Statistical Process Control

Transcription

1 Statistical Process Control

2 Outline Statistical Process Control (SPC) Process Capability Acceptance Sampling 2

3 Learning Objectives When you complete this supplement you should be able to : S6.1 Explain the purpose of a control chart S6.2 Explain the role of the central limit theorem in SPC S6.3 Build x-charts and R-charts S6.4 List the five steps involved in building control charts 3

4 Learning Objectives When you complete this supplement you should be able to : S6.5 Build p-charts and c-charts S6.6 Explain process capability and compute C p and C pk S6.7 Explain acceptance sampling 4

5 Statistical Process Control The objective of a process control system is to provide a statistical signal when assignable causes of variation are present 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 Quickens appropriate actions to eliminate assignable causes 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" 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 8

9 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 # # # # # # # # # # # # # # # # Each of these represents one sample of five boxes of cereal # # # # # # # # # # Figure S6.1 Weight 9

10 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 Frequency The solid line represents the distribution Figure S6.1 Weight 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 (bellshaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Frequency Central tendency Variation Shape Figure S6.1 Weight Weight Weight 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 present, the output of a process forms a distribution that is stable over time and is predictable Frequency Prediction Weight Figure S6.1 12

13 Samples (e) To measure the process, we take samples and analyze the sample statistics following these steps If assignable causes are present, the process output is not stable over time and is not predicable Frequency Weight?????????????????? Prediction Figure S6.1 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 14

15 Process Control Lower control limit Frequency (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.) Figure S6.2 15

16 Control Charts for Variables Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables x-chart tracks changes in the central tendency R-chart indicates a gain or loss of dispersion 16

17 Control Charts for Variables Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables x-chart tracks changes in the central tendency R-chart indicates a gain or loss of dispersion 17

18 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 will be the same as the population mean 2) The standard deviation of the sampling distribution ( ) will x equal the population standard deviation ( ) divided by the square root of the sample size, n = x x n 18

19 Population and Sampling Distributions Population distributions Distribution of sample means Beta Normal = Mean of sample means = x Standard deviation of the sample means x n Uniform 3 x 2 x 1 x = x 1 x 2 x 3 x 95.45% fall within ±2 x 99.73% of all x fall within ± 3 x Figure S6.3 19

20 Sampling Distribution Figure S6.4 Sampling distribution of means Process distribution of means = x = (mean) 20

21 Sampling Distribution Figure S6.4 As the sample size increases, the sampling distribution narrows n = 100 n = 50 n = 25 Mean of process 21

22 Setting Chart Limits For x-charts when we know Upper control limit (UCL) x z x Lower control limit (LCL) x z x = Where x = 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 (process) standard deviation n = sample size 22

23 Setting Control Limits Randomly select and weigh nine (n = 9) boxes each hour Average weight in the first sample ounces WEIGHT OF SAMPLE WEIGHT OF SAMPLE WEIGHT OF SAMPLE HOUR (AVG. OF 9 BOXES) HOUR (AVG. OF 9 BOXES) HOUR (AVG. OF 9 BOXES)

24 Setting Control Limits Average mean of 12 samples = x 12 i 1 Avg of 9 boxes 12 x 16 ounces n 9 z 3 1 ounce Number of samples = 12 24

25 Setting Control Limits Average mean of 12 samples = x 12 i 1 Avg of 9 boxes 12 = UCL x z x x 17 ounces 9 3 = LCL x z x x 15 ounces 9 3 x 16 ounces n 9 z 3 1 ounce Number of samples = 12 25

26 Setting Control Limits 17 = UCL Control Chart for samples of 9 boxes Out of control Variation due to assignable causes 16 = Mean Variation due to natural causes 15 = LCL Sample number Variation due to assignable causes Out of control 26

27 Setting Chart Limits For x-charts when we don't know = UCL x x A 2 R = LCL x x A 2 R where k R i i 1 R average range of the samples k A 2 = control chart factor found in Table S6.1 = x = mean of the sample means R i = range for sample i k = total number of samples 27

28 Control Chart Factors TABLE S6.1 SAMPLE SIZE, n Factors for Computing Control Chart Limits (3 sigma) MEAN FACTOR, UPPER RANGE, A 2 D LOWER RANGE, D

29 Setting Control Limits Super Cola example labeled as "net weight 12 ounces" Process average = 12 ounces Average range =.25 ounces Sample size = 5 = x UCL x A 2 R 12 (.577)(.25) ounces = x LCL x A 2 R ounces From Table S6.1 UCL = Mean = 12 LCL =

30 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 30

31 Setting Chart Limits For R-Charts Upper control limit (UCL R ) = D 4 R Lower control limit (LCL R ) = D 3 R where UCL R upper control chart limit for the range LCL R lower control chart limit for the range D 4 and D 3 values from Table S6.1 31

32 Restaurant Control Limits For salmon fillets at Darden Restaurants Sample Mean Sample Range x Bar Chart Range Chart UCL = = = x LCL = UCL = R = LCL = 0 32

33 Setting Control Limits Average range = 8 minutes Sample size = 4 From Table S6.1 D 4 = 2.282, D 3 = 0 UCL R D 4 R (2.282)(8) minutes LCL R D 3 R (0)(8) 0 minutes UCL = Mean = 8 LCL = 0 33

34 Mean and Range Charts (a) These sampling distributions result in the charts below Figure S6.5 x-chart R-chart UCL LCL UCL LCL (Sampling mean is shifting upward, but range is consistent) (x-chart detects shift in central tendency) (R-chart does not detect change in mean) 34

35 Mean and Range Charts (b) These sampling distributions result in the charts below Figure S6.5 x-chart R-chart UCL LCL UCL LCL (Sampling mean is constant, but dispersion is increasing) (x-chart indicates no change in central tendency) (R-chart detects increase in dispersion) 35

36 Steps In Building Control Charts 1. Collect 20 to 25 samples, often of n = 4 or n = 5 observations each, from a stable process, and compute the mean and range of each 2. Compute the overall means ( and R), set appropriate control limits, usually at the 99.73% level, and calculate the preliminary upper and lower control limits If the process is not currently stable and in control, use = the desired mean,, instead of to calculate limits. = x x 36

37 Steps In Creating Control Charts 3. Graph the sample means and ranges on their respective control charts and determine whether they fall outside the acceptable limits 4. Investigate points or patterns that indicate the process is out of control try to assign causes for the variation, address the causes, and then resume the process 5. Collect additional samples and, if necessary, revalidate the control limits using the new data 37

38 Setting Other Control Limits TABLE S6.2 DESIRED CONTROL LIMIT (%) Common z Values Z VALUE (STANDARD DEVIATION REQUIRED FOR DESIRED LEVEL OF CONFIDENCE)

39 Control Charts for Attributes For variables that are categorical Defective/nondefective, good/bad, yes/no, acceptable/unacceptable Measurement is typically counting defectives Charts may measure 1. Percent defective (p-chart) 2. Number of defects (c-chart) 39

40 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 p is estimated by UCL p p z p LCL p p z p ˆ p p 1 p n where p mean fraction (percent) defective inthe samples z number of standard deviations p standard deviation of the sampling distribution n number of observations in each sample 40

41 p-chart for Data Entry SAMPLE NUMBER NUMBER OF ERRORS FRACTION DEFECTIVE SAMPLE NUMBER NUMBER OF ERRORS FRACTION DEFECTIVE

42 p-chart for Data Entry SAMPLE NUMBER p NUMBER OF ERRORS FRACTION DEFECTIVE SAMPLE NUMBER NUMBER OF ERRORS FRACTION DEFECTIVE ˆ p Total number of errors Total number of records examined 80 (100)(20).04 (.04)(1.04).02 (rounded up from.0196) UCL p p z ˆ p.04 3(.02) LCL p p z ˆ p.04 3(.02)

43 p-chart for Data Entry Fraction defective Sample number UCL p = 0.10 p = 0.04 LCL p =

44 p-chart for Data Entry Possible assignable causes present Fraction defective Sample number UCL p = 0.10 p = 0.04 LCL p =

45 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 c mean number of defects per unit c standard deviation of defects per unit Control limits (99.73%) c 3 c 45

46 c-chart for Cab Company c 54 complaints/9 days 6 complaints/day UCL c c 3 c LCL c c 3 c Number defective Day 6 7 UCL c = c = 6 LCL c =

47 Managerial Issues and Control Charts Three major management decisions: Select points in the processes that need SPC Determine the appropriate charting technique Set clear and specific SPC policies and procedures 47

48 Which Control Chart to Use TABLE S6.3 Helping You Decide Which Control Chart to Use VARIABLE DATA USING AN x-chart AND R-CHART 1. Observations are variables 2. Collect 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 48

49 Which Control Chart to Use TABLE S6.3 ATTRIBUTE DATA USING A P-CHART Helping You Decide Which Control Chart to Use 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 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; flaws in a bolt of cloth; crimes in a year; broken seats in a stadium; typos in a chapter of this text; or complaints in a day 49

50 Patterns in Control Charts Upper control limit Target Lower control limit Normal behavior. Process is "in control." Figure S6.7 50

51 Patterns in Control Charts Upper control limit Target Figure S6.7 Lower control limit One plot out above (or below). Investigate for cause. Process is "out of control." 51

52 Patterns in Control Charts Upper control limit Target Figure S6.7 Lower control limit Trends in either direction, 5 plots. Investigate for cause of progressive change. 52

53 Patterns in Control Charts Upper control limit Target Figure S6.7 Lower control limit Two plots very near lower (or upper) control. Investigate for cause. 53

54 Patterns in Control Charts Upper control limit Target Lower control limit Run of 5 above (or below) central line. Investigate for cause. Figure S6.7 54

55 Patterns in Control Charts Upper control limit Target Lower control limit Erratic behavior. Investigate. Figure S6.7 55

56 Patterns in Control Charts Run test Identify abnormalities in a process Runs of 5 or 6 points above or below the target or centerline suggest assignable causes may be present Process may not be in statistical control There are a variety of run tests 56

57 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 57

58 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 =

59 Process Capability Ratio Insurance claims process Process mean x = minutes Process standard deviation =.516 minutes Design specification = 210 ± 3 minutes C p = Upper Specification - Lower Specification 6 59

60 Process Capability Ratio Insurance claims process Process mean x = minutes Process standard deviation =.516 minutes Design specification = 210 ± 3 minutes C p = Upper Specification - Lower Specification 6 = = (.516) 60

61 Process Capability Ratio Insurance claims process Process mean x = minutes Process standard deviation =.516 minutes Design specification = 210 ± 3 minutes C p = Upper Specification - Lower Specification = = (.516) Process is capable 61

62 Process Capability Index Upper C pk = minimum of Specification x,, Limit Lower x Specification Limit 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 62

63 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 63

64 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) (3)

65 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 (.251) C pk = minimum of, (3) (.249) (3)

66 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 (.251) C pk = minimum of, (3) (.249) (3).0005 Both calculations result in.001 C pk = = New machine is NOT capable 66

67 Interpreting C pk Figure S6.8 C pk = negative number C pk = zero C pk = between 0 and 1 C pk = 1 C pk > 1 Lower specification limit Upper specification limit 67

68 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 68

69 Acceptance Sampling Form of quality testing used for incoming materials or finished goods Rejected lots can be: Take samples at random from a lot (shipment) of items 1. Returned to the supplier Inspect each of the items in the sample 2. Culled for defectives Decide whether to reject (100% the inspection) whole lot based on the inspection results 3. May be re-graded to a lower specification Only screens lots; does not drive quality improvement efforts 69

70 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 70

71 The "Perfect" OC Curve Keep whole shipment P(Accept Whole Shipment) Return whole shipment Cut-Off % Defective in Lot 71

72 An OC Curve = 0.05 producer's risk for AQL Figure S6.9 Probability of Acceptance = 0.10 Consumer's risk for LTPD AQL LTPD Good Indifference lots zone Bad lots Percent defective 72

73 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 73

74 Producer's and Consumer's Risks Producer's risk ( ) Probability of rejecting a good lot Probability of rejecting a lot when the fraction defective is at or above the AQL Consumer's risk ( ) Probability of accepting a bad lot Probability of accepting a lot when fraction defective is below the LTPD 74

75 OC Curves for Different Sampling Plans n = 50, c = 1 n = 100, c = 2 75

76 Average Outgoing Quality AOQ = (P d)(p a )(N n) N where P d P a = true percent defective of the lot = probability of accepting the lot N = number of items in the lot n = number of items in the sample 76

77 Average Outgoing Quality 1. If a sampling plan replaces all defectives 2. If we know the true 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 limit (AOQL) 77

78 Automated Inspection Modern technologies allow virtually 100% inspection at minimal costs Not suitable for all situations 78

79 SPC and Process Variability Lower specification limit Upper specification limit (a) Acceptance sampling (Some bad units accepted; the "lot" is good or bad) (b) Statistical process control (Keep the process "in control") Process mean, (c) C pk > 1 (Design a process that is in within specification) Figure S