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- 1. 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
- 2. 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© 2008 Prentice Hall, Inc. S6 – 2
- 3. Outline – Continued Process Capability Process Capability Ratio (Cp) Process Capability Index (Cpk ) Acceptance Sampling Operating Characteristic Curve Average Outgoing Quality© 2008 Prentice Hall, Inc. S6 – 3
- 4. 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© 2008 Prentice Hall, Inc. S6 – 4
- 5. 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© 2008 Prentice Hall, Inc. S6 – 5
- 6. 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 – 6
- 7. 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 – 7
- 8. 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 – 8
- 9. Samples To measure the process, we take samples and analyze the sample statistics following these steps Each of these represents one (a) Samples of the sample of five product, say five boxes of cereal boxes of cereal taken off the filling Frequency # # machine line, vary # # # from each other in # # # # weight # # # # # # # # # # # # # # # # # Figure S6.1 Weight© 2008 Prentice Hall, Inc. S6 – 9
- 10. Samples To measure the process, we take samples and analyze the sample statistics following these steps The solid line represents the (b) After enough distribution samples are taken from a stable process, they form a Frequency pattern called a distribution Figure S6.1 Weight© 2008 Prentice Hall, Inc. S6 – 10
- 11. 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 Frequency Central tendency Variation Shape Weight Weight Weight© 2008 Prentice Hall, Inc. S6 – 11
- 12. Samples To measure the process, we take samples and analyze the sample statistics following these steps (d) If only natural causes of variation are Frequency present, the Prediction output of a process forms a distribution that Tim e is stable over Weight time and is Figure S6.1 predictable© 2008 Prentice Hall, Inc. S6 – 12
- 13. Samples To measure the process, we take samples and analyze the sample statistics following these steps ? ? ?? ? (e) If assignable ? ? ? ? causes are ? ? ? ? ?? ? ?? present, the Frequency ? process output is Prediction not stable over time and is not predicable Tim e Weight Figure S6.1© 2008 Prentice Hall, Inc. S6 – 13
- 14. 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 – 14
- 15. Process Control (a) In statistical control and capable of producing within Frequency control limits Lower control limit Upper control limit (b) In statistical control but not capable of producing within control limits (c) Out of control Size (weight, length, speed, etc.) Figure S6.2© 2008 Prentice Hall, Inc. S6 – 15
- 16. Types of Data Variables Attributes Characteristics that Defect-related can take any real characteristics value Classify products May be in whole or as either good or in fractional bad or count numbers defects Continuous random Categorical or variables discrete random variables© 2008 Prentice Hall, Inc. S6 – 16
- 17. 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 x=µ as the population mean µ 2. The standard deviation of the sampling distribution (σ x) will σ equal the population standard σx = n deviation (σ ) divided by the square root of the sample size, n© 2008 Prentice Hall, Inc. S6 – 17
- 18. Population and Sampling Distributions Three population Distribution of distributions sample means Mean of sample means = x Beta Standard deviation of σ the sample = σx = Normal n means Uniform | | | | | | | -3σ x -2σ x -1σ x x +1σ x +2σ x +3σ x 95.45% fall within ± 2σ x 99.73% of all x fall within ± 3σ x Figure S6.3© 2008 Prentice Hall, Inc. S6 – 18
- 19. Sampling Distribution Sampling distribution of means Process distribution of means x=µ (mean) Figure S6.4© 2008 Prentice Hall, Inc. S6 – 19
- 20. 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 – 20
- 21. Setting Chart Limits For x-Charts when we know σ Upper control limit (UCL) = x + zσ x Lower control limit (LCL) = x - zσ x where x = mean of the sample means or a target value set for the process z = number of normal standard deviations σx = standard deviation of the sample means = σ/ n© 2008 Prentice Hall, Inc. σ = population standard S6 – 21
- 22. Setting Control Limits Hour 1 Hour Mean Hour Mean Sample Weight of 1 16.1 7 15.2 Number Oat Flakes 2 16.8 8 16.4 1 17 3 15.5 9 16.3 2 13 4 16.5 10 14.8 3 16 5 16.5 11 14.2 4 18 6 16.4 12 17.3 n=9 5 17 6 16 For 99.73% control limits, z = 3 7 15 8 17 UCLx = x + zσ x = 16 + 3(1/3) = 17 ozs 9 16 Mean 16.1 LCLx = x - zσ x = 16 - 3(1/3) = 15 ozs σ= 1© 2008 Prentice Hall, Inc. S6 – 22
- 23. Setting Control Limits Control Chart for sample of Variation due Out of to assignable 9 boxes control causes 17 = UCL Variation due to 16 = Mean natural causes 15 = LCL Variation due | | | | | | | | | | | | to assignable 1 2 3 4 5 6 7 8 9 10 11 12 Out of causes Sample number control© 2008 Prentice Hall, Inc. S6 – 23
- 24. Setting Chart Limits For x-Charts when we don’t know σ 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© 2008 Prentice Hall, Inc. S6 – 24
- 25. 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© 2008 Prentice Hall, Inc. S6 – 25
- 26. Setting Control Limits Process average x = 12 ounces Average range R = .25 Sample size n = 5© 2008 Prentice Hall, Inc. S6 – 26
- 27. 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 – 27
- 28. Setting Control Limits Process average x = 12 ounces Average range R = .25 Sample size n = 5 UCLx = x + A2R UCL = 12.144 = 12 + (.577)(.25) = 12 + .144 Mean = 12 = 12.144 ounces LCLx = x - A2R LCL = 11.857 = 12 - .144 = 11.857 ounces© 2008 Prentice Hall, Inc. S6 – 28
- 29. 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 – 29
- 30. 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 – 30
- 31. Setting Control Limits Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D4 = 2.115, D3 = 0 UCLR = D4R UCL = 11.2 = (2.115)(5.3) = 11.2 pounds Mean = 5.3 LCLR = D3 R LCL = 0 = (0)(5.3) = 0 pounds© 2008 Prentice Hall, Inc. S6 – 31
- 32. Mean and Range Charts (a) These (Sampling mean is sampling shifting upward but distributions range is consistent) result in the charts below UCL (x-chart detects x-chart shift in central tendency) LCL UCL (R-chart does not R-chart detect change in mean) LCL Figure S6.5© 2008 Prentice Hall, Inc. S6 – 32
- 33. Mean and Range Charts (b) These (Sampling mean sampling is constant but distributions dispersion is result in the increasing) charts below UCL (x-chart does not x-chart detect the increase in dispersion) LCL UCL (R-chart detects R-chart increase in dispersion) LCL Figure S6.5© 2008 Prentice Hall, Inc. S6 – 33
- 34. 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 – 34
- 35. Manual and Automated Control Charts© 2008 Prentice Hall, Inc. S6 – 35
- 36. 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 – 36
- 37. 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 + zσ p ^ p(1 - p) σp = ^ n LCLp = p - zσ p ^p = mean fraction defective in the samplez = number of standard deviationsσp = standard deviation of the sampling distribution ^n = sample size © 2008 Prentice Hall, Inc. S6 – 37
- 38. p-Chart for Data Entry 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 80 (.04)(1 - .04) p= = .04 σp = = .02 (100)(20) ^ 100© 2008 Prentice Hall, Inc. S6 – 38
- 39. p-Chart for Data Entry UCLp = p + zσ p = .04 + 3(.02) = .10 ^ LCLp = p - zσ p = .04 - 3(.02) = 0 ^ .11 – .10 – UCLp = 0.10 .09 – Fraction defective .08 – .07 – .06 – .05 – .04 – p = 0.04 .03 – .02 – .01 – LCLp = 0.00 .00 – | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 Sample number© 2008 Prentice Hall, Inc. S6 – 39
- 40. p-Chart for Data Entry UCLp = p + zσ p = .04 + 3(.02) = .10 ^ Possible LCLp = p - zσ p = .04 - 3(.02) = 0 ^ assignable causes present .11 – .10 – UCLp = 0.10 .09 – Fraction defective .08 – .07 – .06 – .05 – .04 – p = 0.04 .03 – .02 – .01 – LCLp = 0.00 .00 – | | | | | | | | | | 2 4 6 8 10 12 14 16 18 20 Sample number© 2008 Prentice Hall, Inc. S6 – 40
- 41. 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 c = mean number defective in the sample© 2008 Prentice Hall, Inc. S6 – 41
- 42. c-Chart for Cab Company c = 54 complaints/9 days = 6 complaints/day UCLc = c + 3 c 14 – UCLc = 13.35 14 Number defective =6+3 6 12 – = 13.35 10 – 8 – 6 – c= 6 LCLc = c - 3 c 4 – =6-3 6 2 – LCLc = 0 0 – | | | | | | | | | =0 1 2 3 4 5 6 7 8 9 Day© 2008 Prentice Hall, Inc. S6 – 42
- 43. 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 – 43
- 44. 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 – 44
- 45. 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 – 45
- 46. 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 – 46
- 47. Patterns in Control Charts Upper control limit Target Lower control limit Normal behavior. Process is “in control.” Figure S6.7© 2008 Prentice Hall, Inc. S6 – 47
- 48. Patterns in Control Charts Upper control limit Target Lower control limit One plot out above (or below). Investigate for Figure S6.7 cause. Process is “out of control.”© 2008 Prentice Hall, Inc. S6 – 48
- 49. Patterns in Control Charts Upper control limit Target Lower control limit Trends in either direction, 5 plots. Figure S6.7 Investigate for cause of progressive change.© 2008 Prentice Hall, Inc. S6 – 49
- 50. Patterns in Control Charts Upper control limit Target Lower control limit Two plots very near lower (or upper) Figure S6.7 control. Investigate for cause.© 2008 Prentice Hall, Inc. S6 – 50
- 51. Patterns in Control Charts Upper control limit Target Lower control limit Run of 5 above (or below) central line. Figure S6.7 Investigate for cause.© 2008 Prentice Hall, Inc. S6 – 51
- 52. Patterns in Control Charts Upper control limit Target Lower control limit Erratic behavior. Investigate. Figure S6.7© 2008 Prentice Hall, Inc. S6 – 52
- 53. 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© 2008 Prentice Hall, Inc. S6 – 53
- 54. Process Capability Ratio Upper Specification - Lower Specification Cp = 6σ 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© 2008 Prentice Hall, Inc. S6 – 54
- 55. Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation σ = .516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification Cp = 6σ© 2008 Prentice Hall, Inc. S6 – 55
- 56. Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation σ = .516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification Cp = 6σ 213 - 207 = = 1.938 6(.516)© 2008 Prentice Hall, Inc. S6 – 56
- 57. Process Capability Ratio Insurance claims process Process mean x = 210.0 minutes Process standard deviation σ = .516 minutes Design specification = 210 ± 3 minutes Upper Specification - Lower Specification Cp = 6σ 213 - 207 = = 1.938 Process is 6(.516) capable© 2008 Prentice Hall, Inc. S6 – 57
- 58. Process Capability Index Upper Lower Cpk = minimum of Specification - x , x - Specification Limit Limit 3σ 3σ 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© 2008 Prentice Hall, Inc. S6 – 58
- 59. Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation σ = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches© 2008 Prentice Hall, Inc. S6 – 59
- 60. Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation σ = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches (.251) - .250 Cpk = minimum of , (3).0005© 2008 Prentice Hall, Inc. S6 – 60
- 61. Process Capability Index New Cutting Machine New process mean x = .250 inches Process standard deviation σ = .0005 inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches (.251) - .250 .250 - (.249) Cpk = minimum of , (3).0005 (3).0005 Both calculations result in New machine is .001 Cpk = = 0.67 NOT capable .0015© 2008 Prentice Hall, Inc. S6 – 61
- 62. Interpreting Cpk Cpk = negative number Cpk = zero Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Figure S6.8© 2008 Prentice Hall, Inc. S6 – 62
- 63. 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© 2008 Prentice Hall, Inc. S6 – 63
- 64. Acceptance Sampling Form of quality testing used for incoming materials or finished goods Take samples at random from a lot Rejected lots can be: (shipment) of items Inspect each of theReturnedthethe items in to sample supplier Decide whether to reject the whole lot Culled for based on the inspection results defectives Only screens lots; does not drive (100% inspection) quality improvement efforts© 2008 Prentice Hall, Inc. S6 – 64
- 65. 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© 2008 Prentice Hall, Inc. S6 – 65
- 66. The “Perfect” OC Curve Keep whole shipment P(Accept Whole Shipment) 100 – 75 – Return whole 50 – shipment 25 – Cut-Off 0 – | | | | | | | | | | | 0 10 20 30 40 50 60 70 80 90 100 % Defective in Lot© 2008 Prentice Hall, Inc. S6 – 66
- 67. An OC Curve Figure S6.9 100 – 95 – α = 0.05 producer’s risk for AQL 75 – Probability of 50 – Acceptance 25 – 10 – β = 0.10 0 |– | | | | | | | | Percent 0 1 2 3 4 5 6 7 8 defective AQL LTPD Consumer’s risk for LTPD Good Indifference Bad lots lots zone© 2008 Prentice Hall, Inc. S6 – 67
- 68. 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© 2008 Prentice Hall, Inc. S6 – 68
- 69. Producer’s and Consumer’s Risks Producers risk (α ) Probability of rejecting a good lot Probability of rejecting a lot when the fraction defective is at or above the AQL Consumers risk (β ) Probability of accepting a bad lot Probability of accepting a lot when fraction defective is below the LTPD© 2008 Prentice Hall, Inc. S6 – 69
- 70. OC Curves for Different Sampling Plans n = 50, c = 1 n = 100, c = 2© 2008 Prentice Hall, Inc. S6 – 70
- 71. Average Outgoing Quality (Pd)(Pa)(N - n) AOQ = 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© 2008 Prentice Hall, Inc. S6 – 71
- 72. 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)© 2008 Prentice Hall, Inc. S6 – 72
- 73. Automated Inspection Modern technologies allow virtually 100% inspection at minimal costs Not suitable for all situations© 2008 Prentice Hall, Inc. S6 – 73
- 74. SPC and Process Variability Lower Upper specification specification limit 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, µ Figure S6.10© 2008 Prentice Hall, Inc. S6 – 74

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