55 35.5 258.22.562 22.84.535 26.69 59.84.253 5.72.24 72.9 26.73.56 96.22.38 55.93 2.95.2 9.24.9 75 46.49 5.99.88 87.48.2 8 4.45 2.4.72 86..8 85 36.33.37.62 85.3.6 33.25 99.5.55 84.57.4 95 3.9 98.55.49 84..3 29.4 98.9.45 84.2.2 (PPM) 6 65 7 9.5 ˆ (PPM) 5 25 35 4 45 55 6 65 7 75 8 85 9 95.46.47.328.44.227.4.47.36.9.29.48.23.2.3.7.6.4.2..5.22.3.36.23.52.3.7.35.88.38.3.4.9.45.37.43.75.383.557.299.42.233.296.79.95.3..86.53.47.2.9..6.3.5.58.4.88.7.23.94.6..93.2.225.3.256.39.285.4.82.564.69.47.546.382.42.296.273.22.63.45.8.78.32.32.6..47.25.88.65.34.4.82.54.23.79.274.98.36.23.355.225.39.232.882.66.773.585.636.497.486.396.339..29.2.6.7.42.44.23.4.63.36.5.89.72.55.232.2.29.243.34.27.387.288.432.34.47.37.97.77.829.662.74.585.554.48.396.37.25.25.29.36.52.55.29.7.78.45.4.3.27.96.276.26.34.33.396.336.446.358.494.378.534.392.939.75.87.74.756.65.6.552.445.434.287.299.5.62.62.67.34.2.92.55.63.36.238.234.33.3.384.359.443.396.495.422.545.442.587.457.976.834.955.833.899.82.79.769.626.659.432.49.24.278.4.8.59.38.54.97.26.234.363.393.458.54.542.57.67.64.664.64.7.657.749.668.986.884.978.885.953.884.882.864.745.789.54.62.35.369.4.6.83.54.24.33.336.39.455.54.557.628.644.696.7.73.76.749.82.756.836.76.99.98.987.99.975.92.933.92.825.864.629.75.383.445.75.2.4.67.248.64.398.37.527.585.633.7.79.773.78.798.826.89.862.8.888.82.994.942.99.942.984.943.96.94.88.9.7.78.442.57.29.236.23.8.287.9.452.42.587.644.693.768.775.823.832.843.872.85.9.849.92.85.995.959.993.958.989.959.976.959.98.94.759.83.495.56.24.269.42.9.323.23.499.46.637.689.742.8.82.86.87.876.94.88.927.879.944.88.997.97.995.97.992.97.985.972.945.96.86.869.543.66.269.298.59..357.233.54.494.68.727.782.843.854.887.899.92.928.94.946.92.958.93.998.98.996.979.994.98.99.98.963.973.843.897.588.647.297.326.75..387.252.577.523.79.757.88.869.883.99.922.92.945.923.959.92.968.922
- 5 25.9 efectve Percent (PPM) 35.8 Coverage Probablty.6 45 2.97.5 2.75.4 2.5.3 2.25 2..2.75 2 4 6 8 2 4 6 8 22 24 Value.5.35. 26 28 32 Average Cardnalty, 5, CS t : loge L t loge L ˆ loge L ˆ.353,.5,.75,2.,2.25,2.5,2.75,2.97 9.9 8 75 efectve Percent (PPM) 7 65.8 6 Coverage Probablty 4.7.7.6 95 55 85 2.97.5 2.75.4 2.5 2.25.3 2..75.2 Value.5.35. 2 4 6 8 2 4 6 8 22 24 26 28 32 34 36 38 Average Cardnalty, 5, Confdence Set
5 25.9 efectve Percent (PPM) 35 Coverage Probablty.8 4.7 45.6 2.97.5 2.75 2.5.4 2.25.3 2..75.2.5 Value.35. 2 4 6 8 2 4 6 8 22 24 26 Average Cardnalty, 5PPM,,.5.5, efectve Percent (PPM) 9.9 8 7.8 [] 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.,2.25,2.5,2.75,2.97 65 6.7.6 95 55 85 2.97.5 75 2.75 2.5.4 2.25 2..3.75 [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. 875-883. [3] Pgnatello, J., Samuel, R., Estmaton of the Change Pont of a Normal Mean n SPC Applcatons..5.2.35 Value. 2 4 6 8 2 4 6 8 22 24 26 28 32 34 36 38 4 42 44 46 48 5 52 54 Average Cardnalty, 5,,.5
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