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# Introduction - PowerPoint PPT Presentation

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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• 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

• 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

• 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

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 with parameters n  0 and 0 < p < 1, is given by

• The mean and variance of the binomial distribution are

Control Charts for Attributes

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

• The sample fraction nonconforming is given as

where is a random variable with mean and variance

Control Charts for Attributes

Standard Given

• If a standard value of p is given, then the control limits for the fraction nonconforming are

Control Charts for Attributes

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

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

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

n = 100, m = 10

Control Charts for Attributes

Control Limits are:

Control Charts for Attributes

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Example4

Control Charts for Attributes

• Sample size

• Frequency of sampling

• Width of the control limits

Control Charts for Attributes

• 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

• 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

• 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

• Suppose we want the probability of at least one nonconforming unit in the sample to be at least 0.95.

Control Charts for Attributes

• 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

• 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

• 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

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

Control Charts for Attributes

• 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

• 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 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

• 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

1.Variable Width Control Limits

2.Control Limits Based on Average Sample Size

3.Standardized Control Chart

Control Charts for Attributes

P Charts with Variable Sample Size Size— Example

Control Charts for Attributes

• 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 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

• 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

• 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 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

• 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

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 Size

• Standard Given:

• No Standard Given:

Control Charts for Attributes

• 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

• Variable Width Control Limits

• Control Limits Based on Average Sample Size

• Standardized Control Chart

Control Charts for Attributes

• 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 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

• 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

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

Demerit Schemes Size2

Control Charts for Attributes

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

• Number of demerits per unit:

where n = number of inspection units

D =

Control Charts for Attributes

where

and

Control Charts for Attributes

• 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

• 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

• 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

• 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

• 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

• 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