#### Transcript Statistical Process Control - University of Hawaii at Hilo

Operations Management Supplement 6 – Statistical Process Control PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 7e Operations Management, 9e © 2008 Prentice Hall, Inc. S6 – 1 Statistical Process Control (SPC) Variability is inherent in every process Natural or common causes Special or assignable causes Provides a statistical signal when assignable causes are present Detect and eliminate assignable causes of variation © 2008 Prentice Hall, Inc. S6 – 2 Natural Variations Also called common causes Affect virtually all production processes Expected amount of variation Output measures follow a probability distribution For any distribution there is a measure of central tendency and dispersion If the distribution of outputs falls within acceptable limits, the process is said to be “in control” © 2008 Prentice Hall, Inc. S6 – 3 Assignable Variations Also called special causes of variation Generally this is some change in the process Variations that can be traced to a specific reason The objective is to discover when assignable causes are present Eliminate the bad causes Incorporate the good causes © 2008 Prentice Hall, Inc. S6 – 4 Samples To measure the process, we take samples and analyze the sample statistics following these steps Figure S6.1 © 2008 Prentice Hall, Inc. Frequency (a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Each of these represents one sample of five boxes of cereal # # # # # # # # # # # # # # # # # # # # # # # # # # Weight S6 – 5 Samples To measure the process, we take samples and analyze the sample statistics following these steps Figure S6.1 © 2008 Prentice Hall, Inc. Frequency (b) After enough samples are taken from a stable process, they form a pattern called a distribution The solid line represents the distribution Weight S6 – 6 Samples To measure the process, we take samples and analyze the sample statistics following these steps Frequency (c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Figure S6.1 Central tendency Weight © 2008 Prentice Hall, Inc. Variation Weight Shape Weight S6 – 7 Samples (d) If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable © 2008 Prentice Hall, Inc. Frequency To measure the process, we take samples and analyze the sample statistics following these steps Prediction Weight Figure S6.1 S6 – 8 Samples To measure the process, we take samples and analyze the sample statistics following these steps Frequency (e) If assignable causes are present, the process output is not stable over time and is not predicable ? ?? ?? ? ? ? ? ? ? ? ? ? ?? ? ? ? Prediction Weight Figure S6.1 © 2008 Prentice Hall, Inc. S6 – 9 Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes © 2008 Prentice Hall, Inc. S6 – 10 Process Control Frequency Lower control limit (a) In statistical control and capable of producing within control limits Upper control limit (b) In statistical control but not capable of producing within control limits (c) Out of control Size (weight, length, speed, etc.) © 2008 Prentice Hall, Inc. Figure S6.2 S6 – 11 Types of Data Variables Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables © 2008 Prentice Hall, Inc. Attributes Defect-related characteristics Classify products as either good or bad or count defects Categorical or discrete random variables S6 – 12 Central Limit Theorem Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve 1. The mean of the sampling distribution (x) will be the same as the population mean m 2. The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n © 2008 Prentice Hall, Inc. x=m sx = s n S6 – 13 Population and Sampling Distributions Three population distributions Distribution of sample means Mean of sample means = x Beta Standard s deviation of the sample = sx = n means Normal Uniform | | | | -3sx -2sx -1sx x | | +1sx +2sx +3sx 95.45% fall within ± 2sx 99.73% of all x fall within ± 3sx © 2008 Prentice Hall, Inc. | Figure S6.3 S6 – 14 Control Charts for Variables For variables that have continuous dimensions Weight, speed, length, strength, etc. x-charts are to control the central tendency of the process R-charts are to control the dispersion of the process These two charts must be used together © 2008 Prentice Hall, Inc. S6 – 15 Setting Chart Limits For x-Charts when we know s Upper control limit (UCL) = x + zsx Lower control limit (LCL) = x - zsx where © 2008 Prentice Hall, Inc. x = mean of the sample means or a target value set for the process z = number of normal standard deviations sx = standard deviation of the sample means = s/ n s = population standard deviation n = sample size S6 – 16 Setting Control Limits Hour 1 Sample Weight of Number Oat Flakes 1 17 2 13 3 16 4 18 n=9 5 17 6 16 7 15 8 17 9 16 Mean 16.1 s= 1 © 2008 Prentice Hall, Inc. Hour 1 2 3 4 5 6 Mean 16.1 16.8 15.5 16.5 16.5 16.4 Hour 7 8 9 10 11 12 Mean 15.2 16.4 16.3 14.8 14.2 17.3 For 99.73% control limits, z = 3 UCLx = x + zsx = 16 + 3(1/3) = 17 ozs LCLx = x - zsx = 16 - 3(1/3) = 15 ozs S6 – 17 Setting Control Limits Control Chart for sample of 9 boxes Variation due to assignable causes Out of control 17 = UCL Variation due to natural causes 16 = Mean 15 = LCL | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 Sample number © 2008 Prentice Hall, Inc. Out of control Variation due to assignable causes S6 – 18 Setting Chart Limits For x-Charts when we don’t know s Upper control limit (UCL) = x + A2R Lower control limit (LCL) = x - A2R where © 2008 Prentice Hall, Inc. R = average range of the samples A2 = control chart factor found in Table S6.1 x = mean of the sample means S6 – 19 Control Chart Factors Sample Size n Mean Factor A2 Upper Range D4 Lower Range D3 2 3 4 5 6 7 8 9 10 12 1.880 1.023 .729 .577 .483 .419 .373 .337 .308 .266 3.268 2.574 2.282 2.115 2.004 1.924 1.864 1.816 1.777 1.716 0 0 0 0 0 0.076 0.136 0.184 0.223 0.284 Table S6.1 © 2008 Prentice Hall, Inc. S6 – 20 Setting Control Limits Process average x = 12 ounces Average range R = .25 Sample size n = 5 © 2008 Prentice Hall, Inc. S6 – 21 Setting Control Limits Process average x = 12 ounces Average range R = .25 Sample size n = 5 UCLx = x + A2R = 12 + (.577)(.25) = 12 + .144 = 12.144 ounces From Table S6.1 © 2008 Prentice Hall, Inc. S6 – 22 Setting Control Limits Process average x = 12 ounces Average range R = .25 Sample size n = 5 UCLx LCLx © 2008 Prentice Hall, Inc. = x + A2R = 12 + (.577)(.25) = 12 + .144 = 12.144 ounces UCL = 12.144 = x - A2R = 12 - .144 = 11.857 ounces LCL = 11.857 Mean = 12 S6 – 23 R – Chart Type of variables control chart Shows sample ranges over time Difference between smallest and largest values in sample Monitors process variability Independent from process mean © 2008 Prentice Hall, Inc. S6 – 24 Setting Chart Limits For R-Charts Upper control limit (UCLR) = D4R Lower control limit (LCLR) = D3R where R = average range of the samples D3 and D4 = control chart factors from Table S6.1 © 2008 Prentice Hall, Inc. S6 – 25 Setting Control Limits Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D4 = 2.115, D3 = 0 UCLR = D4R = (2.115)(5.3) = 11.2 pounds UCL = 11.2 LCLR LCL = 0 © 2008 Prentice Hall, Inc. = D3R = (0)(5.3) = 0 pounds Mean = 5.3 S6 – 26 Mean and Range Charts (a) (Sampling mean is shifting upward but range is consistent) These sampling distributions result in the charts below UCL (x-chart detects shift in central tendency) x-chart LCL UCL (R-chart does not detect change in mean) R-chart LCL Figure S6.5 © 2008 Prentice Hall, Inc. S6 – 27 Mean and Range Charts (b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) UCL (x-chart does not detect the increase in dispersion) x-chart LCL UCL (R-chart detects increase in dispersion) R-chart LCL Figure S6.5 © 2008 Prentice Hall, Inc. S6 – 28 Steps In Creating Control Charts 1. Take samples from the population and compute the appropriate sample statistic 2. Use the sample statistic to calculate control limits and draw the control chart 3. Plot sample results on the control chart and determine the state of the process (in or out of control) 4. Investigate possible assignable causes and take any indicated actions 5. Continue sampling from the process and reset the control limits when necessary © 2008 Prentice Hall, Inc. S6 – 29 Manual and Automated Control Charts © 2008 Prentice Hall, Inc. S6 – 30 Control Charts for Attributes For variables that are categorical Good/bad, yes/no, acceptable/unacceptable Measurement is typically counting defectives Charts may measure Percent defective (p-chart) Number of defects (c-chart) © 2008 Prentice Hall, Inc. S6 – 31 Control Limits for p-Charts Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics UCLp = p + zsp^ sp = ^ LCLp = p - zsp^ where © 2008 Prentice Hall, Inc. p z sp^ n = = = = p(1 - p) n mean fraction defective in the sample number of standard deviations standard deviation of the sampling distribution sample size S6 – 32 p-Chart for Data Entry Sample Number 1 2 3 4 5 6 7 8 9 10 Number of Errors Fraction Defective 6 5 0 1 4 2 5 3 3 2 .06 .05 .00 .01 .04 .02 .05 .03 .03 .02 80 p = (100)(20) = .04 © 2008 Prentice Hall, Inc. Sample Number Number of Errors 11 6 12 1 13 8 14 7 15 5 16 4 17 11 18 3 19 0 20 4 Total = 80 sp^ = Fraction Defective .06 .01 .08 .07 .05 .04 .11 .03 .00 .04 (.04)(1 - .04) = .02 100 S6 – 33 p-Chart for Data Entry UCLp = p + zsp^ = .04 + 3(.02) = .10 Fraction defective LCLp = p - zsp^ = .04 - 3(.02) = 0 .11 .10 .09 .08 .07 .06 .05 .04 .03 .02 .01 .00 – – – – – – – – – – – – UCLp = 0.10 p = 0.04 | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 LCLp = 0.00 Sample number © 2008 Prentice Hall, Inc. S6 – 34 p-Chart for Data Entry UCLp = p + zsp^ = .04 + 3(.02) = .10 Fraction defective Possible LCLp = p - zsp^ = .04 - 3(.02) = 0 assignable causes present .11 .10 .09 .08 .07 .06 .05 .04 .03 .02 .01 .00 – – – – – – – – – – – – UCLp = 0.10 p = 0.04 | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 LCLp = 0.00 Sample number © 2008 Prentice Hall, Inc. S6 – 35 Control Limits for c-Charts Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics UCLc = c + 3 c where © 2008 Prentice Hall, Inc. LCLc = c - 3 c c = mean number defective in the sample S6 – 36 c-Chart for Cab Company c = 54 complaints/9 days = 6 complaints/day LCLc = c - 3 c =6-3 6 =0 UCLc = 13.35 14 – 12 – Number defective UCLc = c + 3 c =6+3 6 = 13.35 10 8 6 4 2 – – – – – c= 6 0 – | | 1 2 | 3 | 4 | 5 | 6 | 7 | 8 LCLc = 0 | 9 Day © 2008 Prentice Hall, Inc. S6 – 37 Managerial Issues and Control Charts Three major management decisions: Select points in the processes that need SPC Determine the appropriate charting technique Set clear policies and procedures © 2008 Prentice Hall, Inc. S6 – 38 Which Control Chart to Use Variables Data Using an x-chart and R-chart: Observations are variables Collect 20 - 25 samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart Track samples of n observations each © 2008 Prentice Hall, Inc. S6 – 39 Which Control Chart to Use Attribute Data Using the p-chart: Observations are attributes that can be categorized in two states We deal with fraction, proportion, or percent defectives Have several samples, each with many observations © 2008 Prentice Hall, Inc. S6 – 40 Which Control Chart to Use Attribute Data Using a c-Chart: Observations are attributes whose defects per unit of output can be counted The number counted is a small part of the possible occurrences Defects such as number of blemishes on a desk, number of typos in a page of text, flaws in a bolt of cloth © 2008 Prentice Hall, Inc. S6 – 41 Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 © 2008 Prentice Hall, Inc. Normal behavior. Process is “in control.” S6 – 42 Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 © 2008 Prentice Hall, Inc. One plot out above (or below). Investigate for cause. Process is “out of control.” S6 – 43 Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 © 2008 Prentice Hall, Inc. Trends in either direction, 5 plots. Investigate for cause of progressive change. S6 – 44 Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 © 2008 Prentice Hall, Inc. Two plots very near lower (or upper) control. Investigate for cause. S6 – 45 Patterns in Control Charts Upper control limit Target Lower control limit Figure S6.7 © 2008 Prentice Hall, Inc. Run of 5 above (or below) central line. Investigate for cause. S6 – 46 Patterns in Control Charts Upper control limit Target Lower control limit Erratic behavior. Investigate. Figure S6.7 © 2008 Prentice Hall, Inc. S6 – 47