Operations Management Supplement 6 – Statistical Process Control PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 7e Operations Management, 9e

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

Outline – Continued Process Capability Acceptance Sampling Process Capability Ratio (Cp) Process Capability Index (Cpk ) Acceptance Sampling Operating Characteristic Curve Average Outgoing Quality

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

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

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.

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”

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

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

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

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

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

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

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.

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

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.

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 =

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

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

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

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

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

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

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

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

Setting Control Limits Process average x = 12 ounces Average range R = .25 Sample size n = 5

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

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

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

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

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

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

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

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

Manual and Automated Control Charts

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)

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 ^ p(1 - p) n sp = ^ Instructors may wish to point out the calculation of the standard deviation reflects the binomial distribution of the population LCLp = p - zsp ^ where p = mean fraction defective in the sample z = number of standard deviations sp = standard deviation of the sampling distribution n = sample size ^

p-Chart for Data Entry p = = .04 sp = = .02 1 6 .06 11 6 .06 Sample Number Fraction Sample Number Fraction Number of Errors Defective Number of Errors Defective 1 6 .06 11 6 .06 2 5 .05 12 1 .01 3 0 .00 13 8 .08 4 1 .01 14 7 .07 5 4 .04 15 5 .05 6 2 .02 16 4 .04 7 5 .05 17 11 .11 8 3 .03 18 3 .03 9 3 .03 19 0 .00 10 2 .02 20 4 .04 Total = 80 p = = .04 80 (100)(20) (.04)(1 - .04) 100 sp = = .02 ^

p-Chart for Data Entry UCLp = p + zsp = .04 + 3(.02) = .10 ^ LCLp = p - zsp = .04 - 3(.02) = 0 ^ .11 – .10 – .09 – .08 – .07 – .06 – .05 – .04 – .03 – .02 – .01 – .00 – Sample number Fraction defective | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 UCLp = 0.10 LCLp = 0.00 p = 0.04

Possible assignable causes present p-Chart for Data Entry UCLp = p + zsp = .04 + 3(.02) = .10 ^ Possible assignable causes present LCLp = p - zsp = .04 - 3(.02) = 0 ^ .11 – .10 – .09 – .08 – .07 – .06 – .05 – .04 – .03 – .02 – .01 – .00 – Sample number Fraction defective | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 UCLp = 0.10 LCLp = 0.00 p = 0.04 There is always a focus on finding and eliminating problems. But control charts find any process changed, good or bad. The clever company will be looking at Operator 3 and 19 as they reported no errors during this period. The company should find out why (find the assignable cause) and see if there are skills or processes that can be applied to the other operators.

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 LCLc = c - 3 c Instructors may wish to point out the calculation of the standard deviation reflects the Poisson distribution of the population where the standard deviation equals the square root of the mean where c = mean number defective in the sample

c-Chart for Cab Company c = 54 complaints/9 days = 6 complaints/day UCLc = c + 3 c = 6 + 3 6 = 13.35 | 1 2 3 4 5 6 7 8 9 Day Number defective 14 – 12 – 10 – 8 – 6 – 4 – 2 – 0 – UCLc = 13.35 LCLc = 0 c = 6 LCLc = c - 3 c = 6 - 3 6 = 0

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

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

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

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

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

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

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

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

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

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

Process Capability The natural variation of a process should be small enough to produce products that meet the standards required A process in statistical control does not necessarily meet the design specifications Process capability is a measure of the relationship between the natural variation of the process and the design specifications

Process Capability Ratio Cp = Upper Specification - Lower Specification 6s A capable process must have a Cp of at least 1.0 Does not look at how well the process is centered in the specification range Often a target value of Cp = 1.33 is used to allow for off-center processes Six Sigma quality requires a Cp = 2.0

Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6s

Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6s = = 1.938 213 - 207 6(.516)

Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6s = = 1.938 213 - 207 6(.516) Process is capable

Process Capability Index Cpk = minimum of , Upper Specification - x Limit 3s Lower x - Specification Limit A capable process must have a Cpk of at least 1.0 A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes

Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches

Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches Cpk = minimum of , (.251) - .250 (3).0005

Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation s = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches Cpk = minimum of , (.251) - .250 (3).0005 .250 - (.249) Both calculations result in New machine is NOT capable Cpk = = 0.67 .001 .0015

Interpreting Cpk Cpk = negative number Cpk = zero Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Figure S6.8

Acceptance Sampling Form of quality testing used for incoming materials or finished goods Take samples at random from a lot (shipment) of items Inspect each of the items in the sample Decide whether to reject the whole lot based on the inspection results Only screens lots; does not drive quality improvement efforts Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required.

Acceptance Sampling Form of quality testing used for incoming materials or finished goods Take samples at random from a lot (shipment) of items Inspect each of the items in the sample Decide whether to reject the whole lot based on the inspection results Only screens lots; does not drive quality improvement efforts Rejected lots can be: Returned to the supplier Culled for defectives (100% inspection) Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required.

Operating Characteristic Curve Shows how well a sampling plan discriminates between good and bad lots (shipments) Shows the relationship between the probability of accepting a lot and its quality level You can use this and the next several slides to begin a discussion of the “quality” of the acceptance sampling plans. You will find additional slides on “consumer’s” and “producer’s” risk to pursue the issue in a more formal manner in subsequent slides.

The “Perfect” OC Curve Keep whole shipment P(Accept Whole Shipment) % Defective in Lot P(Accept Whole Shipment) 100 – 75 – 50 – 25 – 0 – | | | | | | | | | | | 0 10 20 30 40 50 60 70 80 90 100 Return whole shipment Cut-Off

Probability of Acceptance Consumer’s risk for LTPD An OC Curve Figure S6.9 Probability of Acceptance Percent defective | | | | | | | | | 0 1 2 3 4 5 6 7 8 100 – 95 – 75 – 50 – 25 – 10 – 0 –  = 0.05 producer’s risk for AQL LTPD AQL  = 0.10 Consumer’s risk for LTPD Bad lots Indifference zone Good lots

AQL and LTPD Acceptable Quality Level (AQL) Poorest level of quality we are willing to accept Lot Tolerance Percent Defective (LTPD) Quality level we consider bad Consumer (buyer) does not want to accept lots with more defects than LTPD Once the students understand the definition of these terms, have them consider how one would go about choosing values for AQL and LTPD.

Producer’s and Consumer’s Risks Producer's risk () Probability of rejecting a good lot Probability of rejecting a lot when the fraction defective is at or above the AQL Consumer's risk (b) Probability of accepting a bad lot Probability of accepting a lot when fraction defective is below the LTPD This slide introduces the concept of “producer’s” risk and “consumer’s” risk. The following slide explores these concepts graphically.

OC Curves for Different Sampling Plans n = 50, c = 1 n = 100, c = 2 This slide presents the OC curve for two possible acceptance sampling plans.

Average Outgoing Quality AOQ = (Pd)(Pa)(N - n) N where Pd = true percent defective of the lot Pa = probability of accepting the lot N = number of items in the lot n = number of items in the sample It is probably important to stress that AOQ is the average percent defective, not the average percent acceptable.

Average Outgoing Quality If a sampling plan replaces all defectives If we know the incoming percent defective for the lot We can compute the average outgoing quality (AOQ) in percent defective It is probably important to stress that AOQ is the average percent defective, not the average percent acceptable. The maximum AOQ is the highest percent defective or the lowest average quality and is called the average outgoing quality level (AOQL)

Automated Inspection Modern technologies allow virtually 100% inspection at minimal costs Not suitable for all situations

SPC and Process Variability Lower specification limit Upper specification limit Process mean, m (a) Acceptance sampling (Some bad units accepted) (b) Statistical process control (Keep the process in control) This may be a good time to stress that an overall goal of statistical process control is to “do it better,” i.e., improve over time. (c) Cpk >1 (Design a process that is in control) Figure S6.10