7.4 The c-chart (fixed sample size)

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Jasmin Bishop
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1 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

2 7.4.1 c is Known Let C i be the number of nonconformities in the i th inspection unit. Because C i Poisson(c), we know µ Ci = c and σc 2 i = c. The control limits for the c-chart are: UCL = c + 3 Centerline = c (16) If LCL < 0, them reset LCL = 0. LCL = c 3. As long as c i remains within control limits for each unit and no systematic pattern is evident, we conclude the process is in control at level c. If c i 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 c is Unknown The control limits can be established by testing the validity of trial control limits based on m preliminary samples. The estimate of c used in constructing the trial limits is c = The trial control limits are: m i=1 c i m. 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 c i 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

3 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 $ 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 $ 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

in New Trucks (c unknown) Test 1 Test 2 Test 6 Test Descriptions One point beyond Zone A (outside control

4 SAS output for c chart (c known) c Chart for Paint Defects on New Trucks (given c=7) Test 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) Test 1 Test 2 Test 6 Test Descriptions One point beyond Zone A (outside control limits) Nine points in a row on one side of center line Four out of five points in a row in Zone B or beyond 106

5 SAS output for c chart (c known) c Chart for Paint Defects on New Trucks (given c=7) c Chart Summary for defects Subgroup -3 Sigma Limits with n=1 for Count- Special Sample Lower Subgroup Upper Tests truckid Size Limit Count Limit Signaled C C C C C C C C E E E E A A A Q Q Q Q Q Test 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) c Chart Summary for defects Subgroup -3 Sigma Limits with n=1 for Count- Special Sample Lower Subgroup Upper Tests truckid Size Limit Count Limit Signaled B B B B B D D D D D M M L L L Z Z Z Z Z Test 1 Test 2 Test 6 Test Descriptions One point beyond Zone A (outside control limits) Nine points in a row on one side of center line Four out of five points in a row in Zone B or beyond 107

6 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 C i be the total number of nonconformities and n i be the number of inspection units in the i th sample. Assume C i Poisson(n i u). Then U i = C i /n i is the average number of nonconformities per inspection unit in the i th sample. Hence, E(U i ) = var(u i ) = Note: n i is not necessarily an integer. For example, if the inspection unit is a square foot, then n can be 1.5 square feet u is Known The control Limits for the u chart are UCL = u + 3 Centerline = u (18) LCL = u 3 As long as u i remains within control limits for each unit and no systematic pattern is evident, we conclude the process is in control at level u. If u i 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 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: m i=1 u = C i Total number of nonconformities m i=1 n = i Total number of inspection units 108

7 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 u i 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 LINES; ; 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

8 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 LINES; ; 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

9 SAS output for u-chart (u known) u Chart for Fabric Defects per Square Meter (given u=.20) SAS output for u-chart (u unknown) u Chart for Fabric Defects per Square Meter (u unknown) 111

10 SAS output for c chart (c known) u Chart for Fabric Defects per Square Meter (given u=.20) u Chart Summary for defects -3 Sigma Limits for Count per Unit- Subgroup Subgroup Sample Lower Count Upper roll Size Limit per Unit Limit SAS output for c chart (c unknown) u Chart for Fabric Defects per Square Meter (u unknown) u Chart Summary for defects ---3 Sigma Limits for Count per Unit-- Subgroup Subgroup Sample Lower Count Upper roll Size Limit per Unit Limit

11 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. β c = P (LCL < C i < UCL c) = P (C i < UCL c) P (C i LCL c) = P (C i < UCL c) P (C i LCL c) = UCL i= LCL e c c i i! where C i Poisson(c), and UCL is the largest integer UCL and UCL 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 U i = C i /n: β u = P (LCL < U i < UCL u) = P (C i < nucl u) P (C i nlcl u) = P (C i < nucl c) P (C i nlcl c) = nucl i= nlcl e nu (nu) i i! 113

ISyE 512 Chapter 7. Control Charts for Attributes. Instructor: Prof. Kaibo Liu. Department of Industrial and Systems Engineering UW-Madison

ISyE 512 Chapter 7. Control Charts for Attributes. Instructor: Prof. Kaibo Liu. Department of Industrial and Systems Engineering UW-Madison ISyE 512 Chapter 7 Control Charts for Attributes Instructor: Prof. Kaibo Liu Department of Industrial and Systems Engineering UW-Madison Email: kliu8@wisc.edu Office: Room 3017 (Mechanical Engineering