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Operations Management Statistical Process Control Supplement 6

Operations Management Statistical Process Control Supplement 6. Outline. STATISTICAL PROCESS CONTROL (SPC) Control Charts for Variables The Central Limit Theorem Setting Mean Chart Limits Setting Range Chart Limits ( R -Charts) Using Mean and Range Charts Control Charts for Attributes

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Operations Management Statistical Process Control Supplement 6

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  1. Operations ManagementStatistical Process ControlSupplement 6 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  2. Outline • STATISTICAL PROCESS CONTROL (SPC) • Control Charts for Variables • The Central Limit Theorem • Setting Mean Chart Limits • Setting Range Chart Limits (R-Charts) • Using Mean and Range Charts • Control Charts for Attributes • Managerial Issues and Control Charts © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  3. Outline • PROCESS CAPABILITY • Process Capability Ratio (Cp) • Process Capability Index (Cpk • ACCEPTANCE SAMPLING • Operating Characteristic (OC) Curves • Average Outgoing Quality © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  4. Learning Objectives When you complete this chapter, you should be able to Identify or Define: • Natural and assignable causes of variation • Central limit theorem • Attribute and variable inspection • Process control • LCL and UCL • p-charts and C-charts © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  5. Learning Objectives - Continued When you complete this chapter, you should be able to Identify or Define: • Acceptance sampling • OC curve • AQL and LTPD • AOQ • Producer’s and consumer’s risk © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  6. Learning Objectives - Continued When you complete this chapter, you should be able to Describe or explain: • The role of statistical quality control © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  7. Statistical Quality Control (SPC) • Measures performance of a process • Uses mathematics (i.e., statistics) • Involves collecting, organizing, & interpreting data • Objective: provide statistical signal when assignable causes of variation are present • Used to • Control the process as products are produced • Inspect samples of finished products © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  8. Statistical Quality Control Acceptance Sampling Process Control Variables Charts Attributes Charts Types of Statistical Quality Control © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  9. Natural and Assignable Variation © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  10. Variables Attributes Quality Characteristics • Characteristics for which you focus on defects • Classify products as either ‘good’ or ‘bad’, or count # defects • e.g., radio works or not • Categorical or discrete random variables • Characteristics that you measure, e.g., weight, length • May be in whole or in fractional numbers • Continuous random variables © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  11. Statistical Process Control (SPC) • Statistical technique used to ensure process is making product to standard • All process are subject to variability • Natural causes: Random variations • Assignable causes: Correctable problems • Machine wear, unskilled workers, poor material • Objective: Identify assignable causes • Uses process control charts © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  12. (a) In statistical control and capable of producing within control limits. A process with only natural causes of variation and capable of producing within the specified control limits. Frequency Lower control limit Upper control limit (b) In statistical control, but not capable of producing within control limits. A process in control (only natural causes of variation are present) but not capable of producing within the specified control limits; and (c) Out of control. A process out of control having assignable causes of variation. Size (Weight, length, speed, etc. ) Process Control: Three Types of Process Outputs © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  13. Three population distributions Distribution of sample means Beta Standard deviation of the sample means Normal Uniform (mean) The Relationship Between Population and Sampling Distributions © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  14. Sampling distribution of the means Process distribution of the sample Sampling Distribution of Means, and Process Distribution © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  15. Process Control Charts © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  16. Control Chart Purposes • Show changes in data pattern • e.g., trends • Make corrections before process is out of control • Show causes of changes in data • Assignable causes • Data outside control limits or trend in data • Natural causes • Random variations around average © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  17. Central Limit Theorem As sample size gets large enough, sampling distribution becomes almost normal regardless of population distribution. Theoretical Basis of Control Charts © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  18. Theoretical Basis of Control Charts Central Limit Theorem Standard deviation Mean © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  19. Control Chart Types Continuous Numerical Data Categorical or Discrete Numerical Data Control Charts Variables Attributes Charts Charts R P C X Chart Chart Chart Chart © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  20. No Produce Good Start Provide Service Can we assign causes? Take Sample Yes Inspect Sample Stop Process Create Find Out Why Control Chart Statistical Process Control Steps © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  21. X Chart • Type of variables control chart • Interval or ratio scaled numerical data • Shows sample means over time • Monitors process average • Example: Weigh samples of coffee & compute means of samples; Plot © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  22. Variation due to assignable causes 17=UCL 16=Mean Variation due to natural causes 15=LCL Variation due to assignable causes 1 2 3 4 5 6 7 8 9 10 11 12 Sample Number Out of control Control Chart for Samples of 9 Boxes © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  23. From Table S6.1 Range for sample i Mean for sample i # Samples X Chart Control Limits © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  24. Factors for Computing Control Chart Limits © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  25. R Chart • Type of variables control chart • Interval or ratio scaled numerical data • Shows sample ranges over time • Difference between smallest & largest values in inspection sample • Monitors variability in process • Example: Weigh samples of coffee & compute ranges of samples; Plot © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  26. From Table S6.1 Range for Sample i # Samples R Chart Control Limits © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  27. Steps to Follow When Using Control Charts Collect 20 to 25 samples of n=4 or n=5 from a stable process and compute the mean. Compute the overall means, set approximate control limits,and calculate the preliminary upper and lower control limits.If the process is not currently stable, use the desired mean instead of the overall mean to calculate limits. Graph the sample means and ranges on their respective control charts and determine whether they fall outside the acceptable limits. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  28. Steps to Follow When Using Control Charts - continued • Investigate points or patterns that indicate the process is out of control. Assign causes for the variations. • Collect additional samples and revalidate the control limits. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  29. Mean and Range Charts Complement Each Other © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  30. p Chart • Type of attributes control chart • Nominally scaled categorical data • e.g., good-bad • Shows % of nonconforming items • Example: Count # defective chairs & divide by total chairs inspected; Plot • Chair is either defective or not defective © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  31. z = 2 for 95.5% limits; z = 3 for 99.7% limits # Defective Items in Sample i Size of sample i p Chart Control Limits © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  32. P-Chart for Data Entry Example UCLp=0.10 LCLp=0.00 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  33. c Chart • Type of attributes control chart • Discrete quantitative data • Shows number of nonconformities (defects) in a unit • Unit may be chair, steel sheet, car etc. • Size of unit must be constant • Example: Count # defects (scratches, chips etc.) in each chair of a sample of 100 chairs; Plot © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  34. c Chart Control Limits Use3 for 99.7% limits # Defects in Unit i # Units Sampled © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  35. Patterns to Look for in Control Charts © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  36. Deciding Which Control Chart to Use • Using an X 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. • Using the P-Chart: • We deal with fraction, proportion, or percent defectives • Observations are attributes that can be categorized in two states • Have several samples, each with many observations • Assume a binomial distribution unless the number of samples is very large – then assume a normal distribution. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  37. Deciding Which Control Chart to Use • Using a C-Chart: • Observations are attributes whose defects per unit of output can be counted • The number counted is often a small part of the possible occurrences • Assume a Poisson distribution • Defects such as: number of blemishes on a desk, number of typos in a page of text, flaws in a bolt of cloth © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  38. Process Capability Ratio, Cp © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  39. Process Capability Cpk • Assumes that the process is: • under control • normally distributed © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  40. Cpk = negative number Cpk = zero Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Meanings of Cpk Measures © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  41. What Is Acceptance Sampling? • Form of quality testing used for incoming materials or finished goods • e.g., purchased material & components • Procedure • Take one or more 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 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  42. What Is an Acceptance Plan? • Set of procedures for inspecting incoming materials or finished goods • Identifies • Type of sample • Sample size (n) • Criteria (c) used to reject or accept a lot • Producer (supplier) & consumer (buyer) must negotiate © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  43. Operating Characteristics Curve • Shows how well a sampling plan discriminates between good & bad lots (shipments) • Shows the relationship between the probability of accepting a lot & its quality © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  44. 0 1 2 3 4 5 6 7 8 9 10 OC Curve100% Inspection P(Accept Whole Shipment) 100% Keep whole shipment Return whole shipment 0% Cut-Off % Defective in Lot © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  45. P(Accept Whole Shipment) Probability is not 100%: Risk of keeping bad shipment or returning good one. 100% Keep whole shipment Return whole shipment 0% Cut-Off 0 1 2 3 4 5 6 7 8 9 10 % Defective in Lot OC Curve with Less than 100% Sampling © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  46. AQL & LTPD • Acceptable quality level (AQL) • Quality level of a good lot • Producer (supplier) does not want lots with fewer defects than AQL rejected • Lot tolerance percent defective (LTPD) • Quality level of a bad lot • Consumer (buyer) does not want lots with more defects than LTPD accepted © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  47. Producer’s & Consumer’s Risk • Producer's risk () • Probability of rejecting a good lot • Probability of rejecting a lot when fraction defective is AQL • Consumer's risk (ß) • Probability of accepting a bad lot • Probability of accepting a lot when fraction defective is LTPD © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  48. 100 95 75 50 25 10 0  = 0.05 producer’s risk for AQL Probability of Acceptance = 0.10 Percent Defective Consumer’s risk for LTPD 0 1 2 3 4 5 6 7 8 AQL LTPD Good lots Bad lots Indifference zone An Operating Characteristic (OC) Curve Showing Risks © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  49. P(Accept Whole Shipment) n = 50, c = 1 100% n = 100, c = 2 0% AQL LTPD 0 1 2 3 4 5 6 7 8 9 10 % Defective in Lot OC Curves for Different Sampling Plans © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

  50. Average Outgoing Quality 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 © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458

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