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STATISTICAL PROCESS CONTROL AND QUALITY MANAGEMENT 1 Quality • Fitness for use, acceptable standard • Based on needs, expectations and customer requests Types of Quality • Quality of design • Quality of conformance • Quality of performance 2 Quality of Design • Differences in quality due to design differences, intentional differences Quality of Conformance • Degree to which product meets or exceeds standards Quality of Performance • Long term consistent functioning of the product, reliability, safety, serviceability, maintainability 3 Quality and Productivity • Improved quality leads to lower costs and increased profits 4 Statistics and Quality Management • Statistical analysis is used to assist with product design, monitor the production process, and check quality of the finished product 5 Checking Finished Product Quality • Random samples selected from batches of finished product can be used to check for product quality 6 Assisting With Production Design • A variety of experimental design techniques are available for improving the production process 7 Monitoring the Process • Control charts is used to monitor the process as it unfolds • Sampling from the production line to see if variation in product quality is consistent with expectations 8 The Control Chart • A special type of sequence plot which is used to monitor a process • Throughout the process measurements are taken and plotted in a sequence plot • Plot also contains upper and lower control limits indicating the expected range of the process when it is behaving properly 9 Examples PROCESS CONTROL CHART 53 51 50 49 48 97 93 89 85 81 77 73 69 65 61 57 53 49 45 41 37 33 29 25 21 17 13 9 5 47 1 BATCH MEAN 52 BATCH NUMBER MEAN TRUE MEAN LOWER UPPER 10 Variation • There is no two natural items in any category are the same. • Variation may be quite large or very small. • If variation very small, it may appear that items are identical, but precision instruments will show differences. 3 Categories of variation • Within-piece variation – One portion of surface is rougher than another portion. • Apiece-to-piece variation – Variation among pieces produced at the same time. • Time-to-time variation – Service given early would be different from that given later in the day. Source of variation • Equipment – Tool wear, machine vibration, … • Material – Raw material quality • Environment – Temperature, pressure, humadity • Operator – Operator performs- physical & emotional Control Chart Viewpoint Variation due to Common or chance causes Assignable causes Control chart may be used to discover “assignable causes” Control chart functions • Control charts are powerful aids to understanding the performance of a process over time. Input PROCESS What’s causing variability? Output Control charts identify variation • Chance causes - “common cause” – inherent to the process or random and not controllable – if only common cause present, the process is considered stable or “in control” • Assignable causes - “special cause” – variation due to outside influences – if present, the process is “out of control” Control charts help us learn more about processes • Separate common and special causes of variation • Determine whether a process is in a state of statistical control or out-of-control • Estimate the process parameters (mean, variation) and assess the performance of a process or its capability Control charts to monitor processes • To monitor output, we use a control chart – we check things like the mean, range, standard deviation • To monitor a process, we typically use two control charts – mean (or some other central tendency measure) – variation (typically using range or standard deviation) Types of Data • Variable data – Product characteristic that can be measured • Length, size, weight, height, time, velocity • Attribute data Product characteristic evaluated with a discrete choice • Good/bad, yes/no Types of Control Charts • Control Chart For The Mean(X chart) • Control Chart For The Range(R chart) • Control Chart For A Proportion(p chart) • Control Chart For Attribute Measures(c chart) 20 Control Chart For The Sample Mean • Assuming that the sample mean is approximately normal with mean and standard deviation , a control chart for the mean usually consists of three horizontal lines • The vertical axis is used to plot the magnitude of observed sample means while the horizontal axis represents time or the order of the sequence of observed means 21 Control Chart For The Sample Mean • The center line is at the mean, and upper and lower control limits are at • +3 / n and -3 / n • Since (standard deviation)is usually unknown the • term / n is usually replaced by an estimator based on the sample range 22 Control Chart For The Sample Mean The formula is given by x A2 R where R = average value of the range k = Ri / k i 1 k = number of samples The values of A2 depend on the sample size 23 Control Chart For The Sample Mean • LCL = x R A2 • UCL = x R A2 24 Control Chart For The Range • Designed to monitor variability in the product • Range easier to determine than standard deviation • Distribution of sample range assumes product measurement is normally distributed 25 Control Chart For The Range • Upper and lower control limits and center line obtained from R and the values of D3, D4 • according to the formulae • LCL = D3 R • UCL = D4 R • The values of D3, D4 are based on sample size 26 Control Chart For The Sample Proportion • Population proportion • The sample mean is now a mean proportion given by p • Where p = total number of objects in sample with characteristic divided by total sample size 27 Control Chart For The Sample Proportion • Using the central limit theorem the control limits are given by: (1 ) p3 n 28 Control Chart For The Sample Proportion • Since the true proportion is usually unknown we replace it by the average proportion p3 p (1 p ) n • If the sample size varies then the upper and lower limits will vary and the equations become: p 3 p (1 p ) ni • where ni = sample size in sample i 29 Control Chart For Attribute Measures • Alternative method of counting good and bad items. • Defects are measured by merely counting the no. of defects. • Where c = total number of defects in a sample 30 Control Chart For Attribute Measures • Process average or central line c= • UCL: c 3 • LCL: c 3 c n p c c 31 Example: Control Charts for Variable Data Sample 1 2 3 4 5 6 7 8 9 10 Slip Ring Diameter (cm) 1 2 3 4 5.02 5.01 4.94 4.99 5.01 5.03 5.07 4.95 4.99 5.00 4.93 4.92 5.03 4.91 5.01 4.98 4.95 4.92 5.03 5.05 4.97 5.06 5.06 4.96 5.05 5.01 5.10 4.96 5.09 5.10 5.00 4.99 5.14 5.10 4.99 5.08 5.01 4.98 5.08 5.07 5 4.96 4.96 4.99 4.89 5.01 5.03 4.99 5.08 5.09 4.99 X R Example: Control Charts for Variable Data Sample 1 2 3 4 5 6 7 8 9 10 Slip Ring Diameter (cm) 1 2 3 4 5.02 5.01 4.94 4.99 5.01 5.03 5.07 4.95 4.99 5.00 4.93 4.92 5.03 4.91 5.01 4.98 4.95 4.92 5.03 5.05 4.97 5.06 5.06 4.96 5.05 5.01 5.10 4.96 5.09 5.10 5.00 4.99 5.14 5.10 4.99 5.08 5.01 4.98 5.08 5.07 5 4.96 4.96 4.99 4.89 5.01 5.03 4.99 5.08 5.09 4.99 X 4.98 5.00 4.97 4.96 4.99 5.01 5.02 5.05 5.08 5.03 50.09 R 0.08 0.12 0.08 0.14 0.13 0.10 0.14 0.11 0.15 0.10 1.15 Calculation From Table above: • Sigma X-bar = 50.09 • Sigma R = 1.15 • m = 10 Thus; • X-Double bar = 50.09/10 = 5.009 cm • R-bar = 1.15/10 = 0.115 cm Note: The control limits are only preliminary with 10 samples. It is desirable to have at least 25 samples. Trial control limit • UCLx-bar = X-D bar + A2 R-bar = 5.009 + (0.577)(0.115) = 5.075 cm • LCLx-bar = X-D bar - A2 R-bar = 5.009 - (0.577)(0.115) = 4.943 cm n=5 • UCLR = D4R-bar = (2.114)(0.115) = 0.243 cm • LCLR = D3R-bar = (0)(0.115) = 0 cm For A2, D3, D4: see Table B, Appendix 3-Sigma Control Chart Factors Sample size n 2 3 4 5 6 7 8 X-chart A2 1.88 1.02 0.73 0.58 0.48 0.42 0.37 R-chart D3 0 0 0 0 0 0.08 0.14 D4 3.27 2.57 2.28 2.11 2.00 1.92 1.86 X-bar Chart R Chart Subgroup 18 2 1 12.45 12.39 2 12.55 3 3 4 5 12.40 12.37 12.40 12.38 12.36 12.38 12.44 12.46 12.44 12.30 12.39 12.36 4 12.38 12.39 12.37 12.55 12.37 5 12.37 12.44 12.44 12.37 12.38 6 12.45 12.37 12.36 12.41 12.39 7 12.46 12.38 12.51 12.44 12.55 8 12.44 12.39 12.38 12.39 12.37 9 12.44 12.55 12.41 12.44 12.39 10 12.35 12.38 12.37 12.44 12.38 11 12.36 12.40 12.41 12.35 12.44 12 12.51 12.36 12.41 12.36 12.39 13 12.38 12.30 12.45 12.37 12.44 14 12.41 12.37 12.45 12.45 12.37 15 12.37 12.44 12.45 12.46 12.38 39 Sample No. Number of defects 1 10 2 9 3 8 4 11 5 7 6 12 7 7 8 10 9 13 10 12 11 13 12 14 From a lot of 100 40 Sample No. Number of defects Proportion 1 10 0.10 2 9 0.9 3 8 0.8 4 11 0.11 5 7 0.7 6 12 0.12 7 7 0.7 8 10 0.10 9 13 0.13 10 12 0.12 11 13 0.13 12 14 0.14 From a lot of 100 41 The Normal Distribution = Standard deviation Mean -3 -2 -1 +1 +2 +3 68.26% 95.44% USL 99.74% LSL -3 CL +3 • • • • • 34.13% of data lie between and 1 above the mean (). 34.13% between and 1 below the mean. Approximately two-thirds (68.28 %) within 1 of the mean. 13.59% of the data lie between one and two standard deviations Finally, almost all of the data (99.74%) are within 3 of the mean. Normal Distribution Review Define the 3-sigma limits for sample means as follows: 3 3(0.05) Upper Limit 5.01 5.077 n 5 3 3(0.05) Lower Limit 5.01 4.943 n 5 What is the probability that the sample means will lie outside 3-sigma limits? Note that the 3-sigma limits for sample means are different from natural tolerances which are at 3 Common Causes Process Out of Control • The term out of control is a change in the process due to an assignable cause. • When a point (subgroup value) falls outside its control limits, the process is out of control. Assignable Causes (a) Mean Average Grams Assignable Causes Average (b) Spread Grams Assignable Causes Average Grams (c) Shape Assignable causes likely Control Charts UCL Nominal LCL 1 2 Samples 3 Control Chart Examples Variations UCL Nominal LCL Sample number Control Limits and Errors Type I error: Probability of searching for a cause when none exists (a) Three-sigma limits UCL Process average LCL Control Limits and Errors Type I error: Probability of searching for a cause when none exists (b) Two-sigma limits UCL Process average LCL Control Limits and Errors (a) Three-sigma limits Type II error: Probability of concluding that nothing has changed UCL Shift in process Process average average LCL Control Limits and Errors (b) Two-sigma limits Type II error: Probability of concluding that nothing has changed UCL Shift in process Process average average LCL Achieve the purpose Our goal is to decrease the variation inherent in a process over time. As we improve the process, the spread of the data will continue to decrease. Quality improves!! Improvement Examine the process • A process is considered to be stable and in a state of control, or under control, when the performance of the process falls within the statistically calculated control limits and exhibits only chance, or common causes. Consequences of misinterpreting the process • Blaming people for problems that they cannot control • Spending time and money looking for problems that do not exist • Spending time and money on unnecessary process adjustments • Taking action where no action is warranted • Asking for worker-related improvements when process improvements are needed first Process variation • When a system is subject to only chance causes of variation, 99.74% of the measurements will fall within 6 standard deviations – If 1000 subgroups are measured, 997 will fall within the six sigma limits. Mean -3 -2 -1 +1 +2 +3 68.26% 95.44% 99.74% Chart zones • Based on our knowledge of the normal curve, a control chart exhibits a state of control when: ♥ Two thirds of all points are near the center value. ♥ The points appear to float back and forth across the centerline. ♥ The points are balanced on both sides of the centerline. ♥ No points beyond the control limits. ♥ No patterns or trends. Acceptance Sampling Acceptance sampling is a form of testing that involves taking random samples of “lots,” or batches, of finished products and measuring them against predetermined standards. • A “lot,” or batch, of items can be inspected in several ways, including the use of single, double, or sequential sampling. Single Sampling • Two numbers specify a single sampling plan: They are the number of items to be sampled (n) and a pre specified acceptable number of defects (c). If there are fewer or equal defects in the lot than the acceptance number, c, then the whole batch will be accepted. If there are more than c defects, the whole lot will be rejected or subjected to 100% screening. Double Sampling • Often a lot of items is so good or so bad that we can reach a conclusion about its quality by taking a smaller sample than would have been used in a single sampling plan. If the number of defects in this smaller sample (of size n1) is less than or equal to some lower limit (c1), the lot can be accepted. If the number of defects exceeds an upper limit (c2), the whole lot can be rejected. But if the number of defects in the n1 sample is between c1 and c2, a second sample (of size n2) is drawn. The cumulative results determine whether to accept or reject the lot. The concept is called double sampling. Sequential Sampling • Multiple sampling is an extension of double sampling, with smaller samples used sequentially until a clear decision can be made. When units are randomly selected from a lot and tested one by one, with the cumulative number of inspected pieces and defects recorded, the process is called sequential sampling. OPERATING CHARACTERISTIC (OC) CURVES • The operating characteristic (OC) curve describes how well an acceptance plan discriminates between good and bad lots. A curve pertains to a specific plan, that is, a combination of n (sample size) and c (acceptance level). It is intended to show the probability that the plan will accept lots of various quality levels. • Figure shows a perfect discrimination plan for a company that wants to reject all lots with more than 2 ½ % defectives and accept all lots with less than 2 ½ % defectives. OC Curves for Two Different Acceptable Levels of Defects (c = 1, c = 4) for the Same Sample Size (n = 100). • So one way to increase the probability of accepting only good lots and rejecting only bad lots with random sampling is to set very tight acceptance levels. • OC Curves for Two Different Sample Sizes (n = 25, n = 100) but Same Acceptance Percentages (4%). Larger sample size shows better discrimination. Sampling Terms • Acceptance quality level (AQL): the percentage of defects at which consumers are willing to accept lots as “good” • Lot tolerance percent defective (LTPD): the upper limit on the percentage of defects that a consumer is willing to accept • Consumer’s risk: the probability that a lot contained defectives exceeding the LTPD will be accepted • Producer’s risk: the probability that a lot containing the acceptable quality level will be rejected THE OPERATING-CHARACTERISTIC (OC) CURVE • For a given a sampling plan and a specified true fraction defective p, we can calculate – Pa -- Probability of accepting lot • If lot is truly good, 1 - Pa = a (Producers' Risk) • If lot is truly bad, Pa = b (Consumer ‘s Risk) • A plot of Pa as a function of p is called the OC curve for a given sampling plan The OC curve shows the features of a particular sampling plan, including the risks of making a wrong decision. Construction of OC curve • In attribute sampling, where products are determined to be either good or bad, a binomial distribution is usually employed to build the OC curve. The binomial equation is where n = number of items sampled (called trials) p = probability that an x (defect) will occur on any one trial P(x) = probability of exactly x results in n trials In a Poisson approximation of the binomial distribution, the mean of the binomial, which is np, is used as the mean of the Poisson, which is λ; that is, λ = np Example • Probability of acceptance A shipment of 2,000 portable battery units for microcomputers is about to be inspected by a Malaysian importer. The Korean manufacturer and the importer have set up a sampling plan in which the risk is limited to 5% at an acceptable quality level (AQL) of 2% defective, and the risk is set to 10% at Lot Tolerance Percent Defective (LTPD) = 7% defective. We want to construct the OC curve for the plan of n = 120 sample size and an acceptance level of c ≤ 3 defectives. Both firms want to know if this plan will satisfy their quality and risk requirements. Example N=1000 n = 100 AQL=1% LTPD=5% ß=10% α = 5% C<=2 Does the plan meet the producer’s and consumer’s requirement. AVERAGE OUTGOING QUALITY In most sampling plans, when a lot is rejected, the entire lot is inspected and all of the defective items are replaced. Use of this replacement technique improves the average outgoing quality in terms of percent defective. In fact, given (1) any sampling plan that replaces all defective items encountered and (2) the true incoming percent defective for the lot, it is possible to determine the average outgoing quality (AOQ) in percent defective. AVERAGE OUTGOING QUALITY The equation for AOQ is Example Selected Values of % Defective Mean of Poisson, λ = np P (acceptance) .01 1 0.920 .02 2 0.677 .03 3 0.423 .04 4 0.238 .05 5 0.125 .06 6 0.062 Example • The percent defective from an incoming lot in is 3%. An OC curve showed the probability of acceptance to be .515. Given a lot size of 2,000 and a sample of 120, what is the average outgoing quality in percent defective?