1. S6 - 1© 2011 Pearson Education, Inc. publishing as Prentice Hall S6 Statistical Process Control PowerPoint presentation to accompany Heizer and Render Operations Management, 10e Principles of Operations Management, 8e PowerPoint slides by Jeff Heyl

2. S6 - 2© 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

3. S6 - 3© 2011 Pearson Education, Inc. publishing as Prentice Hall  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 Statistical Process Control (SPC)

4. S6 - 4© 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”

5. S6 - 5© 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

6. S6 - 6© 2011 Pearson Education, Inc. publishing as Prentice Hall Samples To measure the process, we take samples and analyze the sample statistics following these steps (a)Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Frequency Weight # ## # ## ## # ### #### ######### # Each of these represents one sample of five boxes of cereal Figure S6.1

7. S6 - 7© 2011 Pearson Education, Inc. publishing as Prentice Hall Samples To measure the process, we take samples and analyze the sample statistics following these steps (b)After enough samples are taken from a stable process, they form a pattern called a distribution The solid line represents the distribution Frequency Weight Figure S6.1

8. S6 - 8© 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 Weight Central tendency Weight Variation Weight Shape Frequency Figure S6.1

9. S6 - 9© 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 Weight Time Frequency Prediction Figure S6.1

10. S6 - 10© 2011 Pearson Education, Inc. publishing as Prentice Hall Samples To measure the process, we take samples and analyze the sample statistics following these steps (e)If assignable causes are present, the process output is not stable over time and is not predicable Weight Time Frequency Prediction ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Figure S6.1

11. S6 - 11© 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

12. S6 - 12© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Control Figure S6.2 Frequency (weight, length, speed, etc.) Size Lower control limit Upper control limit (a) In statistical control and capable of producing within control limits (b) In statistical control but not capable of producing within control limits (c) Out of control

13. S6 - 13© 2011 Pearson Education, Inc. publishing as Prentice Hall Types of Data  Characteristics that can take any real value  May be in whole or in fractional numbers  Continuous random variables VariablesAttributes  Defect-related characteristics  Classify products as either good or bad or count defects  Categorical or discrete random variables

14. S6 - 14© 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 1.The mean of the sampling distribution (x) will be the same as the population mean  x =   n  x = 2.The standard deviation of the sampling distribution (  x ) will equal the population standard deviation (  ) divided by the square root of the sample size, n

15. S6 - 15© 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 =  x =  n Mean of sample means = x ||||||| -3  x -2  x -1  x x+1  x +2  x +3  x 99.73% of all x fall within ± 3  x 95.45% fall within ± 2  x Figure S6.3

16. S6 - 16© 2011 Pearson Education, Inc. publishing as Prentice Hall Sampling Distribution x =  (mean) Sampling distribution of means Process distribution of means Figure S6.4

17. S6 - 17© 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

18. S6 - 18© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Chart Limits For x-Charts when we know  Upper control limit (UCL) = x + z  x Lower control limit (LCL) = x - z  x wherex=mean of the sample means or a target value set for the process z=number of normal standard deviations  x =standard deviation of the sample means =  / n  =population standard deviation n=sample size

19. S6 - 19© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits Hour 1 SampleWeight of NumberOat Flakes 117 213 316 418 517 616 715 817 916 Mean16.1  =1 HourMeanHourMean 116.1715.2 216.8816.4 315.5916.3 416.51014.8 516.51114.2 616.41217.3 n = 9 LCL x = x - z  x = 16 - 3(1/3) = 15 ozs For 99.73% control limits, z = 3 UCL x = x + z  x = 16 + 3(1/3) = 17 ozs

20. S6 - 20© 2011 Pearson Education, Inc. publishing as Prentice Hall 17 = UCL 15 = LCL 16 = Mean Setting Control Limits Control Chart for sample of 9 boxes Sample number |||||||||||| 123456789101112 Variation due to assignable causes Variation due to natural causes Out of control

21. S6 - 21© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Chart Limits For x-Charts when we don’t know  Lower control limit (LCL) = x - A 2 R Upper control limit (UCL) = x + A 2 R whereR=average range of the samples A 2 =control chart factor found in Table S6.1 x=mean of the sample means

22. S6 - 22© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Chart Factors Table S6.1 Sample Size Mean Factor Upper Range Lower Range n A 2 D 4 D 3 21.8803.2680 31.0232.5740 4.7292.2820 5.5772.1150 6.4832.0040 7.4191.9240.076 8.3731.8640.136 9.3371.8160.184 10.3081.7770.223 12.2661.7160.284

23. S6 - 23© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits Process average x = 12 ounces Average range R =.25 Sample size n = 5

24. S6 - 24© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits UCL x = x + A 2 R = 12 + (.577)(.25) = 12 +.144 = 12.144 ounces Process average x = 12 ounces Average range R =.25 Sample size n = 5 From Table S6.1

25. S6 - 25© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits UCL x = x + A 2 R = 12 + (.577)(.25) = 12 +.144 = 12.144 ounces LCL x = x - A 2 R = 12 -.144 = 11.857 ounces Process average x = 12 ounces Average range R =.25 Sample size n = 5 UCL = 12.144 Mean = 12 LCL = 11.857

26. S6 - 26© 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 ||||||||| 1357911131517 11.5 – 11.0 – 10.5 – Sample Range Range Chart UCL = 0.6943 R = 0.2125 LCL = 0 ||||||||| 1357911131517 0.8 – 0.4 – 0.0 –

27. S6 - 27© 2011 Pearson Education, Inc. publishing as Prentice Hall Restaurant Control Limits Specifications LSL 10 USL 12 Capability Mean = 10.959 Std.dev = 1.88 C p = 1.77 C pk = 1.7 Capability Histogram LSLUSL 10.210.510.811.111.411.712.0

28. S6 - 28© 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

29. S6 - 29© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Chart Limits For R-Charts Lower control limit (LCL R ) = D 3 R Upper control limit (UCL R ) = D 4 R where R=average range of the samples D 3 and D 4 =control chart factors from Table S6.1

30. S6 - 30© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits UCL R = D 4 R = (2.115)(5.3) = 11.2 pounds LCL R = D 3 R = (0)(5.3) = 0 pounds Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D 4 = 2.115, D 3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0

31. S6 - 31© 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) R-chart (R-chart does not detect change in mean) UCL LCL Figure S6.5 x-chart (x-chart detects shift in central tendency) UCL LCL

32. S6 - 32© 2011 Pearson Education, Inc. publishing as Prentice Hall Mean and Range Charts R-chart (R-chart detects increase in dispersion) UCL LCL Figure S6.5 (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

33. S6 - 33© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

34. S6 - 34© 2011 Pearson Education, Inc. publishing as Prentice Hall Manual and Automated Control Charts

35. S6 - 35© 2011 Pearson Education, Inc. publishing as Prentice Hall Managerial Issues and Control Charts  Select points in the processes that need SPC  Determine the appropriate charting technique  Set clear policies and procedures Three major management decisions:

36. S6 - 36© 2011 Pearson Education, Inc. publishing as Prentice Hall Which Control Chart to Use Table S6.3 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 R- chart 3.Track samples of n observations each.

37. S6 - 37© 2011 Pearson Education, Inc. publishing as Prentice Hall Patterns in Control Charts Normal behavior. Process is “in control.” Upper control limit Target Lower control limit Figure S6.7

38. S6 - 38© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts One plot out above (or below). Investigate for cause. Process is “out of control.” Figure S6.7

39. S6 - 39© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7

40. S6 - 40© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Two plots very near lower (or upper) control. Investigate for cause. Figure S6.7

41. S6 - 41© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Run of 5 above (or below) central line. Investigate for cause. Figure S6.7

42. S6 - 42© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Erratic behavior. Investigate. Figure S6.7

43. S6 - 43© 2011 Pearson Education, Inc. publishing as Prentice Hall 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

44. S6 - 44© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Ratio C p = Upper Specification - Lower Specification 6   A capable process must have a C p of at least 1.0  Does not look at how well the process is centered in the specification range  Often a target value of C p = 1.33 is used to allow for off-center processes  Six Sigma quality requires a C p = 2.0

45. S6 - 45© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Ratio C p = Upper Specification - Lower Specification 6  Insurance claims process Process mean x = 210.0 minutes Process standard deviation  =.516 minutes Design specification = 210 ± 3 minutes

46. S6 - 46© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Ratio C p = Upper Specification - Lower Specification 6  Insurance claims process Process mean x = 210.0 minutes Process standard deviation  =.516 minutes Design specification = 210 ± 3 minutes = = 1.938 213 - 207 6(.516)

47. S6 - 47© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Ratio C p = Upper Specification - Lower Specification 6  Insurance claims process Process mean x = 210.0 minutes Process standard deviation  =.516 minutes Design specification = 210 ± 3 minutes = = 1.938 213 - 207 6(.516) Process is capable

48. S6 - 48© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Index  A capable process must have a C pk 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 C pk = minimum of, Upper Specification - x Limit  Lower x -Specification Limit 

49. S6 - 49© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation  =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches

50. S6 - 50© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation  =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches C pk = minimum of, (.251) -.250 (3).0005

51. S6 - 51© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Capability Index New Cutting Machine New process mean x =.250 inches Process standard deviation  =.0005 inches Upper Specification Limit =.251 inches Lower Specification Limit =.249 inches C pk = = 0.67.001.0015 New machine is NOT capable C pk = minimum of, (.251) -.250 (3).0005.250 - (.249) (3).0005 Both calculations result in

52. S6 - 52© 2011 Pearson Education, Inc. publishing as Prentice Hall Interpreting C pk C pk = negative number C pk = zero C pk = between 0 and 1 C pk = 1 C pk > 1 Figure S6.8

53. S6 - 53© 2011 Pearson Education, Inc. publishing as Prentice Hall Automated Inspection  Modern technologies allow virtually 100% inspection at minimal costs  Not suitable for all situations

54. S6 - 54© 2011 Pearson Education, Inc. publishing as Prentice Hall SPC and Process Variability (a)Acceptance sampling (Some bad units accepted) (b)Statistical process control (Keep the process in control) (c)C pk >1 (Design a process that is in control) Lower specification limit Upper specification limit Process mean,  Figure S6.10