Quality Control

This module can be used for the two major areas of statistical quality control - acceptance sampling and control charts and for process capability. For acceptance sampling, both attributes and variables plans can be developed. Attributes plans are used when the measurement is a defective/ nondefective type of measurement, while variables plans are used for taking a numerical result rather than simply a yes/no. In addition, the model can be used to compute the producer's and consumer's risk under a given sampling plan and/or to make a crude plot of the operating characteristic (OC) curve. For control charts it is possible to develop p-charts for the percentage defective, x-bar charts for the mean or c-charts for the number of defects. The screens for the first three options are similar while the screens for the control charts are also similar.

Acceptance Sampling

The exact elements in the data screen depend on whether an attributes sampling plan or a variables sampling plan is selected. In either case, the types of data screens are very similar. We begin with the description of the attributes data screen and present the variables data screen later.

Attributes sampling

A sample screen that includes both the data and solution appears next.

Data

AQL . For acceptance sampling, the Acceptable Quality Level must be given. The AQL must be (strictly) greater than 0 and must be less than 1. The interpretation of .01 is an AQL of 1% defective.

LTPD . The Lot Tolerance Percent Defective must be entered. This has characteristics similar to the AQL . It must be between 0 and 1.

ALPHA - The producer's risk. The probability of a type 1 error can be set using the drop-down box to be either 1% or 5% in attributes sampling. For variables sampling, this entry is numerical, with a maximum allowable value of .99.

BETA - The consumer's risk. The probability of a type 2 error can be set to be 1%, 5%, or 10% for attributes sampling. This entry is numerical, with a maximum value of .1 for variables sampling.

Solution

A sample problem and solution screen appears in the preceding screen. In this example, we are trying to determine the appropriate sampling plan when the AQL is specified at 1%, the LTPD is specified at 5%, alpha is 5%, and beta is 10%.

The sample size. The minimum sample size that meets the requirements above is determined and displayed. In this example, the appropriate size is 82.

The critical value. The maximum number of defective units (attributes sampling) or the maximum variable average (variables sampling) is displayed. In this example, the maximum allowable number of defects in the 82 units is 2.

Two additional output values appear on the right, indicating that the actual risks differ from the specified risks. The program is designed to find the minimum sample size that meets the requirements. The requirements can be more than met due to the integer nature of the sample size and critical value.

NOTE: The computation of the actual risks is based on the binomial distribution.

Actual producer's risk. The producer's risk in the input is the upper level for the allowable producer's risk. The actual producer's risk can be less and is displayed. In this example, it happens to be .0495 which is nearly the same as .05 that was set as input.

Actual consumer's risk. The consumer's risk in the input is the upper level for the allowable risk. The actual consumer's risk can be less and is displayed. In this example, it happens to be .0844 which is less than the setting of .1 that was entered as input.

An Operating Characteristic (OC) Curve can be displayed as illustrated next.

An Average Outgoing Quality ( AOQ ) Curve is also available.

Example 2 - Variables sampling

Next we present the data and output for a variables sampling plan. We want to accept the lot if the mean is 200 pounds but reject the lot if the mean is 180 pounds. The standard deviation of the items produced is 10 pounds. We are using 5% and 10% for alpha and beta respectively. For variables sampling, alpha and beta are numerical rather than preset by the drop-down box. The next example will illustrate this more clearly.

The output is very similar to the output for attributes sampling. In this example, we should sample 3 items, and weigh them; if the average weight is less than 189.8964 pounds, we should reject the lot.

Control Charts

The fourth, fifth, and sixth options from the submenu are used to develop control charts. Option 4 is used when the percentage of defects is of interest, option 5 when there is a variable measurement and an x-bar or Range (R-Bar) chart is required. The last option is for the number of defects (distributed as a Poisson Random Variable). In any of these cases it is necessary to indicate how many samples there are.

Example 3 - a p-bar chart

The module begins by asking for the number of samples. In example 3, which is shown in the following screen, we filled in a column of data indicating the number of defects in each of 10 samples. Also, we have asked for a 3-sigma control chart at the top. The top indicates that the sample size for each of these samples was 150. The program has computed the average percentage of defects which is displayed as 3.8%. The standard deviation of p-bar is shown at the upper right as .0156.

A control chart can also be displayed and is exhibited below.

Example 4 - an x-bar and range chart

After the number of samples is entered, there are two options. Either the raw data can be entered or the mean and range for each sample can be entered. We will display both. In the example, we display the data and output for the mean (x-Bar) and Range (R-bar) charts. Six samples of five items have been taken and their weights have been recorded. The first sample had an average weight of 561.9 pounds and a range of 25.3 pounds. The control charts are set up based on the range. (Some authors set up control charts based on standard deviations rather than ranges.) The Mean and Range chart on the right are based on three standard deviations.

Example 5 - Using raw data in x bar and range charts

The following screen presents the data for an example with raw data.

The software computes the mean and range for each sample and then computes the control charts.

Example 6 - c charts

The following screen contains a sample c-chart. The number of samples is entered followed by the number of defects in each sample. The program computes and displays the defect rate (4.4), its standard deviation (2.0976), and the control limits.

Process capability

For process capability the upper and lower tolerances for a process must be set. Optionally, the mean may be set. If the mean is not set then the center point between the upper and lower tolerances will be used. Finally, a standard deviation must be given. Both an upper and lower index are computed and process capability is the minimum of these two indices.