7.4 The c-chart (fixed sample size) • Often there is greater concern for the total number of nonconformities (or defects) (C) than the fraction of nonconforming (defective) units (p). • An inspection unit is the basic unit inspected for nonconformities or defects. • The inspection unit could be: – An individual item (e.g., a car, refridgerator, laptop, etc.). – Multiple individual items grouped together (e.g., 5 cars, 10 refridgerators, 25 laptops, etc.). – A basic unit of continuous size (e.g., 10 miles of railway or roadway, 100 square feet of fabric, etc.). • Goal: Develop control charts for the total number of nonconformities per inspection unit. • The c-chart is based on the total number of nonconformities or defects in an inspection unit. • A nonconforming unit is an inspection unit not satisfying one or more of the specifications for that product. • A nonconformity is defined to be a specific occurrence on a unit of production that does not meet specifications. • Therefore, any individual unit contained in the inspection unit can have more than one nonconformity. • Recall: a defective unit is a nonconforming unit that is unfit for usage. • Usual assumption: The occurrence of nonconformities in samples of constant size is wellmodeled by the Poisson distribution. This implies: – The number of opportunities or potential locations for nonconformities may be infinitely large. This, in turn, implies that the probability of occurrence of a nonconformity at any location be small and constant. – The definition of an inspection unit is the same for each sample. That is, each inspection unit always represents an identical interval, area, or volume of opportunity for the occurrence of nonconformities. • Assume that each of the inspection units are the same size or contain the same quantity of production units and the occurrence of nonconformities within an inspection unit follows a Poisson distribution with parameter c. • Nonconformity (or defect) data are more informative than fraction nonconforming data because there will usually be several types of nonconformities. • Therefore, we can simultaneously chart each type of nonconformity on individual c-charts as long as the Poisson conditions above are met for each type. • Pareto charts can also accompany the c-charts to summarize defects data. • In practice, the conditions to assume a Poisson distribution are not satisfied exactly. As long as the departures are not severe, the Poisson distribution works reasonably well. 103 7.4.1 c is Known • Let Ci be the number of nonconformities in the ith inspection unit. • Because Ci ∼ Poisson(c), we know µCi = c and σC2 i = c. • The control limits for the c-chart are: UCL = c + 3 Centerline = c (16) LCL = c − 3 . • If LCL < 0, them reset LCL = 0. • As long as ci remains within control limits for each unit and no systematic pattern is evident, we conclude the process is in control at level c. • If ci is outside the control limits or a systematic pattern is evident, we conclude the process has shifted to a new level and is out of control at level c. 7.4.2 c is Unknown • The control limits can be established by testing the validity of trial control limits based on m preliminary samples. Pm ci • The estimate of c used in constructing the trial limits is c = i=1 . m • The trial control limits are: UCL = c + 3 Centerline = c (17) LCL = c − 3 • Testing the validity of the trial limits is identical to the method discussed for the p-chart. • If one or more of the ci are outside of the trial control limits, then assignable causes should be sought. Next, follow the rules used for p-charts. • Records on the different types of nonconformities that occur can easily be kept with this type of control chart to help determine which type of defect is most common. 104 Example for c-chart with known c: Newly painted trucks of the same model are inspected, and the number of paint defects per truck is recorded. When the paint process is in control there are an average of 7 defects per truck. Therefore, we will use c = 7 as a known total number of defects. The data set contains a truck ID and the number of defects observed on that truck. SAS Code for c-chart (c known) DM ’LOG; CLEAR; OUT; CLEAR;’; * ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\CCHART1.PDF’; ODS LISTING ; OPTIONS NODATE NONUMBER LS=76 PS=54; ******************************************************; *** c-chart (c known) ***; *** The response is the number of paint defects ***; *** per truck (inspection unit) ***; ******************************************************; DATA trucks; INPUT truckid $ defects @@; LINES; C1 5 C2 4 C3 4 C4 8 C5 7 C6 12 C7 3 C8 11 E4 8 E9 4 E7 9 E6 13 A3 5 A4 4 A7 9 Q1 15 Q2 8 Q3 9 Q9 10 Q4 8 ; /* Specify Expected Number of Nonconformities using u0= Option */ TITLE ’c Chart for Paint Defects on New Trucks (given c=7)’; SYMBOL1 WIDTH=3 VALUE=DOT; PROC SHEWHART DATA=trucks; CCHART defects*truckid=’1’ / u0 = 7 CSYMBOL = c0 TESTS = 1 to 8 LTESTS = 20 ALLLABEL=(truckid) ZONELABELS TABLETESTS TABLELEGEND; LABEL truckid = ’Truck ID’ defects = ’Defects’; RUN; Example for c-chart with unknown c: Trucks of the same model are to be painted using a new painting process. Twenty trucks are painted using the new process and are then inspected. The number of paint defects per truck is recorded. Twenty trucks are used to establish control limits for the total number of defects per truck (assuming the process is deemed to be in control). The data set contains a truck ID and the number of defects observed on that truck. SAS Code for c-chart (c unknown) DM ’LOG; CLEAR; OUT; CLEAR;’; * ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\CCHART2.PDF’; ODS LISTING; OPTIONS NODATE NONUMBER LS=76 PS=54; ******************************************************; *** c-chart (c unknown) ***; *** The response is the number of paint defects ***; *** per truck (inspection unit) ***; ******************************************************; DATA trucks; INPUT truckid $ defects @@; LINES; B1 12 B2 4 B3 4 B4 3 B5 4 D1 2 D2 3 D3 3 D4 2 D9 4 M2 9 M6 13 L3 5 L4 4 L7 6 Z1 15 Z2 8 Z3 9 Z7 6 Z9 8 ; TITLE1 ’c Chart for Paint Defects in New Trucks (c unknown)’; SYMBOL1 V=dot W=3; PROC SHEWHART DATA=trucks; CCHART defects*truckid=’1’ / TESTS = 1 to 8 LTESTS = 20 ALLLABEL=(truckid) ZONELABELS TABLETESTS TABLELEGEND; LABEL truckid = ’Truck ID’ defects = ’Defects’; RUN; 105 SAS output for c chart (c known) c Chart for Paint Defects on New Trucks (given c=7) The SHEWHART Procedure Test Descriptions Test 1 One point beyond Zone A (outside control limits) SAS output for c chart (c unknown) c Chart for Paint Defects in New Trucks (c unknown) The SHEWHART Procedure Test Descriptions Test 1 One point beyond Zone A (outside control limits) Test 2 Nine points in a row on one side of center line Test 6 Four out of five points in a row in Zone B or beyond 106 SAS output for c chart (c known) c Chart for Paint Defects on New Trucks (given c=7) The SHEWHART Procedure c Chart Summary for defects Subgroup Sample Size truckid C1 C2 C3 C4 C5 C6 C7 C8 E4 E9 E7 E6 A3 A4 A7 Q1 Q2 Q3 Q9 Q4 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 Test 1 -3 Sigma Limits with n=1 for CountLower Subgroup Upper Limit Count Limit 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5.000000 4.000000 4.000000 8.000000 7.000000 12.000000 3.000000 11.000000 8.000000 4.000000 9.000000 13.000000 5.000000 4.000000 9.000000 15.000000 8.000000 9.000000 10.000000 8.000000 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 14.937254 Special Tests Signaled 1 Test Descriptions One point beyond Zone A (outside control limits) SAS output for c chart (c unknown) c Chart for Paint Defects in New Trucks (c unknown) The SHEWHART Procedure c Chart Summary for defects truckid B1 B2 B3 B4 B5 D1 D2 D3 D4 D9 M2 M6 L3 L4 L7 Z1 Z2 Z3 Z7 Z9 Subgroup Sample Size 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 Test 1 Test 2 Test 6 -3 Sigma Limits with n=1 for CountLower Subgroup Upper Limit Count Limit 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Test One point beyond Nine points in a Four out of five 12.000000 4.000000 4.000000 3.000000 4.000000 2.000000 3.000000 3.000000 2.000000 4.000000 9.000000 13.000000 5.000000 4.000000 6.000000 15.000000 8.000000 9.000000 6.000000 8.000000 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 13.669940 Special Tests Signaled 6 2 1 Descriptions Zone A (outside control limits) row on one side of center line points in a row in Zone B or beyond 107 7.5 The u-chart (variable sample size) • If the interval, area, or volume of opportunity for nonconformities varies from sample to sample, we may standardize the unit size and use the u-chart. • The average number of nonconformities per inspection unit is u = c/n where c is the total number of nonconformities and n is the number of inspection units per sample. • Note: For c-charts, the inspection unit is the primary sampling unit. For u-charts, the inspection unit represents the standard unit size. • Let Ci be the total number of nonconformities and ni be the number of inspection units in the ith sample. Assume Ci ∼ Poisson(ni u). • Then Ui = Ci /ni is the average number of nonconformities per inspection unit in the ith sample. Hence, E(Ui ) = var(Ui ) = • Note: ni is not necessarily an integer. For example, if the inspection unit is a square foot, then n can be 1.5 square feet. 7.5.1 u is Known • The control Limits for the u chart are UCL = u + 3 Centerline = u (18) LCL = u − 3 • As long as ui remains within control limits for each unit and no systematic pattern is evident, we conclude the process is in control at level u. • If ui is outside the control limits or a systematic pattern is evident, we conclude the process has shifted to a new level and is out of control at level u. 7.5.2 u is Unknown • When u is not known, it is estimated from observed data. 1. Select m preliminary samples. 2. Our estimate of u is the average of nonconformities per inspection unit: Pm Ci Total number of nonconformities u = Pi=1 = m Total number of inspection units i=1 ni 108 3. The trial control Limits for the u chart are UCL = u + 3 Centerline = u (19) LCL = u − 3 • If one or more of the ui are outside of the trial control limits, then assignable causes should be sought. Next, follow the rules used for p-charts. Example for u-chart with known u: A textile company uses a u-chart to monitor the number of defects per square meter of fabric. The fabric is spooled onto rolls as it is inspected for defects. Each roll of fabric is one meter wide and 30 meters in length, and an inspection unit is defined as one square meter. Thus, there are 30 inspection units per roll of fabric. Three operators were used during data collection. When the process is in control there is an average of 0.20 defects per square meter. Therefore, we will use u = 0.20 as a known average number of defects. The data set contains a roll number, the number of square meters of fabric inspected from that roll (inspection unit), and the total number of defects observed for that inspection unit. SAS Code for u-chart (u known) DM ’LOG; CLEAR; OUT; CLEAR;’; * ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\UCHART1.PDF’; ODS LISTING; OPTIONS NODATE NONUMBER LS=76 PS=54; ****************************************************; *** u-chart (u known) ***; *** The response is the number of fabric defects ***; *** per square meter ***; ****************************************************; DATA fabrics1; INPUT operator roll defects sqmeters @@; LINES; 1 1 7 30.0 2 2 11 27.6 3 3 15 30.4 1 4 6 34.8 2 5 11 26.0 3 6 15 28.6 1 7 5 28.0 2 8 10 30.2 3 9 8 28.2 1 10 3 31.4 2 11 3 30.3 3 12 14 27.8 1 13 3 27.0 2 14 9 30.0 3 15 7 32.1 1 16 6 34.8 2 17 7 26.5 2 18 5 30.0 3 19 14 31.3 1 20 13 31.6 2 21 11 29.4 3 22 6 28.6 1 23 6 27.5 2 24 9 32.6 3 25 11 31.7 ; TITLE ’u Chart for Fabric Defects per Square Meter (given u=.20)’; SYMBOL1 V=dot W=3; PROC SHEWHART DATA=fabrics1 ; UCHART defects*roll=’1’ / SUBGROUPN = sqmeters ALLLABEL=(operator) U0 = 0.20 USYMBOL = u0 TABLE ; LABEL roll = ’Roll of Fabric’; RUN; 109 Example for u-chart with unknown u: The description is the same as above except that u is unknown. Various size inspection units from twenty-five rolls are used to establish control limits for the total number of defects per square meter of fabric (assuming the process is deemed to be in control). SAS Code for u-chart (u unknown) DM ’LOG; CLEAR; OUT; CLEAR;’; ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\UCHART2.PDF’; OPTIONS NODATE NONUMBER; ****************************************************; *** u-chart (u unknown) ***; *** The response is the number of fabric defects ***; *** per square meter ***; ****************************************************; DATA fabrics2; INPUT operator roll defects sqmeters @@; LINES; 1 1 7 30.0 2 2 11 27.6 3 3 15 30.4 1 4 6 34.8 2 5 11 26.0 3 6 15 28.6 1 7 5 28.0 2 8 10 30.2 3 9 8 28.2 1 10 3 31.4 2 11 3 30.3 3 12 14 27.8 1 13 3 27.0 2 14 9 30.0 3 15 7 32.1 1 16 6 34.8 2 17 7 26.5 2 18 5 30.0 3 19 14 31.3 1 20 13 31.6 2 21 11 29.4 3 22 6 28.6 1 23 6 27.5 2 24 9 32.6 3 25 11 31.7 ; TITLE ’u Chart for Fabric Defects per Square Meter (u unknown)’; SYMBOL1 V=dot W=3; PROC SHEWHART DATA=fabrics2 ; UCHART defects*roll=’1’ / SUBGROUPN = sqmeters ALLLABEL=(operator) TABLE ; LABEL roll = ’Roll of Fabric’; RUN; 110 SAS output for u-chart (u known) u Chart for Fabric Defects per Square Meter (given u=.20) The SHEWHART Procedure SAS output for u-chart (u unknown) u Chart for Fabric Defects per Square Meter (u unknown) The SHEWHART Procedure 111 SAS output for c chart (c known) u Chart for Fabric Defects per Square Meter (given u=.20) The SHEWHART Procedure u Chart Summary for defects roll Subgroup Sample Size 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30.0000 27.6000 30.4000 34.8000 26.0000 28.6000 28.0000 30.2000 28.2000 31.4000 30.3000 27.8000 27.0000 30.0000 32.1000 34.8000 26.5000 30.0000 31.3000 31.6000 29.4000 28.6000 27.5000 32.6000 31.7000 -3 Sigma Limits for Count per UnitSubgroup Lower Count Upper Limit per Unit Limit 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.23333333 0.39855072 0.49342105 0.17241379 0.42307692 0.52447552 0.17857143 0.33112583 0.28368794 0.09554140 0.09900990 0.50359712 0.11111111 0.30000000 0.21806854 0.17241379 0.26415094 0.16666667 0.44728435 0.41139241 0.37414966 0.20979021 0.21818182 0.27607362 0.34700315 0.44494897 0.45537696 0.44333213 0.42742941 0.46311741 0.45087260 0.45354628 0.44413654 0.45264558 0.43942607 0.44373334 0.45445668 0.45819889 0.44494897 0.43680111 0.42742941 0.46062335 0.44494897 0.43980823 0.43866719 0.44743583 0.45087260 0.45584086 0.43497813 0.43829044 SAS output for c chart (c unknown) u Chart for Fabric Defects per Square Meter (u unknown) The SHEWHART Procedure u Chart Summary for defects roll Subgroup Sample Size 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30.0000 27.6000 30.4000 34.8000 26.0000 28.6000 28.0000 30.2000 28.2000 31.4000 30.3000 27.8000 27.0000 30.0000 32.1000 34.8000 26.5000 30.0000 31.3000 31.6000 29.4000 28.6000 27.5000 32.6000 31.7000 ---3 Sigma Limits for Count per Unit-Subgroup Lower Count Upper Limit per Unit Limit 0.00000000 0.00000000 0.00000000 0.01511065 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00071343 0.00000000 0.00000000 0.00000000 0.00000000 0.00386365 0.01511065 0.00000000 0.00000000 0.00025480 0.00162417 0.00000000 0.00000000 0.00000000 0.00605141 0.00207630 0.23333333 0.39855072 0.49342105 0.17241379 0.42307692 0.52447552 0.17857143 0.33112583 0.28368794 0.09554140 0.09900990 0.50359712 0.11111111 0.30000000 0.21806854 0.17241379 0.26415094 0.16666667 0.44728435 0.41139241 0.37414966 0.20979021 0.21818182 0.27607362 0.34700315 112 0.58201323 0.59452788 0.58007285 0.56098796 0.60381722 0.58912219 0.59233088 0.58103822 0.59124994 0.57538517 0.58055434 0.59342345 0.59791449 0.58201323 0.57223496 0.56098796 0.60082409 0.58201323 0.57584381 0.57447444 0.58499771 0.58912219 0.59508461 0.57004720 0.57402231 7.5.3 Operating Characteristic Function • The Operating Characteristic Curve (OCC) for both the c-chart and the u-chart are based on the Poisson distribution. • For the c-chart, the OCC is a plot of β against c, the true mean total number of defects per inspection unit. βc = P (LCL < Ci < UCL|c) = P (Ci < UCL|c) − P (Ci ≤ LCL|c) = P (Ci < bUCLc|c) − P (Ci ≤ bLCLc|c) bUCLc = X e−c ci i! i=dLCLe where Ci ∼ Poisson(c), and bUCLc is the largest integer ≤ UCL and dUCLe is the smallest integer ≥ LCL. • For the u-chart, the OCC is a plot of β against u, the true mean number of defects per inspection unit for sample size n. Because Ui = Ci /n: βu = P (LCL < Ui < UCL|u) = P (Ci < nUCL|u) − P (Ci ≤ nLCL|u) = P (Ci < bnUCLc|c) − P (Ci ≤ bnLCLc|c) bnUCLc = X i=dnLCLe e−nu (nu)i i! 113

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