Tata McGraw

Chapter 9AProcess Capability and SPC

Process VariationProcess Capability Process Control ProceduresVariable dataAttribute dataAcceptance SamplingOperating Characteristic CurveOBJECTIVES 9A-*

Basic Forms of VariationAssignable variation is caused by factors that can be clearly identified and possibly managedCommon variation is inherent in the production process Example: A poorly trained employee that creates variation in finished product output.Example: A molding process that always leaves burrs or flaws on a molded item.9A-*

Taguchis View of VariationTraditional view is that quality within the LS and US is good and that the cost of quality outside this range is constant, where Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs.9A-*

Process Capability Index, CpkShifts in Process MeanCapability Index shows how well parts being produced fit into design limit specifications.As a production process produces items small shifts in equipment or systems can cause differences in production performance from differing samples.9A-*

A simple ratio: Specification Width_________________________________________________________ Actual Process Width

Generally, the bigger the better.Process Capability A Standard Measure of How Good a Process Is.9A-*

Process CapabilityThis is a one-sided Capability IndexConcentration on the side which is closest to the specification - closest to being bad9A-*

The Cereal Box ExampleWe are the maker of this cereal. Consumer reports has just published an article that shows that we frequently have less than 16 ounces of cereal in a box.Lets assume that the government says that we must be within 5 percent of the weight advertised on the box.Upper Tolerance Limit = 16 + .05(16) = 16.8 ouncesLower Tolerance Limit = 16 .05(16) = 15.2 ouncesWe go out and buy 1,000 boxes of cereal and find that they weight an average of 15.875 ounces with a standard deviation of .529 ounces.

9A-*

Cereal Box Process CapabilitySpecification or Tolerance LimitsUpper Spec = 16.8 ozLower Spec = 15.2 ozObserved WeightMean = 15.875 ozStd Dev = .529 oz9A-*

What does a Cpk of .4253 mean?An index that shows how well the units being produced fit within the specification limits. This is a process that will produce a relatively high number of defects.Many companies look for a Cpk of 1.3 or better 6-Sigma company wants 2.0!

9A-*

Types of Statistical SamplingAttribute (Go or no-go information)Defectives refers to the acceptability of product across a range of characteristics.Defects refers to the number of defects per unit which may be higher than the number of defectives.p-chart application Variable (Continuous)Usually measured by the mean and the standard deviation.X-bar and R chart applications9A-*

Statistical Process Control (SPC) ChartsUCLLCLSamples over time 1 2 3 4 5 6UCLLCLSamples over time 1 2 3 4 5 6UCLLCLSamples over time 1 2 3 4 5 6Normal BehaviorPossible problem, investigatePossible problem, investigate9A-*

Control Limits are based on the Normal Curvex0123-3-2-1zmStandard deviation units or z units. 9A-*

Control LimitsWe establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations from some x-bar or mean value. Based on this we can expect 99.7% of our sample observations to fall within these limits. LCLUCL99.7%9A-*

Example of Constructing a p-Chart: Required DataSample No.No. ofSamplesNumber of defects found in each sample9A-*

Sheet1

SampleObs 1Obs 2Obs 3Obs 4Obs 5AvgRangenA2D3D4

110.68210.68910.77610.79810.71410.7320.11621.8803.27

210.78710.8610.60110.74610.77910.7550.25931.0202.57

310.7810.66710.83810.78510.72310.7590.17140.7302.28

410.59110.72710.81210.77510.7310.7270.22150.5802.11

510.69310.70810.7910.75810.67110.7240.11960.4802

610.74910.71410.73810.71910.60610.7050.14370.420.081.92

710.79110.71310.68910.87710.60310.7350.27480.370.141.86

810.74410.77910.6610.73710.82210.7480.16290.340.181.82

910.76910.77310.64110.64410.72510.7100.132100.310.221.78

1010.71810.67110.70810.8510.71210.7320.179110.290.261.74

1110.78710.82110.76410.65810.70810.7480.163

1210.62210.80210.81810.87210.72710.7680.250

1310.65710.82210.89310.54410.7510.7330.349

1410.80610.74910.85910.80110.70110.7830.158

1510.6610.68110.64410.74710.72810.6920.103

Averages10.7370.187

A2

SamplenDefectives

11004

21002

31005

41003

51006

61004

71003

81007

91001

101002

111003

121002

131002

141008

151003

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Sheet1

10.7318

10.7546

10.7586

10.727

10.724

10.7052

10.7346

10.7484

10.7104

10.7318

10.7476

10.7682

10.7332

10.7832

10.692

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Avg

Sheet2

0.116

0.259

0.171

0.221

0.119

0.143

0.274

0.162

0.132

0.179

0.163

0.25

0.349

0.158

0.103

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Range

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Statistical Process Control Formulas:Attribute Measurements (p-Chart)Given:Compute control limits: 9A-*

1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sampleExample of Constructing a p-chart: Step 19A-*

Chart1

0.040.10140180630

0.040.10140180630

0.050.10140180630

0.030.10140180630

0.080.10140180630

0.040.10140180630

0.030.10140180630

0.140.10140180630

0.010.10140180630

0.020.10140180630

0.030.10140180630

0.020.10140180630

0.020.10140180630

0.080.10140180630

0.030.10140180630

Observation

p

Sheet1

SampleObs 1Obs 2Obs 3Obs 4Obs 5AvgRangenA2D3D4

110.68210.68910.77610.79810.71410.7320.11621.8803.27

210.78710.8610.60110.74610.77910.7550.25931.0202.57

310.7810.66710.83810.78510.72310.7590.17140.7302.28

410.59110.72710.81210.77510.7310.7270.22150.5802.11

510.69310.70810.7910.75810.67110.7240.11960.4802

610.74910.71410.73810.71910.60610.7050.14370.420.081.92

710.79110.71310.68910.87710.60310.7350.27480.370.141.86

810.74410.77910.6610.73710.82210.7480.16290.340.181.82

910.76910.77310.64110.64410.72510.7100.132100.310.221.78

1010.71810.67110.70810.8510.71210.7320.179110.290.261.74

1110.78710.82110.76410.65810.70810.7480.163

1210.62210.80210.81810.87210.72710.7680.250

1310.65710.82210.89310.54410.7510.7330.349

1410.80610.74910.85910.80110.70110.7830.158

1510.6610.68110.64410.74710.72810.6920.103

Averages10.7370.187

A2

SamplenDefectivespUCLLCL

110040.040.09304928810

210020.020.09304928810

310050.050.09304928810

410030.030.09304928810

510060.060.09304928810

610040.040.09304928810

710030.030.09304928810

810070.070.09304928810

910010.010.09304928810

1010020.020.09304928810

1110030.030.09304928810

1210020.020.09304928810

1310020.020.09304928810

1410080.080.09304928810

1510030.030.09304928810

n-barsum defectsp-bar

100550.0366666667

sp

15000.0187942071

UCL0.0930492881

LCL-0.0197159548

0

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Range

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2. Calculate the average of the sample proportions3. Calculate the standard deviation of the sample proportion Example of Constructing a p-chart: Steps 2&39A-*

4. Calculate the control limitsUCL = 0.0924LCL = -0.0204 (or 0)Example of Constructing a p-chart: Step 49A-*

Example of Constructing a p-Chart: Step 55. Plot the individual sample proportions, the average of the proportions, and the control limits 9A-*

Example of x-bar and R Charts: Required Data9A-*

Sheet: Sheet1

Sheet: Sheet2

Sheet: Sheet3

Sheet: Sheet4

Sheet: Sheet5

Sheet: Sheet6

Sheet: Sheet7

Sheet: Sheet8

Sheet: Sheet9

Sheet: Sheet10

Sheet: Sheet11

Sheet: Sheet12

Sheet: Sheet13

Sheet: Sheet14

Sheet: Sheet15

Sheet: Sheet16

Sample

Obs 1

Obs 2

Obs 3

Obs 4

Obs 5

Avg

Range

n

A2

D3

D4

1.0

10.682

10.689

10.776

10.798

10.714

10.731800000000002

0.11599999999999966

2.0

10.787

10.86

10.601

10.746

10.779

10.7546

0.25899999999999856

10.667

10.838

10.785

10.723

10.758599999999998

0.17099999999999937

4.0

10.591

10.727

10.812

10.726999999999999

0.22100000000000009

5.0

10.693

10.708

10.79

10.758

10.671

10.724

0.11899999999999977

6.0

10.749

10.714

10.738

10.719

10.606

10.705200000000001

0.14300000000000068

7.0

10.791

10.713

10.689

10.877

10.603

10.7346

0.2740000000000009

8.0

10.744

10.779

10.11

10.737

10.75

10.623999999999999

0.6690000000000005

9.0

10.769

10.773

10.641

10.644

10.725

10.7104

0.13199999999999967

10.0

10.718

10.671

10.708

10.85

10.712

10.731800000000002

0.17900000000000027

11.0

10.787

10.821

10.764

10.658

10.708

10.7476

0.16300000000000026

12.0

10.622

10.802

10.818

10.872

10.727

10.768199999999998

0.25

13.0

10.657

10.822

10.893

10.544

10.75

10.7332

0.3490000000000002

14.0

10.806

10.749

10.859

10.801

10.701

10.7832

0.15799999999999947

10.681

10.644

10.747

10.728

10.692

0.10299999999999976

A2

0.58

X-Bar

UCL

LCL

D3

0.0

10.856245333333334

10.600581333333334

Averages

10.728413333333334

0.22039999999999996

D4

2.11

Mean

10.728413333333334

R-Bar

0.4650439999999999

0.0

Range

0.22039999999999996

A2

Sample

n

Defectives

p

0.04

0.04

0.05

0.03

0.08

0.04

0.03

0.16

0.01

0.02

0.03

0.02

0.02

0.08

0.03

1375.0

56.0

0.04072727272727273

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

10.731800000000002

10.7546

10.758599999999998

10.726999999999999

10.724

10.705200000000001

10.7346

10.7484

10.7104

10.731800000000002

10.7476

10.768199999999998

10.7332

10.7832

10.692

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

0.11599999999999966

0.25899999999999856

0.17099999999999937

0.22100000000000009

0.11899999999999977

0.14300000000000068

0.2740000000000009

0.16199999999999903

0.13199999999999967

0.17900000000000027

0.16300000000000026

0.25

0.3490000000000002

0.15799999999999947

0.10299999999999976

Example of x-bar and R charts: Step 1. Calculate sample means, sample ranges, mean of means, and mean of ranges.9A-*

Chart1

Chart2

10.731810.856245333310.6005813333

10.754610.856245333310.6005813333

10.758610.856245333310.6005813333

10.72710.856245333310.6005813333

10.72410.856245333310.6005813333

10.705210.856245333310.6005813333

10.734610.856245333310.6005813333

10.62410.856245333310.6005813333

10.710410.856245333310.6005813333

10.731810.856245333310.6005813333

10.747610.856245333310.6005813333

10.768210.856245333310.6005813333

10.733210.856245333310.6005813333

10.783210.856245333310.6005813333

10.69210.856245333310.6005813333

Sample

Means

Chart3

0.1160.4650440

0.2590.4650440

0.1710.4650440

0.2210.4650440

0.1190.4650440

0.1430.4650440

0.2740.4650440

0.6690.4650440

0.1320.4650440

0.1790.4650440

0.1630.4650440

0.250.4650440

0.3490.4650440

0.1580.4650440

0.1030.4650440

Sample

Ranges

Sheet1

x-barRange

SampleObs 1Obs 2Obs 3Obs 4Obs 5AvgRangeUCLLCLUCLLCLnA2D3D4

110.68210.68910.77610.79810.71410.7320.11610.856245333310.60058133330.465044021.8803.27

210.78710.8610.60110.74610.77910.7550.25910.856245333310.60058133330.465044031.0202.57

310.7810.66710.83810.78510.72310.7590.17110.856245333310.60058133330.465044040.7302.28

410.59110.72710.81210.77510.7310.7270.22110.856245333310.60058133330.465044050.5802.11

510.69310.70810.7910.75810.67110.7240.11910.856245333310.60058133330.465044060.4802

610.74910.71410.73810.71910.60610.7050.14310.856245333310.60058133330.465044070.420.081.92

710.79110.71310.68910.87710.60310.7350.27410.856245333310.60058133330.465044080.370.141.86

810.74410.77910.1110.73710.7510.6240.66910.856245333310.60058133330.465044090.340.181.82

910.76910.77310.64110.64410.72510.7100.13210.856245333310.60058133330.4650440100.310.221.78

1010.71810.67110.70810.8510.71210.7320.17910.856245333310.60058133330.4650440110.290.261.74

1110.78710.82110.76410.65810.70810.7480.16310.856245333310.60058133330.4650440

1210.62210.80210.81810.87210.72710.7680.25010.856245333310.60058133330.4650440

1310.65710.82210.89310.54410.7510.7330.34910.856245333310.60058133330.4650440

1410.80610.74910.85910.80110.70110.7830.15810.856245333310.60058133330.4650440

1510.6610.68110.64410.74710.72810.6920.10310.856245333310.60058133330.4650440

Averages10.7280.220400

A20.58X-BarUCLLCL

D3010.856245333310.6005813333

D42.11

Mean10.728R-Bar0.4650440

Range0.220

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Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and Necessary Tabled ValuesFrom Exhibit TN8.79A-*

Sheet: Sheet1

Sheet: Sheet2

Sheet: Sheet3

Sheet: Sheet4

Sheet: Sheet5

Sheet: Sheet6

Sheet: Sheet7

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Sheet: Sheet10

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Sample

Obs 1

Obs 2

Obs 3

Obs 4

Obs 5

Avg

Range

n

A2

D3

D4

1.0

10.682

10.689

10.776

10.798

10.714

10.731800000000002

0.11599999999999966

2.0

10.787

10.86

10.601

10.746

10.779

10.7546

0.25899999999999856

10.667

10.838

10.785

10.723

10.758599999999998

0.17099999999999937

4.0

10.591

10.727

10.812

10.726999999999999

0.22100000000000009

5.0

10.693

10.708

10.79

10.758

10.671

10.724

0.11899999999999977

6.0

10.749

10.714

10.738

10.719

10.606

10.705200000000001

0.14300000000000068

7.0

10.791

10.713

10.689

10.877

10.603

10.7346

0.2740000000000009

8.0

10.744

10.779

10.66

10.737

10.822

10.7484

0.16199999999999903

9.0

10.769

10.773

10.641

10.644

10.725

10.7104

0.13199999999999967

10.0

10.718

10.671

10.708

10.85

10.712

10.731800000000002

0.17900000000000027

11.0

10.787

10.821

10.764

10.658

10.708

10.7476

0.16300000000000026

12.0

10.622

10.802

10.818

10.872

10.727

10.768199999999998

0.25

13.0

10.657

10.822

10.893

10.544

10.75

10.7332

0.3490000000000002

14.0

10.806

10.749

10.859

10.801

10.701

10.7832

0.15799999999999947

10.681

10.644

10.747

10.728

10.692

0.10299999999999976

Averages

10.736706666666668

0.18659999999999985

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

10.731800000000002

10.7546

10.758599999999998

10.726999999999999

10.724

10.705200000000001

10.7346

10.7484

10.7104

10.731800000000002

10.7476

10.768199999999998

10.7332

10.7832

10.692

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

0.11599999999999966

0.25899999999999856

0.17099999999999937

0.22100000000000009

0.11899999999999977

0.14300000000000068

0.2740000000000009

0.16199999999999903

0.13199999999999967

0.17900000000000027

0.16300000000000026

0.25

0.3490000000000002

0.15799999999999947

0.10299999999999976

Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Values9A-*

Example of x-bar and R charts: Steps 5&6. Calculate R-chart and Plot ValuesUCLLCL9A-*

Basic Forms of Statistical Sampling for Quality ControlAcceptance Sampling is sampling to accept or reject the immediate lot of product at handStatistical Process Control is sampling to determine if the process is within acceptable limits9A-*

Acceptance SamplingPurposesDetermine quality levelEnsure quality is within predetermined levelAdvantagesEconomyLess handling damageFewer inspectorsUpgrading of the inspection jobApplicability to destructive testingEntire lot rejection (motivation for improvement) 9A-*

Acceptance Sampling (Continued)DisadvantagesRisks of accepting bad lots and rejecting good lotsAdded planning and documentationSample provides less information than 100-percent inspection 9A-*

Acceptance Sampling: Single Sampling PlanA simple goal Determine (1) how many units, n, to sample from a lot, and (2) the maximum number of defective items, c, that can be found in the sample before the lot is rejected9A-*

RiskAcceptable Quality Level (AQL)Max. acceptable percentage of defectives defined by producerThe a (Producers risk)The probability of rejecting a good lotLot Tolerance Percent Defective (LTPD)Percentage of defectives that defines consumers rejection pointThe (Consumers risk)The probability of accepting a bad lot9A-*

Operating Characteristic CurveThe OCC brings the concepts of producers risk, consumers risk, sample size, and maximum defects allowed togetherThe shape or slope of the curve is dependent on a particular combination of the four parameters9A-*

Example: Acceptance Sampling ProblemZypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot.

Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel. 9A-*

Example: Step 1. What is given and what is not? In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.What you need to determine is your sampling plan is c and n.9A-*

Example: Step 2. Determine cFirst divide LTPD by AQL.Then find the value for c by selecting the value in the TN7.10 n(AQL)column that is equal to or just greater than the ratio above. So, c = 6.9A-*

Question BowlA methodology that is used to show how well parts being produced fit into a range specified by design limits is which of the following?Capability indexProducers riskConsumers riskAQLNone of the aboveAnswer: a. Capability index9A-*

Question BowlOn a quality control chart if one of the values plotted falls outside a boundary it should signal to the production manager to do which of the following?System is out of control, should be stopped and fixedSystem is out of control, but can still be operated without any concernSystem is only out of control if the number of observations falling outside the boundary exceeds statistical expectationsSystem is OK as isNone of the aboveAnswer: c. System is only out of control if the number of observations falling outside the boundary exceeds statistical expectations9A-*

Question BowlYou want to prepare a p chart and you observe 200 samples with 10 in each, and find 5 defective units. What is the resulting fraction defective?252.50.00250.00025Can not be computed on data aboveAnswer: c. 0.0025 (5/(2000x10)=0.0025)9A-*

Question BowlYou want to prepare an x-bar chart. If the number of observations in a subgroup is 10, what is the appropriate factor used in the computation of the UCL and LCL?1.880.310.221.78None of the aboveAnswer: b. 0.319A-*

Question BowlYou want to prepare an R chart. If the number of observations in a subgroup is 5, what is the appropriate factor used in the computation of the LCL?00.881.882.11None of the aboveAnswer: a. 09A-*

Question BowlYou want to prepare an R chart. If the number of observations in a subgroup is 3, what is the appropriate factor used in the computation of the UCL?0.871.001.882.11None of the aboveAnswer: e. None of the above9A-*

Question BowlThe maximum number of defectives that can be found in a sample before the lot is rejected is denoted in acceptance sampling as which of the following?AlphaBetaAQL cNone of the aboveAnswer: d. c9A-*

1-*

End of Chapter 9A9A-*

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