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quasi-measurement

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quasi-measurement

    Control Charts for AttributesBy the attribute method we mean the measurement of quality through noting the presence or absence of some characteristic in each of the units, and counting how many units do not posses the quality characteristic. The advantage of the attribute method is that a single chart can be set up for several characteristics, whereas a variables chart must be set up for each of the characteristics with an accompanying chart for controlling variability.

    p-Chart for Fraction Nonconforming

    This is also known as the proportion chart. The p-chart configuration is intended to evaluate the process in terms of the proportion or fraction of the total units in a sample in which a designated classification event occurs. This designated classification event

    may be a deviation more than the specified on a measurement scale, quasi-measurement scale, go or not-go gauge, judgment, etc. It could also be a nonconformity, defect, blemish, presence or absence of some characteristic, etc. The classification may also be based on several characteristics. Instead of proportions, if percents are used, then the p-chart will stand for percent chart.

    Let p stands for the fraction nonconforming of the process and be the sample

    fraction nonconforming computed as the ratio of the number of nonconforming units d to the sample size n. That is, = d/n. Let d follows a binomial distribution with

    parameters n and p ie

It is further known that the mean and variance of are p and p(1-p)/n respectively. If

    the true value of p is known, the control limits become

    with the central line being at p. Here p could be a standard value p'. Suppose that the true fraction nonconforming is unknown. As usual, it is assumed that the total number of units tested from the process is subdivided into m rational subgroups consisting of n, n, n...n..n units respectively and a value of the 123i.m

    proportion defective is computed for each subgroup. For convenience, one assumes that the subgroup sizes are all equal. If d is the number of defectives found in the ith i

subgroup, then the estimate of p is = d/n. The average of various values iii

    is . The control limits are set at

For example, consider the following data on the number of defectives obtained for 50

    subgroups of 100 resistors drawn from a process.

    Table 3.12 Nonconforming resistors in various subgroups

    i d i d i d i d i d iiiii iiiii

    1 0 0.00 11 0 0.00 21 1 0.01 31 1 0.01 41 1 0.01

    2 0 0.00 12 2 0.02 22 2 0.02 32 3 0.03 42 1 0.01

    3 2 0.02 13 1 0.01 23 0 0.00 33 0 0.00 43 3 0.03

    4 0 0.00 14 1 0.01 24 1 0.01 34 1 0.01 44 2 0.02

    5 1 0.01 15 1 0.01 25 1 0.01 35 2 0.02 45 1 0.01

    6 0 0.00 16 0 0.00 26 0 0.00 36 2 0.02 46 1 0.01

    7 2 0.02 17 0 0.00 27 0 0.00 37 0 0.00 47 0 0.00

    8 2 0.02 18 0 0.00 28 1 0.01 38 2 0.02 48 0 0.00

    9 1 0.01 19 2 0.02 29 0 0.00 39 2 0.02 49 0 0.00

    10 1 0.01 20 3 0.03 30 0 0.00 40 1 0.01 50 2 0.02

    The table given below also gives the values. The value of is 0.01. The control i

    limits are found as

    .

    Figure 3.44 provides the MINITAB p-chart output for the above data.

    Figure 3.44 MINITAB p-chart Output

    If the computed value for LCL is negative, it is set at zero. This means that there is no 'control' exercised to detect any quality improvement. If the subgroup sizes are unequal, then p is estimated as

and the (varying) control limits are given by

Alternatively, an 'average' sample size could be used. One can also

    plot the standardised value of p against the control limits ?3. i

    Choice of Subgroup Size

    Let p be the process average fraction defective (considered as acceptable) and 1

    suppose that we wish to detect a k level shift in a p-chart. That is, the process

    fraction defective shifts to an unacceptable level p = p + k. Let us further suppose 21

    that we wish to fix the probability of detecting the shift at 0.5. Under the normal approximation to binomial, the upper control limit of the

    p-chart based on p = p will coincide with p. Hence one has 12

From this relation, one obtains a formula for n as:

    For example, if the shift of the process to an unacceptable level of 0.02 from the acceptable level of 0.005 must be detected with 50% probability, the subgroup size should be about 200. If small subgroup sizes are used and p is small, the observance of one defective may mean an out-of-control signal. In order to avoid this, it may be desirable to have a large sample size for the p-chart.

    Sometimes it may be desirable to have a lower control limit greater than zero in order to look for samples that contain no defectives or to detect quality improvement. That is, it is desired that

    If p is small, obviously, the value of n should be very large. For example, for p = 0.01, the minimum subgroup size becomes 891.

Operating Characteristic Curve

    1. With only an upper control limit:

    The OC function giving the probability that a value of p=p (or d) falls below the UCL ii

    [or int(nUCL)] is given by

where n is the subgroup size.

    2. With both lower and upper control limits:

    The OC function giving the probability that p (or d) falls within the LCL and UCL is ii

    given by

    For very large subgroup sizes, LCL is also used as an action limit since a very low number of defectives may imply the inaccurate use of gages etc. The LCL is necessary to detect a quality improvement. If only quality degradation need be detected, the OC function based on the upper control limit should suffice. MINITAB macros can be used to draw the OC curves of a p-chart. For example, the OC curves of the p-chart having n=50, LCL=1 and UCL=18 are given in Figures 3.45 and 3.46.

    Figure 3.45 OC Curve With Both UCL and LCL

    Figure 3.46 OC Curve With Only UCL

    np-chart

    The np-chart is intended to evaluate the process in terms of the total number of units in a sample in which a given classification event occurs.

    The np-chart is essentially a p-chart, the only difference being the observed number of defectives is directly plotted instead of the observed proportion defective. If p is the proportion defective, then d, the number of defectives in the subgroup size n, follows a binomial distribution whose expected value is np with standard

    . Here p could be a standard value. When no standards are deviation

    available, one uses the estimate and draws the control limits

    at . The central line is drawn at n . The OC function of

    the np chart is similar to that of the p-chart and on the x-axis one plots the d values instead of p values.

    c-chart for Counts

    By the term area of opportunity, we mean a unit or a portion of material, process,

    product or service in which one or more designated events occur. The term is synonymous with the term unit and is usually preferred where there is no natural unit, eg continuous length of cloth.

    By defect, we mean the departure of a characteristic from its prescribed specification level that will render the product or service unfit to meet the normal usage requirements. By nonconformity, we mean a product which may not meet the

    specification requirement but may meet the usage requirements. For example, dirt in a

    block of cheese is a defect but underweight is a nonconformity.

    By c or count, we mean the number of events of a given classification occurring in a sample. More than one such event may occur in a desired area of opportunity.

    The c-chart or count chart is a configuration designed to evaluate the process in terms of the count of events (eg. defect or nonconformity) of a given classification occurring in a sample.

    We assume that the number of nonconformities d follows a Poisson distribution whose mean and variance are equal to the parameter c. That is, d follows:

    .

    The control chart for count d with three sigma limits is therefore

    c 3c,

    the central line being c. If LCL is less than zero, it is set at zero. Here c could be a standard value. In its absence, c is estimated as the average number of nonconformities in a sample, say , and the control limits are set at 3 .

    Consider the following table showing the number of nonconformities observed in a sample of 20 subgroups of cellular phones of five each. Here C1 stands for subgroup number and C2 stands for the number of events (defects) that occured in the given area of opportunity (phone).

    Table 3.13 Number of Nonconformities in various subgroups

    Row C1 C2 Row C1 C2 Row C1 C2 Row C1 C2 Row C1 C2

    1 1 3 21 5 0 41 9 1 61 13 1 81 17 1

    2 1 0 22 5 1 42 9 1 62 13 2 82 17 1

    3 1 1 23 5 2 43 9 1 63 13 1 83 17 2

    4 1 1 24 5 2 44 9 2 64 13 2 84 17 0

    5 1 0 25 5 1 45 9 2 65 13 1 85 17 2

    6 2 1 26 6 0 46 10 0 66 14 0 86 18 2

    7 2 0 27 6 0 47 10 1 67 14 0 87 18 0

    8 2 0 28 6 2 48 10 0 68 14 2 88 18 1

    9 2 1 29 6 1 49 10 1 69 14 1 89 18 2

    10 2 0 30 6 2 50 10 0 70 14 0 90 18 2

    11 3 0 31 7 0 51 11 3 71 15 1 91 19 0

    12 3 2 32 7 0 52 11 0 72 15 2 92 19 0

    13 3 0 33 7 0 53 11 2 73 15 1 93 19 2

    14 3 1 34 7 0 54 11 0 74 15 1 94 19 0

    15 3 2 35 7 0 55 11 1 75 15 1 95 19 4

    16 4 2 36 8 1 56 12 1 76 16 0 96 20 1

    17 4 0 37 8 0 57 12 0 77 16 1 97 20 0

    18 4 0 38 8 1 58 12 0 78 16 0 98 20 0

    19 4 0 39 8 0 59 12 0 79 16 0 99 20 1

    20 4 2 40 8 0 60 12 0 80 16 1 100 20 0

    The value of is 84/20 = 4.2. The control limits are then found as

    4.2 3,?;?;;;; or 0 to 10.4

    and the central line is set at 4.2. One plots the total number of defects found in each subgroup thereafter in the c-chart. While using a c-chart, a signal for a special cause may require further analysis using a cause and effect diagram.

    Figure 3.48 is the MINITAB output of the c-chart for the above data. It should be noted that MINITAB has no provision to input subgroup codes for drawing a c-chart.

    Figure 3.48 MINITAB c-chart Output

    OC Function

    1. With UCL only:

    If action is taken only with the upper control limit (ie d exceeding UCL), the OC function giving the probability of d being less than UCL is

    since d follows the Poisson distribution with parameter c. To obtain the OC curve, one computes P as above for a given value of c, and a plot of (c, P) will yield the OC aa

    curve. It can be noted that the above OC function is similar to the OC function of a single sampling plan (discussed in Chapter 4) with acceptance number Ac = int(UCL). The sample size of the single sampling plan cannot be interpreted directly. 2. With both LCL and UCL:

    The OC function of the c-chart with both LCL and UCL giving the probability of d falling within LCL and UCL is

    .

    It is easy to evaluate the OC function of the c-chart as

While using calculators to find P, the recurrence relation or normal approximation a

    may be useful.

    Figures 3.49 and 3.50 show the OC curves of a c-chart obtained using MINITAB macros.

    Figure 3.49 OC Curve With Only UCL

    Figure 3.50 OC Curve With Both UCL and LCL

    u-chart

    The u-chart or count per unit chart is a configuration to evaluate the process in terms of average number of events of a given classification per unit area of opportunity. The u-chart is convenient for a product composed of units whose inspection covers more than one characteristic such as dimension checked by gages, electrical and mechanical characteristics checked by tests, and visual defects observed by eye. Under these conditions, independent defects may occur in one unit of product and a preferred quality measure is to count all defects observed and divide by the number of units inspected to give a value for defects per unit, rather than a value for the fraction defective. Here only the independent defects are to be counted. The u-chart is

    particularly useful for products such as textiles, wire, sheet materials, etc, which are continuous and extensive. Here the opportunity for defects/nonconformities is large even though the chance of a defect at one spot is small.

    The total number of units tested is subdivided into m rational subgroups of size n each. For each subgroup, a value of u, the defects per unit, is computed. The average number of defects is found as

    Assuming that the number of defects follows the Poisson distribution, the control limits of the u-chart are given by,

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