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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 l.jpg
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


Type of attributes control chart l.jpg
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


Control charts for fraction nonconforming l.jpg
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


The characteristic of binomial distribution l.jpg
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


The binomial distribution l.jpg
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


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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


Development of the fraction nonconforming control chart 2 l.jpg
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


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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


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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


Development of the fraction nonconforming control chart 5 l.jpg
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


Example 1 l.jpg
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


Example 2 l.jpg
Example2

n = 100, m = 10

Control Charts for Attributes


Example 3 l.jpg
Example3

Control Limits are:

Control Charts for Attributes


Example 4 l.jpg

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Example4

Control Charts for Attributes


Design of the fraction nonconforming control chart l.jpg
Design of the Fraction Nonconforming Control Chart

  • Sample size

  • Frequency of sampling

  • Width of the control limits

Control Charts for Attributes


Sample size of fraction nonconforming control chart l.jpg
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


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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


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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


Example 219 l.jpg
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


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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


Example l.jpg
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


Sample size of fraction nonconforming control chart 3 l.jpg
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


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Example

  • If p=0.05 and three-sigma limits are used, the sample size must be

Control Charts for Attributes


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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


Interpretation of p chart l.jpg
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


The np control chart l.jpg
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


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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


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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


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P Charts with Variable Sample Size Size— Example

Control Charts for Attributes


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Variable Width Control Limits Size

  • 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



Control limits based on an average sample size l.jpg
Control Limits Based on an Average Sample Size 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



Slide34 l.jpg
平均管制界限與個別不良率之分析 Size1

  • 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


Slide35 l.jpg
平均管制界限與個別不良率之分析 Size2

  • 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


The standardized control chart l.jpg
The Standardized Control Chart Size

  • 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



Control charts for nonconformities defects 1 l.jpg
Control Charts for Nonconformities (Defects) Size1

  • 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


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Control Charts for Nonconformities (Defects) Size2

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


C chart l.jpg
c-chart Size

  • Standard Given:

  • No Standard Given:

Control Charts for Attributes


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Choice of Sample Size: The u Chart Size

  • 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


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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


Variable width control limits43 l.jpg
Variable Width Control Limits Size

  • 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


Control limits based on an average sample size44 l.jpg
Control Limits Based on an Average Sample Size 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


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The Standardized Control Chart Size

  • 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


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Demerit Systems Size1

  • 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


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Demerit Schemes Size2

Control Charts for Attributes


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Demerit Schemes Size3

  • 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


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Demerit Systems—Control Chart Development Size1

  • Number of demerits per unit:

    where n = number of inspection units

    D =

Control Charts for Attributes


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Demerit Systems—Control Chart Development Size2

where

and

Control Charts for Attributes


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Dealing with Low-Defect Levels Size1

  • When defect levels or count rates in a process become very low, say under 1000 occurrences per million, then there are long periods of time between the occurrence of a nonconforming unit.

  • Zero defects occur

  • Control charts (u and c) with statistic consistently plotting at zero are uninformative.

Control Charts for Attributes


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Dealing with Low-Defect Levels Size2

  • Chart the time between successive occurrences of the counts – or time between events control charts.

  • If defects or counts occur according to a Poisson distribution, then the time between counts occur according to an exponential distribution.

Control Charts for Attributes


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Dealing with Low-Defect Levels Size3

  • Exponential distribution is skewed.

  • Corresponding control chart very asymmetric.

  • One possible solution is to transform the exponential random variable to a Weibull random variable using x = y1/3.6 (where y is an exponential random variable) – this Weibull distribution is well-approximated by a normal.

  • Construct a control chart on x assuming that x follows a normal distribution.

  • See Example 6-6, page 304.

Control Charts for Attributes


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Dealing with Low-Defect Levels Size4

  • Data transformation: defect (Poisson distribution) → time between event (Exponential distribution) → x=y1/3.6=y0.2777 → X bar – MR control chart

Control Charts for Attributes


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Choice Between Attributes and Variables Control Charts Size

  • Each has its own advantages and disadvantages

  • Attributes data is easy to collect and several characteristicsmay be collected per unit.

  • Variables data can be more informative since specific information about the process mean and variance is obtained directly.

  • Variables control chartsprovide an indication of impending trouble (corrective action may be taken before any defectives are produced).

  • Attributes control charts will not react unless the process has already changed (more nonconforming items may be produced).

Control Charts for Attributes


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Guidelines for Implementing Control Charts Size

  • Determine which process characteristics to control.

  • Determine where the charts should be implemented in the process.

  • Choose the proper type of control chart.

  • Take action to improve processes as the result of SPC/control chart analysis.

  • Select data-collection systems and computer software.

Control Charts for Attributes








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