IES 331 Quality Control Chapter 6 Control Charts for Attributes Week 7-8 July 19-28, 2005

Attribute Data If item does not conform to standard on one or more of these characteristics, it is classified as nonconforming Conforming / Nonconforming units Non-defective / Defecting units Good / Bad Pass / Fail Nonconforming unit will contain at least on nonconformity Nonconformities / Defects Each specific point at which a specification is not satisfied ex: scratch, chip, dirty spots, accident

6-2 Control Chart for Fraction Nonconforming Fraction of Nonconforming: Ratio of the number of nonconforming items in a population to the total number of items in that population Fraction of nonconforming ~ Binomial Distribution p Probability that any unit will not conform to specifications n Random sample of n unit. Sample size D The number of units of products that are nonconforming

The Control Chart for Fraction Nonconforming Sample fraction nonconforming: ratio of the number of nonconforming units in the sample, D, to the sample size n Mean and Variances

The Control Chart for Fraction Nonconforming With specified standard value

The Control Chart for Fraction Nonconforming When p is not known, it must be estimated from collected data Average of these individual sample fractions nonconforming Fraction Nonconforming control chart: No Standard Given “Trial Control Limit”

Ex 1 (Exercise 6-1) Also see Example 6-1 Sample # Number of Noncomforming Assemblies 1 7 2 4 3 5 6 8 10 9 11 12 15 13 14 16 17 18 19 20 The data that follow give the number of nonconforming bearing and seal assemblies in sample size of 100 Construct a fraction nonconforming control chart for these data. If any points plot out of control, assume that assignable causes can be found and determine the revised control limits

The np Control Chart Alternative to p Control Chart Based on the number nonconforming rather than the fraction nonconforming If standard value p is not known, use the estimator Revisit Ex 1

Variable Sample Size 3 approaches to deal with variable sample size Variable-width control limits: to determine control limits for each individual sample that are based on specific sample size Control limits based on average sample size: to obtain an approximate set of control limits (constant control limits) The standardized control chart: The points are plotted in standard deviation units Center line at zero Upper and lower control limits  +3 and - 3

Ex 2 (Exercise 6-3) Also see Example 6-2 Day Units Inspected Nonconforming units Fraction Nonconforming 1 80 4 0.050 2 110 7 0.064 3 90 5 0.056 75 8 0.107 130 6 0.046 120 70 0.057 125 0.040 9 105 0.076 10 95 0.074 The following data represent the results of inspecting all units of a personal computer produced for the last 10 days. Does the process appear to be in control?

The Operating-Characteristic Function OC curve provides a measure of the sensitivity of the control chart Ability to detect the shift in the process fraction nonconforming from the nominal value to some other value Probability of incorrectly accepting the hypothesis of statistical control (i.e., type II or β-error)

Average Run Length (ARL) ARL for any Shewhart control chart If the process in control, ARL0 If the process is out of control, ARL1

6-3 Control Charts for Nonconformities (Defects) We can develop control charts for Total number of nonconformities in a unit, or Average number of nonconformities per unit Defects or nonconformities ~ Poisson Distribution Where x is the number of nonconformities and C > 0 c chart: same sample size

Ex 3 (Exercise 6-41) Also see Example 6-3 Sample Number Number of Nonconformities 1 2 3 4 7 5 8 6 10 13 9 19 11 24 12 14 15 16 17 18 20 21 22 Ex 3 (Exercise 6-41) Also see Example 6-3 The data represent the number of nonconformities per 1000 meters in telephone cable. From analysis of these data, would you conclude that the process is in control?

Choice of Sample Size: u chart Average number of nonconformities per inspection unit It may requires several inspection units in the sample u chart: setting up the control chart based on the average number of nonconformities per inspection unit

Ex 4 (Exercise 6-44) Also see Example 6-4 Sample Number Number of Nonconformities 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 An automobile manufacturer wishes to control the number of nonconformities in a subassembly area producing manual transmissions. The inspection unit is defined as four transmissions, and data from 16 samples (each of size 4) are shown here. Set up a control chart for nonconformities per unit

Control charts for nonconformities with variable sample size c chart will be difficult to interpret – varying center line and control limits Alternative approaches are: u chart (nonconformities per unit) Constant center line Control limit will vary inversely with the n1/2 Use control limits based on an average sample size Use a standardized control chart (preferred)

Demerit Systems for Defects

The Operating Characteristic Function OC curves for c and u charts can be obtained from the Poisson distribution

Action taken to improve a process