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Statistical Process Control MGMT 3057 © 2011 Pearson Education S 6 - 1 Statistical Process Control MGMT 3057 © 2011 Pearson Education S 6 - 1

Outline u Statistical Process Control (SPC) u Control Charts for Variables u The Central Outline u Statistical Process Control (SPC) u Control Charts for Variables u The Central Limit Theorem u Setting Mean Chart Limits (x-Charts) u Setting Range Chart Limits (R-Charts) u Using Mean and Range Charts u Control Charts for Attributes u Managerial Issues and Control Charts © 2011 Pearson Education S 6 - 2

Outline – Continued u Process Capability Ratio (Cp) u Process Capability Index (Cpk ) Outline – Continued u Process Capability Ratio (Cp) u Process Capability Index (Cpk ) u Acceptance Sampling u Operating Characteristic Curve u Average Outgoing Quality © 2011 Pearson Education S 6 - 3

Learning Objectives When you complete this supplement you should be able to: 1. Explain Learning Objectives When you complete this supplement you should be able to: 1. Explain the use of a control chart 2. Explain the role of the central limit theorem in SPC 3. Build x-charts and R-charts 4. List the five steps involved in building control charts © 2011 Pearson Education S 6 - 4

Learning Objectives When you complete this supplement you should be able to: 5. Build Learning Objectives When you complete this supplement you should be able to: 5. Build p-charts and c-charts 6. Explain process capability and compute Cp and Cpk 7. Explain acceptance sampling 8. Compute the AOQ © 2011 Pearson Education S 6 - 5

Statistical Process Control The objective of a process control system is to provide a 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 S 6 - 6

Statistical Process Control (SPC) u Variability is inherent in every process u Natural or Statistical Process Control (SPC) u Variability is inherent in every process u Natural or common causes u Special or assignable causes u Provides a statistical signal when assignable causes are present u Detect and eliminate assignable causes of variation © 2011 Pearson Education S 6 - 7

Natural Variations u Also called common causes u Affect virtually all production processes u Natural Variations u Also called common causes u Affect virtually all production processes u Expected amount of variation u Output measures follow a probability distribution u For any distribution there is a measure of central tendency and dispersion u If the distribution of outputs falls within acceptable limits, the process is said to be “in control” © 2011 Pearson Education S 6 - 8

Assignable Variations u Also called special causes of variation u Generally this is some Assignable Variations u Also called special causes of variation u Generally this is some change in the process u Variations that can be traced to a specific reason u The objective is to discover when assignable causes are present u Eliminate the bad causes u Incorporate the good causes © 2011 Pearson Education S 6 - 9

Samples To measure the process, we take samples and analyze the sample statistics following Samples To measure the process, we take samples and analyze the sample statistics following these steps Figure S 6. 1 © 2011 Pearson Education 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 S 6 - 10

Samples To measure the process, we take samples and analyze the sample statistics following Samples To measure the process, we take samples and analyze the sample statistics following these steps Figure S 6. 1 © 2011 Pearson Education 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 S 6 - 11

Samples To measure the process, we take samples and analyze the sample statistics following 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 S 6. 1 Central tendency Weight © 2011 Pearson Education Variation Weight Shape Weight S 6 - 12

Samples (d) If only natural causes of variation are present, the output of a 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 © 2011 Pearson Education Frequency To measure the process, we take samples and analyze the sample statistics following these steps Prediction ime T Weight Figure S 6. 1 S 6 - 13

Samples To measure the process, we take samples and analyze the sample statistics following Samples To measure the process, we take samples and analyze the sample statistics following these steps Prediction Frequency (e) If assignable causes are present, the process output is not stable over time and is not predicable ? ? ? ? ? ime T Weight Figure S 6. 1 © 2011 Pearson Education S 6 - 14

Control Charts Constructed from historical data, the purpose of control charts is to help Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes © 2011 Pearson Education S 6 - 15

Process Control Frequency Lower control limit (a) In statistical control and capable of producing 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. ) © 2011 Pearson Education Figure S 6. 2 S 6 - 16

Types of Data Variables u Characteristics that can take any real value u May Types of Data Variables u Characteristics that can take any real value u May be in whole or in fractional numbers u Continuous random variables © 2011 Pearson Education Attributes u Defect-related characteristics u Classify products as either good or bad or count defects u Categorical or discrete random variables S 6 - 17

Central Limit Theorem Regardless of the distribution of the population, the distribution of sample 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 © 2011 Pearson Education x=m sx = s n S 6 - 18

Population and Sampling Distributions Three population distributions Distribution of sample means Mean of sample 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 | | -3 sx -2 sx -1 sx x | | | +1 sx +2 sx +3 sx 95. 45% fall within ± 2 sx 99. 73% of all x fall within ± 3 sx © 2011 Pearson Education Figure S 6. 3 S 6 - 19

Sampling Distribution Sampling distribution of means Process distribution of means x=m (mean) © 2011 Sampling Distribution Sampling distribution of means Process distribution of means x=m (mean) © 2011 Pearson Education Figure S 6. 4 S 6 - 20

Control Charts for Variables u For variables that have continuous dimensions u Weight, speed, Control Charts for Variables u For variables that have continuous dimensions u Weight, speed, length, strength, etc. u x-charts are to control the central tendency of the process u R-charts are to control the dispersion of the process u These two charts must be used together © 2011 Pearson Education S 6 - 21

Setting Chart Limits For x-Charts when we know s Upper control limit (UCL) = 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 S 6 - 22

Setting Control Limits using Samples: The weights of boxes of Oat Flakes within a Setting Control Limits using Samples: The weights of boxes of Oat Flakes within a large production lot are sampled each hour. Managers want to set control limits that include 99. 73% of the sample means. SD = 1 © 2011 Pearson Education S 6 - 23

Setting Control Limits Hour 1 Sample Weight of Number Oat Flakes 1 17 2 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 © 2011 Pearson Education Hour 1 2 3 4 5 6 Mean 16. 1 16. 8 15. 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 S 6 - 24

Setting Control Limits Control Chart for sample of 9 boxes Variation due to assignable 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 © 2011 Pearson Education Out of control Variation due to assignable causes S 6 - 25

Setting Chart Limits For x-Charts when we don’t know s Upper control limit (UCL) Setting Chart Limits For x-Charts when we don’t know s Upper control limit (UCL) = x + A 2 R Lower control limit (LCL) = x - A 2 R where R =average range of the samples A 2 =control chart factor found in Table S 6. 1 x =mean of the sample means © 2011 Pearson Education S 6 - 26

Control Chart Factors Sample Size n Mean Factor A 2 Upper Range D 4 Control Chart Factors Sample Size n Mean Factor A 2 Upper Range D 4 Lower Range D 3 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. 076 0. 136 0. 184 0. 223 0. 284 © 2011 Pearson Education Table S 6. 1 S 6 - 27

Setting Limits using Table Values: Super Cola bottles soft drinks labeled “net weight 12 Setting Limits using Table Values: Super Cola bottles soft drinks labeled “net weight 12 ounces. ” Indeed, an overall process average of 12 ounces has been found by taking many samples, in which each sample contained 5 bottles. The average range of the process is 0. 25 ounce. The OM team wants to determine the upper and lower control limits for averages in this process. © 2011 Pearson Education S 6 - 28

Setting Control Limits Process average x = 12 ounces Average range R =. 25 Setting Control Limits Process average x = 12 ounces Average range R =. 25 Sample size n = 5 © 2011 Pearson Education S 6 - 29

Setting Control Limits Process average x = 12 ounces Average range R =. 25 Setting Control Limits Process average x = 12 ounces Average range R =. 25 Sample size n = 5 UCLx = x + A 2 R = 12 + (. 577)(. 25) = 12 +. 144 = 12. 144 ounces From Table S 6. 1 © 2011 Pearson Education S 6 - 30

Setting Control Limits Process average x = 12 ounces Average range R =. 25 Setting Control Limits Process average x = 12 ounces Average range R =. 25 Sample size n = 5 UCLx LCLx = x + A 2 R = 12 + (. 577)(. 25) = 12 +. 144 = 12. 144 ounces UCL = 12. 144 = x - A 2 R = 12 -. 144 = 11. 857 ounces LCL = 11. 857 © 2011 Pearson Education Mean = 12 S 6 - 31

Restaurant Control Limits For salmon filets at Darden Restaurants Sample Mean x Bar Chart Restaurant Control Limits For salmon filets at Darden Restaurants Sample Mean x Bar Chart 11. 5 – UCL = 11. 524 11. 0 – x – 10. 959 10. 5 – | | | | | 1 3 5 7 9 11 13 15 17 LCL – 10. 394 Sample Range Chart 0. 8 – UCL = 0. 6943 0. 4 – R = 0. 2125 0. 0 – | 1 © 2011 Pearson Education | | | | 3 5 7 9 11 13 15 17 LCL = 0 S 6 - 32

Restaurant Control Limits Capability Histogram LSL USL Specifications LSL 10 USL 12 © 2011 Restaurant Control Limits Capability Histogram LSL USL Specifications LSL 10 USL 12 © 2011 Pearson Education 0 12. 7 11. 4 11. 1 11. 8 10. 5 10. 2 Capability Mean = 10. 959 Std. dev = 1. 88 Cp = 1. 77 Cpk = 1. 7 S 6 - 33

R – Chart u Type of variables control chart u Shows sample ranges over R – Chart u Type of variables control chart u Shows sample ranges over time u Difference between smallest and largest values in sample u Monitors process variability u Independent from process mean © 2011 Pearson Education S 6 - 34

Setting Chart Limits For R-Charts Upper control limit (UCLR) = D 4 R Lower Setting Chart Limits For R-Charts Upper control limit (UCLR) = D 4 R Lower control limit (LCLR) = D 3 R where R =average range of the samples D 3 and D 4 =control chart factors from Table S 6. 1 © 2011 Pearson Education S 6 - 35

Setting Range Limits using Table Values: The average range of a product at Clinton Setting Range Limits using Table Values: The average range of a product at Clinton Manufacturing is 5. 3 pounds. With a sample size of 5, owner Roy Clinton wants to determine the upper and lower control chart limits. © 2011 Pearson Education S 6 - 36

Setting Control Limits Average range R = 5. 3 pounds Sample size n = Setting Control Limits Average range R = 5. 3 pounds Sample size n = 5 From Table S 6. 1 D 4 = 2. 115, D 3 = 0 UCLR = D 4 R = (2. 115)(5. 3) = 11. 2 pounds UCL = 11. 2 LCLR LCL = 0 © 2011 Pearson Education = D 3 R = (0)(5. 3) = 0 pounds Mean = 5. 3 S 6 - 37

Mean and Range Charts (a) (Sampling mean is shifting upward but range is consistent) 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 (x-chart detects shift in central tendency) LCL UCL R-chart LCL (R-chart does not detect change in mean) Figure S 6. 5 © 2011 Pearson Education S 6 - 38

Mean and Range Charts (b) These sampling distributions result in the charts below (Sampling Mean and Range Charts (b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) UCL x-chart (x-chart does not detect the increase in dispersion) LCL UCL R-chart LCL (R-chart detects increase in dispersion) Figure S 6. 5 © 2011 Pearson Education S 6 - 39

Steps In Creating Control Charts 1. Take samples from the population and compute the 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 © 2011 Pearson Education S 6 - 40

Manual and Automated Control Charts © 2011 Pearson Education S 6 - 41 Manual and Automated Control Charts © 2011 Pearson Education S 6 - 41

Control Charts for Attributes u For variables that are categorical u Good/bad, yes/no, acceptable/unacceptable Control Charts for Attributes u For variables that are categorical u Good/bad, yes/no, acceptable/unacceptable u Measurement is typically counting defectives u Charts may measure u Percent defective (p-chart) u Number of defects (c-chart) © 2011 Pearson Education S 6 - 42

Control Limits for p-Charts Population will be a binomial distribution, but applying the Central 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 p z ^ sp n © 2011 Pearson Education p(1 - p) n =mean fraction defective in the sample =number of standard deviations =standard deviation of the sampling distribution =sample size S 6 - 43

Setting Control Limits for Percent Defective: Clerks at Mosier Data Systems key in thousands Setting Control Limits for Percent Defective: Clerks at Mosier Data Systems key in thousands of insurance records each day for a variety of client firms. CEO Donna Mosier wants to set control limits to include 99. 73% of the random variation in the data entry process when it is in control. © 2011 Pearson Education S 6 - 44

p-Chart for Data Entry Sample Number p= Fraction Defective Sample Number 6 5 0 p-Chart for Data Entry Sample Number p= Fraction Defective Sample Number 6 5 0 1 4 2 5 3 3 2 1 2 3 4 5 6 7 8 9 10 Number of Errors . 06. 05. 00. 01. 04. 02. 05. 03. 02 11 12 13 14 15 16 17 18 19 20 80 (100)(20) © 2011 Pearson Education =. 04 sp = ^ Number of Errors Fraction Defective 6 1 8 7 5 4 11 3 0 4 Total = 80 (. 04)(1 -. 04) 100 . 06. 01. 08. 07. 05. 04. 11. 03. 00. 04 =. 02 S 6 - 45

p-Chart for Data Entry ^ UCLp = p + zsp =. 04 + 3(. 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 LCLp = 0. 00 20 Sample number © 2011 Pearson Education S 6 - 46

p-Chart for Data Entry ^ UCLp = p + zsp =. 04 + 3(. 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 LCLp = 0. 00 20 Sample number © 2011 Pearson Education S 6 - 47

Control Limits for c-Charts Population will be a Poisson distribution, but applying the Central 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 © 2011 Pearson Education LCLc = c - 3 c c =mean number defective in the sample S 6 - 48

Setting Control Limits for number of Defects: Red Top Cab Company receives several complaints Setting Control Limits for number of Defects: Red Top Cab Company receives several complaints per day about the behaviour of its drivers. Over a 9 -day period (where days are the units of measure), the owner, Gordon Hoft, received the following numbers of calls from irate passengers: 3, 0, 9, 9, 6, 7, 4, 9, 8, for a total of 54 complaints. Hoft wants to computs 99. 73% control limits. © 2011 Pearson Education S 6 - 49

c-Chart for Cab Company UCLc = c + 3 c =6+3 6 = 13. c-Chart for Cab Company UCLc = c + 3 c =6+3 6 = 13. 35 LCLc = c - 3 c =6 -3 6 =0 Number defective c = 54 complaints/9 days = 6 complaints/day UCLc = 13. 35 14 – 12 – 10 – 8 6 4 2 0 – – – | c= 6 LCLc = 0 | | | | 1 2 3 4 5 6 7 8 9 Day © 2011 Pearson Education S 6 - 50

Managerial Issues and Control Charts Three major management decisions: u Select points in the Managerial Issues and Control Charts Three major management decisions: u Select points in the processes that need SPC u Determine the appropriate charting technique u Set clear policies and procedures © 2011 Pearson Education S 6 - 51

Which Control Chart to Use Variables Data Using an x-Chart and R-Chart 1. Observations Which Control Chart to Use Variables Data Using an x-Chart and R-Chart 1. Observations are variables 2. 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 Rchart 3. Track samples of n observations each. Table S 6. 3 © 2011 Pearson Education S 6 - 52

Which Control Chart to Use Attribute Data Using the p-Chart 1. Observations are attributes Which Control Chart to Use Attribute Data Using the p-Chart 1. Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states. 2. We deal with fraction, proportion, or percent defectives. 3. There are several samples, with many observations in each. For example, 20 samples of n = 100 observations in each. Table S 6. 3 © 2011 Pearson Education S 6 - 53

Which Control Chart to Use Attribute Data Using a c-Chart 1. Observations are attributes Which Control Chart to Use Attribute Data Using a c-Chart 1. Observations are attributes whose defects per unit of output can be counted. 2. We deal with the number counted, which is a small part of the possible occurrences. 3. 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 S 6. 3 © 2011 Pearson Education S 6 - 54

Patterns in Control Charts Upper control limit Target Lower control limit Normal behavior. Process Patterns in Control Charts Upper control limit Target Lower control limit Normal behavior. Process is “in control. ” Figure S 6. 7 © 2011 Pearson Education S 6 - 55

Patterns in Control Charts Upper control limit Target Lower control limit Figure S 6. Patterns in Control Charts Upper control limit Target Lower control limit Figure S 6. 7 © 2011 Pearson Education One plot out above (or below). Investigate for cause. Process is “out of control. ” S 6 - 56

Patterns in Control Charts Upper control limit Target Lower control limit Figure S 6. Patterns in Control Charts Upper control limit Target Lower control limit Figure S 6. 7 © 2011 Pearson Education Trends in either direction, 5 plots. Investigate for cause of progressive change. S 6 - 57

Patterns in Control Charts Upper control limit Target Lower control limit Figure S 6. Patterns in Control Charts Upper control limit Target Lower control limit Figure S 6. 7 © 2011 Pearson Education Two plots very near lower (or upper) control. Investigate for cause. S 6 - 58

Patterns in Control Charts Upper control limit Target Lower control limit Run of 5 Patterns in Control Charts Upper control limit Target Lower control limit Run of 5 above (or below) central line. Investigate for cause. Figure S 6. 7 © 2011 Pearson Education S 6 - 59

Patterns in Control Charts Upper control limit Target Lower control limit Erratic behavior. Investigate. Patterns in Control Charts Upper control limit Target Lower control limit Erratic behavior. Investigate. Figure S 6. 7 © 2011 Pearson Education S 6 - 60

Process Capability u The natural variation of a process should be small enough to Process Capability u The natural variation of a process should be small enough to produce products that meet the standards required u A process in statistical control does not necessarily meet the design specifications u Process capability is a measure of the relationship between the natural variation of the process and the design specifications © 2011 Pearson Education S 6 - 61

Process Capability Ratio Upper Specification - Lower Specification Cp = 6 s u A Process Capability Ratio Upper Specification - Lower Specification Cp = 6 s u A capable process must have a Cp of at least 1. 0 u Does not look at how well the process is centered in the specification range u Often a target value of Cp = 1. 33 is used to allow for off-center processes u Six Sigma quality requires a Cp = 2. 0 © 2011 Pearson Education S 6 - 62

Process Capability Ration (Cp) In a GE insurance claims process, x bar = 210. Process Capability Ration (Cp) In a GE insurance claims process, x bar = 210. 0 minutes, and S. D = 0. 516. The design specification to meet customer expectations is 210 +/- 3 minutes. So the upper specification is 213 minutes and the lower specification is 207 minutes. The OM manager wants to compute the process capability ration. © 2011 Pearson Education S 6 - 63

Process Capability Ratio Insurance claims process Process mean x = 210. 0 minutes Process Process Capability Ratio Insurance claims process Process mean x = 210. 0 minutes Process standard deviation s =. 516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification Cp = 6 s © 2011 Pearson Education S 6 - 64

Process Capability Ratio Insurance claims process Process mean x = 210. 0 minutes Process Process Capability Ratio Insurance claims process Process mean x = 210. 0 minutes Process standard deviation s =. 516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification Cp = 6 s 213 - 207 = = 1. 938 6(. 516) © 2011 Pearson Education S 6 - 65

Process Capability Ratio Insurance claims process Process mean x = 210. 0 minutes Process Process Capability Ratio Insurance claims process Process mean x = 210. 0 minutes Process standard deviation s =. 516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification Cp = 6 s 213 - 207 = = 1. 938 6(. 516) © 2011 Pearson Education Process is capable S 6 - 66

Process Capability Index Upper Lower Cpk = minimum of Specification - x , x Process Capability Index Upper Lower Cpk = minimum of Specification - x , x - Specification Limit 3 s 3 s u A capable process must have a Cpk of at least 1. 0 u A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes © 2011 Pearson Education S 6 - 67

Process Capability Index (Cpk): You are the process improvement manager and have developed a Process Capability Index (Cpk): You are the process improvement manager and have developed a new machine to cut insoles for the company’s top-of-the-line running shoes. You are excited because the company’s goal is no more than 3. 4 defects per million and this machine may be the innovation you need. The insoles cannot be more than =/- 0. 001 of an inch from the required thickness of 0. 250’’. You want to know if you should replace the existing machine, which has a Cpk of 1. 0. © 2011 Pearson Education S 6 - 68

Process Capability Index New Cutting Machine New process mean x =. 250 inches Process 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 © 2011 Pearson Education S 6 - 69

Process Capability Index New Cutting Machine New process mean x =. 250 inches Process 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 (. 251) -. 250 Cpk = minimum of , (3). 0005 © 2011 Pearson Education S 6 - 70

Process Capability Index New Cutting Machine New process mean x =. 250 inches Process 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 (. 251) -. 250 - (. 249) Cpk = minimum of , (3). 0005 Both calculations result in Cpk = . 001 = 0. 67. 0015 © 2011 Pearson Education New machine is NOT capable S 6 - 71

Interpreting Cpk = negative number Cpk = zero Cpk = between 0 and 1 Interpreting Cpk = negative number Cpk = zero Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Figure S 6. 8 © 2011 Pearson Education S 6 - 72

Acceptance Sampling u Form of quality testing used for incoming materials or finished goods Acceptance Sampling u Form of quality testing used for incoming materials or finished goods u Take samples at random from a lot (shipment) of items u Inspect each of the items in the sample u Decide whether to reject the whole lot based on the inspection results u Only screens lots; does not drive quality improvement efforts © 2011 Pearson Education S 6 - 73

Acceptance Sampling u Form of quality testing used for incoming materials or finished goods Acceptance Sampling u Form of quality testing used for incoming materials or finished goods u Take samples at random from a lot (shipment) of items u Rejected lots can be: u Inspect each of the items into the u Returned the sample supplier u Decide whether to reject the whole lot based on the inspection for defectives u Culled results u Only screens lots; (100% not drive does inspection) quality improvement efforts © 2011 Pearson Education S 6 - 74

Operating Characteristic Curve u Shows how well a sampling plan discriminates between good and Operating Characteristic Curve u Shows how well a sampling plan discriminates between good and bad lots (shipments) u Shows the relationship between the probability of accepting a lot and its quality level © 2011 Pearson Education S 6 - 75

P(Accept Whole Shipment) The “Perfect” OC Curve Keep whole shipment 100 – 75 – P(Accept Whole Shipment) The “Perfect” OC Curve Keep whole shipment 100 – 75 – Return whole shipment 50 – 25 – Cut-Off | | | | | 0 |– | 0 10 20 30 40 50 60 70 80 90 100 % Defective in Lot © 2011 Pearson Education S 6 - 76

An OC Curve Figure S 6. 9 100 – 95 – = 0. 05 An OC Curve Figure S 6. 9 100 – 95 – = 0. 05 producer’s risk for AQL 75 – Probability of 50 – Acceptance 25 – 10 – = 0. 10 0 |– 0 Consumer’s risk for LTPD © 2011 Pearson Education | 1 Good lots | 2 AQL | 3 | 4 | 5 Indifference zone | 6 | 7 LTPD | 8 Percent defective Bad lots S 6 - 77

AQL and LTPD u Acceptable Quality Level (AQL) u Poorest level of quality we AQL and LTPD u Acceptable Quality Level (AQL) u Poorest level of quality we are willing to accept u Lot Tolerance Percent Defective (LTPD) u Quality level we consider bad u Consumer (buyer) does not want to accept lots with more defects than LTPD © 2011 Pearson Education S 6 - 78

Producer’s and Consumer’s Risks u Producer's risk ( ) u Probability of rejecting a Producer’s and Consumer’s Risks u Producer's risk ( ) u Probability of rejecting a good lot u Probability of rejecting a lot when the fraction defective is at or above the AQL u Consumer's risk ( ) u Probability of accepting a bad lot u Probability of accepting a lot when fraction defective is below the LTPD © 2011 Pearson Education S 6 - 79

OC Curves for Different Sampling Plans n = 50, c = 1 n = OC Curves for Different Sampling Plans n = 50, c = 1 n = 100, c = 2 © 2011 Pearson Education S 6 - 80

Average Outgoing Quality AOQ = (Pd)(Pa)(N - n) N where Pd = true percent 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 © 2011 Pearson Education S 6 - 81

Average Outgoing Quality 1. If a sampling plan replaces all defectives 2. If we Average Outgoing Quality 1. If a sampling plan replaces all defectives 2. If we know the incoming percent defective for the lot We can compute the average outgoing quality (AOQ) in percent defective The maximum AOQ is the highest percent defective or the lowest average quality and is called the average outgoing quality level (AOQL) © 2011 Pearson Education S 6 - 82

Automated Inspection u Modern technologies allow virtually 100% inspection at minimal costs u Not Automated Inspection u Modern technologies allow virtually 100% inspection at minimal costs u Not suitable for all situations © 2011 Pearson Education S 6 - 83

SPC and Process Variability Lower specification limit Upper specification limit (a) Acceptance sampling (Some SPC and Process Variability Lower specification limit Upper specification limit (a) Acceptance sampling (Some bad units accepted) (b) Statistical process control (Keep the process in control) (c) Cpk >1 (Design a process that is in control) Process mean, m Figure S 6. 10 © 2011 Pearson Education S 6 - 84