A Comparative Study of the Monitoring Performance for Weighted Control
Charts
Shuenn-Ren Cheng
1, Yan-Hung Yeh
2, and Ming-Hung Shu
31
Department of Business Administration, Cheng Shiu University, Taiwan, R.O.C.
2,3
Department of Industrial Engineering & Management, National Kaohsiung University of
Applied Sciences, Taiwan, R.O.C.
E-mail :
1[email protected],
2[email protected] w,
3[email protected]
Abstract
Statistical process control (SPC) is the method that monitors process quality characteristic. Through control charts, one can detect whether the present process malfunction. However, some annoyances may arise while the engineers need to choose appropriately control chart under different levels of process malfunction. The main objective of this research is using step-by-step procedures to present a comparative study of the monitoring performance for Shewhart, cumulative sum (CUSUM), exponentially weighted moving average (EWMA) and generally weighted moving average (GWMA) control charts. While the process means or the standard deviations were changing in different levels, the performance of each weighted control chart can be then compared by using average run length (ARL). A rule of thumb for selecting better control schemes is provided as a truthfully reference to help engineers in choosing the more appropriate control charts immediately when the assignable causes occurred.
1. Introduction
An important objective of statistical process control (SPC) is to detect quickly the occurrence of assignable causes of process variations, so that the process can be investigated and any required corrective action can be taken before many nonconforming units are manufactured. The corrective action is the elimination of variability in the process. It may be impossible to completely eliminate variability, but the control chart is an effective tool in reducing variability as much as possible.
The difference among control charts Shewhart, cumulative sum (CUSUM), exponentially weighted moving average (EWMA) [2][3][4][5] and generally weighted moving average (GWMA) [11] are to do with each charting technique using the data generated
by the production process. The weighted function of four control charts was showed in figures 1(a)-1(d).
However, some annoyances may arise while the engineers need to choose appropriately control chart under different levels of process variation. The main objective of this research is to do a comparative study of the monitoring performance for weighted control charts. According to a fixed in-control average run length (ARL), Monte Carlo numerical simulation is used to determine the parameters of control charts. While the process mean or standard deviation was changing in different levels, the performance of each weighted control chart based on ARL can be then compared, each run ends when either control limit is exceeded and each ARL is average number of 100,000 runs.
Among the comparing procedures, the setting of the parameters in each control chart is displayed by the profile so that it could be immediately implement during online process monitoring by engineers. With the simulation results for each weighted control chart, we use a better control scheme for monitoring the manufacturing process application.
Figure 1. Data weighting for control charts
causes existing in the process. Finally, a real-word example on chip resistors manufacturing process is investigated to illustrate the applicability of the results.
6. References
[1] S. V. Crowder, “A simple method for studying run-length distributions of exponentially weighted moving average charts”, Technometrics, 29, 1987, pp. 401-407.
[2] S. V. Crowder, “Design of exponentially weighted moving average schemes”, Journal of Quality Technology, 21, 1989, pp. 155-162.
[3] F. F. Gan, “An optimal design of quality control charts”,
Journal of Quality Technology, 23, 1991, pp. 279-286.
[4] J. S. Hunter, “The exponentially weighted moving average”, Journal of Quality Technology, 18, 1986, pp. 203– 210.
[5] J. M. Lucas, “The design and use of cumulative sum quality control schemes”, Journal of Quality Technology, 8, 1976, pp. 1-12.
[6] J. M. Lucas, and M. S. Saccucci, “Exponentially weighted moving average control schemes: properties and enhancements”, Technometrics, 32, 1990, pp. 1-29.
[7] Montgomery, D. C., Introduction to Statistical Quality
Control, Five Edition, John Wiley & Sons, New York, 2005.
[8] E. S. Page, “Continuous inspection schemes”, Biometrika, 41, 1954, pp. 100-114.
[9] S. W. Roberts, “Control chart tests based on geometric moving averages”, Technometrics, 1, 1959, pp. 239-250. [10] P. B. Robinson, and T. Y. Ho, “Average run lengths of geometric moving average charts by numerical methods”,
Technometrics, 20, 1978, pp. 85–93.
[11] S. H. Sheu, and T. C. Lin, “The generally weighted moving average control chart for detecting small shifts in the process mean”, Quality Engineering, 16(2), 2003, pp. 209-231.
[12] Shewhart, W. A., Economic Control of Quality, D. Van Nostran Co., New York, NY, 1931.
Table 1. Values of ARL with various process means shifted
ʳ Shewhartʳ CUSUMʳ EWMA
ʳ ʳ ʳ ʳ k 0.25 k 0.50ʳ k 0.75ʳ k 1.00 k 1.25ʳ k 1.50ʳ O 0.10 O 0.20 O 0.25 O 0.50 ̆˻˼˹̇ʳ ʳ L 3.092ʳ ʳ L 8.576 L 5.074ʳ L 3.539 L 2.666 L 2.105 L 1.710 L 2.826 L 2.966 L 3.004 L 3.072 0.00Vʳ 499.910ʳ 499.618 499.994ʳ 499.659ʳ 499.147ʳ 499.625ʳ 499.572ʳ 500.441 499.977 499.610 500.408 0.25Vʳ 375.920ʳ 93.818ʳ 145.199ʳ 200.717ʳ 250.519ʳ 292.167ʳ 327.728ʳ 77.740 143.364 167.501 255.020 0.50Vʳ 202.333ʳ 30.025ʳ 37.742ʳ 56.255ʳ 80.893ʳ 109.508ʳ 140.074ʳ 5.531 31.589 41.952 87.287 0.75Vʳ 103.819ʳ 16.515ʳ 16.337ʳ 21.157ʳ 29.955ʳ 42.439ʳ 58.417ʳ 2.000 8.312 13.171 34.274 1.00Vʳ 54.9090ʳ 11.150ʳ 9.490ʳ 10.543ʳ 13.620ʳ 18.837ʳ 26.263ʳ 1.966* 2.966 4.953 15.854 1.50Vʳ 18.010ʳ 6.579ʳ 4.828ʳ 4.458ʳ 4.784ʳ 5.716ʳ 7.358ʳ 1.907* 1.932 2.023 5.093 2.00Vʳ 7.271ʳ 4.540ʳ 3.054ʳ 2.556ʳ 2.459ʳ 2.620ʳ 3.003ʳ 2.552 1.842 1.832 1.795* 3.00Vʳ 2.154ʳ 2.682ʳ 1.607ʳ 1.280ʳ 1.194ʳ 1.182*ʳ 1.207ʳ 1.540 1.501 1.486 1.431 GWMA ʳ ʳ q 0.50ʳ ʳ q 0.75ʳ q 0.80ʳ q 0.90ʳ 0.50 w ʳ w 0.75ʳ w 0.80ʳ w 0.90ʳʳw 0.50ʳw 0.75 w 0.80 w 0.90 w 0.50 w 0.75 w 0.80 w 0.90 w 0.50ʳw 0.75ʳw 0.80 w 0.90 ̆˻˼˹̇ʳ ʳL 3.084ʳL 3.077ʳL 3.077ʳL 3.074ʳʳL 3.063ʳL 3.028 L 3.021 L 3.008 L 3.050 L 3.001L 2.993L 2.978 L 2.998 L 2.882ʳL 2.864 L 2.839 0.00V 500.615 499.252 500.832 500.635 500.563 500.132 499.611 499.718 500.096 499.523 500.365 499.777 499.322 500.189 500.845 500.589 0.25V ʳ 227.400ʳ 240.582ʳ 244.376ʳ 249.564ʳ 101.948ʳ 134.567 140.628 152.953 72.246 108.812 116.095 129.688 11.825*ʳ 39.407ʳ 46.604 61.846 0.50V ʳ 70.774ʳ 78.865ʳ 80.779ʳ 84.063ʳ 17.751ʳ 29.386 31.779 36.597 9.446 19.262 21.652 26.537 2.059*ʳ 2.392ʳ 2.670 3.675 0.75V ʳ 26.301ʳ 30.244ʳ 31.084ʳ 32.644ʳ 4.755ʳ 8.209 9.135 11.029 2.791 4.430 5.057 6.565 1.988ʳ 1.984ʳ 1.983* 1.985 1.00Vʳ 11.558ʳ 13.764ʳ 14.221ʳ 15.011ʳ 2.446ʳ 3.186 3.457 4.115 2.043 2.201 2.286 2.554 1.977ʳ 1.970ʳ 1.969 1.967* 1.50Vʳ 3.758ʳ 4.403ʳ 4.547ʳ 4.817ʳ 1.945ʳ 1.953 1.958 1.979 1.940 1.934 1.933 1.932 1.933ʳ 1.916ʳ 1.913 1.910* 2.00Vʳ 2.176ʳ 2.334ʳ 2.375ʳ 2.461ʳ 1.855ʳ 1.847 1.846 1.843 1.853 1.841 1.839 1.835 1.840ʳ 1.810ʳ 1.805 1.798* 3.00V ʳ 1.535ʳ 1.535ʳ 1.536ʳ 1.538ʳ 1.525ʳ 1.511 1.508 1.503 1.520 1.500 1.497 1.491 1.499ʳ 1.453ʳ 1.446 1.436
*Indicates a smaller value of each level of process mean shifted among four control charts. Table 2. A rule of thumb for selecting better
control schemes in various range of mean shifted
Range of mean
shifted Proposed scheme
0.25 ~ 0.50 GWMA (q 0.90, w 0.50 and L 2.998) 0.75 GWMA (q 0.90, w 0.80 and L 2.864) 1.00 ~ 1.50 EAMA (GWMA (O 0.10 and L 2.826)
0.90, 0.90
q w and L 2.839) 2.00 EAMA (GWMA (O 0.50 and L 3.072)
0.90, 0.90
q w and L 2.839) 3.00 CUSUM (k 1.25 and L 2.105)
Table 3. A rule of thumb for selecting better control schemes in various ranges of standard
deviation changed Range of standard
deviation changed Proposed scheme 1.25 ~ 1.50 Shewhart (L 3.092)
1.75 ~ 3.00 GWMA Shewhart (L 3.092)
