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MD 021 - Management and Operations Statistical Process Control Outline Definition of statistical process control (SPC) Process variation Characteristics of control charts Types of control charts Choosing an SPC method 1 Definition of Statistical Process Control Statistical process control (SPC) is the application of statistical techniques to determine whether the output of a process conforms to the product or service design. SPC is implemented via control charts that are used to monitor the output of the process and indicate the presence of problems requiring further action. 2 Process Variation All processes exhibit variation. For example, a machine that fills cereal boxes will not put exactly the same amount in each box. If we were to collect data over time on the amount of cereal in each box, a plot of the data would be described as a distribution. A distribution is characterized by its: Mean Cereal Production Spread 700 Range 600 No. of boxes 500 Standard deviation 400 300 Shape 200 Symmetric 100 Skewed 0 9.7 9.75 9.8 9.85 9.9 9.95 10 10.1 10.1 10.2 10.2 10.3 10.3 Average weight (oz.) 3 Quality Measurements Control charts can be used to monitor processes where output is measured as either variables or attributes. Variables measures - Characteristics of a product or service that can be measured on a continuous scale. Examples include length, width, and time. Attributes measures - Characteristics of a product or service that can be quickly counted for acceptable quality. Examples include the number of defects in a product or service. 4 Sources of Process Variation Sources of process variation can be categorized as: Common causes - Random, or unavoidable, sources of variation within a process. A process with only common causes of variation is stable (i.e. the mean and spread do not change over time). Such a process is said to be “in a state of statistical control” or “in-control”. Assignable causes - Any cause of variation that can be identified and eliminated, originating from outside the normal process 5 Characteristics of Control Charts A control chart is a time-ordered diagram to monitor a quality characteristic, consisting of: A nominal value, or center line – The average of several past samples Two control limits used to judge whether action is required, an upper control limit (UCL) and a lower control limit (LCL) Data points, each consisting of the average measurement calculated from a sample taken from the process, ordered over time. By the Central Limit Theorem, regardless of the distribution of the underlying individual measurements, the distribution of the sample means will follow a normal distribution. The control limits are set based on the sampling distribution of the quality measurement. 6 Purpose of Control Charts Control charts are intended to reflect only common causes of variation in order to detect assignable causes of variation. Question: If, at the time we are constructing a control chart, there are assignable causes of variation in the process, how can we construct a meaningful control chart? Answer: By carefully choosing our sample size so that only common causes are found within a sample. The control limits are set to: Usually detect when the process has gone out of control (narrow control limits work better) Usually not overreact to random variation (wider control limits work better) The control limits are set to strike a balance between these competing priorities. 7 Control Charts for Variables Standard Deviation of the Process, , Known Control charts for variables (with the standard deviation of the process, , known ) monitor the mean, X , of the process distribution. The control limits are: UCL X z X LCL X z X where: X = center line of the chart and the average of several past sample means, z is the standard normal deviate (number of standard deviations from the average), X / n and is the standard deviation of the distribution of sample means, and n is the sample size 8 2. An automatic filling machine is used to fill 1-liter bottles of cola. The machine’s output is approximately normal with a mean of 1.0 liter and a standard deviation of 0.01 liter. Output is monitored using means of samples of 25 observations. a. Determine upper and lower control limits that will include roughly 97 percent of the sample means when the process is in control. x = 1.0 liter, = .01 liter, n = 25 a. Control limits : x 2 , [z = 2.17 for 97%] n .01 UCL is 1.0 2.17 1.0043 25 .01 LCL is 1.0 2.17 .9957 25 b. Given the sample means: 1.005, 1.001, 0.998, 1.002, 0.995 and 0.999, is the process in control? 1.006 out b. 1.0043 * UCL 1.002 1.000 (liters) .998 Mean .9957 .994 LCL * 9 out Control Charts for Variables Standard Deviation of the Process, , Unknown Control charts for variables monitor the mean ( X chart) and variability (R chart) of the process distribution. R chart: To calculate the range of the data, subtract the smallest from the largest measurement in the sample. The control limits are: UCLR D4 R and LCLR D3 R where: R = average of several past R values and is the central line of the control chart, and D3 , D4 = constants that provide three standard deviation (three-sigma) limits for a given sample size X chart: The control limits are: UCL X X A2 R and LCL X X A2 R where: X = central line of the chart and the average of past sample means, and A2 = constant to provide three sigma limits for the process mean. 10 Control Chart for Variables Example X , R Charts Webster Chemical Company produces mastics and caulking for the construction industry. The product is blended in large mixers and then pumped into tubes and capped. Webster is concerned whether the filling process for tubes of caulking is in statistical control. The process should be centered on 8 ounces per tube. Several samples of eight tubes are taken and each tube is weighed in ounces. Tube number Sample 1 2 3 4 5 6 7 8 1 7.98 8.34 8.02 7.94 8.44 7.68 7.81 8.11 2 8.23 8.12 7.98 8.41 8.31 8.18 7.99 8.06 3 7.89 7.77 7.91 8.04 8.00 7.89 7.93 8.09 4 8.24 8.18 7.83 8.05 7.90 8.16 7.97 8.07 5 7.87 8.13 7.92 7.99 8.10 7.81 8.14 7.88 6 8.13 8.14 8.11 8.13 8.14 8.12 8.13 8.14 a) Assuming that taking only 6 samples is sufficient, use the data in the table to construct three-sigma R-chart and X -chart control limits. Is the process in statistical control? b) The process variability for the first and sixth samples appear to be out of control. Webster looks for assignable causes and quickly notes that the weighing scale was gummed up with caulking. Apparently, a tube was not properly capped. The sticky scale did not correctly read the variation in weights for the sixth sample. Delete that data and recalculate R , UCLR , and LCLR . Is the process in statistical control? 11 Solution to Control Chart for Variables Example a) Tube number Sample 1 2 3 4 5 6 7 8 Avg. Range 1 7.98 8.34 8.02 7.94 8.44 7.68 7.81 8.11 8.040 0.76 2 8.23 8.12 7.98 8.41 8.31 8.18 7.99 8.06 8.160 0.43 3 7.89 7.77 7.91 8.04 8.00 7.89 7.93 8.09 7.940 0.32 4 8.24 8.18 7.83 8.05 7.90 8.16 7.97 8.07 8.050 0.41 5 7.87 8.13 7.92 7.99 8.10 7.81 8.14 7.88 7.980 0.33 6 8.13 8.14 8.11 8.13 8.14 8.12 8.13 8.14 8.130 0.03 8.050 0.38 X = 8.050, R = 0.38, n = 8 From Table 7.1: UCLR D4 R 1.864(0.38) 0.708 LCLR D3 R 0.136(0.38) 0.052 UCLX X A2 R 8.050 0.373(0.38) 8.192 LCLX X A2 R 8.050 0.373(0.38) 7.908 b) We delete the sixth observation and recalculate the control limits. The ranges, including the range for the first sample are now all within the revised control limits, and the process average for the second sample now falls just inside of the revised control limits. (0.76 0.43 0.32 0.41 0.33) R 0.45 5 8.040 8.160 7.940 8.050 7.980 X 8.034 5 UCLR D4 R 1.864(0.45) 0.839 LCLR D3 R 0.136(0.45) 0.061 UCLX X A2 R 8.034 0.373(0.45) 8.202 LCLX X A2 R 8.034 0.373(0.45) 7.866 12 Control Charts for Attributes p-Chart A p-chart is a commonly used control chart for attributes, whereby the quality characteristic is counted, rather than measured, and the entire item or service can be declared good or defective. The standard deviation of the proportion defective, p, is: p p(1 p) / n , where n = sample size, and p = average of several past p values and central line on the chart. Using the normal approximation to the binomial distribution, which is the actual distribution of p, UCL p p z p and LCL p p z p where z is the normal deviate (number of standard deviations from the average). 13 Control Chart for Attributes Example p-Chart A sticky scale brings Webster’s attention to whether caulking tubes are being properly capped. If a significant proportion of the tubes aren’t being sealed, Webster is placing their customers in a messy situation. Tubes are packaged in large boxes of 144. Several boxes are inspected and the following number of leaking tubes are found: Sample Tubes Sample Tubes Sample Tubes 1 3 8 6 15 5 2 5 9 4 16 0 3 3 10 9 17 2 4 4 11 2 18 6 5 2 12 6 19 2 6 4 13 5 20 1 7 2 14 1 Total 72 Calculate p-chart three-sigma control limits to assess whether the capping process is in statistical control. 72 Solution: n = 144, p 0.025 20(144) p(1 p) 0.025(1 0.025) p 0.013 n 144 UCL p p z p 0.025 3(0.013) 0.064 LCL p p z p 0.025 3(0.013) 0.014 (adjusted to zero) The highest proportion of defectives occurs in sample #10, but is still within control limits: p10 9 / 144 0.0625 . Therefore, the process is in statistical control. 14 Control Charts for Attributes c-Chart A c-chart is another type of control chart for attributes, whereby the quality characteristic is counted as the number of defects/ unit. Using the normal approximation to the Poisson distribution, which is the actual distribution of c, UCLc c z c and LCLc c z c where c is the average number of defects/unit and the center line of the c-chart. 15 Control Chart for Attributes Example c-Chart At Webster Chemical, lumps in the caulking compound could cause difficulties in dispensing a smooth bead from the tube. Testing for the presence of lumps destroys the product, so Webster takes random samples. The following are the results of the study: Tube # Lumps Tube # Lumps Tube # Lumps 1 6 5 6 9 5 2 5 6 4 10 0 3 0 7 1 11 9 4 4 8 6 12 2 Determine the c-chart two-sigma upper and lower control limits for this process. Solution: ( 6 5 0 4 6 4 1 6 5 0 9 2) c 4 12 c c 4 2 UCLc c z c 4 (2)(2) 8 LCLc c z c 4 (2)(2) 0 The eleventh tube has too many lumps (9), so the process is probably out of control. 16 Process Capability Exercise Webster Chemical’s nominal weight for filling tubes of caulk is 8.00 ounces 0.60 ounces. The target process capability ratio is 1.33. The current distribution of the filling process is centered on 8.054 ounces with a standard deviation of 0.192 ounces. Compute the process capability index to assess whether the filling process is capable and set properly. Solution: Process capability ratio: Upper specification - Lower specification 8.6 7.4 Cp 10417 . 6 . 6(0192) Process capability index: X - Lower specification Upper specification - X C pk miminum of , 3 3 8.054 7.400 8.600 - 8.054 1135, . 0.948 . 3(0192) . 3(0192) C pk 0.948 The process is not capable of consistently meeting specifications according to the minimum capability level set by Webster. 17 Supplementary Material: 1. Calculating P xi pi n p i = proportion of defective products x i = number of the defected products n = sample size m = number of samples m ( x1 x2 ... xm ) i x1 x2 x ... m x ( p1 p 2 ... p m ) n n n p i 1 m m mn mn total number of defected products p total number of products 18