1. © 2006 Prentice Hall, Inc.S6 – 1 Operations Management Supplement 6 – Statistical Process Control © 2006 Prentice Hall, Inc. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e

2. © 2006 Prentice Hall, Inc.S6 – 2 Outline  Statistical Process Control (SPC)  Control Charts for Variables  Setting Mean Chart Limits (x-Charts)  Setting Range Chart Limits (R-Charts)  Process Capability  Process Capability Ratio (C p )  Process Capability Index (C pk )

3. © 2006 Prentice Hall, Inc.S6 – 3  Variability is inherent in every process  Natural or common causes  Special causes  SPC charts provide statistical signals when assignable causes are present  SPC approach support the detection and elimination of assignable causes of variation Statistical Process Control (SPC)

4. © 2006 Prentice Hall, Inc.S6 – 4 Common Cause Variations  Also called common causes  Affect virtually all production processes  Generally this required some change at the systemic level of an organization  Managerial action is often necessary  The objective is to discover when avoidable common causes are present  Eliminate the root causes of the common variations

5. © 2006 Prentice Hall, Inc.S6 – 5 Assignable Variations  Also called special causes of variation  Generally this is some change in the local activity or process  Variations that can be traced to a specific reason at a localized activity  The objective is to discover when special causes are present  Eliminate the root causes of the special variations  Incorporate the good causes

6. © 2006 Prentice Hall, Inc.S6 – 6 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. © 2006 Prentice Hall, Inc.S6 – 7 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. © 2006 Prentice Hall, Inc.S6 – 8 Attributes of Distributions (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. © 2006 Prentice Hall, Inc.S6 – 9 Identifying Presence of Common Sources of Variation (d)If only common 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. © 2006 Prentice Hall, Inc.S6 – 10 Identifying Presence of Special Sources of Variation (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. © 2006 Prentice Hall, Inc.S6 – 11 Data Used for Quality Judgments  Characteristics that can take any real value  May be in whole or in fractional numbers  Continuous random variables, e.g. weight, length, duration, etc. VariablesAttributes  Defect-related characteristics  Classify products as either good or bad or count defects  Categorical or discrete random variables

12. © 2006 Prentice Hall, Inc.S6 – 12 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 =x =x =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

13. © 2006 Prentice Hall, Inc.S6 – 13 Interpreting SPC Charts 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 statistical control and incapable of producing within limits

14. © 2006 Prentice Hall, Inc.S6 – 14 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xx+1x+2x+3x-3x-2x-1xx+1x+2x+3x|||||||-3x-2x-1xx+1x+2x+3x-3x-2x-1xx+1x+2x+3x 99.73% of all x fall within ± 3  x 95.45% fall within ± 2  x Figure S6.3

15. © 2006 Prentice Hall, Inc.S6 – 15 Sampling Distribution x =  (mean) Sampling distribution of means Process distribution of means Figure S6.4

16. © 2006 Prentice Hall, Inc.S6 – 16 Steps In Creating Control Charts 1.Take representative sample from output of a process over a long period of time, e.g. 10 units every hour for 24 hours. 2.Compute means and ranges for the variables and calculate the control limits 3.Draw control limits on the control chart 4.Plot a chart for the means and another for the mean of ranges on the control chart 5.Determine state of process (in or out of control) 6.Investigate possible reasons for out of control events and take corrective action 7.Continue sampling of process output and reset the control limits when necessary

17. © 2006 Prentice Hall, Inc.S6 – 17 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. © 2006 Prentice Hall, Inc.S6 – 18 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. © 2006 Prentice Hall, Inc.S6 – 19 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. © 2006 Prentice Hall, Inc.S6 – 20 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. © 2006 Prentice Hall, Inc.S6 – 21 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. © 2006 Prentice Hall, Inc.S6 – 22 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. © 2006 Prentice Hall, Inc.S6 – 23 Setting Control Limits Process average x = 16.01 ounces Average range R =.25 Sample size n = 5

24. © 2006 Prentice Hall, Inc.S6 – 24 Setting Control Limits UCL x = x + A 2 R = 16.01 + (.577)(.25) = 16.01 +.144 = 16.154 ounces Process average x = 16.01 ounces Average range R =.25 Sample size n = 5 From Table S6.1

25. © 2006 Prentice Hall, Inc.S6 – 25 Setting Control Limits UCL x = x + A 2 R = 16.01 + (.577)(.25) = 16.01 +.144 = 16.154 ounces LCL x = x - A 2 R = 16.01 -.144 = 15.866 ounces Process average x = 16.01 ounces Average range R =.25 Sample size n = 5 UCL = 16.154 Mean = 16.01 LCL = 15.866

26. © 2006 Prentice Hall, Inc.S6 – 26 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

27. © 2006 Prentice Hall, Inc.S6 – 27 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

28. © 2006 Prentice Hall, Inc.S6 – 28 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

29. © 2006 Prentice Hall, Inc.S6 – 29 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) UCLLCL Figure S6.5 x-chart (x-chart detects shift in central tendency) UCLLCL

30. © 2006 Prentice Hall, Inc.S6 – 30 Mean and Range Charts R-chart (R-chart detects increase in dispersion) UCLLCL 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) UCLLCL

31. © 2006 Prentice Hall, Inc.S6 – 31 Automated Control Charts

32. © 2006 Prentice Hall, Inc.S6 – 32 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)

33. © 2006 Prentice Hall, Inc.S6 – 33 Patterns in Control Charts Normal behavior. Process is “in control.” Upper control limit Target Lower control limit Figure S6.7

34. © 2006 Prentice Hall, Inc.S6 – 34 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

35. © 2006 Prentice Hall, Inc.S6 – 35 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

36. © 2006 Prentice Hall, Inc.S6 – 36 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

37. © 2006 Prentice Hall, Inc.S6 – 37 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

38. © 2006 Prentice Hall, Inc.S6 – 38 Process Capability Ratio Cp =Cp =Cp =Cp = 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

39. © 2006 Prentice Hall, Inc.S6 – 39 Process Capability Ratio Cp =Cp =Cp =Cp = Upper Specification - Lower Specification 6  Insurance claims process Process mean x = 210.0 minutes Process standard deviation  =.516 minutes Design specification = 210 ± 3 minutes

40. © 2006 Prentice Hall, Inc.S6 – 40 Process Capability Ratio Cp =Cp =Cp =Cp = 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)

41. © 2006 Prentice Hall, Inc.S6 – 41 Process Capability Ratio Cp =Cp =Cp =Cp = 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

42. © 2006 Prentice Hall, Inc.S6 – 42 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 

43. © 2006 Prentice Hall, Inc.S6 – 43 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

44. © 2006 Prentice Hall, Inc.S6 – 44 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

45. © 2006 Prentice Hall, Inc.S6 – 45 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

46. © 2006 Prentice Hall, Inc.S6 – 46 Process Capability Comparison 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 p = = 0.66.251 -.249.0030 New machine is NOT capable Cp =Cp =Cp =Cp = Upper Specification - Lower Specification 6 

47. © 2006 Prentice Hall, Inc.S6 – 47 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

48. © 2006 Prentice Hall, Inc.S6 – 48 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