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- 1. Chapter 7. Control Charts for Attributes
- 2. Control Chart for Fraction Nonconformingg Fraction nonconforming is based on the binomial distribution. n: size of populationp p p: probability of nonconformance D: number of products not conforming Successive products are independent. Mean of D = np Variance of D = np(1-p)
- 3. Sample fraction nonconformance ˆMean of p:Mean of p: ˆVariance of p:
- 4. w: statistics for quality Mean of w: µwMean of w: µw Variance of w: σw 2 L: distance of control limit from center line (in standard deviation units) If p is the true fraction nonconformance:
- 5. If p is not know, we estimate it from samples. m: samples, each with n units (or observations) D: number of nonconforming units in sample iDi: number of nonconforming units in sample i Average of all observations:
- 7. Example 6-1. 6-oz cardboard cans of orange juice
- 8. Design of Fraction Nonconforming ChartDesign of Fraction Nonconforming Chart Three parameters to be specified:p p 1. sample size 2. frequency of sampling 3 width of control limits3. width of control limits Common to base chart on 100% inspection of all process output over time. Rational subgroups may also play role in determining sampling frequency.
- 10. Variable Sample Size Variable-Width Control Limits ( ) ( )( ) ( )1- 1- UCL 3 LCL 3 i i p p p p p p n n = + = −
- 12. Variable Sample Size Control Limits Based on an Average Sample Size U l i F i lUse average sample size. For previous example:
- 14. Variable Sample Size Standard Control Chart - Points are plotted in standard deviation units. UCL = 3UCL = 3 Center line = 0 LCL = -3
- 16. Operating Characteristic Function andOperating Characteristic Function and Average Run Length Calculations Probability of type II error { } { } { } { } ˆ ˆUCL | LCL | UCL | LCL | P p p P p p P D n p P D n p β = < − ≤ < ≤{ } { }UCL | LCL |P D n p P D n p= < − ≤
- 17. C t l Ch t f N f iti ( D f t )Control Charts for Nonconformities (or Defects) Procedures with Constant Sample Size x: number of nonconformities c > 0: parameter of Poisson distribution Set to zero if negative
- 18. If no standard is given, estimate c then use the following parameters: Set to zero if negativeg
- 21. Choice of Sample Size: µ Chart x: total nonconformities in n inspection units u: average number of nonconformities per inspection unit : obserd average number of nonconformities per inspection unitu: obserd average number of nonconformities per inspection unitu
- 23. Control Charts for Nonconformities Procedure with Variable Sample SizeProcedure with Variable Sample Size
- 25. Control Charts for Nonconformities Demerit Systems: not all defects are of equal importance
- 26. ciA: number of Class A defects in ith inspection units Similarly for ciB, ciC, and ciD for Classes B, C, and D. di: number of demerits in inspection unit i Constants 100 50 10 and 1 are demerit weightsConstants 100, 50, 10, and 1 are demerit weights. : inspection units b f d it it n : number of demerits per unit where i n i i u D u D d= = ∑1 i i in = ∑
- 27. µi: linear combination of independent Poisson variables is average number of Class A defects per unit, etc.Aµ
- 28. Control Charts for Nonconformities Operating Characteristic Function x: Poisson random variable c: true mean value β: type II error probabilityβ ype e o p obab y
- 29. For example 6-3 Number of nonconformities is integer.
- 31. Control Charts for Nonconformities • If defect level is low, <1000 per million, c and u charts become ineffective Dealing with Low Defect Levels ineffective. • The time-between-events control chart is more effective. • If the defects occur according to a Poisson distribution, the probability distribution of the time between events is the exponentialp y p distribution. • Constructing a time-between-events control chart is essentially equivalent to control charting an exponentially distributed variable. • To use normal approximation translate exponential distribution to• To use normal approximation, translate exponential distribution to Weibull distribution and then approximate with normal variable : normal approximation for exponential variablex y 1 0.27773.6 x y y= =
- 35. Guidelines for Implementing Control Charts Applicable for both variable and attribute control
- 36. Determining Which Characteristics and Where to Put Control ChartsWhere to Put Control Charts
- 38. Choosing Proper Type of Control Chart
- 41. Actions Taken to Improve Process