Statistical_Process_Control

Toni White March 28, 2010 OPS/571 Week 5 – Statistical Process Controls Statistical Process Control Total Quality Management is a philosophy that stresses three principles for achieving high levels of process performance and quality: Customer Satisfaction, Employee Involvement, and Continuous Improvement in Performance. This paper will address a practical type of continuous improvement - the use of statistical process control. One definition of statistical process control (SPC) is “the application of statistical techniques to determine whether a process is delivering what the customer wants” (Goetsch & Davis, 2006). SPC primarily involves using control charts to detect defective services or products or to indicate that the process has changed and that the same services or products will deviate from their design specifications unless corrections are made. For example, in this paper we will examine • A decrease in the average number of calories consumed on a daily basis, Control Charts SPC may be broadly broken down into three sets of activities: understanding the process; understanding the causes of variation; and elimination of the sources of special cause variation. In understanding a process, the process is typically mapped out and monitored using control charts. Control charts are used to identify variation that may be due to special causes, and to eliminate concern over variation due to common causes. When, through the control charts, variation that is due to special causes is identified, or the process capability is found lacking, additional effort is exerted to determine causes of that variance and eliminate it. The tools used include Ishikawa diagrams, designed experiments and Pareto charts (Wheeler, 1999). Once the causes of variation have been determined, elimination of those causes that are both statistically and practically significant must occur. Typically, the elimination process includes development of standard work, error-proofing and training, however, additional measures can also be used. Detecting Special Causes According to Shewart, since control is not defined as the complete absence of variation, we do want to identify and eliminate special causes of variation (Pyzdek, 1980). The following is a list of special causes: 1. A special cause is indicated when a single point falls outside a control limit. 2. A special cause is indicated when two out of three successive values are: a) on the same side of the centerline, and b) more than two standard deviations from the centerline. 3. A special cause is indicated when eight or more successive values fall on the same side of the centerline. 4. A special cause is indicated by a trend of six or more values in a row steadily increasing or decreasing. Control Charts for Variables There are different types of control charts for use. They include: I chart. Used to construct a confidence interval that moves through time, then by tracking how each new period’s data falls within that range to help in making judgment about how a process is performing. Used when individual data is unknown. X-Bar and S-Chart. Used when each subgroup has more than one observation to account for the sample variations. In the example below, we have the average caloric intake using a sample of 5 days for 5 consecutive weeks. |  |Wk 1 |Wk 2 |Wk 3 |Wk 4 |Wk 5 | |count |5 |5 |5 |5 |5 | |Xbar |1,182.80 |1,100.60 |1,114.60 |1,082.20 |1,184.80 | |variance |2,627.70 |10,111.30 |3,804.80 |11,694.70 |1,279.70 | |sdev |51.26 |100.55 |61.68 |108.14 |35.77 | |minimum |1121 |1000 |1021 |900 |1143 | |maximum |1250 |1250 |1189 |1189 |1232 | |range |129 |250 |168 |289 |89 | |  |  |  |  |  |  | |confidence interval 95.% lower |1,119.15 |975.74 |1,038.01 |947.92 |1,140.38 | |confidence interval 95.% upper |1,246.45 |1,225.46 |1,191.19 |1,216.48 |1,229.22 | Xbar is the average for each day, sdev is the standard deviation for each day. If we take the average of the average (Xbarbar) we get 1133, and if we take the standard deviation of all the sample means (Sbar) we get 190.0. Now we can construct the UCL and LCL as: UCL = Xbarbar + 5*Sbar/sqrt(n) LCL = Xbarbar – 5*Sbar/sqrt(n) Where n is the size of the sample from each day – so if the sample sizes are the same for each period the UCL and LCL will be the same across the chart, but if the sample sizes vary, then the UCL and LCL will also vary. Doing this and graphing gives us the Xbar chart: [pic] Note that things look rather stable: there are no observations outside the 3-sigma limits. The two sigma limits are 1242.98 and 1023.02 and no observations are outside those limits. There are not eight successive values above or below the centerline, and there is not a trend of six or more. We may also observe what is happening to the variance over time in an sbar chart. First we construct the average of the standard deviations and then the standard deviation of the standard deviations. Then use 5 times this standard deviation to construct the UCL and LCL. Note that if the LCL is calculated to be negative, we set it equal to zero since negative values do not make sense. [pic] Again things look pretty stable. In practice one would first want to look at the s-chart to make sure the process was stable, and then go to the xbar chart, but both can help identify abnormalities in the process. The xbar chart looks at variations over time and the s chart looks at variations within groups (Wise & Fair, 1998). Control Charts for Count or Attribute Data Charts for count or attribute data include the following: P-chart. Easiest to deal with – used when the data contains percentages or proportions being tracked over time. The American Society for Testing and Materials (ASTM) has set guidelines for p-charts: The lower limit on subgroup size for p-charts may not yield reliable information when: 1. Subgroups have less than 25 in the denominator, or 2. the subgroup size n multiplied by Pbar is less than one. U-Chart. Used when we have count data and different sample sizes for each period – where there is an unequal area of opportunity. C-Chart. An alternative to the U-chart, but when there is equal opportunity for defects (or when the opportunity is unknown). The U-chart generally is more powerful than the C-chart since it has more information in it. Similarly the Xbar chart is more powerful than the I chart, however, the choice of chart may be determined by the amount of information or lack of information. Statistical process control helps mangers achieve and maintain a process distribution that does not change in terms of its mean and variance. In summation, the control limits on the control charts signal when the mean or variability of the process changes enabling managers to make decisions in regards to processes. REFERENCES Goetsch, David L., & Davis, Stanley. Quality Management -5th Edition. Prentice Hall, 2006. Pyzdek, Thomas. Economic Control of Quality of Manufactured Product. (1980) ASQC Quality Press. Wheeler, Donald J. (1999). Understanding Variation: The Key to Managing Chaos - 2nd Edition. SPC Press, Inc. Wise, Stephen A. & Fair, Douglas C (1998). Innovative Control Charting: Practical SPC Solutions for Today's Manufacturing Environment. ASQ Quality Press. www.astm.org