Statistical Process Control

Transcription

1 FH MAINZ MSC. INTERNATIONAL BUSINESS Statistical Process Control Application of Classical Shewhart Control Charts February Amelia Curry Matrikel-Nr.: Prepared for: Prof. Daniel Porath Due Date: January 6, 2010

2 2 Table of Contents I. Introduction... 3 II. Statistical Process Control... 3 III. Control Charts... 4 IV. Application of Control Charts... 6 IV.1. Background... 6 IV.2. Assumptions... 6 IV.3. Subgroup Analysis... 9 IV.3.1. Shewhart s and Control Charts... 9 IV.3.2. R and Control Charts...13 V. Conclusion Bibliography Appendix 1. Overall Measurement Data Appendix 2. Subgroup Data and Calculation... 23

3 3 I. Introduction Recent economic recession has necessitated hard and soft savings for many organizations. Yet, some see this as an opportunity for quality engineers to establish the effects of good quality management (Nichols & Houry, 2009). Strategically, it can assist organizations in orientating itself to the changing external environment; specifically by focusing on customer needs through continuous process improvement. Continuous process improvement in manufacturing involves defect reduction (Arbogast, 1997) which can be achieved by employing scientific method in quality and process control. Quality is defined as characteristics that a product or service must have (Anderson & et.al, 2007). Quality control is a series of inspections and measurements to determine whether quality standards are being met (Anderson & et.al, 2007). It has had a long history; however the effective application of statistic to quality control just began in the 1920s as a consequence of the development of sampling theory (NIST/SEMATECH). The general consensus is that quality in the manufacturing industry is not limited to the products; even more important is the quality of the processes involved in producing the goods. In order to control the process, several tools of Statistical Process Control (SPC), namely histograms, check sheets, Pareto charts, cause and effect diagrams, scatter diagrams and control charts (NIST/SEMATECH) are employed to determine whether the process is in control or out of control. This paper attempts to illustrate the application of SPC method, specifically of different control charts of measurement variables in manufacturing process. II. Statistical Process Control Process control is the continuous adjustment of the process based on the information supplied by the monitoring tools such as the SPC (NIST/SEMATECH). These tools are applied to examine the

4 4 variations in the output quality. By assuming that the production process is a continuous one (Anderson & et.al, 2007), the variation can be categorized as (Lind, Marchal, & Wathen, 2008): (1) Assignable variation which is not a random one and may be the result of worn out tools, improper setting of machinery, poor quality raw materials or human error. As the consequence, the process is labeled as out of control process. (2) Chance variation which is random and cannot be completely eliminated since it is caused by random variations in manufacturing processes such as temperature, pressure, humidity, etc. Such processes are considered to be in statistical control. The testing methodology in statistical process control is summarized in the following table: The outcomes of statistical process control State of production process Decision H 0 True - Process in Control H 0 False Process out of Control Continue Process Correct decision Type II error (allowing an out-ofcontrol process to continue) Adjust Process Type I error (adjusting an incontrol process) Correct decision Table 1. The Outcomes of Statistical Process Control (Anderson & et.al, 2007) III. Control Charts Fundamentally, control chart is a plot of the selected sample with its associated measurement to determine whether the process involved in producing the said sample is within the specification limits of an in control process. In other words, every time a point is plotted on the control chart, we are carrying out a hypothesis test to determine whether the process is in control (Lind, Marchal, & Wathen, 2008). A population model and the associated Upper Control Limit (UCL) and Lower Control Limit (LCL) are determined from the historical data of how the process typically performed (NIST/SEMATECH).

5 5 Measurements that fall outside the control limits are examined to see if they belong to the same population as the specified model (NIST/SEMATECH); in other words, we attempt to separate the assignable variation from the chance variation. There are two basic types of control charts (NIST/SEMATECH): univariate control chart which is based on one quality characteristic and multivariate control chart which represents more than one quality characteristics. Univariate control charts include variable control chart which depicts measurements and requires the interval or the ratio scale of measurements and attribute control chart which classifies a product or service as either acceptable or unacceptable (Lind, Marchal, & Wathen, 2008). Variable control charts can be subdivided into (NIST/SEMATECH): (1) Shewhart s and R charts for subgroup measurement, (2) Moving range for individual measurement, (3) Cumulative Sum (CUSUM) control chart, and (4) Exponentially Weighted Moving Average (EWMA) control chart. For each of the control chart, there are two main features: (1) the center line which corresponds to the target value when the process is in control (2) the Upper Control Limit (UCL) and Lower Control Limit (LCL) which determine whether the process is in control or out of control. The basic configuration of control chart can be illustrated as follows: Figure 1. The Construction of a Simple Control Chart (Moameni & Zinck, 1997)

6 6 It can be seen, that UCL and LCL are basically the critical values which represent the probability of a data point falling beyond the limits by assuming that only chance variation is present (NIST/SEMATECH). ±3σ limits indicate that for in control process, the probability of data falling within the limits is 99.73% (Levinson, 1999) or a data point is expected to fall beyond the limits every 370 times. The general rule of thumb for control charts is that an in control process is indicated by data points fall within the control limits and exhibits random pattern (NIST/SEMATECH). IV. Application of Control Charts IV.1. Background The data selected for the application of control charts in this paper is 450 continuous random variables from lithography process in semiconductor industry (Source: (NIST/SEMATECH)) 1. The quality characteristics applied in control charts are the width of lines measured from five different sites of a single wafer. There are 30 lots, each with three wafers. The charts designed in the analysis are Shewhart variable control charts. IV.2. Assumptions One of the assumptions in applying control charts is rational sub grouping of the samples, which means that the sampling is performed consecutively from the process output (Frank, 2003). It is therefore presumed that the line width measurements were taken from the lithography process in sequence. Another assumption that has to be accepted due to the limitation of the paper is the noncorrelativity of the measurements. To apply 3σ limit, the underlying assumption is that the chance variations are normally distributed (NIST/SEMATECH). To test whether this assumption holds, normal probability plot is generated using SPSS. The result can be seen below: 1 See Appendix for Data

7 Expected Cum Prob 7 Normal P-P Plot of Line Width 1,0 0,8 0,6 0,4 0,2 0,0 0,0 0,2 0,4 0,6 0,8 1,0 Observed Cum Prob Figure 2. Normal Probability Plot It appears that the overall line width is normally distributed and as a consequence, the ±3σ control limits can be applied. The normality of the overall distribution is also important to establish the applicability of central limit theory in the subsequent subgroup analysis. It has been suggested that effective application of normal probability distribution in estimating the population parameters for subgroups with small size applies for primary distribution that does not differ significantly from normality (Levinson, 1999). The next step is to investigate the shape of the distribution. Skewness increases the risk of finding a chance variation above the UCL and below the LCL (NIST/SEMATECH). The histogram of the data is illustrated below:

8 Frequency Mean =2, Std. Dev. =0, N = , , , , , , , Line Width Figure 3. Histogram Plot The primary distribution is somewhat positively skewed with coefficient To separate chance variations from assignable variations, some studies have attempted to determine the level of significance for skewness coefficient. Levinson (1999) has listed the values of coefficients which are considered to be within the random statistical variations with 95% and 99% confidence intervals for sample size of 25 to 100. However, the size of the overall measurement in this analysis is 450 data points; therefore it cannot be concluded whether the skewness of distribution results from significant assignable variations.

9 9 IV.3. Subgroup Analysis Lithographic processes consist of inherent variations resulted from variations in materials, environmental parameters which may affect equipments, and human error (Levinson, 1999). From the histogram in the previous section, it is not conclusive whether the variations in line width measurements are due to these chance variations or that the process is out of control and therefore further investigation and adjustments are warranted. Shewhart control charts are designed to answer this question. The sub-grouping of the 450 data points can be carried out in three different ways: (1) single measurements of line width i.e. subgroups with sample size (n) = 1, (2) subgroup of 90 wafers with n = 5 i.e. measurements from different sites in a single wafer, and (3) subgroup of 30 lots with n = 15. Due to the limitation of the paper, the subsequent analysis is based on sub-grouping of 90 wafers. IV.3.1. Shewhart s and Control Charts Since the population σ is unknown, an unbiased estimator of standard deviation is calculated using: Equation 1 The average of the m subgroups standard deviations: Equation 2 With s i represents the standard deviation of ith subgroup and the constant: Equation 3 The control limits and the center line of the s chart are:

10 10 Equation 4 Equation 5 Equation 6 The population mean μ also has to be estimated by a target or the average of subgroup means, i.e. grand mean: Equation 7 is the mean of the ith subgroup. The control limits and center line of the chart are: Equation 8 Equation 9 Equation 10

11 11 Based on the equations above, the results of the calculation are given in the table below: Parameters Results m C n UCL x LCL x UCL s LCL s Table 2. The Parameters for Shewhart s and Control Charts The subgroup estimation of standard deviation ( = ) is smaller than the overall standard deviation ( ) shown in histogram because the overall standard deviation also contains the between-wafer variations (Levinson, 1999). The control charts generated by SPSS are shown below:

12 12 Figure 4. Control Chart Based on Standard Deviation Figure 5. Shewhart s Control Chart

13 13 IV.3.2. R and Control Charts The R and control charts can assist in determining whether the variability in a process is in control or whether shifts are occurring over time (Berenson & Levine, 1999), i.e. the shift of current parameters from the initial values. The control limits and central line for R chart are given as follows: Equation 11 Equation 12 Equation 13 The average range is: Equation 14 The range of ith subgroup is and the control limits and center line of the chart are: Equation 15 Equation 16 Equation 17 A 2, d 2, and d 3 are constants which values depend on the size of the subgroup and can be found in statistic table (See (NIST/SEMATECH)). The results of the calculation for R and control charts are:

14 14 Parameters Results n 5 A D 3 0 D UCL x LCL x UCL R LCL R 0 Table 3. The Parameters for and R Control Charts Using SPSS, the control charts calculated using average range can be seen below: Figure 6. Control Chart Based on Range

15 15 Figure 7. R Control Chart R and s control charts measure the within- subgroup variation (May & Spanos, 2006), i.e. the variability of line width from different sites in a single wafer. From the charts above, all of data points are within the standard deviation and range control limits. This can be interpreted as the variation within a single wafer is in statistical control, which means the variations present are of the chance or processinherent nature. R control chart is regarded to be effective for small sample size (n 10). For n = 5 as in the case with the line width measurements in this paper, the relative efficiency of range approach to standard approach is (NIST/SEMATECH). On the other hand, both control charts examine the between-subgroup variability (May & Spanos, 2006) i.e. the variability of line width from 90 sequences of wafer. It can be observed that several data points fall beyond the control limits of both charts. Based on Western Electric Rules

16 16 (WECO), this can be interpreted as the lithography process to be out of control and shifts of the population parameters have occurred over time (NIST/SEMATECH). It is also detected that there are more than 8 consecutive points (subgroup 20 30) that fall below the target value. The probability for this pattern to happen is 0.39% and therefore signals mean shifts (Levinson, 1999). From the results, it appears that the initial assumption of rational sub grouping holds for these control charts because the indication of rational sampling is the minimizing of within-subgroup variation and maximizing of between-subgroup variation in the presence of assignable causes (May & Spanos, 2006). V. Conclusion The analysis of wafer subgroups indicates that the lithography process is out of control and investigation of materials, equipment and operators is necessary to find the assignable causes. Nevertheless, the analysis is limited to wafer sub-grouping which represents the within-lot variation. It is recommended to design control charts based on between-lot variation (NIST/SEMATECH) with larger standard deviation and wider distance between control limits and the target value. Additionally, the application of Shewhart control charts is assigned to non-correlated variables. For further study, it is crucial to perform autocorrelation test on the data set. Since the charts employ ±3σ limits, the sensitivity of the charts in detecting the shifts of the parameters also needs to be determined by generating OC Curve.

17 17 Bibliography Anderson, D. R., & et.al. (2007). Statistics for Business and Economics. London: Thomson Learning. Arbogast, G. W. (1997). A Case Study: Statistical Analysis in a Production Quality Improvement Project. Journal of Quality Management: 2(2), Berenson, M. L., & Levine, D. M. (1999). Basic Business Statistics: Concepts and Application. Prentice Hall. Frank, P. (2003). Control Charts in Quality Control - Shewhart Charts Application. Retrieved January 02, 2010, from The Faculty of Electrical Engineering and Communication - Brno University of Technology: Kurekova, E. (2001). Measurement Process Capability: Trends and Approaches. Measurement Science Review: 1(1), Levinson, H. J. (1999). Lithography Process Control. SPIE Press Book. Lind, D. A., Marchal, W. G., & Wathen, S. A. (2008). Statistical Techniques in Business & Economics with Global Data Sets. McGraw-Hill Irwin. May, G. S., & Spanos, C. J. (2006). Fundamentals of Semiconductor Manufacturing and Process Control. John Wiley & Sons Inc. Moameni, A., & Zinck, J. A. (1997). Application of SQC Charts and Geostatistics to Soil Quality Assessment in a Semi-Arid Environment of South-Central Iran. ITC Journal: 3(4), 28p. Nam, K. H., Kim, D. K., & Park, D. H. (2001). Large-Sample Interval Estimators for Process Capability Indices. Quality Engineering: 14(2), Nichols, M. D., & Houry, K. (2009). Adapting to Troubled Times: Versatility is Key if Quality is to Come to the Forefront. Quality Progress: January 2009, 8-9. NIST/SEMATECH. (n.d.). 6. Process or Product Monitoring and Control. Retrieved November 21, 2009, from e-handbook of Statistical Methods:

18 18 Appendix. 1 Overall Measurement Data Lot Wafer Site Line Width Lot Wafer Site Line Width 1 1 Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left

19 4 2 Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right

20 8 1 Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top

21 12 1 Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center Right Right Bottom Bottom Top Top Left Left Center Center

22 15 3 Right Right Bottom Bottom

23 23 Appendix. 2 Subgroup Data and Calculation Wafer

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