Statistical Process Control PowerPoint presentation to accompany Heizer and Render Operations Management, 10e Principles of Operations Management, 8e PowerPoint slides by Jeff Heyl © 2011 Pearson Education, Inc. publishing as Prentice Hall

Outline Statistical Process Control (SPC) Control Charts for Variables The Central Limit Theorem Setting Mean Chart Limits (x-Charts) Setting Range Chart Limits (R-Charts) Using Mean and Range Charts Control Charts for Attributes Managerial Issues and Control Charts © 2011 Pearson Education, Inc. publishing as Prentice Hall

Outline – Continued Process Capability Acceptance Sampling Process Capability Ratio (Cp) Process Capability Index (Cpk ) Acceptance Sampling Operating Characteristic Curve Average Outgoing Quality © 2011 Pearson Education, Inc. publishing as Prentice Hall

Learning Objectives When you complete this supplement you should be able to: Explain the use of a control chart Explain the role of the central limit theorem in SPC Build x-charts and R-charts List the five steps involved in building control charts © 2011 Pearson Education, Inc. publishing as Prentice Hall

Learning Objectives When you complete this supplement you should be able to: Build p-charts and c-charts Explain process capability and compute Cp and Cpk Explain acceptance sampling Compute the AOQ © 2011 Pearson Education, Inc. publishing as Prentice Hall

Statistical Process Control The objective of a process control system is to provide a statistical signal when assignable causes of variation are present © 2011 Pearson Education, Inc. publishing as Prentice Hall

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 Points which might be emphasized include: - Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance. - Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units - While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later. © 2011 Pearson Education, Inc. publishing as Prentice Hall

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” © 2011 Pearson Education, Inc. publishing as Prentice Hall

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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Each of these represents one sample of five boxes of cereal Samples To measure the process, we take samples and analyze the sample statistics following these steps Each of these represents one sample of five boxes of cereal (a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Frequency Weight # Figure S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

The solid line represents the distribution Samples To measure the process, we take samples and analyze the sample statistics following these steps The solid line represents the distribution (b) After enough samples are taken from a stable process, they form a pattern called a distribution Frequency Weight Figure S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Samples To measure the process, we take samples and analyze the sample statistics following these steps (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 Weight Central tendency Variation Shape Frequency © 2011 Pearson Education, Inc. publishing as Prentice Hall

Samples To measure the process, we take samples and analyze the sample statistics following these steps (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 Prediction Weight Time Frequency Figure S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Samples To measure the process, we take samples and analyze the sample statistics following these steps Prediction ? (e) If assignable causes are present, the process output is not stable over time and is not predicable Weight Time Frequency Figure S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes Students should understand both the concepts of natural and assignable variation, and the nature of the efforts required to deal with them. © 2011 Pearson Education, Inc. publishing as Prentice Hall

Process Control (a) In statistical control and capable of producing within control limits Frequency Lower control limit Upper control limit (b) In statistical control but not capable of producing within control limits This slide helps introduce different process outputs. It can also be used to illustrate natural and assignable variation. (c) Out of control (weight, length, speed, etc.) Size Figure S6.2 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Types of Data Variables Attributes Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables Defect-related characteristics Classify products as either good or bad or count defects Categorical or discrete random variables Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process. © 2011 Pearson Education, Inc. publishing as Prentice Hall

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 The mean of the sampling distribution (x) will be the same as the population mean m x = m This slide introduces the difference between “natural” and “assignable” causes. The next several slides expand the discussion and introduce some of the statistical issues. 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 s n sx = © 2011 Pearson Education, Inc. publishing as Prentice Hall

Population and Sampling Distributions Three population distributions Beta Normal Uniform Distribution of sample means Standard deviation of the sample means = sx = s n Mean of sample means = x | | | | | | | -3sx -2sx -1sx x +1sx +2sx +3sx 99.73% of all x fall within ± 3sx 95.45% fall within ± 2sx Figure S6.3 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Sampling Distribution Sampling distribution of means Process distribution of means x = m (mean) It may be useful to spend some time explicitly discussing the difference between the sampling distribution of the means and the mean of the process population. Figure S6.4 © 2011 Pearson Education, Inc. publishing as Prentice Hall

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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Setting Chart Limits For x-Charts when we know s Upper control limit (UCL) = x + zsx Lower control limit (LCL) = x - zsx where 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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Setting Control Limits Hour 1 Sample Weight of Number Oat Flakes 1 17 2 13 3 16 4 18 5 17 6 16 7 15 8 17 9 16 Mean 16.1 s = 1 Hour Mean Hour Mean 1 16.1 7 15.2 2 16.8 8 16.4 3 15.5 9 16.3 4 16.5 10 14.8 5 16.5 11 14.2 6 16.4 12 17.3 n = 9 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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Setting Control Limits Control Chart for sample of 9 boxes Variation due to assignable causes Out of control Sample number | | | | | | | | | | | | 1 2 3 4 5 6 7 8 9 10 11 12 17 = UCL 15 = LCL 16 = Mean Variation due to natural causes Out of control © 2011 Pearson Education, Inc. publishing as Prentice Hall

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 R = average range of the samples A2 = control chart factor found in Table S6.1 x = mean of the sample means © 2011 Pearson Education, Inc. publishing as Prentice Hall

Control Chart Factors Sample Size Mean Factor Upper Range Lower Range n A2 D4 D3 2 1.880 3.268 0 3 1.023 2.574 0 4 .729 2.282 0 5 .577 2.115 0 6 .483 2.004 0 7 .419 1.924 0.076 8 .373 1.864 0.136 9 .337 1.816 0.184 10 .308 1.777 0.223 12 .266 1.716 0.284 Table S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Setting Control Limits Process average x = 12 ounces Average range R = .25 Sample size n = 5 © 2011 Pearson Education, Inc. publishing as Prentice Hall

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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Setting Control Limits Process average x = 12 ounces Average range R = .25 Sample size n = 5 UCL = 12.144 Mean = 12 LCL = 11.857 UCLx = x + A2R = 12 + (.577)(.25) = 12 + .144 = 12.144 ounces LCLx = x - A2R = 12 - .144 = 11.857 ounces © 2011 Pearson Education, Inc. publishing as Prentice Hall

Restaurant Control Limits For salmon filets at Darden Restaurants Sample Mean x Bar Chart UCL = 11.524 x – 10.959 LCL – 10.394 | | | | | | | | | 1 3 5 7 9 11 13 15 17 11.5 – 11.0 – 10.5 – Sample Range Range Chart UCL = 0.6943 R = 0.2125 LCL = 0 | | | | | | | | | 1 3 5 7 9 11 13 15 17 0.8 – 0.4 – 0.0 – © 2011 Pearson Education, Inc. publishing as Prentice Hall

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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Setting Control Limits Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D4 = 2.115, D3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0 UCLR = D4R = (2.115)(5.3) = 11.2 pounds LCLR = D3R = (0)(5.3) = 0 pounds © 2011 Pearson Education, Inc. publishing as Prentice Hall

© 2011 Pearson Education, Inc. publishing as Prentice Hall

Mean and Range Charts (a) These sampling distributions result in the charts below (Sampling mean is shifting upward but range is consistent) x-chart (x-chart detects shift in central tendency) UCL LCL R-chart (R-chart does not detect change in mean) UCL LCL Figure S6.5 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Mean and Range Charts (b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) x-chart (x-chart does not detect the increase in dispersion) UCL LCL R-chart (R-chart detects increase in dispersion) UCL LCL Figure S6.5 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Steps In Creating Control Charts Take samples from the population and compute the appropriate sample statistic Use the sample statistic to calculate control limits and draw the control chart Plot sample results on the control chart and determine the state of the process (in or out of control) Investigate possible assignable causes and take any indicated actions Continue sampling from the process and reset the control limits when necessary © 2011 Pearson Education, Inc. publishing as Prentice Hall

Manual and Automated Control Charts © 2011 Pearson Education, Inc. publishing as Prentice Hall

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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

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. Table S6.3 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Which Control Chart to Use Attribute Data Using the p-Chart Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states. We deal with fraction, proportion, or percent defectives. There are several samples, with many observations in each. For example, 20 samples of n = 100 observations in each. Table S6.3 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Which Control Chart to Use Attribute Data Using a c-Chart Observations are attributes whose defects per unit of output can be counted. We deal with the number counted, which is a small part of the possible occurrences. Defects may be: number of blemishes on a desk; complaints in a day; crimes in a year; broken seats in a stadium; typos in a chapter of this text; or flaws in a bolt of cloth. Table S6.3 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Patterns in Control Charts Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Normal behavior. Process is “in control.” Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Patterns in Control Charts Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. One plot out above (or below). Investigate for cause. Process is “out of control.” Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Patterns in Control Charts Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Patterns in Control Charts Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Two plots very near lower (or upper) control. Investigate for cause. Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Patterns in Control Charts Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Run of 5 above (or below) central line. Investigate for cause. Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Patterns in Control Charts Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Erratic behavior. Investigate. Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall