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Introduction

Introduction . Data that can be classified into one of several categories or classifications is known as attribute data . Classifications such as conforming and nonconforming are commonly used in quality control. Another example of attributes data is the count of defects .

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Introduction

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  1. Introduction • Data that can be classified into one of several categories or classifications is known as attribute data. • Classifications such as conforming and nonconformingare commonly used in quality control. • Another example of attributes data is the count of defects. Control Charts for Attributes

  2. Type of Attributes Control Chart • Control Chart for Fraction Nonconforming — p chart • Control Chart for Nonconforming — np chart • Control Chart for Nonconformities — c chart • Control Chart for Nonconformities per unit — u chart Control Charts for Attributes

  3. Control Charts for Fraction Nonconforming • Fraction nonconforming is the ratio of the number of nonconforming items in a population to the total number of items in that population. • Control charts for fraction nonconforming are based on the binomial distribution. Control Charts for Attributes

  4. The characteristic of binomial distribution 1. All trials are independent. 2. Each outcome is either a “success” or “failure”. 3. The probability of success on any trial is given as p. The probability of a failure is 1-p. 4. The probability of a success is constant. Control Charts for Attributes

  5. The Binomial distribution • The binomial distribution with parameters n  0 and 0 < p < 1, is given by • The mean and variance of the binomial distribution are Control Charts for Attributes

  6. Development of the Fraction Nonconforming Control Chart1 Assume • n = number of units of product selected at random. • D = number of nonconforming units from the sample • p= probability of selecting a nonconforming unit from the sample. • Then: Control Charts for Attributes

  7. Development of the Fraction Nonconforming Control Chart2 • The sample fraction nonconforming is given as where is a random variable with mean and variance Control Charts for Attributes

  8. Development of the Fraction Nonconforming Control Chart3 Standard Given • If a standard value of p is given, then the control limits for the fraction nonconforming are Control Charts for Attributes

  9. Development of the Fraction Nonconforming Control Chart4 No Standard Given • If no standard value of p is given, then the control limits for the fraction nonconforming are where Control Charts for Attributes

  10. Development of the Fraction Nonconforming Control Chart5 Trial Control Limits • Control limits that are based on a preliminary set of data can often be referred to as trial control limits. • The quality characteristic is plotted against the trial limits, if any points plot out of control, assignable causes should be investigated and points removed. • With removal of the points, the limits are then recalculated. Control Charts for Attributes

  11. Example1 A process that produces bearing housings is investigated. Ten samples of size 100 are selected. • Is this process operating in statistical control? Control Charts for Attributes

  12. Example2 n = 100, m = 10 Control Charts for Attributes

  13. Example3 Control Limits are: Control Charts for Attributes

  14. P C h a r t f o r C 1 0 . 1 0 3 . 0 S L = 0 . 0 9 5 3 6 n o i t r o 0 . 0 5 p o r P = 0 . 0 3 8 0 0 P 0 . 0 0 - 3 . 0 S L = 0 . 0 0 0 0 1 2 3 4 5 6 7 8 9 1 0 S a m p l e N u m b e r Example4 Control Charts for Attributes

  15. Design of the Fraction Nonconforming Control Chart • Sample size • Frequency of sampling • Width of the control limits Control Charts for Attributes

  16. Sample size of fraction nonconforming control chart • If p is very small, we should choose n sufficiently large so that we have a high probability of finding at least one nonconforming unit in the sample. • Otherwise, we might find that the control limits are such that the presence of only one nonconforming unit in the sample would indicate an out-of control condition. Control Charts for Attributes

  17. Sample size of fraction nonconforming control chart1 • Choose the sample size n so that the probability of finding at least one nonconforming unit per sample is at least γ. Control Charts for Attributes

  18. Example1 • If P=0.01, n=8, then UCL=0.1155 • If there is one nonconforming unit in the sample, then p=1/8=0.1250, and we can conclude that the process is out of control. Control Charts for Attributes

  19. Example2 • Suppose we want the probability of at least one nonconforming unit in the sample to be at least 0.95. Control Charts for Attributes

  20. Sample size of fraction nonconforming control chart2 • The sample size can be determined so that a shift of some specified amount,  can be detected with a stated level of probability (50% chance of detection, Duncan, 1986). If  is the magnitude of a process shift, then n must satisfy: Therefore, Control Charts for Attributes

  21. Example • Suppose that p=0.01, and we want the probability of detecting a shift to p=0.05 to be 0.50. Control Charts for Attributes

  22. Sample size of fraction nonconforming control chart3 • Positive Lower Control Limit • The sample size n, can be chosen so that the lower control limit would be nonzero: and Control Charts for Attributes

  23. Example • If p=0.05 and three-sigma limits are used, the sample size must be Control Charts for Attributes

  24. Width of the control limits • Three-sigma control limits are usually employed on the control chart for fraction nonconforming. • Narrower control limits would make the control chart more sensitive to small shifts in p but at the expense of more frequent “false alarms.” Control Charts for Attributes

  25. Interpretation of p Chart • Care must be exercised in interpreting points that plotbelow the lower control limit. • They often do not indicate a real improvement in process quality. • They are frequently caused by errors in the inspection process or improperly calibrated test and inspection equipment. • Inspectors deliberately passed nonconforming units or reported fictitious data. Control Charts for Attributes

  26. The np control chart • The actual number of nonconforming can also be charted. Let n = sample size, p = proportion of nonconforming. The control limits are: (if a standard, p, is not given, use ) Control Charts for Attributes

  27. Variable Sample Size1 • In some applications of the control chart for the fraction nonconforming, the sample is a 100% inspection of the process output over some period of time. • Since different numbers of units could be produced in each period, the control chart would then have a variable sample size. Control Charts for Attributes

  28. Three Approaches for Control Charts with Variable Sample Size 1.Variable Width Control Limits 2.Control Limits Based on Average Sample Size 3.Standardized Control Chart Control Charts for Attributes

  29. P Charts with Variable Sample Size — Example Control Charts for Attributes

  30. Variable Width Control Limits • Determine control limits for each individual sample that are based on the specific sample size. • The upper and lower control limits are Control Charts for Attributes

  31. Control Charts for Attributes

  32. Control Limits Based on an Average Sample Size • Control charts based on the average sample size results in an approximate set of control limits. • The average sample size is given by • The upper and lower control limits are Control Charts for Attributes

  33. Control Charts for Attributes

  34. 平均管制界限與個別不良率之分析1 • 點落在管制界限內，其樣本大小，小於平均樣本大小時 • 不須計算個別管制界限，該點在管制狀態內。 • 點落在管制界限內，其樣本大小，大於平均樣本大小時 • 計算個別管制界限，再對該點樣本進行判斷。 Control Charts for Attributes

  35. 平均管制界限與個別不良率之分析2 • 點落在管制界限外，其樣本大小，大於平均樣本大小時 • 不須計算個別管制界限，該點在管制狀態外。 • 點落在管制界限外，其樣本大小，小於平均樣本大小時 • 計算個別管制界限，再對該點樣本進行判斷。 Control Charts for Attributes

  36. The Standardized Control Chart • The points plotted are in terms of standard deviation units. The standardized control chart has the follow properties: • Centerline at 0 • UCL = 3 LCL = -3 • The points plotted are given by: Control Charts for Attributes

  37. Control Charts for Attributes

  38. Control Charts for Nonconformities (Defects)1 • There are many instances where an item will contain nonconformities but the item itself is not classified as nonconforming. • It is often important to construct control charts for the total number of nonconformities or the average number of nonconformities for a given “area of opportunity”. The inspection unit must be the same for each unit. Control Charts for Attributes

  39. Control Charts for Nonconformities (Defects)2 Poisson Distribution • The number of nonconformities in a given area can be modeled by the Poisson distribution. Let c be the parameter for a Poisson distribution, then the mean and variance of the Poisson distribution are equal to the value c. • The probability of obtaining x nonconformities on a single inspection unit, when the average number of nonconformities is some constant, c, is found using: Control Charts for Attributes

  40. c-chart • Standard Given: • No Standard Given: Control Charts for Attributes

  41. Choice of Sample Size: The u Chart • If we find c total nonconformities in a sample of n inspection units, then the average number of nonconformities per inspection unit is u = c/n. • The control limits for the average number of nonconformities is Control Charts for Attributes

  42. Three Approaches for Control Charts with Variable Sample Size • Variable Width Control Limits • Control Limits Based on Average Sample Size • Standardized Control Chart Control Charts for Attributes

  43. Variable Width Control Limits • Determine control limits for each individual sample that are based on the specific sample size. • The upper and lower control limits are Control Charts for Attributes

  44. Control Limits Based on an Average Sample Size • Control charts based on the average sample size results in an approximate set of control limits. • The average sample size is given by • The upper and lower control limits are Control Charts for Attributes

  45. The Standardized Control Chart • The points plotted are in terms of standard deviation units. The standardized control chart has the follow properties: • Centerline at 0 • UCL = 3 LCL = -3 • The points plotted are given by: Control Charts for Attributes

  46. Demerit Systems1 • When many different types of nonconformities or defects can occur, we may need some system for classifying nonconformities or defects according to severity; or to weigh various types of defects in some reasonable manner. Control Charts for Attributes

  47. Demerit Schemes2 Control Charts for Attributes

  48. Demerit Schemes3 • Let ciA, ciB, ciC, and ciD represent the number of units in each of the four classes. • The following weights are fairly popular in practice: • Class A-100, Class B - 50, Class C – 10, Class D - 1 di = 100ciA + 50ciB + 10ciC + ciD di - the number of demerits in an inspection unit Control Charts for Attributes

  49. Demerit Systems—Control Chart Development1 • Number of demerits per unit: where n = number of inspection units D = Control Charts for Attributes

  50. Demerit Systems—Control Chart Development2 where and Control Charts for Attributes

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